/
Text
Nucleation
BASIC THEORY WITH APPUCATIONS
• *••■"■: •-'•
DIMO KASHCHIEV
B
Nucleation
Basic Theory with Applications
For Borislav, Peter, Dimiter and Nikoleta
and for Mimi
Nucleation
Basic Theory with Applications
Dimo Kashchiev
Institute of Physical Chemistry, Bulgarian Academy of Sciences,
Sofia, Bulgaria
UTT ERWORTH
JJE 1 N E M A N N
OXFORD AMSTERDAM BOSTON LONDON NEWYORK PARIS
SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO
Butterworth-Heinemann
An imprint of Elsevier Science
Linacre House, Jordan Hill, Oxford 0X2 8DP
200 Wheeler Road, Burlington, MA 01803
First published 2000
Transferred to digital printing 2003
Copyright © 2000, Dimo Kashchiev. All rights reserved
The right of Dimo Kashchiev to be identified as the author of this
work has been asserted in accordance with the Copyright, Designs
and Patents Act 1988
No part of this publication may be reproduced in any material form (including
photocopying or storing in any medium by electronic means and whether
or not transiently or incidentally to some other use of this publication) without
the written permission of the copyright holder except in accordance with the
provisions of the Copyright, Designs and Patents Act 1988 or under the terms of
a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road,
London, England WIT 4LP. Applications for the copyright holder's written
permission to reproduce any part of this publication should be addressed
to the publisher
British Library Cataloguing in Publication Data
A catalogne record for this book is available from the British Library
Library of Congress Cataloguing in Publication Data
A catalogue record for this book is available from the Library of Congress
ISBN 07506 4682 9
For information on all Butterworth-Heinemann
publications visit our website at wwwbh.com
Printed and bound in Great Britain by Antony Rowe Ltd, Eastbourne
Contents
Preface ix
Symbols and abbreviations xiii
Part 1 Thermodynamics of nucleation 1
1 First-order phase transitions 3
2 Driving force for nucleation 9
3 Work for cluster formation 17
3.1 Homogeneous nucleation 20
3.2 Heterogeneous nucleation 30
3.3 General formulae 40
3.4 Absence of one-dimensional nucleation 42
4 Nucleus size and nucleation work 45
4.1 General formulae 45
4.2 Homogeneous nucleation 46
4.3 Heterogeneous nucleation 50
4.4 Atomistically small nuclei 55
5 Nucleation theorem 58
5.1 Phenomenological proof 59
5.2 Thermodynamic proof 64
5.3 Generalizations 65
5.4 Integral form 67
6 Properties of clusters 70
6.1 Inside pressure 70
6.2 Chemical potential 71
6.3 Vapour pressure 73
6.4 Solubility 77
6.5 Melting point 78
6.6 Specific surface energy 79
7 Equilibrium cluster size distribution 83
7.1 Equilibrium concentration of clusters 84
7.2 Equilibrium concentration of nuclei 93
8 Density-functional approach 97
8.1 General considerations 97
8.2 Gradient approximation 100
8.3 Hard-sphere approximation 102
8.4 Quasi-thermodynamics 104
8.5 Quasi-thermodynamic formulation 107
Part 2 Kinetics of nucleation 113
9 Master equation 115
9.1 General formulation 115
9.2 Nucleation stage 124
9.3 Coalescence stage 130
9.4 Ageing stage 132
10 Transition frequencies 136
10.1 Monomer attachment frequency 136
10.2 Monomer detachment frequency 157
10.3 Multimer attachment frequency 165
10.4 Multimer detachment frequency 171
10.5 General formulae 172
11 Nucleation rate 174
12 Equilibrium 178
13 Stationary nucleation 184
13.1 Stationary cluster size distribution 184
13-2 Stationary rate of nucleation 192
13.3 Particular cases 204
13.4 Concentration of supernuclei 214
13.5 Comparison with experiment 214
14 First application of the nucleation theorem 224
15 Non-stationary nucleation 231
15.1 Non-stationary cluster size distribution 232
15.2 Non-stationary rate of nucleation 243
15.3 Time lag of nucleation 249
15.4 Delay time of nucleation 258
15.5 Concentration of supernuclei 267
15.6 Suggestion 270
15.7 Finding the equilibrium concentration of nuclei 270
16 Second application of the nucleation theorem 274
17 Nucleation at variable supersaturation 279
17.1 Quasi-stationary cluster size distribution 280
17.2 Quasi-stationary rate of nucleation 283
17.3 Condition for quasi-stationarity 285
Part 3 Factors affecting nucleation 291
18 Seed size 293
19 Line energy 300
20 Strain energy 309
21 Electric field 315
21.1 General formulae 315
21.2 Nucleation on ions 317
21.3 Nucleation in external electric field 323
22
Carrier-gas pressure
330
Contents vii
23 Solution pressure 338
24 Pre-existing clusters 346
24.1 Non-stationary cluster size distribution 346
24.2 Non-stationary rate of nucleation 353
24.3 Concentration of supemuclei 356
24.4 Delay time of nucleation 362
25 Active centres 366
Part 4 Applications 371
26 Overall crystallization 373
26.1 General formulae 373
26.2 Polynuclear mechanism 377
26.3 Mononuclear mechanism 383
26.4 Two-stage crystallization 387
27 Crystal growth 391
27.1 Continuous growth 391
27.2 Nucleation-mediated growth 396
27.3 Spiral growth 404
28 Third application of the nucleation theorem 410
29 Induction time 413
30 Fourth application of the nucleation theorem 428
31 Metastability limit 430
32 Maximum number of supemuclei 436
32.1 General formulae 437
32.2 Instantaneous nucleation 440
32.3 Progressive nucleation 440
33 Size distribution of supemuclei 446
33.1 General formulae 447
33.2 Particular cases 452
34 Growth of thin films 468
35 Rupture of amphiphile bilayers 480
Appendices 489
Al Exact formula for the non-stationary nucleation rate 489
A2 Approximate formula for the non-stationary nucleation rate 491
A3 Initial concentration of supemuclei in previously supersaturated systems 492
References 495
Author index 515
Subject index
525
This Page Intentionally Left Blank
Preface
Nucleation is the process with which the formation of new phases begins and
is thus a widely spread phenomenon in both nature and technology.
Condensation and evaporation, crystal growth, deposition of thin films and
overall crystallization are only a few of the processes in which nucleation
plays a prominent role. Nucleation is nowadays believed to be involved in
such apparently different phenomena as, e.g. volcano eruption [Blander 1979],
electron condensation in solids [Chakraverty 1970], formation of black holes
in the Universe [Kapusta 1984], development of decompression sickness in
deep-sea divers [Yount and Hoffman 1986], irradiation-induced formation
of voids in nuclear reactors [Wiedersich and Katz 1979], rupture of foam,
membrane and emulsion bilayers [Exerowa and Kashchiev 1986; Exerowa
et al. 1992], formation of electron-hole liquid in semiconductors [Keldysh
1986; Fishman 1988], earthquake [Rundle 1989], appearance of turbulence
in liquid crystals subjected to strong electric fields [Kai et al. 1989], formation
of particulate matter in space [Rossi and Maciel 1984; Hasegawa and Kozasa
1988], crack-mediated fracture of stressed solids [Rundle and Klein 1989;
Golubovic and Feng 1991], and various cosmological phase transitions [Hogan
1983; Kampfer 1988; Mendell and Hiscock 1989; La and Steinhardt 1989;
Steinhardt 1990; Fischler et al. 1990]. The above list is by far incomplete
and it should not be a surprise if it proves that even the Big Bang was a
nucleation phenomenon.
At present, nucleation is an established area of research and technology:
the first paper on the kinetics of nucleation was published by Volmer and
Weber in 1926, but already Gibbs in his thermodynamic works from the end
of the nineteenth century obtained basic theoretical results in the area.
Fundamentally, nucleation is a topic in the physics of the phase transitions of
first order. Applicationally, it is involved in the modern production of traditional
and new materials and coatings for the needs of various technologies, which
is why nucleation theory, experiment and practice is an interdisciplinary
topic. It is an ingredient in the more general university courses on
thermodynamics, solid state physics, atmospheric physics, geophysics, physical
chemistry, colloid and surface science, biophysics, etc. Knowledge about
nucleation is indispensable for a better control over the performance of
many common and high technologies in the industries based on materials
science and engineering. Clearly, a sufficiently detailed book on the basics
of nucleation and its applications could be of great help to students, researchers
and engineers alike.
With the above idea in mind, I decided to write this book as a detailed and
systematic account of the basic principles, developments and applications of
the theory of nucleation. The book is aimed at being both an introduction to
x Preface
the theory of nucleation and a survey of the main developments in it. For that
reason, on one hand, practically all considered theoretical results are derived
in the book itself and, on the other hand, the book contains an extensive list
of references to original papers, review articles and books. This makes the
book particularly useful for students and newcomers in the field of nucleation
and growth of new phases. Understanding the results is facilitated also by
the sufficiently large number of figures. These display mostly theoretical
dependences, but also illustrate basic experimental findings and the
correspondence between theory and experiment. Despite its pedagogical
character, the book is not just an account of the status quo in nucleation
theory. It includes latest developments in the theory and throughout I extended
and generalized existing results and obtained new ones in order to achieve a
more coherent presentation. All this makes the book of interest also for
researchers and experts in the field.
While working on the book, my wish was to make it self-contained: I did
not want the studious reader to be forced to look for the sources referred to
in which it is demonstrated how a given formula is obtained. That is why the
commonly used 'it can be shown that. ..' is practically absent in the book.
During the many years of my work on various aspects of nucleation I was
asked scores of times by students and colleagues to recommend them a book
in which they could find an orderly, systematic and unified derivation of the
basic theoretical results describing the process. The problem is that, apart
from only a few specialized books on nucleation, most of these results are
dispersed in numerous original articles, reviews and chapters in books on
related topics. In this book 1 did my best to present in such an orderly and
systematic way both the fundamentals and the progress of the theory of
nucleation. For further reading, here is a list of books and reviews dealing
with nucleation: books [Volmer 1939; Gibbs 1928; Frenkel 1955; Dufour
and Defay 1963; Aleksandrov et al. 1963; Hirth and Pound 1963; Nielsen
1964; Defay and Prigogine 1966; Vetter 1967; Rusanov 1967, 1978;
Zettlemoyer(ed.) 1969;Lyubov 1969,1975; Krastanov 1970;Skripov 1972;
Abraham 1974a; Christian 1975; Lewis and Anderson 1978; Kaischew 1980;
Skripov and Koverda 1984; Doremus 1985; Sohnel and Garside 1992;
Mullin 1993; Gutzow and Schmelzer 1995; Markov 1995; Baidakov 1995;
Budevski et al. 1996; Debenedetti 1996]; reviews [McDonald 1962, 1963;
Ftdaetal. 1966; Kaischew and Budevski 1967; Kahlweit 1970;Zinsmeister
1970; Gutzow and Toschev 1971; Toschev 1973a; Venables 1973; Stowell
1974a; Venables and Price 1975; Binder and Stauffer 1976; Lewis and Halpem
1976; Skripov 1977; Zettlemoyer(ed.) 1977,1979; Kem etal. 1979; Stoyanov
1979; Budevski et al. 1980; Chernov 1980; Russell 1980; Gutzow 1980;
Kotake and Glass 1981;StoyanovandKashchiev 1981;James 1982;Kashchiev
1984a; Venables et al. 1984; Hodgson 1984; Gutzow et al. 1985;
Fishman 1988; Ickert and Schneider 1990; Kelton 1991; Milchev 1991;
Oxtoby 1992a, b, 1998; Zinke-Almang et al. 1992; Mutaftschiev 1993;
Jakubczyk and Sangwal 1994; Laaksonen et al. 1995; Kukushkin and Osipov
1998].
Preface xi
The book has four parts which are devoted to the thermodynamics of
nucleation, the kinetics of nucteation, the effect of various factors on nucleation
and the application of the theory to other processes which involve nucleation.
The first two parts describe in detail the two basic approaches in nucleation
theory - the thermodynamic and the kinetic ones. They contain derivations
of the basic and most important formulae of the theory and discuss their
limitations and possibilities for improvement. The third part deals with
some often encountered factors that can affect nucleation and is a natural
continuation of the first two parts. The last part is devoted to the application
of the theory to processes and problems of practical importance such as melt
crystallization and polymorphic transformation, crystal growth, growth of
thin solid films and size distribution of droplets and crystallites in condensation
and crystallization.
Typically, a chapter in a part formulates the corresponding problem, derives
the basic theoretical result(s), discusses the approximations involved, applies
the derived result(s) to particular cases and refers the reader to original and/
or recent articles which might be of help for further study. In this way, the
book is both a kind of textbook and a survey of nucleation theory and after
studying it (or at least its first two parts) the reader will be an educated
researcher in the field. The mathematical derivations are sufficiently simple
to be followed even by non-theorists and emphasize the physical meaning of
the results obtained. The reader who is not interested in these derivations
need only pay attention to the formulation of the problem and the final
expression(s) representing its solution. Chapters 13-17, 24-29, 31, 32 and
35 present available experimentally and numerically obtained data for various
nucleation quantities and discuss the correspondence between theory and
experiment.
It is already more than thirty years since I joined the Institute of Physical
Chemistry at the Bulgarian Academy of Sciences. This institute is
internationally recognized as one of the leaders in the research on nucleation,
crystal growth, dispersed systems and electrochemical phase formation. Its
high status is largely due to the fact that until recently it was headed by
Professor Rostislav Kaischew who, along with Professor Ivan Stranski, is a
pioneer in nucleation theory and a founder of the Bulgarian physicochemical
school. In my work I had and still have the fortune to be in contact with
Professor Kaischew and other colleagues who are prominent experts in the
above fields. 1 have benefited a lot from these contacts in enriching my
knowledge about nucleation and with this book I am trying to convey the
essence of this knowledge to the nucleation community worldwide. A point
I would like to make, too, is that, in the book I refer to and use a certain
number of papers in Bulgarian and also Russian which are important, but
generally unknown to the nucleation community. It is my belief that my
book will thus contribute to bridging the information gap which was created
between the nucleation research in West and East during the dark years of
the recently ended Cold War.
This book would not see light if during the years long work on it I did not
xii Preface
have the ceaseless and staunch support of my wife. My gratitude is to you,
Mimi, for this indispensable help which I consider as your tacit contribution
to science.
Dimo Kashchiev
Sofia, June 1999
Symbols and abbreviations
The number of the equation in which the symbol is first used is given in
parentheses.
A kinetic parameter in stationary nucleation rate (13.39)
A* area of surface of contact between nucleus and substrate
(4.46)
A' kinetic parameter in stationary nucleation rate (13.41)
A" kinetic parameter in stationary nucleation rate (13,51)
A\, area of surface of amphiphile bilayer (35.22)
ACt„ area of surface of contact between n-sized cluster and old
phase (10.51)
A( area of surface of crystal face (27.14)
Am kinetic parameter in bilayer lifetime (35.23)
A„ area of surface of contact between n-sized cluster and
substrate (3.76)
Ap kinetic parameter in bilayer lifetime (35.27)
As area of substrate surface (3.51)
Az area of nucleation-exclusion zone (32.11)
a activity of solute (2.11)
a shape factor for 3D nucleation (3.20)
ae equilibrium activity of solute (2.11)
ac factor in coefficient of light extinction (29.24)
aef effective molecular area (3.70)
aQ molecular area (3.74)
B thermodynamic parameter in stationary nucleation rate
(13.48)
B* thermodynamic parameter in stationary nucleation rate
(13.66)
Bz parameter characterizing the effect of nucleation-
exclusion zones (32.16)
b shape factor for 2D nucleation (3.71)
C concentration of solute (2.14)
C* equilibrium concentration of nuclei (7.37)
Ce equilibrium concentration of solute (2.14)
C„, C(n) equilibrium cluster size distribution (7.3)
Cn(t), C(n, t) quasi-equilibrium cluster size distribution (9.22)
Co concentration of nucleation sites (7.3)
C\ equilibrium concentration of monomers (7.5)
Cie equilibrium concentration of monomers at A/i = 0 (7.17)
xiv Symbols and abbreviations
c shape factor for 3D nucleation (3.18)
c* capture number (10.36)
c% shape factor for growth of supernucleus (26,7)
c'g shape factor for growth of monolayer supernucleus
(27.16)
cz shape factor for growth of nucleation-exclusion zone
(32.2)
ce parameter defined by eq. (21.27)
D vector of electric displacement (21.1)
D coefficient of volume diffusion of monomer (10.11)
Dn coefficient of volume diffusion of n-sized cluster (10.21)
Ds coefficient of surface diffusion of monomer (10.25)
Ds „ coefficient of surface diffusion of n-sized cluster (10.94)
d dimensionality of growth (26.7)
dr differentially small volume (8.1)
dQ molecular diameter (8.14)
E vector of electric field (21.1)
E strength of external electric field (21.21)
E* binding energy of nucleus (4.45)
£des activation energy for desorption (10.29)
En binding energy of n-sized cluster (3.32)
E* 'substrate' binding energy of nucleus (7.55)
£sd activation energy for surface diffusion (10.25)
Esn 'substrate' binding energy of n-sized cluster (7-31)
Ev activation energy for viscous flow (10.56)
Ex value of En at n = 1 (7.31)
e base of natural logarithms (7.15)
£-0 charge of electron (2.27)
F(R, t) distribution function of supernuclei (33.2)
F2 Helmholtz free energy of old phase after cluster
formation (3.6)
/ Helmholtz free energy per molecule (8.1)
/* frequency of monomer attachment to nucleus (11.8)
/* value of/* at A^/ = 0 (13.42)
/e.s value of/s at Ap = 0 (27.6)
fmAn-< 0 frequency of monomer attachment to n-sized cluster
(9-16)
/„„ frequency of transition of n-sized cluster into m-sized
cluster (9.1)
/s frequency of monomer attachment to growth site (27.2)
/0* value of/* at A^/ = 0 (13.43)
G Gibbs free energy (1.4)
G radial growth rate of supernucleus (26.7)
Gc growth constant of supernucleus (26.15)
Gc growth rate of crystal (27.1)
Gcz growth constant of nucleation-exclusion zone (32.12)
Symbols and abbreviations xv
Gex excess energy of cluster (3.3)
Gf growth rate of thin film (34.2)
G, Gibbs free energy of old phase before cluster formation
(3.1)
Gt(R) radius-dependent factor in radial growth rate of
supemucleus (33.7)
G2 Gibbs free energy of old phase after cluster formation
(3.2)
G2(r) time-dependent factor in radial growth rate of
supemucleus (33.7)
g„, g(n, t) frequency of monomer detachment from n-sized cluster
(9.16)
gs frequency of monomer detachment from growth site
(27.3)
h height of cluster (3.69)
h* height of nucleus (4.37)
hf mean thickness of thin film (34.1)
hi thickness of /th layer of thin film (34.1)
ftp Planck constant (10.59)
I impingement rate (2.9)
/e equilibrium impingement rate (2.9)
J rate ofnucleation (11.1)
J rate of non-stationary nucleation (15.1)
J' rate of detectable nucleation (11.11)
Ja rate of non-stationary nucleation per active centre (25.10)
yas rate of stationary nucleation per active centre (25.3)
yqs rate of quasi-stationary nucleation (17.1)
Js rate of stationary nucleation (13.1)
ys c critical rate of stationary nucleation (31.1)
j* net flux through nucleus size (11.5)
jn,j(n, r) net flux through size n (9.6)
K pre-exponential factor (15.121)
£c adsorption constant (10.46)
K, K„ Kvd parameters in induction time (30.1), (29.53). (29.16)
Kp adsorption constant (10.43)
K0, K,, Ki parameters in induction time (29.20), (29.56), (29.18)
k Boltzmann constant (1.1)
kc coefficient of light extinction (29.23)
ky, /to, k^ parameters in rate of crystal growth (27.37), (27.39),
(27.31)
Ls length of path travelled by light beam in scattering
medium (29.23)
M total number of molecules in system (1.1)
rriQ molecular mass (10.1)
N number of supernuclei (25.1)
Na number of nucleation-active centres (7.9)
xvi Symbols and abbreviations
Nex extended number of supemuclei (25.16)
/Vm maximum number of supemuclei (25.9)
Ns number of adsorption sites on substrate (7.8)
n number of molecules in cluster (3.2)
n* number of molecules in nucleus (4.1)
ri number of molecules in smallest detectable cluster
(11.12)
ns Gibbs excess number of molecules in cluster (3.85)
n* Gibbs excess number of molecules in nucleus (5.13)
fi! left end of nucleus region (7.41)
n2 right end of nucleus region (7.42)
P pressure of carrier gas (22.1)
P pressure of solution (23.2)
Pni probability for non-ingestion of nucleation-active centre
(32.3)
Pna probability for non-occupation of nucleation-active centre
(32.3)
Pi probability for appearance of at least one nucleus (26.37)
p pressure in system (2.4)
p pressure of vapours (2.5)
p* pressure inside nucleus (4.3)
p„ equilibrium pressure (1.9)
pef effective pressure (8.24)
pn pressure inside n-sized cluster (3.6)
Q charge of ion (21.8)
q KJMA exponent (26.34)
qi kinetic index of filling of ith layer of thin film (34.4)
R radius of cluster (3.22)
/?* radius of nucleus (4.9)
R' radius of smallest detectable supernucleus (33.23)
Ray average radius of detectable supemuclei (33.23)
Rb radius of biggest supernucleus (33.10)
Rt radius of ion (21.6)
Rs radius of seed (18.1)
RT radius of nucleation-exclusion zone (32.2)
/¾ molecular radius (10.1)
/?2 radius of resized supernucleus (33.6)
r position vector (8.1)
S supersaturation ratio (7.19)
Sc critical supersaturation ratio (31.5)
snew entropy per molecule in new phase (5.39)
snew entropy per molecule in nucleus (5.37)
Add entropy per molecule in old phase (5.37)
T absolute temperature (1.1)
TA absolute temperature of divergence of viscosity (10.56)
Te absolute equilibrium (melting) temperature (2.19)
Symbols and abbreviations xvii
T0 absolute pre-existing temperature (24.17)
t time (9.1)
fav average lifetime of metastable system (26.9)
tb lifetime of amphiphile bilayer (35.23)
fg growth time of n2-siztd supemucleus to smallest
detectable size (33.27)
t, induction time (29.1)
t[c critical induction time (31.12)
t'R' moment of appearance of supemucleus which has radius
fl'at time /(33.25)
ty mean time for appearance of at least one nucleus (26.39)
V volume of system (1.1)
V* volume of nucleus (4.5)
Vgj extended volume (26.4)
Vn volume of n-sized cluster (3.6)
Vz volume of nucleation-exclusion zone (32.2)
v(n, t) growth rate of n-sized cluster (9.21)
I/I, growth rate of hole in amphiphile bilayer (35.22)
vs growth constant of monolayer supemucleus (27.17)
vs velocity of propagation of monolayer step (27.23)
v0 molecular volume (2.4)
W work for cluster formation (3.4)
W* work for nucleus formation (4.2)
Wi value of W at n = 1 (7.5)
X* stationary concentration of nuclei (13.23)
X'* derivative of X(n) with respect to n at n = n* (24.10)
Xny X(n) stationary cluster size distribution (13.3)
Z* concentration of nuclei (11.4)
Z* non-stationary concentration of nuclei (15.54)
Z„, Z(n, t) cluster size distribution (9.1)
Z„, Z(n, t) non-stationary cluster size distribution (15.2)
Z„0t Z0(n) initial cluster size distribution (9.2)
Zq pre-existing concentration of nuclei (24.10)
Z<5* derivative of Zq(«) with respect to n at n = n* (24.10)
Zj concentration of monomers (9.32)
z Zeldovich factor (13.33)
Z\ valency of ion (2.27)
a fraction of crystallized volume (26.1)
cCd detectable fraction of crystallized volume (29.8)
<Xi coverage of /th layer of thin film (34.1)
p Zeldovich coefficient (7.38)
/Jd detectable relative decrement of intensity of transmitted
light (29.47)
r gamma-function (26.19)
y* coefficient of monomer sticking to nucleus (13.44)
xviii Symbols and abbreviations
yn coefficient of monomer sticking to n-sized cluster (10.2)
ys coefficient of monomer sticking to growth site (27.7)
A* width of nucleus region (7.43)
An* excess number of molecules in nucleus (5.21)
A/? underpressure (4.14)
Aps spinodal underpressure (4.23)
AS* excess entropy of nucleus (5.37)
Ase difference in molecular entropy at phase equilibrium
(2.21)
AT undercooling (2.21)
Arc critical undercooling (31.9)
Aue difference between volume of molecule in melt and
crystal (10.79)
A/i supersaturation (2.1)
Aii rate of change of supersaturation (17.11)
Afic critical supersaturation (31.1)
Afis spinodal supersaturation (4.13)
Ay0 pre-existing supersaturation (24.17)
Aa effective specific surface energy (3.70)
A<p overvoltage (2.27)
<5l> Dirac delta-function (15.71)
Cc dielectric constant of cluster (21.5)
£„! dielectric constant of medium (21.4)
£q permittivity of empty space (21.4)
C, concentration of supernuclei (11.1)
£" concentration of detectable clusters (11-11)
£a concentration of nucleation-active centres (33.56)
£m maximum concentration of supernuclei (33.32)
£o initial concentration of supernuclei (11.10)
7J viscosity (10.54)
9 delay time of nucleation (15.99)
9' delay time of detectable nucleation (33.67)
0p delay time of nucleation at pre-existing clusters (24.60)
9» wetting angle (3.52)
i? time constant of overall crystallization (26.18)
1¾ time constant of filling of ith layer of thin film (34.4)
k specific edge energy of 2D cluster (3.69)
k' specific line energy of cap-shaped cluster (19.1)
A molecular heat of phase transition (3.31)
A; eigenvalue (15.9)
^ mean surface diffusion distance (10.30)
V* chemical potential of molecules in nucleus (4.4)
£/e equilibrium chemical potential (1.10)
jinew chemical potential of molecules in new phase (2.1)
£/newn chemical potential of molecules in n-sized cluster (3.6)
£/okl chemical potential of molecules in old phase (2.1)
Symbols and abbreviations xix
v growth exponent of supernucleus (26.15)
v, growth exponent of 2D supernucleus in ;th layer of thin
film (34.6)
71^ equilibrium surface pressure of insoluble monolayer
(35.8)
7C, surface pressure of insoluble monolayer (35.4)
p number density of molecules (8.1)
pnew number density of molecules in new phase (5.9)
ft*w number density of molecules in nucleus (5.10)
p0id number density of molecules in old phase (5.8)
a specific surface energy of interface between new and old
phase (3.18)
oj, specific surface energy of amphiphile bilayer (35.2)
c% effective specific surface energy (4.38)
<7j specific surface energy of interface between new phase
and substrate (3.53)
a, specific surface energy of interface between substrate and
old phase (3.51)
T time lag of nucleation (15.54)
Td mean time of desorption (10.28)
0 effective excess energy of cluster (3.86)
#* effective excess energy of nucleus (4.6)
#s effective surface energy of cluster (3.82)
&* effective surface energy of nucleus (4.5)
#i value of #at n = 1 (7.34)
0 total surface energy of interface between cluster and old
phase (3.6)
0* total surface energy of interface between nucleus and old
phase (4.12)
tj>s total surface energy of interface between substrate and
old phase (3.46)
tpsa effective strain energy per molecule of cluster (20.6)
X difference between actual and pre-existing
supersaturation (24.20)
*P activity factor of nucleation (4.42)
y/ Volmer function of wetting angle (3.52)
parameter of quasi-stationarity (17.8)
conm, co(«, m, t) frequency of coalescence between n-sized and m-sized
clusters (9.34)
EDS equimolecular dividing surface
HEN heterogeneous nucleation
HON homogeneous nucleation
IN instantaneous nucleation
KJMA Kolmogorov-Johnson-Mehl-Avrami
PN progressive nucleation
XX Symbols and abbreviations
ST surface of tension
TDE thermodynamic equilibrium
VDW van der Waals
ID one-dimensional
2D two-dimensional
3D three-dimensional
Part 1
Thermodynamics of nucleation
This Page Intentionally Left Blank
Chapter 1
First-order phase transitions
The physical nature of the phase transitions of first order can be illustrated
sufficiently well with the aid of an analysis of the behaviour of the van der
Waals (VDW) fluid which is characterized by the following equation of state
[Guggenheim 1957; Landau and Lifshitz 1976]
(P + M2a'IV2){V - Mb') = MkT. (1.1)
Here P and V are, respectively, the pressure and the volume of the fluid, M
is the number of molecules (or atoms) in this volume, T is the absolute
temperature, k is the Boltzmann constant, and a and b' are material constants
accounting, respectively, for the molecular interactions and the molecular
volume. These two constants can be expressed in terms of the critical pressure
PCT, volume Vcr and temperature 7cr of the VDW fluid by using the relations
[Guggenheim 1957; Landau and Lifshitz 1976]
PCY = a'121b'2, Vcl = 3Mb', TQI = %a'l21kb'. (1.2)
For this reason, with the help of the reduced pressure P' - P/Ptr, volume V
- V7Vcr and temperature T - TITCT eq. (1.1) passes into the reduced VDW
equation
(P'+ 3/V2)(3V - 1) = 81' (1.3)
which, according to the law of corresponding states, is universal in the sense
that it does not contain explicitly the material constants of the fluid
[Guggenheim 1957; Landau and Lifshitz 1976].
Let us now consider the VDW fluid when it is kept at constant temperature
T and pressure p, the respective reduced pressure being p = p/PCI. The fluid
will be in stable thermodynamic equilibrium (TDE) only when it occupies a
volume at which its Gibbs free energy G = F + pV is minimum. By definition
[Guggenheim 1957], the Helmholtz free energy F is given by F = - P(V)dV
so that using eq. (1.3) leads to
G(V,T') = GKf(j') + MkTCY[3p'V'i% - 9/8V - T In (3V - 1)] (1.4)
where Gref is a reference energy. From the conditions for minimum
{dGldV)T = {dFidV)T+p = -P(V) +/7 = 0
{d2G!dv\ = {^FidV2)r = -{dPidV)T> 0
it follows that the volume at which the fluid is in stable TDE is the solution
of the algebraic equation
P{V) = P' (1-5)
4 Nuclealion: Basic Theory with Applications
corresponding to such a portion of the VDW isotherm (1.3) for which (dPl
dV)T < 0. If the solution of eq. (1.5) falls in the range of volumes for which
(dP/dV')1> 0, it corresponds to unstable TDE (then G is maximum) and is
physically irrelevant.
Figure 1.1a shows P(V) isotherms of the VDW fluid calculated from eq.
(1.3) at a given subcritical temperature (T = 0.85, the solid curve) and at the
1.5
1.0 -
0.5.
0
0.55
o
;0.45
l
0.35
(5
■
•
■ liquid
a
- b
d
■
-
-
1 c
• /s s •■.
0 111f/^
\fe
v--«------
c
//\—^~^^-
a L#/
bf/
d*
liquid
. . . . i . .
.*;-<;
. 1 . .
-''
, 1 . ,
i
. . i .
gas
i
..I.
(a)
gas"
j """
(b)
O540
___^-
0.505
i
0.470
i i i
12 3 4
V
Fig. 1.1 Dependence of (a) the reduced pressure and (b) the Gibbs free energy of
VDW fluid on the reduced volume of the fluid: solid curves - eqs (1.3) and (1.4) at
p' = 0.470, 0.505, 0.540 (as indicated) and V = 0.85; dashed curves - eqs (1.3) and
(1.4) atp'- 1 and T'= 1. Nucleation-mediated first-order phase transition occurs
only for P\ V values corresponding to the shaded area between the binodal and
spinodal (the dotted curves B and S, respectively). The equality of the hatched areas
'bdef and fg hi' defines the position of the dot-dashed line 'bfi' of gas/liquid
coexistence.
Firsl-order phase transitions 5
critical temperature (T' = 1, the dashed curve). The respective G(V)
dependences at three different subcritical pressures ipr - 0.470, 0.505 and
0.540, the solid curves) and at the critical pressure (p'=i, the dashed curve)
are drawn in Fig. 1.1b according to eq. (1.4) with an arbitrarily chosen Gref
= 2MkTa. The descending branches of the subcritical P'(V) isotherm at
smaller and larger volumes and the minima of G on the left and on the right
correspond to more and less condensed fluid which can be called liquid and
gas, respectively. The thermodynamically unstable states of the fluid are
limited by the minimum e and the maximum g of the subcritical P(V)
isotherm. At the critical temperature points e and g merge into a single point
c (i.e. the two extrema of the f(V) dependence degenerate into a single
one), and connecting all points e and g for the subcritical temperatures
results in a dome-shaped curve called spinodal (the dotted curve S). The
ordinates of points e and g represent the values of the spinodal pressures
/>i'j(7") aI>d />g.s(7") of the liquid and the gas, respectively.
As seen from Fig. 1.1a, for values of p between the spinodal pressures
p'i,s(T') and p's,,(T') eq. (1.5) has three solutions, two of which are physically
interesting (points a and h, b and i, d and j), since they correspond to a liquid
or a gas in stable TDE. The minima of G for the liquid and the gas state,
however, are equally deep (see curve 0.505 in Fig. 1.1b) only at a certain
pressure />S(T), the respective reduced pressure being p'e = pJP„. That is
why, at p -pe the liquid and the gas can coexist as separate phases if they are
in contact with each other. The liquid and the gas are then in phase equilibrium,
and/7e is the equilibrium (more precisely, the phase-equilibrium) pressure of
the fluid. The reduced phase-equilibrium volumes V|'e (T) of the liquid and
Vg'_t(7") of the gas are those solutions of eq. (1.5) at p'= /¾ that correspond
to points b and i in Fig. 1.1a, and connecting all these points for the subcritical
temperatures results in another dome-shaped curve called binodal (the dotted
curve B).
In the general case when p'* p'e, but satisfies the condition p{r,<p' < p'ss,
the two states of the fluid in stable TDE (see points a and h and points d and
j in Fig. 1.1a, which represent graphically the respective two solutions of eq.
(1.5)) correspond to the two minima of G which, however, are of different
depth (Fig. 1.1 b). This means that while in the state of the deeper minimum
the fluid is in truly stable TDE; in the other state it is only in metastahle
TDE. The corresponding transition from metastable to truly stable
thermodynamic state is known as phase transition of first order, and the
usual process by which this transition begins is nucleation of the new phase
within the old (or the parent) phase. Since the metastable states of the VDW
fluid are in the region between the binodal and the spinodal (the shaded area
between curves B and S in Fig. 1.1a), clearly, it is meaningful to speak of a
first-order phase transition and, hence, of nucleation only when the p', V and
T values correspond to this region.
For the sake of completeness, we must note the well-known thermodynamic
definition of the first-order phase transitions according to which at the point
of phase equilibrium the chemical potentials of the old and the new phases
6 Nucleation: Basic Theory with Applications
are equal, but their derivatives of first order are not [Ehrenfest 1933]. Figure
1.2 depicts the pressure dependence of the chemical potential p = GlM of the
VDW fluid and its derivative (oju/o^»')r calculated at T - 1 (the dashed
curves) and T = 0.85 (the solid curves) from eq. (1.4) with the help of (1.3)
and the already used Gref = 2MkTcv. Indeed, it is seen that when T = 0.85, the
conditions for first-order phase transition are fulfilled at p = p'c = 0.505 (jig
and pi are tne chemical potentials of the gas and the liquid, respectively):
0.6
o
£ 0.4
-^.
i
0.2
0
1.5
Q.
"a 1.0
I-
~3. 0.5
■5"
n
-
,e
gas
gas\
9
/b, i
' /
,'
/'
\ \
'a ^
! 79
b
1
c ,
°*
. . 1 ,
(a)
liquid
(b)
liquid
0.5
1.0
1.5
Fig. 1.2 Pressure dependence of (a) the chemical potential and (b) the pressure
derivative of the chemical potential for VDW fluid at V = 0.85 (solid curves) and
7" = 1 (dashed curves) according to eqs (1.3) and (1.4). The dot-dashed line 'bi'
illustrates the difference between the volumes occupied by a VDW molecule in the
gas and liquid at phase equilibrium.
First-order phase transitions 7
Since the pg and p\ derivatives are equal to the volumes occupied at p'c by
a molecule in the gas and in the liquid, respectively [Guggenheim 1957;
Huang 1963], the above inequality means physically that at the point of first-
order phase transition the density of the VDW fluid changes abruptly. This
is illustrated by the dot-dashed line connecting points i and b in Fig. 1.2b.
Figure 1.2 shows also that at T = 1 both p and (dpldp')r are continuous
functions of />', which reflects mathematically the known fact that at or
above the critical temperature occurrence of first-order phase transition is
impossible.
Clearly, the equilibrium pressure pc is an important thermodynamic
characteristic of the fluid, for it determines the direction of the phase transition:
while for p > pc the gas-to-liquid transition takes place (the gas is then
metastable (Fig. 1.1b) andpis> p\ (Fig. 1.2a)), when p < pc the liquid-to-gas
transition occurs (the metastable phase then is the liquid (Fig. 1.1b) and pt >
p% (Fig. 1.2a)). To find /;c we can use the condition for equality of the values
of the two minima of G at p' - p',., which is equivalent to the condition
Vgi Pc) = P\( Pc )■ Setting equal the right-hand sides of G from (1.4) at />,, Vg'e
and p'c, V,'c, we obtain the exact formula
pi = (VI, - V{ey' ((8/3) 7"In [(3¾ - 1)/(3¾ - 1)]
-3(1/¾-1/¾)}. (1.6)
In view of (1.3), this equation can be represented in the equivalent form
p'c=(Kc-Kcr' \"p'{V)dv (1.7)
which expresses analytically the known Maxwell rule [Guggenheim 1957;
Huang 1963] for the equality of the hatched areas 'bdef and 'fghi' on the
P( V) diagram in Fig. 1.1a. Equation (1.7) is a general thermodynamic relation
defining the equilibrium pressure of whatever phase with a given equation of
state in the form of P(V) isotherm with VDW loops.
Equations (1.6) and (1.7) give p'e implicitly, since Vg'e and V(e themselves,
being the solutions of (1.5) at p = p'e, depend on p', and V. However, when
V is sufficiently less than 1 (i.e. for low enough subcritical temperatures),
we have V,'e« Vg'e = 8773p'e (the gas is nearly ideal) and V{fi = constant
= 1/2 (the liquid is almost incompressible and its reduced volume is close to
1/2, see Fig. 1.1a). Equation (1.6) then leads to the approximate expression
p't(T') = 167' exp [-(1 + 9/47')]. (1.8)
In view of (1.2), this expression can be given the form
pc(T) = (IkTIb') exp [-(1 + 2a'/3b'kT)] (1.9)
which allows approximate calculation of the phase-equilibrium pressure of
the VDW fluid when the material constants a and b' are known. Equation (1.9)
corresponds to the integrated Clapeyron-Clausius formula for the temperature
dependence of the vapour pressure of condensed phases [Glasstone 1956].
8 Nucleation: Basic Theory with Applications
The above considerations should be sufficient to illustrate the physical
nature of the phase transitions of first order. Although the conclusions and
the formulae obtained are the result of an analysis of a concrete system, the
VDW fluid, most of them are of general validity. Worth remembering is,
especially, that (i) first-order phase transition occurs when an old metastable
phase transforms into a new truly stable one, (ii) the thermodynamic limit of
metastability is at the spinodal point (if this exists) and it is, therefore,
meaningless to speak of a first-order phase transition and, hence, nucleation
when the values of the parameters used to characterize the state of the old
phase are not between the corresponding binodal and spinodal values, and
(iii) phase equilibrium or coexistence between the old and the new phases is
possible when the corresponding minima of the free energy of these phases
are equally deep, i.e. when
j"old,e = A'new.c — We- (1-10)
Here /j0id,e an£* ^new,e are> respectively, the chemical potentials of the old and
the new phases at phase equilibrium, and ^/e, the equilibrium chemical potential,
is the abbreviated notation for either of them.
Chapter 2
Driving force for nucleation
In the preceding Chapter we have seen that when p *■ p^ (but is limited
between p[s and />g-s), the VDW fluid can be in two states with differently
deep minima of G. Obviously, the transition from metastable (e.g. gas) to
truly stable (e.g. liquid) state occurs because of the necessity for the fluid to
occupy a lower-energy state. The same holds true for any other metastable
phase and for this reason, in general, the thermodynamic driving force for
the first-order phase transition and, hence, for the nucleation process, is the
quantity
A/i s (Gold - Gnew)/W s ji0|d - /imw (2.1)
known as supersaturation. Physically, the supersaturation A/y is the gain in
free energy per molecule (or atom) associated with the passage of the phase
from the minimum with higher Gibbs free energy Gola to the minimum with
lower Gibbs free energy Gnew (/Joid and /inew are, respectively, the chemical
potentials of the old and the new phases at the corresponding minima).
Equation (2.1) allows expressing Ap through experimentally controllable
parameters in various particular cases. As an example, let us consider the
transition of the VDW fluid from gas (old phase) to liquid (new phase),
which occurs at reduced pressure p' satisfying the condition p'e<p'< /?gs
(this corresponds to transition from point h to point a in Fig. 1.1). Using eq.
(1.4) yields
Gold = Grel (7") + Mtr„[3p'Vg78 - 9/8Vg'- 7"ln(3Vg'- 1)]
G«w = C„,(T') + MkT„[3p'V,'/8 - 918V,'- 7"ln(3V,'- 1)].
Here Vg (/>', T') and V{ (p', 7") are, respectively, the reduced volumes of the
gas and the liquid, at which G has a minimum (i.e. the abscissae of points h
and a in Fig. 1.1). Substitution of G0id and G„ew from above into (2.1) leads
to the exact formula
A/i = kT„{T'ln [(3VI'- l)/(3Vg' - 1)]
+ (9/8)(l/V,'- 1/Vg') + (3/8)//(Vg' - v;)}. (2.2)
With the help of (1.6) this formula can be given the equivalent form
A^ = *7"„(rin [(3¾ - 1)(3V,'- l)/(3Vg' - 1)(3¾ - 1)]
+ (9/8)(l/V,'- UV{e + 1/Vg'c - 1/Vj')
+ (3/8)(/7-^ - p'cVI, + pWc - p'V,')} (2.3)
10 Nucleation: Basic Theory with Applications
which shows that Ap - 0 at p' - p^. Indeed, at phase equilibrium there exists
no driving force for first-order phase transition and, hence, for nucleation
and then it is said that the old phase is saturated. Obviously, nucleation is
impossible as well when Ap < 0: the old phase is then undersaturated (its G
minimum is lower than that of the new phase and poid < ^new).
If T' is kept constant, Ap can be controlled through the pressure p' or the
volume V of the VDW fluid. In the former case the explicit dependence of
Ap on p' is obtained from (2.2) or (2.3) upon expressing Ve' and V,' as
functions of p', and Vg',, and V{f as functions of p'e by using the equation of
state, eq. (1.3), of the VDW fluid. In the latter case eq. (1.3) has again to be
used, but for representing p' and p'c in (2.2) and (2.3) as functions of Vg' and
Vg e, respectively. Tt is worth having an approximate formula for Ap valid for
low enough subcritical temperatures and for pressures sufficiently below the
spinodal pressure p'gs. Under these conditions 3Vg'» 1, 31^',,» 1,
V\ ~ Vi'e = Mv0/Vc, (the liquid is almost incompressible) and Vgp' =* Ve'e/>e
(8/3) 7" (the gas is nearly ideal). In view of the relation 3MkTcr = 8PcrV„,
it then follows from (2.3) that, approximately,
Ap = kTln (p/pe) - v0(p - pc)
= kT In (Vg,e/Vs) - knvUeIVgs)(VgJVs- 1) (2.4)
where v0 is the molecular volume.
Let us now see how Ap can be determined in some important cases of
nucleation during first-order phase transitions in real systems (see also [Bohm
1981]). In finding the respective formulae for A^i we shall use eq. (2,1) and
known expressions for the chemical potentials ^0id and pnew of the old and
the new phases.
(a) Condensation of vapours (p > pe)
In this case the dependence of the chemical potentials of the old phase
(vapours) and the new phase (liquid or solid) on the actual pressure/) of the
vapours is given by [Guggenheim 1957]
Aioid(P) = Ve + kT In <j>lpc) (2.5)
Px»tP) = P<!-i-V(ip-pc) (2.6)
wherepe(T) is the phase-equilibrium pressure (i.e. the pressure of the saturated
vapours of the liquid or the solid), and pe = p0u(pc) = p„ew(pc) is the chemical
potential of the vapours and of the liquid or the solid at phase equilibrium.
The above two expressions hold true provided the vapours behave as ideal
gas, and the liquid or the solid is incompressible. Combining (2.1), (2.5) and
(2.6) leads to (e.g. [Toschev 1973a])
Ap(p, T) = kT In (p/pc) -v0(p-pc) (2.7)
which in most cases can be approximated sufficiently accurately by
[Zettlemoyer 1969]
Ap(p, T) = kTln[p/pe(T)], (2.8)
since usually Vnpc « kT and the second summand in (2.7) is negligible.
Driving force for nuclealion II
Worth noting is the coincidence of eqs (2.4) and (2.7). It is not accidental,
since these two equations are derived under the same assumptions for the gas
and the liquid phases and since in many respects the VDW fluid is an acceptable
model for real fluids. That is why, if for the determination of Ap during
condensation of vapours into a liquid it is necessary to account for the
effects of gas non-ideality and/or liquid compressibility, instead of (2.7) or
(2.8) one can use the full equations (2,2) or (2.3), which allow for these
effects in the scope of the VDW model. The vapour non-ideality can be
taken into account [Guggenheim 1957] also by replacing the p/pe ratio in the
logarithm of eqs (2.7) and (2.8) with the ratio/v//ve of the actual and the
equilibrium fugacities/v and/ve of the vapours.
In condensation of molecular beams onto a substrate the experimentally
controllable parameter usually is the impingement rate / of molecules (per
second per m2) rather than the pressure p. As I = p/(2Km0kT)L'2, in this case
Ap is given approximately by (2.8) in the form [Sigsbee 1969; Lewis and
Anderson 1978]
A^(/, T) = kT\n[IlU{T)] (2.9)
where /e = p^i{2KmQkT)[!1 is the equilibrium value of the impingement rate,
m0 and T being, respectively, the mass of a molecule and the substrate
temperature. This formula is valid when thermal equilibration of the deposited
molecules with the substrate takes a sufficiently short time [Sigsbee 1969].
(b) Boiling, evaporation or sublimation (0 <p < pe)
In this case the old phase is the liquid or the solid, and the new phase is the
vapour. Accordingly, eqs (2.5) and (2.6) are to be used in (2.1) with exchanged
subscripts 'old' and 'new'. This leads again to Ap from eq. (2.7), but with
changed sign of its r.h.s.:
Apip, T) = kTIn {pjp) - v0(pe-p), (2.10)
p being the actual pressure of the liquid in boiling or of the vapour in
evaporation or sublimation. When u0 pe « kT, Ap can be approximated by
Ap(p, T)^kT\n{p,{T)ipl
(c) Condensation of solute (a > ae)
In this case the old phase is constituted of dissolved molecules which along
with a solvent (treated as an inert medium) represent a liquid or solid solution,
and the new phase is the liquid or solid condensate of the dissolved molecules.
Rather than the pressure, the experimentally controllable parameter usually
is the actual solute activity a, and the dependence of p0\d on a is of the form
[Guggenheim 1957]
fJoia(a) = fie + kT\ii(a/ae) (2.11)
where ae(T) is the equilibrium activity, i.e. the activity at which the solute
and the condensate are in phase equilibrium. Since pnew is practically a-
independent, approximately,
12 Nuclealion: Basic Theory with Applications
^new(a) = ^ncw(ae) = Vc- (2- > 2)
Combining the above two expressions with eq. (2.1) yields [Walton 1969a;
Sohnel and Garside 1992]
A/i (a, T) = kT In [alac(T)L (2.13)
For sufficiently dilute solutions a and ae can be replaced, respectively, with
the actual and the equilibrium concentrations C and Ce(T) of the solute (Ce
is also known as solubility) and (2.13) becomes [Nielsen 1964; Sohnel and
Garside 1992]
A/i(C, 7) = Win [C/QT)]. (2.14)
It is worth noting that eqs (2.13) and (2.14) cover also the case of decay
of solids by 'condensation' of atomic vacancies (the old 'phase') 'dissolved'
in them [Hirth and Pound 1963], This 'condensation' leads to the appearance
of macroscopic cavities (the new 'phase') in the solid. Then a, ae and C, Ce
are the actual and the equilibrium activities and concentrations of the vacancies
in the solid.
Equations (2.13) and (2.14) are applicable when the solute molecules in
the solution are not dissociated into ions. In many cases, however, this is not
the ease and then, according to the condition for chemical equilibrium
[Guggenheim 1957; Landau and Lifshitz 1976],
i"oid= Vi/i, + v2ti2+ . ..+ v«ut (2.15)
where vr is the number of ith species in the molecule (i = 1, 2, . . ., k), and
/i; is the chemical potential of the !th species in the solution. Using (2.11) in
the form
Hi(a,) = /jie + kT In (at/aie)
leads to the following expression for/iotd:
/1M = /ie + kTln [(a.Ke)'1 (a2la2,c)^ . . . (aklak^ ].
Here af is the actual activity of the ith species in the solution, aj(. and fi, e are
the equilibrium activity and chemical potential of the /th species, and
Pe = V|/ll,e + V2/l2,e +... + ^11^.
Thus, with the help of eq. (2.12) and the above expression for/i0id, it follows
from (2.1) that
A/i(n, 7) = /trin[n/ne(7)] (2.16)
where n = a,1"' aj2 ... a^1 and ne = fljjajl ■ ■ ■ "H are, respectively, the
actual and equilibrium activity products of the solute [Nielsen 1964; Sohnel
and Garside 1992],
As seen, eq. (2.16) is a generalization of eqs (2.13) and (2.14) (for sufficiently
dilute solutions n and Y\c are the products of the corresponding concentrations
of ith species, ne being the so-called solubility product). An interesting point
concerning eq. (2.16) is that even when the solution is supersaturated with
Driving force for nucleation 13
respect to only one of the species in the solution, but is saturated with respect
to the others (e.g. a1/ale> 1, a2/alx = a3la3l.= ... = aklak,.= 1), there exists
a driving force for condensation of the solute, since then Ay > 0. Moreover,
sufficiently high supersaturation of the solution with regard to one (e.g. axl
flie » 1) or more species can create a driving force (Ay > 0) even if the
solution is undersaturated with regard to one (e.g. a2laitti < 1) or more of the
rest of the species. Detailed considerations concerning Ay for crystallization
from solutions can be found elsewhere [van Leeuwen 1979; van Leeuwen
and Blomen 1979; SiShnel and Garside 1992].
(d) Dissolution (a < ae)
Analogously to the case of boiling, evaporation or sublimation, the driving
force for dissolution is that for condensation of solute, but taken with opposite
sign, as now the old and the new phases are reversed. That is why, according
to eqs (2.13), (2.14) and (2.16), for dissolution Ay is given by
Ay(a, 7) = Win [ac(T))a] = Win [Ce(T)/C] (2.17)
and, more generally, by
Ay(U, T) = Win [11,.(7)/11]. (2.18)
(e) Crystallization of melt or polymorphic transformation by cooling (T< 7e)
In this case the old phase is a melt or a crystal with a given modification, and
the new phase is a crystal (which has another modification in polymorphic
transformation). The commonly used parameter to control ^0]d experimentally
is the temperature rather than the pressure and, for that reason, it is convenient
to express Ay as a function of T. Isobarically, from thermodynamics
[Guggenheim 1957],
y(T) = y„+j' s(T')6T' (2.19)
where s is the entropy per molecule (or atom), and Te is the absolute phase-
equilibrium temperature (the melting point in melt crystallization). Using
(2.19) twice, for yM and ym„, and recalling (2.1) yields [Volmer 1939]
Ap.(7) = f ' As(T')r\T' (2.20)
where
As(7) = sM(T) - s„ew(T).
Obviously, the main problem in finding Ay is to know how the entropies
Sold "nd s„e„ of the old and the new phases depend on T. In the absence of
such a knowledge, following Bohm [1981], it is convenient to use the Taylor
expansion
Ay(T) = - Ase(7- 7"e) - (\l2)(dAs/dT)c(T- Tc)2
- (1/6)(^.5^),,(7-- 7C)3
of Ay from (2.20) in the vicinity of T= Tc where Ase, (dA47d7% and (d2As/
14 Nucleation: Basic Theory with Applications
d7"2)e are the values of As and its derivatives at T = Te. Recalling the known
relationship [Guggenheim 1957]
HT)= [ [cp(ryr]dr
Jo
between the entropy and the heat capacity (per molecule) cp at constant
pressure, we can employ it twice (for the old and the new phase) to find that
dAs/dT = Acp(D/T
tfAsldf1 = Ac; (T)IT - AcpfD/T2
where Acp = cpi0[d- cp>new, A cp = dAcp/dT, and cp oid and cp new are the molecular
heat capacities of the old and the new phase, respectively. With these expressions
for the As derivatives, a truncation of the above expansion of Aft results in
A^T) = AScAT~(AcvJ2Tc)Af1+ [(TcAc'pt - Acpx)/6T?] AT3. (2.21)
Here
AT=Tt~T (2.22)
is the undercooling, Ase = *0idC^e) ~ ^new(^e) = ^^e is the difference in the
molecular entropies of the old and the new phase at T- Te, X is the latent heat
(per molecule) of crystallization or polymorphic transformation, Acpc = cp,oid(^c)
- Cp.newfT'e) and Acp cs dAcp/dT at T = Tc.
Equation (2.21) reduces to the widely used formula [Volmer 1939]
A/((D = AseAT (2.23)
when the undercooling satisfies the condition AT « 27'eAse/Acp c = 2A/AcPJ>,
which is usually met in crystallization of metal melts or in polymorphic
transformation. For glass-forming, polymer and other more complex melts
eq. (2.23) is not always sufficiently correct, but for them it often suffices to
use (2.21) in the simpler form [Bohm 1981]
Af,(T) = AseAT-(AcpJ2Te)AT2. (2.24)
This equation is very similar to the one given by Jones and Chadwick [1971].
A number of other Afi(T) dependencies are also known in the literature (see,
e.g., [Hoffman 1958; Gutzow et al. 1985; Toner et al 1990; Kelton 1991;
Gutzow and Schmelzer 1995]).
Figure 2.1 shows the temperature dependence of A^i for Li20 ■ 2Si02 melt
for which 7e= 1306 K, Aje = 5.26¾ and AcPil!=1.56i [Kelton 1991], The
circles are the exact values computed by Kelton [1991] with the help of
measured Ac?(T) data, and the dashed and solid lines are drawn, respectively,
according to eqs (2.23) and (2.24). It is seen that in this case (2.24) describes
considerably better than (2.23) the actual dependence of Ap on AT.
(f) Melting or polymorphic transformation by heating (T > Te)
In this case all equations from case (e) remain in force taking into account
that now the old phase is the crystal and the new phase is the melt or the
Driving force for nucleation 15
Fig. 2.1 Dependence of the supersaturation on the undercooling for Li20.2Si02
melt: circles - experimental data [Kelton 1991]; circular contour - eq. (2.23);
solid line - eq. (2.24).
crystal with another modification. This means that everywhere in these
equations Ase, Acpe and AcJe have to be taken with opposite sign. Since
melting and polymorphic transformation often occur at T*= Te, in many cases
the approximation (2.23) is good enough:
AMr)=-AIcAT=AJe(T-re).
(2.25)
(g) Electrochemical deposition (<p < %)
In this case the old phase is ionic solute which along with a liquid or solid
solvent forms electrolytic solution, and the new phase is a liquid or solid
with a given Galvani potential (p. The role of the chemical potentials of the
old and new phases is now played by the respective electrochemical potentials
JiM and jjnw [Lange and Nagel 1935; Guggenheim 1957] so that rather
than by eq. (2.1), A^i is determined by
A" = Mold - Mnew (2.26)
The dependence of Ji„m on <p has the form [Lange and Nagel 1935;
Guggenheim 1957]
Mn™(<0) = Me + Z\e0(<p- <p,)
where zL is the valency of the solute ions, e0 is the electronic charge, and <pc
and Jic(T) are the phase-equilibrium Galvani and electrochemical potentials,
respectively. Since Jig]li is practically (^-independent, i.e.
16 Nucleation: Basic Theory with Applications
using the last two equations in (2.26) leads to the approximate formula
[Volmer 1939; Vetter 1967]
Afi(<p) = ZieoA<p (2.27)
where
Aq>= q>c~ q> (2.28)
is the overvoltage.
(h) Electrochemical dissolution {<p > <pc)
Analogously to non-electrochemical dissolution (case (d)), in this case /7oid
and ,unew have to be exchanged in eq. (2.26) and for that reason the
supersaturation is again given by eq. (2.27), but with opposite sign of its
r.h.s.:
Av(<p) = zM<p-<Pe)- (2-29)
Summarizing, we see that A^/ is determined always by eq. (2.1) or its
appropriate generalization (e.g. eq. (2.26)). The difficulties in the determination
of A^/ arise from incomplete knowledge of the exact dependences of ^0id and
H„ew on the respective experimentally controllable parameters. It must be
emphasized that the correct determination of A^/ in each concrete case is of
great importance for the reliable confrontation of theory with experiment.
For instance, in the case of melt crystallization the use of different approximate
formulae for A^/ can lead to considerable differences in the calculated values
of some nucleation parameters [Kelton 1991].
Chapter 3
Work for cluster formation
Setting the old phase in supersaturated state is a necessary condition for the
occurrence of first-order phase transition, but it is not yet a guarantee that
such a transition will really take place within a given time interval. The
ability of the supersaturated old phase to remain a certain time in the state of
metastableTDEis one of the most remarkable features of the phase transitions
of first order and is due to the fact that the metastable state is separated from
the truly stable one by an energy barrier. Besides, a differently high barrier
corresponds to each concrete path along which the transition may proceed
and it might be expected that the actual path of the process will be the one
requiring the lowest energy expenses. The questions thus arise: what is this
path? and what is the height of the respective energy barrier?
A most natural path for the phase transition may seem the one corresponding
to the uniform change of the density of the old phase into the density of the
new phase. This situation is illustrated in Fig. 1.1 in the case of VDW fluid.
If the old phase is a supersaturated (i.e. metastable) VDW gas under pressure
p' = 0.54 at temperature V = 0.85, it occupies a volume corresponding to
point h. In order to pass into the stable state of VDW liquid under the same
pressure, the gas must diminish its volume up to the volume corresponding
to point a. If this is to occur (as assumed) along the path of spatially uniform
change of the gas density, the gas volume should decrease and thus cause a
variation of the Gibbs free energy G of the fluid in accordance with eq. (1.4)
represented by the uppermost curve in Fig. 1.1b. Initially, however, the
transition will be impeded by an increase of G. Only on the left of the
maximum of G will the transition be spontaneous, since then the volume
decrease is accompanied by a lowering of G. For that reason, the difference
between the maximum value of G and the value of G at point h is the energy
barrier to the phase transition along the path of uniform change of density.
Figure 1.1b shows that at the chosen values of/?' and T' this energy barrier
is approximately 0.03M£7"cr or, more generally, about MA/i, since A/i ~ 0.03kTC!.
at/?' = 0.54 (see Fig. 1.2a). This large amount of energy, some 2% of the total
kinetic energy {3/2)MkTcrT' of the VDW gas at T = 0.85, must be spent by
the system only because all M molecules in it take part in the phase transition.
It is clear, therefore, that it is highly improbable for the system to follow the
path of spatially uniform change of its density because of the energetically
high price of this path.
Another conceivable path for the phase transition, which in the light of
the aforesaid seems energetically much less expensive, corresponds to a
non-uniform change of the density of the old phase into the density of the
18 Nucleation: Basic Theory with Applications
new one. Suppose that such a change occurs locally as a density fluctuation
in a spatial region occupied by a small number of molecules (say, up to a few
hundreds). Then, if «* « M is the characteristic number of molecules in the
spatial region of locally changed density, the expectation based on the above
considerations (and confirmed by the analysis in Chapter 4) is that the energy
barrier for the phase transition will be of the order of n*A/i. This is much less
than the energy barrier of about XfAp for the path corresponding to uniformly
changing density and means that, in general, first-order phase transitions are
much more likely to occur by local density fluctuations than by uniform
change of the density of the old phase as a whole. Numerous everyday
observations (recall rains and snowfalls) and experimental investigations
lead categorically to the conclusion that this indeed is the path followed in
reality by a first-order phase transition: nanoscopical formations with density
close to that of the new phase appear randomly in the old phase and the
overgrowth of these precursors of the new phase to larger (usually macroscopic)
sizes brings to an end the phase transition. For this reason, our next
considerations will be devoted solely to phase transitions occurring by local
changes in the density of the old phase. The analysis will be focused on the
initial or the nucleation stage of the phase transition, during which the precursors
of the new phase appear in the system as a result of density fluctuations.
Nucleation itself is the process of random generation of those nanoscopically
small formations of the new phase that have the ability for irreversible
overgrowth to macroscopic sizes.
Given the path along which the phase transition takes place, we must now
turn to the question about the energy barrier (if any) along this path. This
question can be answered, however, only upon adopting a concrete approach
for describing the tiny precursors of the new phase. Different approaches can
be used for such a description, but the two most known of them are the
cluster approach and the density-functional approach. In the cluster approach
the nanoscopically small formation of the new phase is considered as a
cluster of a certain number n of molecules (or atoms) in it (Fig. 3.1a). The
cluster itself is regarded as separated from the old phase by a phase boundary
and that makes it possible to say which of all M molecules in the system are
still in the old phase and which of them already belong to the new phase. In
the density-functional approach the state of the system is described with the
aid of the number density p (m~3) of the molecules (or the atoms) as a
function of the space coordinates (the solid curve in Fig. 3.1b), No phase
boundary is assumed to separate the spatial regions with higher and lower
density and, for that reason, it is not known whether a given molecule belongs
to the old or the new phase. The cluster approach is very perspicuous and has
been used already by the pioneers of the nucleation theory [Gibbs 1928;
Volmer and Weber 1926; Farkas 1927; Kaischew and Stranski 1934a; Becker
and Dbring 1935]. It has been extensively developed in numerous studies on
nucleation and has led to virtually all basic results known hitherto. The
density-functional approach is relatively new and was applied to nucleation
first by Cahn and Hilliard [1959]. It is considerably more formal, but is free
Work for cluster formation 19
(a)
■.■■m
phase boundary
(b)
phase boundary
Fig. 3.1 Nanoscopical formation of new condensed phase according to
(a) the cluster, and (b) the density-functional approach. The dashed line
visualizes the stepwise change of the molecular density at the arbitrarily
positioned phase boundary (the dashed circle) between the cluster and the
old phase. The solid circles schematize molecules.
of some weak points of the cluster approach and, apparently, has the capacity
to produce important new results. We shall limit all further considerations
entirely in the scope of the cluster approach and will describe the density-
functional approach only briefly in Chapter 8. Also, unless especially noted,
the considerations will be restricted to one-component nucleation. Results
20 Nucleation: Basic Theory with Applications
concerning the thermodynamics of multicomponent nucleation can be found
elsewhere (e.g. Neumann and Doting [1940]; Reiss [1950]; Sigsbee [1969];
Nishioka and Kusaka [1992]; Wilemski and Wyslouzil [1995]).
3.1 Homogeneous nucleation
Let us now consider the so-called homogeneous nucleation (HON) occurring
when the clusters of the new phase are in contact only with the old phase and
with no other phases and/or molecular species. Nucleation of droplets in the
bulk of ideally pure supersaturated vapours is a classical example of HON.
Figure 3.2 depicts schematically an old phase of M molecules (or atoms)
in its initial state 1 when it is of uniform density and has Gibbs free energy
G, = MfiM
(3.1)
STATE 1
STATE 2
Fig. 3.2 Old phase ofM molecules before (state 1) and after (state 2) homogeneous
formation of a cluster ofn molecules.
and in its final state 2 when it contains a cluster of n molecules (n = 1,2,
. ..) and has Gibbs free energy G2. Obviously, assigning a definite value of
the cluster size n is related to a concrete choice of the position of the so-
called dividing surface between the cluster and the old phase (see the circular
contours in Figs 3.1 a and 3.2). This surface is a mathematical device introduced
first by Gibbs [1928] to account for the presence of the density profile
appearing as a result of the difference in the densities of the old and the new
phases (Fig. 3.1b). There exist various possibilities to choose the position of
the dividing surface [Gibbs 1928; Guggenheim 1957; Rusanov 1967, 1978;
Ono and Kondo 1960; Toschev 1973a], but once the choice is made and the
uniform densities on the two sides of the surface are fixed, n becomes a well-
defined quantity. As noted by Toschev [1973a], most convenient for analysing
the thermodynamics of one-component nucleation is the so-called
equimolecular dividing surface (EDS) This surface is the one which is
positioned to satisfy the condition for equality of the sum of the number of
molecules inside and outside the cluster to the total number of molecules in
the system (in Fig. 3.1b this means equality of the areas under the dashed
Work for cluster formation 21
and solid lines). Whatever the choice of the dividing surface, however, G2
can always be represented as (n = I, 2, . . .)
G2(«) = (il/- nfcioW + G<«). (3.2)
This formula was used by Kaischew [1951] in the particular case of crystal
clusters. In it the first summand is the Gibbs free energy of the remainder of
the old phase surrounding the cluster. This is so because we consider a
system at constant temperature Tand pressure/? and, hence, with unchanged
chemical potential jiold of the old phase in states 1 and 2. The quantity G(n)
is the Gibbs free energy of the n-sized cluster and takes into account the
energy changes in the system accompanying the formation of the cluster. In
many cases, it is more convenient to refer G{n) to the Gibbs free energy
rifinew which the cluster would have had if it were a part of the bulk new
phase at the given Tand/?. This means expressing G{n) in the form (n = 1,
2, . . .)
G(n) («) (3.3)
where Gcx(n), the cluster excess energy, remains to be determined in each
particular case.
Using the definition equality
Win) = G2(n) - G, (3.4)
and eqs (2.1), (3.1)-(3.3), we can now represent the work W(n) for
homogeneous formation of an n-sized cluster by the following general formula
(n =1,2,...)
Win) = -nAv + Gex(«). (3.5)
As seen, knowing the cluster excess energy Gex is of key importance for
the determination of W, which is the quantity of interest in this chapter. For
that reason, finding Gex is one of the major problems in the theory of nucleation
and we shall now see how this problem can be tackled in the limiting cases
of large enough or small (down to n = 1) clusters. Before doing that, however,
two points have to be noted. First, as it is more convenient, the cluster of size
n - 1 will be treated hereafter formally as distinguishable from the monomer
molecule of the old phase. This means that in (3.5) W(l) * 0. Second, it
should be kept in mind that given the size n of the cluster, Gex depends also
on the shape of the cluster and on A^/, since these affect, e.g., the area and the
structure of the cluster/old phase interface and, thereby, the energy contribution
of this interface to Gex. In all considerations that follow we shall ignore the
dependence of Gex on the cluster shape, assuming that for a given size n, the
cluster has only one shape, not necessarily coinciding with its shape for
another size (e.g. size n - 1 or n + 1). For concreteness, it can be thought that
the specified shape is the statistically most probable one, i.e. the shape which
minimizes Gex. By definition, this shape is known as the equilibrium shape
and is the one which is most widely used in the theory of nucleation [Kaischew
1950, 1951; Zettlemoyer 1969; Toschev 1973b; Mutaftschiev 1993],
22 Nucleation: Basic Theory with Applications
Let us now see how Gex can be determined in the limiting case of sufficiently
large cluster size n. In this case it is possible to arrive at a general analytical
Gex(«) dependence with the help of thermodynamic considerations, since the
cluster may be viewed as a distinct, albeit small, phase with well-defined
thermodynamic properties. Gibbs [1928], Volmer and Weber [1926], Farkas
[1927], Kaischew andStranski [1934a] and Becker and Doring [1935] were
the pioneers who initiated the theory of nucleation by analysing this limiting
case. The theoretical results concerning this limiting case form what is nowadays
known as the classical theory of nucleation.
To find out what physical quantities contribute to the cluster excess
energy Gex we need an explicit expression for W in order to be able to
compare it with W from eq. (3.5). We first invoke the thermodynamic relation
[Guggenheim 1957]
G2(n) = F2(n) + pV
where F2 and V are, respectively, the Helmholtz free energy and the volume
of the system in the final state 2 (Fig. 3.2), and p is the fixed pressure of the
old phase. Then, as it is most convenient, we choose the EDS as a dividing
surface between the cluster and the old phase and represent F2 in the form
[Kaischew and Mutaftschiev 1962; Rusanov 1967, 1978; Toschev 1973a;
Abraham 1974a]
F2(n) = (M-n)pM -pVM_„ + npmv,„-pnVn + <j>(V„). (3.6)
Here the first couple of summands is the Helmholtz free energy of what has
been left of the old phase after the cluster appearance {VM_„ is the volume
occupied by the M - n molecules still in the old phase), the second pair of
summands is the Helmholtz free energy of the n-sized cluster (p„ and V„ are,
respectively, the n-dependent pressure and volume of the cluster, p-„c„j,(p„)
is the chemical potential of the molecules in it), and 0 is the total free energy
of the cluster/old phase interface. Combining the above two equations and
taking into account that V - VM_„ + V„ yields
G2(n) = (M- n)pM + npmv - (pn - p)V„ + ^(V„). (3.7)
Thus, upon substituting this expression for G2 in (3.4) and allowing for (3.1)
we find that in the case of HON under isothermal-isobaric conditions the
work for formation of an EDS-defined n-sized cluster is given most generally
by [Kaischew and Mutaftschiew 1962; Rusanov 1967, 1978; Toschev 1973a;
Abraham 1974a]
W(n) = - (pn-p)Vn + Ow, - A-oid)" + 0(V„). (3.8)
Recalling eq. (2.1), we can now compare the right-hand sides of this
equation and of eq. (3.5) to obtain finally that in the case of HON
Ges(«) = ¢(V„) - (p„ -p)V„ + (pac„,„ - /<new)«. (3.9)
This general formula is valid when the EDS is the dividing surface between
the cluster and the old phase and for such large values of n and Vn which
Work for cluster formation 23
ensure thermodynamically well-defined pressure pn in the cluster and chemical
potential ^incw, of the molecules in it. It shows that the cluster excess energy
GeK accounts for three effects: (i) the existence of interface between the
cluster and the old phase, (ii) the changed pressure in the volume Vn after the
appearance of the cluster, and (iii) the difference in the chemical potential of
the molecules in the cluster and in the bulk (i.e. infinitely large) new phase.
The first effect reflects the change of the molecular interactions in the interface
region, and the last two are due to the fact that the cluster is a phase of finite
size (see Chapter 6). In the final reckoning, however, all these effects can be
quantified in terms of the cluster total surface energy 0. Indeed, from the
condition dG2lt)Vn = 0 for mechanical equilibrium of the system in state 2,
with G2 from eq. (3.7) there follows the generalized Laplace equation [Toschev
1973a]
p„ = p + d^/dV„. (3.10)
Since by definition [Guggenheim 1957]
/W«(/>») = ,"new(/0+ (!/«) |"V„(P)dP, (3.11)
in view of (3.10) the difference of the chemical potentials in (3.9) also
proves to be a function of tj> through its derivative d0/dV„. Thus, finally, we
arrive at the following dependence of Gex on n (through p„ from (3.10) and
G„(n) = 0(V„)-(p„-p)V.+ P V„(P) dP. (3.12)
JP
This relationship tells us that for the determination of the cluster excess
energy GeK it is necessary to know not only the cluster total surface energy
¢, but also the equation of state, V„ = V„(p„, T), of the cluster. Since this
equation is different for condensed (liquid or solid) and gas phases, eq.
(3.12) leads to two different expressions for G6X in these two basic cases.
Indeed, if the cluster is a condensed phase, it can be regarded as practically
incompressible so that the volume occupied by a molecule in the cluster is
approximately independent of the pressure p„ inside the cluster and equal to
the molecular volume u0, i.e.
Vn(Pn) = nv0. (3.13)
If the cluster is a gas phase, in ideal-gas approximation V„ depends on p„
according to the equation of state
V„(pn) = "kT/pn. (3.14)
Using these two equations in the integral in (3.12) yields
Gex(«) = 0(V„) (3.15)
for clusters of condensed phases and
24 Nucleation: Basic Theory with Applications
GJjt) = 0(V„) + nWrin (pjp) - (1 -plpn)} (3.16)
for clusters of gas phases ipn depends on 0 according to (3.10)).
As seen, while in the formation of liquid or solid clusters (e.g. droplets or
crystallites), to a good approximation, Gex is merely equal to the cluster total
surface energy 0 because of the virtually complete cancellation of the last
two summands in (3.12), in the formation of gas clusters (e.g. bubbles) Ge>
depends in a much more complicated way on 0. The explicit determination
of the Ga(n) dependence requires a concrete model for 0 and this is the last
step that remains to be done in the scope of the classical theory of nucleation.
For a given choice of the dividing surface (and for the EDS in particular),
it is always possible to consider <p as proportional to the area Sn of the
dividing surface (i.e. of the surface of the n-sized cluster). Physically, the
proportionality factor <Jn (J m-2) is the orientationally averaged [Dunning
1969; Walton 1969a] specific surface energy of the cluster/old phase interface.
If the clusters are assumed to be regularly (e.g. spherically or polyhedrally)
shaped, we have S„ = c„ V„2'3 and 0 can be written down as
0(V„) = c„o„ V," (3.17)
where c„ is a numerical shape factor. The idea to employ <7„ for the determination
of 0 is the reason for which the classical theory is also called the capillarity
theory of nucleation. Equation (3.17) is easy to use within the approximations
cn = constant = c (independently of its size n, the cluster has the same shape)
and a„ = constant = <J(o„ is identified with the specific surface energy aof
the planar new phase/old phase interface, i.e. with the limiting value of a„ at
n —> °o, which is independent of the choice of the dividing surface [Ono and
Kondo 1960]). It must be noted that since thermodynamics predicts a change
of a„ with decreasing n (see Section 6.6), the latter approximation may be
rather poor for clusters of less than, say, a few scores of molecules. With the
above approximations for c„ and a„ eq. (3.17) simplifies to
0(V„) = Ctrl',2'3 (3.18)
where c = (36;r)"3 for spheres, c = 6 for cubes, etc. As a is independent of
n (and, hence, of V„), the derivative in (3.10) is readily calculated with the
help of (3.18) and the pressure inside the n-sized cluster is found to increase
with decreasing Vn according to
pn = p + 2ca/iv"\ (3.19)
If the cluster total surface energy 0 has to be expressed explicitly as a
function of n, eq. (3.18) shows that it is necessary to know the dependence
of Vn on n. This dependence is very simple for clusters of condensed phases
and is given by eq. (3.13). Upon using it in eq. (3.18) it follows that in this
case
0(n) = aonm (3.20)
where a = cd2/3 equals numerically the surface area of the smallest cluster
of size n - 1 provided this has formally the shape assumed for the larger
Work for cluster formation 25
clusters (e.g. a = (36.ru2)"3 for spheres, a = 6v^n for cubes, etc.). In the
case of gas-phase clusters, however, V„ can be expressed simply as a function
of n only implicitly upon combining eqs (3.14) and (3.19):
n = (p)kT)V„ + (2ca/3CT) V„2'3. (3.21)
That is why, in this case a simple representation of 0 from (3.18) as an
explicit function of n is impossible.
It is worth noting that for spherical clusters, no matter whether the new
phase is condensed or gaseous, <p from (3.18) can be expressed in the same
very simple way as a function of the cluster radius R. Indeed, as for spheres
c = (36tf)"3 and
V„ = (4/3)tcr\ (3.22)
eq. (3.18) transforms into the known formula [Zettlemoyer 1969]
0(«) = 4xoR2. (3.23)
Again for spherical clusters of both condensed and gas phases, using (3.22)
in (3.19) results in the familiar Laplace equation [Laplace 1806]
/>„=/> + la/R. (3.24)
Knowing tj> and p„, we can now write down the cluster excess energy Gex
in the scope of the classical theory under the limitations of n-independent
shape factor c„ and specific surface energy C7M. According to eqs (3.15),
(3.18), (3.20) and (3.23), for clusters of condensed phases
Gcx(Vn) = ccjV213 (3.25)
Gex(n) = acm213, (3.26)
and if the clusters are spherical,
Ga(R) = 4KOR2. (3.27)
For clusters of gas phases, from eqs (3.14), (3.16), (3.18), (3.19) and
(3.22)-(3.24) it follows that
Gex(V„) = (1/3)«tV„2'3 +pV„(\ + 2ca/ip V„"3) In (1 + 2ca/ip V„"3)
(3.28)
Gsx(n) = (l/3)c(T(A:7//7„)2'3n2'3 + nkT In {p„/p), (3.29)
and if the clusters are spherically shaped,
Ges(R) = (4nS)ciR2+ (An/i)pR\\ + lalpR) In (1 + lalpR). (3.30)
The explicit Gex(«) dependence is rather complicated and can be obtained by
using eqs (3.19) and (3.21) to express p„ in (3.29) as a function of n.
The above formulae for the cluster excess energy Gex are valid for sufficiently
large values of n (i.e. for n —> °°). Let us now see how Gex can be determined
in the opposite limiting case of n —» 1. In the range of these small sizes many
thermodynamic concepts and results are inapplicable and the determination
26 Nucleation; Basic Theory with Applications
of Gex requires model considerations at the molecular level. For that reason
it is hard to arrive at a general analytical dependence of Gcx on n for n —» 1.
Walton [1962, 1969b] was the first to use such model considerations in
formulating the so-called atomistic theory of nucleation whose roots can be
traced in the pioneering works of Stranski and Kaischew [1934], Kaischew
and Stranski [1934a], Stranski [1936] and Volmer [1939]. This theory was
developed further by Milchev et al [ 1974] in an analysis of electrochemical
nucleation.
Thus, the question is: what is the analogue of eqs (3.15) and (3.16) in the
framework of a general atomistic theory of HON? The answer to this question
is not known for gas-phase clusters, but for clusters of condensed phases it
can be shown that (3.15) remains approximately valid also for n —^ 1. Indeed,
let «0,d and wnew be the potential energies of a molecule in the bulk of the old
and the new phase, respectively,«, be the potential energy of the ith molecule
in the n-sized cluster, and U„ = Z «,- be the potential energy of the n-sized
cluster as a whole. Then, it is convenient to introduce the quantities X > 0 and
En > 0 defined by (n = 1,2, . ..)
X = «old -«new (3.31)
E,i = ««0|d - U„ = Z (wold - «,). (3.32)
For liquid or solid clusters, to a good approximation, A is the work to transfer
a molecule from the bulk of the new phase into the bulk of the old one, and
E„, the binding energy of the n-sized cluster, is the work for dissociating the
cluster into n single molecules in the old phase. While A is approximately,
e.g., the molecular heat of evaporation or sublimation in condensation of
vapours into a liquid or solid, the heat of dissolution in formation of condensed
phases from solutions, the heat of melting in crystallization of melts, etc., E„
is an unknown function of n, which can be found by model considerations at
the molecular level [Stranski and Kaischew 1934; Kaischew and Stranski
1934a; Volmer 1939; Kaischew 1965; Walton 1962,1969b; Lewis and Anderson
1978; Stoyanov 1979]. Regardless of the model for E„, however, in HON we
have Ul = uold so that from eq. (3.32) it follows that Ex = 0 (a single molecule
of the old phase cannot be dissociated into more molecules).
Let us now consider a given n-sized cluster of condensed phase (i) as a
real cluster in the bulk of the old phase and (ii) as an imaginary 'cluster' in
the bulk of the new phase. If the cluster is dissociated into n single molecules
in the old phase, the work done will be En and nX in the former and the latter
case, respectively. As in the latter case there is no phase boundary between
the imaginary 'cluster' and the new phase (the 'cluster' is just a part of the
large new phase), the difference between the above two works will be due
mainly to the existence of the cluster/old phase interface of the real cluster.
For that reason, approximately, the total surface energy 0 of the cluster will
be given by the formula (n = 1, 2, . . .)
0(n) = nA-E)( (3.33)
proposed first by Stranski [1936] for crystal clusters. Furthermore, since for
Work for cluster formation 27
a liquid or solid n-sized cluster the difference in the energies due to the
vibrations of the molecules in the cluster and of n molecules in the bulk new
phase can be neglected in respect to the difference U„ - n«new in the potential
energies of the cluster and these n molecules [Volmer 1939; Kaischew 1965],
to a first approximation,
G(n) - n^i„ew = U„ - n«„ew.
Hence, thanks to eqs (3.31)-(3.33), this formula can be rewritten as (« = 1,
2, . . .)
G(n) = n^new + 0(n) (3.34)
which, compared with eq. (3.3), leads to (n = 1, 2, . . .)
Gss(«) = <l>(n). (3.35)
This equality tells us that for clusters of condensed phases identification
of Gex with 0 down to the smallest cluster sizes is really an acceptable
approximation. Thus, for such clusters, from eqs (3.33) and (3.35) it follows
that in the framework of the atomistic theory Gex can be expressed as (n - 1,
2, . . .)
G„(n) = nX- E„. (3.36)
It should be emphasized that while eq. (3.26) of the classical theory is valid
for n —> °°, eq. (3.36) holds true not only for n —> 1, but for any number n of
molecules in the cluster. Therefore, comparison of the right-hand sides of
these equations shows that for n —» °o the cluster binding energy En is a
function of n of the following general form:
E„ = An- aan2,i. (3.37)
For crystal clusters, this dependence of £„ on n has been derived by Kaischew
[1937] with the help of model considerations at the molecular level. The
same dependence can be extracted from results of Abraham [1974a] concerning
the potential energy of spherical clusters.
Finally, using in eq. (3.5) the so-obtained expressions for Gex yields the
sought work W for cluster formation in HON. According to the classical
theory, in view of eqs (3.25)-(3.27), for clusters of condensed phases the
result is [Zettlemoyer 1969]
W(V„) = -(Ati/v0)Vn + coV™ (3.38)
W(n) = -nAn + aan211 (3.39)
and, if the clusters are spherical,
W(R) = - (4^/311,,)^3 + 4xoR2. (3.40)
For clusters of gas phases the respective dependences are considerably more
complicated; according to eqs (3.28)-(3.30) it follows that [Kaischew and
Mutaftschiev 1962; Blander 1979]
28 Nucleation: Basic Theory with Applications
W(Vn) = -[(A\iJkT) - In (1 + 2c<yBPvr)](p + 2ecJ/3V„"3) V„
+ (1/3) coV™ (3.41)
W(n) = -n[An - kl In (j>„)p)] + (l/3)ca(A:T/p„)Mn2'3 (3.42)
W(R) = -(4n/i)[(AtilkT) - In (1 + 2a/pR)](p + 2alR)Ri
+ (4n/3)oR2. (3.43)
The last formula is for spherical clusters, and the explicit W(n) dependence
can be obtained from eq. (3.42) upon expressing p„ as a function of n with
the aid of eqs (3.19) and (3.21). When concrete expressions for the
supersaturation A^i (see, e.g., Chapter 2) are used in them, eqs (3.38)-(3.43)
describe various particular cases of HON.
Analogously, in the framework of the atomistic theory, from eqs (3.5) and
(3.36) we find that (n = 1, 2 )
W(n) = -(Au - X)n - E„, (3.44)
but only for clusters of condensed phases. For gas-phase clusters the result
is not known, but it may be expected to be more complicated because of the
complexity of eq. (3.42) in comparison with eq. (3.39). Equation (3.44) is
the general atomistic formula for W{n) in HON of liquid or solid phases.
With properly defined supersaturation A^i (see, e.g., Chapter 2), this formula
is applicable to HON in vapours, solutions, melts, etc.
The above considerations show clearly that the determination of W(n)
hinges on our knowledge of the cluster excess energy Gex(n). Different attempts
were made to find Ga(n) by modifying the classical and the atomistic theories,
but we shall only note the works of Lothe and Pound [1962] and Fisher
[1967a, b]. Following Frenkel [1955], Lothe and Pound [1962] accounted
for the rotational and translational motions of the n-sized cluster in the old
phase and introduced a corresponding energy contribution into the classical
equation (3.26) for G„. They found that this contribution is quite large for
condensed-phase clusters in vapours. Presently, there exist various pros and
cons regarding the Lothe-Pound theory and the issue is still debated (see,
e.g. Zettlemoyer [1969, 1977]; Mutaftschiev [1993]; Reiss et al. [1997];
Ford [1997a]). In analysing condensation near the critical temperature Tc„
Fisher [1967a, b] introduced another energy contribution in the classical
expression for G„. This contribution characterizes the so-called Fisher droplet
model which was used for improving the predictive ability of the classical
theory and for extending the applicability of this theory to nucleation at
temperatures close to the critical temperature [Eggington et al. 1971; Kiang
et al. 1971; Stauffer and Kiang 1977; Dillmann and Meier 1989, 1991;
Delale and Meier 1993; Ford et al. 1993; Kalikmanov and van Dongen 1993,
1995; Laaksonen et al. 1994; Ford 1997b],
Summarizing, we see that given the value of the supersaturation A^i in the
system, the classical theory makes it possible to calculate W as a function of
n or V„ if one knows only one physical parameter - the specific surface
energy o of the planar old phase/new phase interface (of course, the cluster
shape and density must also be known). This is the principal advantage of
Work for cluster formation 29
this theory and its application cannot run into particular problems when
nucleation is mediated by large enough clusters. In nearly all practical cases
of nucleation, however, the W(n) dependence is of importance for rather
small values of n, e.g. for n < 100 and often even for n < 10. For such small
cluster sizes it is correct to use the atomistic theory, but this theory has the
great disadvantage of offering no analytical W(n) dependence: according to
eq. (3.44), it merely replaces one unknown function, W(n), with another
unknown function, En. It is clear, therefore, that finding a sufficiently general
analytical formula for En, which passes into (3.37) for n —» «\ is an important
problem of the future development not only of the atomistic theory per se,
but also of the nucleation theory as a whole.
The curves in Fig. 3.3 illustrate (as indicated) the classical dependences
(3.39) and (3.42) of Won n for HON of water droplets in vapours at T= 293 K
100 | ,
n
Fig, 3.3 Dependence of the work for cluster formation on the cluster size: curve
'droplet' - eq. (3.39) for HON of water droplets in vapours at T = 293 K and
p/pe = 4; curve 'bubble' - eq. (3.42) for HON of steam bubbles in water at
T= 583 K and p^p = 4.
and/? = 4pe and of steam bubbles in water at T= 583 Kand/7 = (l/4)/?e. The
calculations are carried out with the help of the parameter values listed in
Tables 3.1 and 3.2, respectively. According to eqs (2.7) and (2.10), the above
p values correspond to A/; = \AkT. A point of greatest importance is that for
both droplets and bubbles W passes through a maximum at a given cluster
size n*. Physically, the value W* = W(n*) of this maximum is the energy
barrier for occurrence of the first-order phase transition along the path of
local, non-uniform change of the density of the old phase. Therefore, in the
scope of the classical nucleation theory we can now answer quantitatively
the question at the beginning of this section about the height of this barrier.
30 Nucleation: Basic Theory with Applications
Table 3.1 Values of various quantities used for calculation of different dependences
for nucleation of water droplets in vapours.
Quantity
«0 (kg)
»„ (nm3)
d0 (nm)'
ft (kPa)
a (mj/m2)
T.
r=259K
3 x iO~26
0.03
0.39
0.20
77.6
1
r=273 K
3x 10-26
0.03
0.39
0.61
76
i
7=293 K
3x 10^26
0.03
0.39
2.3
73
i
'calculated from d0= (6uo/ii)1/3
Table 3.2 Values of various quantities used for calculation of different
dependences for nucleation of steam bubbles in water at T = 583 K.
m0(kg) fo(nm3) PeCMPa) o(mJ/m2) y„
3xlO~26 0.043 9.9 12 1
As can be read from Fig. 3.3, despite the difference in the respective values
of n* and W*, for both droplets and bubbles W* < n*A/i. Comparison of this
energy with the energy WA/i estimated as necessary for the phase transition
to follow the path of uniform change of the old-phase density shows that it
is indeed highly improbable for the process to proceed along this path.
Physically, this result is not unexpected: rather than to appear in the whole
volume of the old phase, occupied by M molecules, it suffices for the new
phase to come into being only locally in a much smaller volume occupied
only by n* « M molecules. The cluster constituted of n* molecules (or
atoms) is called the nucleus (sometimes critical nucleus), and the energy
barrier W* to the phase transition is known as nucleation work. First-order
phase transitions thus occur by nucleation mechanism, and n* and W* are
basic thermodynamic quantities in the theory of nucleation.
3.2 Heterogeneous nucleation
We shall now consider the so-called heterogeneous nucleation (HEN) which
takes place when the old phase contacts and/or contains other phases and/or
molecular species. HEN is much more widespread in both nature and
technology than is HON, because the old phase practically always has in
itself foreign molecules, microscopic particles, bubbles, etc. or is in contact
with other phases which limit it spatially. The surfaces of these phases and
of the various particles may be a place on which the formation of the clusters
of the new phase is preferred. It is thus important to find W(n) also in the
case of HEN and in doing that we shall follow the procedure already used in
Section 3.1.
Work for cluster formation 31
Figure 3.4 depicts schematically the old phase in the initial and final
states 1 and 2, respectively, before and after the formation of a cluster of n
molecules on the surface of another phase called substrate. As in HON, the
old phase is at constant pressure p and temperature T and contains a fixed
number M of molecules (or atoms). If the Gibbs free energy of the substrate
is Gs, and the total surface energy of the substrate/old phase interface is,
respectively, (¾^ and 0s(n) in the absence and in the presence of the n-sized
cluster, instead of eqs (3.1) and (3.2) we shall have (n = 1, 2,. . .)
G, = WAioid + G, + <SS,0 (3.45)
G2(.n) = (A/ - «Kld + G(n) + Gs + <Ss(n). (3.46)
M
M-n
STATE 1 STATE 2
Fig. 3.4 Old phase ofM molecules before (state 1) and after (state 2) heterogeneous
formation of a cluster ofn molecules on the surface of a substrate.
Since the cluster Gibbs free energy G(ri) can again be expressed by eq. (3.3),
from the definition equality (3.4) it follows that in the case of HEN (n = 1,
2,..-)
W(n) = -nAn + Gtx(n) + <6s(n) - 0S>O. (3.47)
This equation says that in this case, in addition to the cluster excess
energy GCK(n), it is necessary to know also the difference 0s(n) - ¢^ in the
total surface energy of the substrate after and before the cluster formation.
Furthermore, comparison of eqs (3.5) and (3.47) leads to the conclusion that
HEN is energetically favoured only when the substrate affects the excess
energy of the n-sized cluster formed on it in such a way that the sum Gc%(n)
+ 0s(n) - 0s,o *s 'ess than the excess energy that the cluster would have had
if it were formed homogeneously. We note as well that, as in the case of
HON, W(n) from (3.47) has a limiting value W(l) * 0, because the cluster of
size n = 1 is treated as distinguishable from a molecule of the old phase.
Unlike in HON, however, this distinguishability is real (and not only formal),
since the smallest heterogeneously formed cluster of the new phase is in fact
an old-phase molecule adsorbed on the substrate.
Let us now see how W(n) from (3.47) can be determined in the framework
of the classical nucleation theory, i.e. for large enough values of n, provided
that the EDS is the dividing surface between the three phases (cluster, old
32 Nucleation: Basic Theory with Applications
phase and substrate) in the system. This will make it possible also to reveal
what is the cluster excess energy Gcx in the case of HEN. Analogously to
(3.6), the Helmholtz free energy F2(n) of the system in state 2 will be given
by
F2(n) = (M-n)nM-pVM_„ + nn„^„-p„V„ + 0(V„) + Gs + 0S(V„)
where 0s(Vn) = 0s(n), and 0(1^) is again the total surface energy of the n-
sized cluster of volume Vn, but now accounting for the presence of the
cluster/substrate interface. As from thermodynamics [Guggenheim 1957]
Gi(ri) = F2(n) + pV where V = VM_„+ Vn, using F2(n) from the above
expression yields
G2(n) = (M - n)nM + n^ev -(pn-p)V„ + 0(V„) + Gs + fc( V„). (3.48)
From the condition dG2ldVn = 0 for mechanical equilibrium it follows that
rather than by eq. (3.10), the pressure p„ inside an n-sized cluster on a
substrate is determined by the more general Laplace-type formula
p„ = p + d(0 + 0s)/dV„. (3.49)
Substituting Gt and G2 from (3.45) and (3.48) in eq. (3.4), we find that for
a heterogeneously formed n-sized cluster defined by the EDS W(n) has the
following most general form:
W(n) =-(/>„- p)V„ + fo^ - nM)n + 0( V„) + 0S(V„) - 0sO. (3.50)
We can now compare this equation for Win) with eq. (3.47) in order to
conclude that in HEN Gex is again determined by eq. (3.9) and, hence, by eq.
(3.12), i.e.
Ga(n) = <tK.Vn)-(pn-p)Vn+ V" V„(P)dP.
Jp
This means that for heterogeneously formed clusters of condensed or gas
phases Gex is given also by eqs (3.15) and (3.16), respectively, but 0 now
accounts for the presence of the cluster/substrate interface, and pn depends
on both 0 and 0s through their derivatives with respect to V„ (see eq. (3.49)).
That is why, for the explicit determination of the G^in) dependence in HEN
there is a need of concrete models for 0 and 0S. We shall confine the following
considerations to the widespread models of clusters with the shape of cap,
lens or disk on a planar atomically smooth surface (Fig. 3.5). Results concerning
other possible cluster shapes and substrate geometries can be found elsewhere
[Kaischew 1950; Hirth and Pound 1963; Zettlemoyer 1969; Krastanov 1970;
Blander 1979; Russell 1980]. In addition, for the total surface energy of the
substrate before the cluster formation we shall write
0s.o=°Vts (3.51)
where as (J m-2) is the specific surface energy of the substrate/old phase
interface, and As is the area of the substrate surface.
It is appropriate to note at this point that depending on the considered
shape, the heterogeneously formed cluster may have different spatial
Work for cluster formation 33
.' ~^ ° . °
<al (b) (c)
Fig. 3.5 Crass section of (at cap-shaped, (b) lens-shaped, and (c) disk-shaped
cluster ofn molecules on the surface of a substrate.
dimensionality. For instance, when the cluster is cap or lens shaped (Figs.
3.5a and 3.5b), it changes size in three directions and, accordingly, nucleation
is called three dimensional (3D). In the case of a disk-shaped cluster (Fig.
3.5c), the cluster height remains fixed, the cluster is mathematically in a
two-dimensional space and one speaks of two-dimensional (2D) nucleation.
(a) Cap-shaped clusters (Fig. 3.5a)
When the substrate is solid, the shape of a cluster on it can be approximated
by that of a spherical cap [Volmer 1939; Zettlemoyer 1969]. From elementary
geometry, the lateral area, the base area and the volume of a cap of radius R
are known to be, respectively, 2tzR2(\ - cos 6W), 7tR2sin2 6W and
V„ = V(0w)(4/3)?rtf3. (3.52)
As in the case of HON, we shall use the classical approximation <T„ —
constant = c and o-_„ = constant = o-. atJl and a[ being, respectively, the
specific surface energies of the substrate/n-sized cluster and substrate/bulk
new phase interface. Then, for the total surface energy 0 of an n-sized cluster
on the substrate and for the total surface energy 0S of the substrate with such
a cluster on it we can write
<j>(R) = <j2nR2{\ - cos 6W) + 0{xR2 sin2 6W (3.53)
&(/?) = as(As - nR2 sin2 ew). (3.54)
In the above relationships, for clusters of condensed phases 6W is the
equilibrium wetting angle defined from 0 to ;rby the Young equation [Young 1805]
cos 0w = (o-s-o-L)/o\ (3.55)
and the numerical factor y^has values between 0 (at 6W = 0) and 1 (at 0W ~
7t) (Fig. 3.6) and depends on 6W according to [Volmer 1939]
y(ew) = (1/4)(2 + cos 0W)(1 - cos 6W)2. (3.56)
The cases of 6W = 0 and 6W - naie usually referred to as complete wetting
and complete non-wetting, respectively. For clusters of gas phases the
terminology is the opposite, for then 6W is in fact the angle of non-wetting:
then 6W = 0 and 6W = K correspond to complete non-wetting and complete
wetting, respectively. It is important to remember that, according to eq.
(3.55), it is physically sound to approximate the cluster shape by that of cap
only if - o ^ o~fi - Oj ^ (J, as only then is | cos 6W | < 1.
34 Nucleatiun: Basic Theory with Applications
1.0
0.8
0.6
0.4
0.2
0
30
150
180
60 90 120
ew (deg)
Fig. 3.6 Dependence of the factor y/on the wetting angle 6W according to eq. (3.56).
Using eqs (3.53) and (3.54) in (3.49) and accounting for (3.52) and (3.56)
leads to the known result that the pressure pn inside an n-sized cap-shaped
cluster is again given by the Laplace equation (3.24). Also, substituting 0
from (3.53) in eqs (3.15) and (3.16) and using (3.14), (3.24) and (3.52) for
the elimination of n, we find analogously to (3.27) that
Gn(R) = [2o-(l - cos 0W) + o, sin2 6w]7tR2
(3.57)
for clusters of condensed phases and analogously to (3.30) that
G„(K) = [2o(l - cos ej + o- sin2 6W - (8/3)cnj/(ew)];rfi2
+ y(ew)(4Ji/3)/>fi3(l + 2alpR) In (1 + 2a/pR) (3.58)
for clusters of gas phases. As seen, the homogeneous formation of spherical
clusters can be considered as a limiting case of heterogeneous formation of
cap-shaped clusters at 6W = K, since then the caps are spheres and the last two
equations pass into (3.27) and (3.30), respectively.
Knowing Gcx, ¢, and 0S 0, we can now obtain the work W for formation of
cap-shaped clusters in HEN on a foreign substrate. With the help of eqs
(3.51), (3.54) and (3.57), in view of (3.13), (3.52) and (3.56), from (3.47) we
find that for clusters of condensed phases [Zettlemoyer 1969]
(3.59)
W(R) = v<ew)[-(4mV/3u0)fi3 + inoR2]
or, in terms of the number n of molecules in the cap-shaped cluster,
(3.60)
Work for cluster formation 35
where a = (36xv$)"\ Similarly, using eqs (3.51), (3.54) and (3.58) in (3.47)
and accounting for (3.14), (3.52) and (3.56) yields for clusters of gas phases
[Kaischew and Mutaftschiev 1962; Blander 1979]
W(R) = iir(6w){-(4K/i)[(Afi/kT) - In (1 + 2a/pR)](p + 2alR)Ri
+ (4nrt)oR2) (3.61)
W(n) = -n[Afi - kT In (pjp)] + (l/3)<//"3(ew)co(/l77/>„)2'3n1'3. (3.62)
In the last equation c = (367t)"3, and the dependence of p„ on n is given
implicitly by the expression
P„ = p+ 2G[4Ky(6w)pJ3kTn]m (3.63)
which follows from (3.14), (3.24) and (3.52).
Comparison of eqs (3.59)-(3.62) with eqs (3.39), (3.40), (3.42) and (3.43)
leads to a conclusion of great practical importance: the presence of a substrate
for which a, > Oj - a (then 8W < ;rand, hence, \j/< 1) decreases the work for
formation of a cap-shaped cluster on it with respect to the work for
homogeneous formation of a spherical cluster with the same size. According
to (3.52), this decrease is determined by the ratio 1//(9,,) between the volumes
of the cap and the sphere. Thus, the strong stimulating effect that other
phases (e.g. container walls, colloid particles, etc.) in contact or within the
old phase exert often on the nucleation process finds an explanation. As
expected, in the limit of 6W - % (complete non-wetting or wetting in the case
of a cluster of condensed or gas phase, respectively) the substrate has no
influence on the work for cluster formation: then y/= 1 and eqs (3.59)-(3.62)
pass into eqs (3.39), (3.40), (3.42) and (3.43) describing HON.
(b) Lens-shaped clusters (Fig. 3.5b)
In some cases (e.g. when the substrate is liquid, and the old phase is fluid),
the shape of the cluster can be approximated by that of a spherical lens
[Gibbs 1928]. According to elementary geometry, the lens volume is expressed
as
Vn = [VW + (sin 6w/sin 6^1^)1(4/3)^3 (3.64)
where 0W and 0S are defined by [Jarvis et a/. 1975; Blander 1979]
cos 6W = (ol+ a2 - a2)llasa
cos 6S = (a2+ a1 - ofyiOsOi,
and y/(9w) and 1^(¾) are given by (3.56).
Confining the analysis only to clusters of condensed phases, for 0 and Gex
we shall have
Gex(fi) = 0(R)
= o!kR\\ - cos 9W) + CT;2.irfi2(l - cos 0s)(sin 0w/sin 9S)2
and since 0S is again given by (3.54), from (3.47) it follows that
(a = (367tv2)<B)
36 Nucleation: Basic Theory with Applications
W(R) = [yKSw) + (sin 6„/sin 8sfyK8s)][ -(4jrAji/3i>0)«3 + 4xoR2] (3.65)
W(n) = -n&n + [y/(ew) + (sin 6w/sin 6sfyt6s)]"3aon2li. (3.66)
Analogously to (3.59) and (3.60), the above two equations say that the
substrate can decrease the work for formation also of lens-shaped clusters.
According to eq. (3.64), this decrease is again determined by the ratio between
the volumes of the lens and the corresponding homogeneously formed spherical
cluster and depends on the values of a, os and a, in relation to each other.
Indeed, it may be shown [Blander 1979] that the first bracketed factor in
(3.64) is less than unity only if o, - os < o < o"; + <ys.
(c) Disk-shaped clusters (Fig. 3.5c)
When nucleation is 2D, the shape of the cluster can be approximated by that
of a disk of fixed height [Volmer 1939; Zettlemoyer 1969], The disk-shaped
model is physically acceptable in the cases of nucleation, e.g., (i) on a
foreign substrate which is nearly completely wetted (then Sw= 0), (ii) on a
foreign substrate which is wetted 'better' than completely (then os > a+ q
and Young's 0W from (3.55) loses its usual meaning), and (iii) on a substrate
of the new phase itself, called own substrate (then (Ts = o and Oj = 0 so that
6„ = 0). While in the first case the cap-shaped model is not adequate if the
cluster height is about or less than the molecular diameter dQ, in the other
two cases this model is altogether inapplicable.
If R is the disk radius, considering again only clusters of condensed phases,
for 0S, 0 and Gex we shall have
<j>s(R) = o,(As - xR1) (3.67)
Gex(fi) = <p(R) = onR2 + OjicR2 + k„1kR. (3.68)
The last term in eq. (3.68) represents the total surface energy of the lateral
phase boundary between the cluster and the old phase, and K„ (J m"1) is the
specific edge energy of the cluster. Since K„ is a 2D analogue of a„ in (3.17),
its dependence on n is also affected by the choice of the lateral dividing
surface which is projected as a dividing line on the substrate surface [Rusanov
1967, 1978]. In conformity with the definition of the various r/s in all
preceding formulae, we shall consider K„ in eq. (3.68) as defined by the EDS
as a lateral dividing surface. In the classical theory, the problem with the
possible size dependence of the cluster specific edge energy is solved simply
by assuming that, approximately, Kn — constant = K, k being the specific edge
energy of the planar lateral cluster/old phase interface, i.e. the limiting value
of Kn for n —> m. If the disk is of height h which is fixed and corresponds to
one or more molecular monolayers, to a first approximation [Stranski and
Kaischew 1934],
K=ah, (3.69)
but the actual K(h) dependence may be considerably more complicated
especially for h —> dG.
Since the disk volume is V„ = KhR1, with the help of eqs (3,13), (3.51),
Work for cluster formation 37
(3.67) and (3.68), from (3.47) it thus follows that [Hirth and Pound 1963;
Zettlemoyer 1969]
W(R) - -(Aji/a,.f - Aa)7tR2 + 2kkR (3.70)
W(n) = -(A/( - uefAa)n + bKn"2. (3.71)
Here Aa (J rrf2) is an effective specific surface energy defined by
Aa- a + Oj - as, (3.72)
aef = vq/H is an effective molecular area (aef equals the molecular area a0
when the cluster is of monolayer height h = d0), and b = 2(7cacf)U2. It is
important to note that eq. (3.71) remains in force also for prismatically
shaped clusters, but then vis appropriately averaged over the orientations of
the prism faces, and b accounts for the shape of the prism base (e.g. b =
4aef for clusters with the shape of square prism). Combining eqs (3.55)
and (3.72) leads to the formula [Kaischew 1950]
Aa = a(\ -cos 6W) (3.73)
which connects Aa and 8W and shows that Aa > 0, Act = 0 and Aa < 0
correspond, respectively, to incomplete, complete and 'better'-than-complete
wetting. Rather than through 0W, in some cases it is more convenient to
express Aa through the specific adhesion energy ft (J m~2) characterizing
the contact between the new phase and the substrate. This can be done with
the help of the formula [Kaischew 1950]
Aa=2a-ft
which follows from (3.72) and the Dupre relation [Dupre- 1869]
0,= 0 + <rs-ft.
Equations (3.70) and (3.71) reveal that in the case of 2D HEN, depending
on the sign of Ao", i.e. on the choice of the substrate, the supersaturation An
can be effectively increased or decreased. This is due to the energy gain or
loss associated with the replacement of a part of the substrate/old phase
interface by the two interfaces, the substrate/new phase and the new phase/
old phase ones, of the disk-shaped cluster. When Aa< 0 (i.e. when wetting
is 'better' than complete), the supersaturation is effectively higher and 2D
nucleation is possible not only at saturation (then A^t = 0 and - aefAais the
driving force of the process), but also at undersaturation, i.e for negative Afi
values in the range of aefAa < A^t < 0. In the Aa > 0 case, i.e. for incomplete
wetting, the supersaturation is effectively lowered and 2D nucleation can
occur only when A// > acfAa: then it is the alternative of 3D nucleation, e.g.
of caps or lenses. When Aa = 0 (i.e. at complete wetting), eqs (3.70) and
(3.71) find application in the important particular case of 2D nucleation on
own substrate. As already noted, in this case the substrate and the cluster are
the same phase (as = a), there exists no interface between them (a, = 0) and
the creation of the lateral phase boundary remains the only energy impediment
for the cluster formation. Taking into account also that then the cluster is
38 Nuclealion: Basic Theory with Applications
Of monolayer height h = da simplifies (3.70) and (3.71) to the equations
[Brandes 1927]
W(R) = ~( jiA^a0)R2 + 2mcR (3.74)
W(n) = -nA/u + bKnm (3.75)
which are the 2D analogues of (3.39) and (3.40). The last formula is valid
not only for disks (then b = 2(na0)"2), but also for prismatically shaped
clusters with values of b accounting for the cluster shape: for example,
b = 4^02 for square-shaped clusters.
Equations (3.60), (3.66), (3.71) and (3.75) are basic results of the classical
theory and with concrete expressions for the supersaturation A/j they apply
to HEN in vapours, solutions, melts, etc. The general and most important
message from them is that regardless of the cluster shape, as in the case of
HON, the work W to form a cluster on a substrate passes through a maximum
of value W* at n = n*. Under otherwise equal conditions, the stimulating role
of the substrate is, in general, to lessen both the nucleus size n* and the
nucleation work W* and, in some cases, to make possible 2D nucleation in
the undersaturation range. Figure 3.7 illustrates this role by depicting the
W(n) dependence for HON, 3D HEN and 2D HEN of water droplets in
vapours at T = 293 K and p = 4/>e, the droplets having the shape of spheres,
caps and monolayer disks (as indicated). The curves are drawn according to
eqs (3.39), (3.60) and (3.71), respectively, with A^i from (2.8), y/ = 0.16 (this
-10 r. i ■ 11' .' ~n I
1 10 20 30 40 50 60 70 80 90 100
n
Fig. 3.7 Dependence of the work for cluster formation on the cluster size in the case
of water droplets in vapours at T = 293 K and p/pe = 4: curve 'sphere' - eq. (3.39)
for HON; curve 'cap' - eq. {3.60) for HEN of caps on a substrate with 6W = n/3;
curve 'disk' - eq. (3.71) for HEN of disks on a substrate with Ao = {l/2)<y.
Work for clusrer formation 39
means 0W = Kfl in eq. (3.56)), b = 2(JIH0)"2, aef = a0 = (^4)^,K= orf0 and
Afj= cv2 (which corresponds also to 8W = Ttfi in eq. (3.73)). The values of
d0, v0, o"and/>e are those listed in Table 3.1. Comparing the W(n) dependences
for caps and disks, we see that in the range of smaller cluster sizes the
formation of the caps and the disks requires a comparable work. This is an
indication that for incompletely wetted substrates the mode of cluster formation
may change from 3D to 2D with decreasing cluster size n (see Section 4.3).
The above considerations show what is W{n) from eq. (3.47) in the scope
of the classical nucleation theory which applies to large enough clusters,
since it presumes that macroscopic quantities such as surface energy, edge
energy and wetting angle can be used for describing the cluster energetics.
For that reason, the question again arises: what are the Ga(n) and tj>s(n)
dependences in (3.47) in the opposite limiting case of n -> 1? This question
can also be answered by the atomistic theory of nucleation and, as might be
anticipated, for clusters of condensed phases Gex is again given by eq. (3.15)
so that, in view of (3.33),
Gex(n) = nX - En.
This result follows from direct repetition of the analysis in Section 3.1, for
eqs (3.31)-(3.34) hold true also for heterogeneously formed clusters of
condensed phases. The only difference now is that U\ *fc Wqu and E\ — u0]^ —
Ui * 0, since when a molecule of the old phase is adsorbed on the substrate
and becomes a cluster of size n - 1, its potential energy is changed. As to
0s(n), analogously to (3.54), it can be represented as
0s(n) = CJs(As-A„) (3.76)
where A„ = arfljin) is the area of the contact surface between the n-sized
cluster and the substrate, and nc is the number of cluster molecules which are
in contact with the substrate (for monolayer clusters nc = n).
Thus, with Gex and 0S from the above two equations and 0sO from (3.51),
eq. (3.47) yields the general atomistic formula for W(n) in HEN of condensed
phases (n = 1, 2, . . .)
W(n) = -{An - X)n -E„- oyl„. (3.77)
This formula is valid for any size and shape of the cluster on the substrate.
In the absence of substrate, cs = 0 and eq. (3.77) passes into the atomistic
formula (3.44) for HON. It is worth noting as well that for 2D formation of
monolayer clusters, as all molecules of the cluster contact the substrate (i.e.
flc = n), A„ - arfl and (3.77) takes the form (n = 1, 2, . . .)
W(n) = -{Aft - A + a0as)n - E„. (3.78)
In the particular case of 2D nucleation on own substrate, <7S = o" so that this
equation becomes (n - 1, 2, . . .)
W(n) = -(Afi - X + aaa)n - E„. (3.79)
With properly defined supersaturation An eqs (3.77)-(3.79) are the atomistic
40 Nucleation: Basic Theory with Applications
formulae for W(n) in HEN of solid or liquid phases from vapours, solutions,
melts, electrolytes, etc. These equations reveal the connection between W(n)
and the cluster binding energy En and are generalizations of known atomistic
formulae for W(n) [Walton 1962, 1969b; Milchev et al 1974]. It must be
emphasized that in eqs (3.77)-(3.79) E„ is referred to the potential energy of
n molecules in the bulk of the old phase (see eq. (3.32)) rather than to the
potential energy of n molecules adsorbed on the substrate. Also, these equations
contain the <rs term which ensures their passing into the classical formulae
for W(n) when «—»«>. For instance, comparison of (3.78) with (3.71) shows
that for large enough 2D clusters of monolayer height on a substrate (then aef
= a0) the cluster binding energy En is a function of n of the following general
form:
E„ = (A - a0o~ aoa0n - bKn,/2. (3.80)
In particular, for 2D clusters on own substrate Oj = 0 and (3.80) reduces to
the equation
E„ = (A - a0o)n - bKnm (3.81)
which results also from comparing (3.79) with (3.75). The above two equations
are the 2D analogues of eq. (3.37) for HON which involves 3D clusters.
They are in full accord with the Stranski formula (3.33), since the quantities
a0(o+ o{)n + bKnm and a0on + bKn1'2 are nothing else but the total surface
energies of a large enough n-sized monolayer cluster on a foreign or own
substrate, respectively.
3.3 General formulae
The existence of different formulae for the work W to form a cluster in the
different cases of nucleation is a considerable obstacle for establishing general,
model-in dependent relationships in the thermodynamics of nucleation. For
that reason, it seems useful to represent W in such general form which holds
true irrespective of the cluster size and of the type of nucleation (HON,
HEN, 3D, 2D, etc.).
Inspection of the expressions for W obtained above shows that, thermo-
dynamically, the work for cluster formation can be represented in terms of «,
the cluster volume Vn, the pressure pn in the cluster and the chemical potential
A'new.n of the molecules in the cluster. The corresponding general formula for
EDS-defined clusters reads
W(n) = -ipn - p)Vn + Ow - ^oid)" + Q(Vn) (3.82)
where 0S(V„) is an effective surface energy given by
¢.(^) = 0(^) + ^(^)-^,0, (3.83)
flne^n is determined by (3-11) with the aid of the equation of state of the
cluster, and pn and <Z>S are related by the Laplace-type equation
Work for cluster formation 41
p„ = p + d*s/dV„. (3.84)
Since the EDS is the surface separating the cluster from the old phase, eq.
(3.82) is valid for such large values of n and Vn for which #s, p„ and ji„ew„
are well-defined quantities. For an arbitrary choice of the dividing surface,
following Toschev [1973a], it can be shown that W takes the most general
form
W(n) = -(p„-p)V„ + (/inew,„ - pM)n + <jis- pM)ns + ¢,(V„) (3.85)
where ns is the Gibbs excess in the number of molecules (formally, the
number of 'surface' molecules of the cluster), and|is is their chemical potential.
Depending on the choice of the dividing surface, ns can be positive, zero or
negative [Ono and Kondo I960]. For the EDS, by definition, ns = 0 and eq.
(3.85) passes into (3.82). For a dividing surface encircling the EDS, ns < 0
for clusters of a denser new phase and ns > 0 if the clusters are less dense
than the old phase. This means that if the dividing surface is positioned much
deeper within the old phase than is the EDS, even for the smallest clusters of
a few molecules ¢,, p„ and /i„ew,» will be thermodynamically well defined.
The conclusion is, therefore, that eq. (3.85) is a correct formula for the work
to form a cluster of any size provided the dividing surface is defined by a
sufficiently large negative or positive value of ns when the new-phase density
is, respectively, higher or lower than that of the old phase.
Alternatively to eq. (3.85), Wean be represented most generally as (n = 1,
2, . . .)
W(fl) =-«Au + 0(n) (3.86)
when it is considered as a function only of the number n of molecules in the
cluster. This formula is valid for arbitrary choice of the dividing surface,
since this choice is taken into account by the effective excess eneTgy <P{n) of
the cluster. When the EDS is chosen as a dividing surface, the general expression
for ¢(/1) is
*(«) = G„(n) + &(n) - 0S-O (3.87)
with Gex(n) given by (3.9) or (3.12). The concrete forms of ¢(11) from this
expression in the various cases of nucleation considered above are presented
in Table 3.3.
According to eq. (3.85), for arbitrary choice of the dividing surface <P(n)
should include the 'surface' term (|is - ^0id)ns. Thus, most generally, <P(n)
can be written down as
*(«) = &is - jicidR + Gc%(n) + <t>s(n) - fafi (3.88)
where now Ges, 0S and 0S 0 account for the concrete choice of the dividing
surface. For the EDS ns = 0 and (3.88) simplifies to (3.87).
Evidently, the conclusion about the applicability of eq. (3.85) to describe
the work to form a cluster even with the smallest possible size remains in
force also for eq. (3.86) with <P(n) from (3.88). In other words, the use either
of eq. (3.85) or of eq. (3.86) coupled with (3.88) has the potential to yield
42 Nucleation: Basic Theory with Applications
T&ble 3.3 Effective excess energy &(n) of EDS-defined clusters according to the classical
theory of nucleation.
Kind of New phase #(n)
nucleation
HON condensed ccv^n2n
HON (spheres) gaseous3 nkT In (pjp) + (4rt3)lBG(kT/Pn)mn2i:s
3D HEN (caps) condensed yfm(ej{36n:)m Gv$3n2'3
3D HEN (caps) gaseous3 nkT In (pjp) + 1/^(0^(4^3)1^^¾)2¾2^
3D HEN (lenses) condensed [y<<?w) + (sin 0w/sin es)3i/<^)]I/3(36^)l/3 CV^nm
2D HEN condensed naefA.G + bKn112
apn is given by eq. (3.24)
results of most general validity in the theory of nucleation (see Chapters 4
and 5). It must be noted as well that eq. (3.86) can be regarded as a
phenomenological formula which gives W in all cases of nucleation if in it
rather than specified by (3.88), <P(n) is defined solely as an excess energy
satisfying the energy balance required by (3.86).
Finally, we note the obvious analogy between W{n) from eq. (3.86) and
W(n) from eqs (3.44) and (3.77)-(3.79). The implication is that eq. (3.86) is
also the most general atomistic formula for W(ri) in which <Z>(«) is given by
(« = 1,2, ...)
0{n) = Xn - En - osAn (3.89)
in the cases of both HON (with os = 0) and HEN (with <rs > 0) of condensed
phases. For gas-phase clusters <P{n) is unknown, but can be represented
formally as (n = 1, 2, . . .)
<*>(«) = (Pnew,* ~ A'new)" ~ipn- P)V„ + 0(«) - OsAn (3.90)
where again cs = 0 and os > 0 for HON and HEN, respectively. This expression
results from (3.87) with the help of (3.9), (3.51) and (3.76) and in it ^new/»
pn and 0(«) keep formally their usual physical meaning, but are unknown
functions of n.
3.4 Absence of one-dimensional nucleation
In Sections 3.1 and 3.2 we considered nucleation in three and two, but not in
one dimension. This was done because, classically, one-dimensional (ID)
nucleation does not exist. We shall now see what is the reason for the absence
of 1D nucleation of condensed phases in the scope of the classical nucleation
theory. Doing that is worthwhile, since in the literature one can read also
about 1D nucleation (e.g. Voronkov [ 1970]; Frank [ 1974]; Zhang and Nancollas
[1990]; Markov [1995]).
We consider a cylindrically or prismatically shaped condensed-phase cluster
Work for cluster formation 43
formed along a line on a substrate (Fig. 3.8). An example of such a cluster
is a row of molecules at the edge of a monomolecular step between two
terraces on an atomically smooth single crystal face. The n-sized cluster is
EDS defined and has a constant area Ac of each of its two end surfaces and
a length L = (v^A^n. Let ij and K"s (J nf') be, respectively, the length and
the specific line energy of the substrate line on which the cluster is formed.
Then, for the total surface energies #, 0 and #,(«) of the substrate before and
after the cluster formation we can write
fan = cyls + iqLs
(3.91)
(3.92)
substrate
-^
Fig. 3-8 Top view of cylindrically or prismatically shaped cluster of length Lona
line (shown dark grey) on the surface of a substrate.
Let us now denote by k„ (J m~') and (Q„ (J m '), respectively, the energy
per unit length of the cluster/old phase interface and of the cluster/substrate
contact line and let £„ (J) be the energy of each of the two end surfaces of the
cluster. In classical approximation these three energies can be treated as
independent of the cluster size n, i.e. they can be represented as K„ = constant
= K, k- „ = constant = k; and £„ = constant = £. Then, as we are considering
condensed-phase clusters for which eq. (3.15) is in force, for the cluster
excess energy Gcx(n) and total surface energy 0(n) we shall have
Gm(n) = <j>(ri) = (k+ kjL + 2e.
(3.93)
Thus, from eqs (3.87), (3.91)-(3.93) it follows that the cluster effective
excess energy is of the form
*(n) = nrfrf A(C + 2e (3.94)
where dcf = Vq/A,, is an effective molecular diameter, and the combination Ak
44 Nucleation: Basic Theory with Applications
= k + k; - ks of specific line energies is analogous to A<7 from (3.72). For
instance, Ak = 0 for ID clusters of monomolecular thickness when they are
formed at the edge of a monomolecular step on own crystalline substrate
(then tcs = *cand kj = 0). Substituting &(n) from (3.94) in (3.86), we finally
find that, classically, the work for formation of ID condensed-phase clusters
is given by
W(n) = -{Ap - dt[AK)n + 2e. (3.95)
This formula shows that W(n) depends linearly on the cluster size (in the
particular case of A/i = deiAK it is even ^-independent) and, hence, does not
possess a maximum at a given value of n. Physically, the absence of such a
maximum is equivalent to the absence of nucleus, because the nucleus size
n* is defined as the position of this maximum (see Figs 3.3 and 3.7). Moreover,
the absence of a maximum implies that there is no energy barrier W* to the
formation of macroscopically large 1D clusters or, in other words, that there
is no ID nucleation. Indeed, from (3.95) we see that when A/z > detAK, W(n)
only decreases with increasing n, which means that work is gained when a
molecule is added to the ID cluster even if this cluster is with the smallest
size of n = 1. Comparison of eq. (3.95) witheqs (3.60) and (3.71) reveals that
the reason for the absence of ID nucleation of condensed phases is that the
cluster end energy 2e, which corresponds to the effective surface energy
i///3(0w)fl(rn2'3 of a 3D cluster and to the peripheral energy b\nxa of a 2D
cluster, is independent of the cluster size n. Hence, the conclusion: thermo-
dynamically, in the scope of the classical approximations £n = constant = £
and Ak„ = constant = Ak: we cannot speak of ID nucleation of condensed
phases. Only when £n and/or Ak„ = K„ + Kj>r? - k; are functions of n could ID
nucleus be defined and ID nucleation be thermodynamically meaningful -
the condition for that to be the case is the dependence of £„ and/or A^onn
to make W(n) from (3.95) display the maximum needed for the determination
of the ID nucleus size n* and the ID nucleation work W*.
Chapter 4
Nucleus size and nucleation work
The considerations in Chapter 3 show that among the clusters of different
size the n*-sized cluster, i.e. the nucleus, is distinguishable with the highest
energy price for its formation. The energy spent on the nucleus formation is
equal to the nucleation work W*. The special importance of the nucleus
comes from the fact that none of the subnuclei (the clusters of size n < «*)
can grow spontaneously, for such a process is associated with an increase of
the free energy of the system. This is seen in Figs 3.3 and 3.7 in which W
increases with n for n < n*. Only the supemuclei (the clusters of size n > «*)
are capable of spontaneous overgrowth: as evidenced by the descending n >
n* branches of the W(n) dependences in Figs 3.3 and 3.7, this process lessens
the free energy of the system. Since it is the nucleus that can grow most
easily into a supemucleus, formation of nuclei in the supersaturated old
phase is a prerequisite for the onset of the first-order phase transition.
Statistically, however, the nucleus formation is a random event with a
probability largely determined by W* and for that reason the nucleus size n*
and the nucleation work W* are basic quantities in the theory of nucleation.
Let us now see how n* and W* can be found in the various cases of HON
and HEN considered in Chapter 3. Wherever necessary for the mathematics,
we shall regard the cluster size n as a continuous variable.
4.1 General formulae
Knowing the W(n) dependence in each concrete case of nucleation allows
the determination of n* with the help of the condition for extremum
(dWdfl)„=„, = 0, (4.1)
and then of W* from the definition equality
W* s W(n*). (4.2)
In some cases, for the determination of n* and W* it is mathematically
more convenient to employ the formulae
p* = p + (d<%/dV„)* (4.3)
H*=foW (4.4)
W* = -(p* - p)V* + <P* (4.5)
rather than eqs (4.1) and (4.2). Here, for brevity, p* s p,,,, /(* = p„ew,,
46 Nucleation: Basic Theory with Applications
V* = V„*. (d<jydVn)* = (d0s/dV„)VB.v*, and &* = <PS(V*) is the nucleus
effective surface energy. These formulae follow upon using W from the
general formula (3.82) first in the extremum conditions (BWfdVR) ^ = 0
and (dW/dn)n=n* = 0, which are the analogue of eq. (4.1), and then in eq.
(4.2). Equations (4.1) and (4.4) are the two known thermodynamic definitions
of the nucleus size n* [Volmer 1939]. The third definition is a kinetic one
and is presented in Chapter 12.
Physically, eqs (4.3) and (4.4) mean that the nucleus is that special cluster
which is simultaneously in mechanical (cf. (4.3) with (3.10), (3.49) and
(3.84)) and chemical equilibrium with the old phase. It should be remembered,
however, that the thermodynamic equilibrium of the nucleus is labile (and
not stable), since at the nucleus size the work W for cluster formation is
maximum rather than minimum. Another very important point to note is that
eq. (4.5) is invariant with respect to the choice of the dividing surface. This
is so because for an arbitrarily chosen dividing surface both the n and ns
terms in eq. (3.85) vanish for the nucleus [Toschev 1973a] and W* is again
given by eq. (4.5). This equation is thus the most general formula for the
nucleation work W* and is, therefore, applicable also to an atomistically
small nucleus provided that, as already discussed, the dividing surface around
it is positioned far enough into the bulk of the old phase.
Clearly, W* can be expressed equivalently in a most general way by the
equation (n* ~ 1, 2, . . .)
W* - -n*Ay + 0* (4.6)
which results from using W(n) from (3.86) in (4.2) and in which #* = <&(n*)
is the nucleus effective excess energy. This equation is also valid for whatever
kind of nucleation (HON, HEN, 3D, 2D, classical, atomistic, etc.) and choice
of the dividing surface.
With the above formulae in mind, we can now turn to the determination
of n* and W* in the various cases of nucleation considered in Chapter 3.
4.2 Homogeneous nucleation
The expressions for the nucleus size n* and the nucleation work W* are
simplest for nuclei of condensed phases. With the aid of W(n) from (3.39),
eqs (4.1) and (4.2) lead easily to [Zettlemoyer 1969]
k* = 8c3 u0V/27 V (4.7)
W*=4c3v^/27Afi2. (4.8)
In the particular case of spherical nuclei c = (36.T)l/3 and, since according to
(3.13) and (3.22) n* = (47tf3u0)/?*3, f°r the nucleus radius R* and for n* and
W* it follows from (4.7) and (4.8) that [Zettlemoyer 1969]
R* = 2v0o/Av (4.9)
n* = 32;tl;0V/3 V (4.10)
Nucleus size and nucleation work 47
IV* = 16tfu02cr73A/<2. (4.11)
These formulae for R* and W* can be obtained directly by using the conditions
(4.1) and (4.2) for maximum of W(R) from (3.40). From the expressions for
n* and W* we see that W* = {\/2)n*Ap which agrees with the expectation in
Chapter 3 that W* = n*Ap.
Equations (4.7), (4.9) and (4.10) are known as equations of Gibbs-Thomson
[Zettlemoyer 1969], and with the help of 0 from (3.20) W* from (4.8) takes
the simple general form [Gibbs 1928]
IV* = (1/3)0* (4.12)
where 0* = 0(n*) = acrn*2'3 is the total surface energy of the nucleus.
Equation (4.12) thus shows that the nucleation work in HON is merely equal
to one-third of this energy provided that the cluster shape and specific surface
energy are, as assumed by the classical theory, size-independent.
Figure 4.1 displays the dependences of R*, n* and IV* on A/( for HON of
water droplets in vapours at 7= 293 K, resulting from eqs (4.9)-(4.11) with
the values of v0 and a listed in Table 3.1. As seen, both the nucleus size and
the nucleation work diminish with increasing A^i. Since detectable HON of
water occurs typically when A^i > kT, it turns out that the nucleus droplet is
of n* < 100 molecules. This result for n* is characteristic for nearly all cases
of HON and makes quite problematic the usage of the classical theory.
Indeed, for such a small number of molecules (or atoms) in the nucleus it is
hard to rely on the basic assumptions of this theory - constancy of the cluster
shape, density and specific surface energy with respect to the cluster size.
Also, of principal importance, eq. (4.8) predicts that W* = 0 only at A^i —>
<*; which is inconsistent with the thermodynamic requirement
H»(A,us) = 0 (4.13)
for annulment of the nucleation work at the supersaturation A^is corresponding
to the spinodal (A^is s^0]ds-^new where ^„ids is the chemical potential of the
old phase at the spinodal point, e.g., point g in Fig. 1.1a).
Turning now to nuclei of gas phases, we note that n* and W* prove much
simpler functions of the underpressure
&psPc-p>0
(the difference between the equilibrium and the actual pressures of the old
phase) than of the supersaturation A^i from eq. (2.10). The formulae for «*
and W* can be obtained most easily with the help of eqs (4.3)-(4.5). Using
(2.10), (3.11) and (3.14) in (4.4) yields [Aleksandrov el al. 1963]
p*=pc exp (-VgAp/kT). (4.14)
According to eqs (3.18) and (4.3) or in view of eq. (3.19), for the pressure
p* inside the gas nucleus there holds also
p*=p + 2ca/3V*
(4.15)
48 Nucleation: Basic Theory with Applications
£
S °-5
*
a:
0
100
50
3
1
60
r-
^ 40
§ 20
\
■ \ (a)
: \^_^^
^——-_
•\ (b)
r \^^
" %.
""'•--J
\
■\ (c)
:\
: ^-^______^
2 3 4 5
An/kT
Fig. 4.1 Supersaluration dependence of {a) the nucleus radius, (b) the nucleus size,
and (c) the nucleation work for HON of water droplets in vapours at T = 293 K
according to eas (4.9)-(4.11). In the range of higher supersaturalions n* changes
discretely (the step-like line) rather than continuously (the dashed line).
From this expression, since due to (3.14) V* = n*kT/p*, for the number n*
of molecules in the homogeneously formed gas nucleus we find
-%cip*o3inkT(p* -pf
(4.16)
where the p dependence of p* is given by (4.14).
Recalling that in HON <Pt = ¢, we can now use eqs (3.18) and (4.15) in eq.
(4.5) to find that W* is again given by eq. (4.12). This means that in HON of
gas phases the nucleation work is also equal to one-third of the total surface
Nucleus size and nucleation work 49
energy of the nucleus [Gibbs 1928]. Owing to eqs (3.18) and (4.15), from
(4.12) it thus follows that
W* = 4cV/27(p* -p)2, (4.17)
p* being a function of p according to (4.14).
Since in many cases peu0/kT « 1, it is worth noting that for pressures in
the range of 0</?</?e, to a good approximation,/?* = /?e. Hence, for/? values
in this range, «* and W* from the last two equations are given approximately
by
n* = 8c3pc03mkTApi (4.18)
W* = 4cV/27A/>2. (4.19)
In the particular case of spherical gas-phase nuclei c = (36¾)10 and as
from (3.14) and (3.22) n* = (4jr/3)(p*lkT)R*~\ for the nucleus radius R* and
for n* and W* it follows from (4.16) and (4.17) that [Gibbs 1928; Aleksandrov
et al. 1963; Hirth and Pound 1963; Skripov 1972; Blander 1979; Baidakov
1995]
R* = 2a/{p* - p) (4.20)
n* = 32np*aV3kT(p* - pf (4.21)
W* = 16^rj3/3(p* - pf. (4.22)
Here p* is a function of p (see eq. (4.14)) and can be approximated by pG
when pjJtJkT « 1 and 0 < p < pc. Equations (4.16), (4.18), (4.20) and (4.21)
are the Gibbs-Thomson equations for gas nuclei in HON. From (4.16) and
(4.17), with the help of (2.10) and (4.14) we see that W* = (l/2)n*W(l -
e-^fi") [Volmer 1939]. This is the analogue of the relationship W* =
(l/2)n*A/< for condensed-phase nuclei and shows that, as expected in Chapter
3, W* = n*A[i provided Ay does not exceed too much kT. We note that when
Ay » kT, W* = (H2)n*kT, i.e. the nucleation work is about 1/3 of the total
kinetic energy of the n* molecules in the gaseous nucleus.
Figure 4.2 shows R*, n* and W* as functions of Ap for HON of steam
bubbles in water at T - 583 K. The curves are drawn by using eqs (4.20)-
(4.22) with p* =pc and the a and pc values listed in Table 3.2. As seen, R*,
n* and W* diminish with increasing Ap and in practice n* < 200, since
perceptible HON of steam bubbles usually occurs when Ap/pe > 0.7. In
contrast to the case of nucleation of condensed phases (see Fig. 4.1a), now
the radius R* of the nucleus bubble is considerably greater and this makes
more reliable the application of the classical theory to gas-phase HON.
However, the fundamental inconsistency of this theory remains in this case,
too: eq. (4.17) predicts W* = 0 only for Ap —> «. This violates the
thermodynamic requirement
W*(Apg) = 0 (4.23)
which is analogous to (4.13) (Aps = ps- ps, ps being the pressure of the old
phase at the spinodal point, e.g. point e in Fig. 1.1a).
SO Nucleation: Basic Theory with Applications
5r~
4 - ^"~~\_ (a)
e 3:
B 2-
* 1 -
0 -
400 -
300 ;
c 200 ;
100 r
-| '
80 -
I- 60 -
r 40-
^ 20 -
01 ■ ' ' ' I
0.5 0.6 0.7 0.8 0.9 1.0
Ap/pe
Fig. 4.2 Underpressure dependence of (a) the nucleus radius, (b) the nucleus size,
and (c) the nucleation work for HON of steam bubbles in water at T = 583 K
according to eqs (4.20)-(4.22).
4.3 Heterogeneous nucleation
The determination of «* and W* for HEN can be done in the same manner
as in the case of HON provided that the W(n) function is known. Employing
the expressions for W(n) from Section 3.2, let us now find n* and W* for
nucleation of caps, lenses and disks.
(a) Cap-shaped clusters (Fig. 3.5a)
We first consider HEN of condensed phases. Using eq. (3.60) in eqs (4.1)
and (4.2) yields [Zettlemoyer 1969)
(b)
c
Nucleus size and nucleation work 51
n* = \^8J32!tv^aV3A^ (4.24)
W* = v<6w) \bizvl <tV3 Aji2. (4.25)
Comparison of these equations with (4.10) and (4.11) shows that in HEN
n* and W are diminished l/i/Mimes with respect to n* and W* in HON. In
other words, the better the wetting of the substrate by the nucleus (i.e. the
smaller the wetting angle 8W), the smaller the nucleus on the substrate and,
hence, the less the work for its formation. For example, for a hemispherical
nucleus (8W = Till) n* and W* are twice smaller, as then 1//= 1/2 (see Fig.
3.6). In the limiting case of complete non-wetting (8W = K so that 1//= 1) eqs
(4.24) and (4.25) pass into eqs (4.10) and (4.11) for HON. It is worth noting
that the radius R* of the heterogeneously formed cap-shaped nucleus is not
affected by the presence of the substrate (i.e. by the factor I//) and is again
given by the Gibbs-Thomson equation (4.9). The relationship W* = (l/2)n*A^i
noted for HON remains in force also for the cap-shaped nucleus.
As in the case of HON, finding n* and W* in HEN of cap-shaped nuclei
of gas phases is more conveniently done with the aid of eqs (4.3)-(4.5).
Using (2.10), (3.11) and (3.14) in eq. (4.4) results in
p* = />e exp (-voAp/kT)
which means that the pressure p* in the cap-shaped nucleus bubble is the
same as in the spherical homogeneously formed one (cf. eq. (4.14)). On the
other hand, since now 0, fa, 0so and Vn are given by (3.51)-(3.54), from eq.
(4.3) we arrive again at the Laplace equation (cf. eq. (3.24))
P* =P + 2a/R*. (4.26)
Recalling that n* = (47t/3)(p*/kT)R*i, from this equation we find that the
radius R* of the nucleus bubble and the number n* of molecules in it are
determined by the formulae [Hirth and Pound 1963; Blander 1979]
R* = 2al(p* - p) (4.27)
n* = v<ew)32^*o3/3A:r(p* - pf (4.28)
in which p* depends on/> according to eq. (4.14) and can be approximated
by /7e if pzVtJkT < < 1 and 0 < p < pe. These two expressions are the Gibbs-
Thomson equations for cap-shaped gas-phase nuclei in HEN and the former
shows that R* is the same as in HON.
The nucleation work W* can be obtained by using eq. (4.5) in combination
with eqs (3.51)-(3.54) for ¢, 0S, 0sO and V„ and with eq. (4.27) which relates
R* to p* - p. After some algebra the result is [Kaischew and Mutaftschiev
1962; Hirth and Pound 1963; Skripov 1972; Blander 1979]
W* = \fi(9.„)l6x03B(p* - pf (4.29)
where p* can be approximated by pe when p^vt^kT « 1 and 0 < p < pe, but
in general is a function of p according to eq. (4.14).
Comparing eqs (4.28) and (4.29) with eqs (4.21) and (4.22), we see that
in HEN of cap-shaped gaseous nuclei the substrate brings about l/i//-fold
52 Nucleation: Basic Theory with Applications
reduction of both the nucleus size and the nucleation work with respect to n*
and W* in HON. For such nuclei 6W ~ % means complete wetting of the
substrate by the old phase and in this limiting case (4.28) and (4.29) pass into
(4.21) and (4.22) for HON, as then y/= 1. In this case the substrate has no
effect on n* and W*. We note, too, that the relationship W* = (U2)n*kT
(1 - c_4,"kT) obtained for HON remains valid also for a cap-shaped nucleus
bubble on a substrate.
(b) Lens-shaped clusters (Fig. 3.5b)
Considering only nucleation of condensed phases, from eqs (4.1) and (4.2)
with the help of (3.66) we find easily that (e.g. [Russell 1980])
n* = [y/(ew) + (sin ew/sin esfy/{es)]327cv2t7s/3&n3 (4.30)
W* = [t//(0w) + (sin djsin e^f^e^dKvl^B^2. (4.31)
As in the case of cap-shaped nuclei, these equations say that the role of
the substrate is again to decrease in the same degree both the nucleus size
and the nucleation work. This decrease is now controlled by the factor 1//( 8W)
+ (sin 0w/sin 8s)3y/(0s) which depends on the two angles 8W and 8S. For that
reason, the formula W* = (l/2)n*A^ which connects W*, n* and A/y in the
already considered cases of nucleation of condensed phases remains unchanged.
(c) Disk-shaped clusters (Fig. 3.5c)
Clusters with the shape of a disk are characteristic for 2D nucleation on
either foreign or own substrate. Again for nucleation of condensed phases,
we can use eqs (3.70), (3.71), (3.74) and (3.75) in (4.1) and (4.2) to find that
the radius R* of the 2D nucleus, n* and W* are given by [Hirth and Pound
1963; Zettlemoyer 1969] (A^i > aefAo)
R* = /fflef/(A/i - aefAcJ)
„* = b2 kV4(A/( - ae!Ao)2
W* = ftV/4(A^i - flefAfj)
for 2D nuclei on foreign substrate (b = 2(naef)>12) and by (A/< > 0)
R* = tcatJAfi (4.34)
„* = feVMAji2 (4.35)
W* = fcV/4A^ (4.36)
for 2D nuclei on own substrate (b = 2(na0)m).
These formulae show that in distinction to 3D nucleation, the nucleation
work in 2D nucleation is related to the nucleus size n* by the expression H7*
= n*A^i which coincides with the estimate in Chapter 3. The known 2D
analogue of eq. (4.12), W* = (\I2)2xR*k [Brandes 1927], is also evident
from these formulae (2kR*k is the total edge energy of the disk-shaped
nucleus). It must be noted as well that with corresponding values of b eqs
(4.32), (4.33), (4.35) and (4.36) are applicable also to prismatically shaped
nuclei: for instance, for square prisms b = 4a",2 in (4.32) and (4.33) and
Nucleus size and nucleation work 53
b = 4ai'2in (4.35) and (4.36). The equations for R* and n* are the Gibbs-
Thomson equations for 2D nucleation of condensed phases.
Let us now compare the results for n* and W* for 3D nucleation of
condensed phases, eqs (4.7), (4.8), (4.24), (4.25), (4.30) and (4.31), with
those for 2D nucleation, eqs (4.32), (4.33), (4.35) and (4.36). It is seen that
for 2D nucleation both n* and W* depend more weakly on a (provided /fis
proportional to a) and are less sensitive to changes of A/i. The latter distinction
is illustrated in Figs 4.3a and 4.3b in which the n*(Afi) and W*(A^i) dependences
100
80
60
*c 40
20
1
40
I-
_*:
«~ 20
*
'■_ \3D
\3D
. ...1.,..1
- \
\
\,2D
\ 2D
A
*t
....1
(a)
(b)
(c)
0.5
1.0
An/kT
1.5
2.0
Fig. 4.3 Supersaturation dependence of (a) the nucleus size, (b) the nucleation work,
and (c) the height of cap-shaped (curves 3D) and disk-shaped (curves 2D) water
nuclei in HEN on a substrate in vapours at T = 293 K: curves 3D - eqs (4.24) and
{4.25) at 0M, = it/3; curves 2D - eqs (4.32) and (4.33) at ACT = (i/2)o; solid-dashed
curve - eq. (4.37) at 0W = %/3; circle -point oj transition from 3D to 2D HEN.
54 Nucleation: Basic Theory with Applications
for cap-shaped (curves 3D) and disk-shaped (curves 2D) water nuclei on a
substrate in contact with vapours at T = 293 K are drawn according to eqs
(4.24), (4.25), (4.32) and (4.33) with k = od0, aef = a0 = Vo/dQ, 1/=0.16 and
Aa = oil (these y/ and Ao values correspond to 0W = 7tf3 in eqs (3.56) and
(3,73)). The other parameter values used are given in Table 3.1. The choice
of aef = fl0 in eqs (4.32) and (4.33) means that curves 2D describe
heterogeneously formed disk-shaped nuclei of monomolecular height
A* = d0.
The curve in Fig. 4.3c depicts the Ap dependence of the height h* of the
cap-shaped nucleus. This dependence is described by the formula
/i* = (1 - cos 0w)[3tvi*/47ry/(ew)]1/3 (4.37)
which follows from elementary geometry and in which 1 - cos 0W and n* are
specified by eqs (3.73) and (4.24). As seen from Fig. 4.3c, at a certain
supersaturation A/it, given by
Afjt= 2(votd0)(l - cos 0w)a= 2(u0/J0)Aa,
the height h* of the 3D nucleus is equal to the molecular diameter d0. It turns
out IToschev et al. 1968; Markov and Kaischew 1976; Kashchiev 1977] that
for substrates with Aa > 0 this result is of general validity, and Apt is the
transition supersaturation at which a change in the mode of HEN occurs: for
Ap < A/it nucleation is 3D, and for Ap > A/it it is 2D. This change occurs
because while for Ap < Apt the nucleus is of height h* > d$ which decreases
with Ap, at A/i = Apt it is already with the monomolecular height h* = d0.
With a further increase of A/i the nucleus remains with the minimal physically
possible height of one molecule (the line above the dashed portion of the
h*(Ap) curve in Fig. 4.3c). Hence, for A/i > Ap( 2D nucleation takes place.
The above formula for Apl is easy to obtain from the condition h*(Apt) = d0
with the help of eq. (4.37). In Fig. 4.3c we read that at the chosen T = 293
K and 0W = idl the transition from 3D to 2D HEN of the water droplets on
the substrate occurs at ApJkT = 1.4 which in view of (2.8) corresponds to
P = 4/7e.
Inspection of the classical expressions for 3D nucleation shows that
regardless of whether nucleation is homogeneous or heterogeneous, W(n),
«* and W* can be represented by unified formulae if a is replaced by an
effective specific surface energy <7ef. The formulae for the nucleus size and
the nucleation work then read:
n* = 8c3vJa^/27A/z3 (4.38)
W* = 8c3vlol(mAv2 (4.39)
for nuclei of condensed phases and
n* = &c3p*(J^/27kT(p*-pf (4.40)
W* = 4c3<73f/27(p* -/7)2 (4.41)
for nuclei of gas phases (c = (36?r)l/3 for spheres, c = 6 for cubes, etc.). The
Nucleus xize and nucleation work 55
quantity aa accounts for the activity of the substrate with respect to reducing
the nucleus size and the nucleation work and for the deviation of the shape
of the heterogeneous nucleus from that of the homogeneous one. The definition
of c% is
Ofef = ftr (4.42)
where the activity factor f is a number between 0 and 1. For example, for
cap-shaped nuclei f = i//"3(8w), and for lens-shaped ones f = [y(#w) + (sin
6w/sin 0s)3y/(0s)]"3. Obviously, the f = 1 limit corresponds to HON, for then
<7ef = o. The various cases of HEN are described by f < 1 and for them (Tef
< o. Since f and <7ef have smaller values when the substrate is more active
(in the sense that n* and W* are smaller), the activity factor for, equivalently,
the effective specific surface energy aef are convenient parameters for
characterizing the substrate activity.
4.4 Atomistically small nuclei
The considerations in Sections 4.2 and 4.3 are in the scope of the classical
nucleation theory which applies to nuclei of large enough size n*. In many
cases of both HON and HEN, however, a score and even only a few molecules
can constitute the nucleus. The question about the Ay dependences of n* and
W* for these atomistically small nuclei can be answered by using the results
for W(n) of the atomistic theory of HON and HEN described in Chapter 3.
The step-like line in Fig. 4. lb illustrates one important general feature of
the dependence of the nucleus size n* on the supersaturation Ay in the high-
supersaturation range. It is seen that there exists a succession of Ay intervals
in which n* remains constant [Walton 1962, 1969b; Stoyanov 1979; Milchev
1991], because in reality the number of molecules in the nucleus can change
only discretely with not less than 1 molecule. As already noted in Chapter 3,
the atomistic theory is convenient for application to this range of high
supersaturations and small nucleus sizes, since in each of the successive Ay
intervals the classical Gibbs-Thomson dependence of n* on Ay is replaced
by
n*(Ay) = constant. (4.43)
Owing to this result, it has been concluded [Stoyanov 1979; Milchev
1991J that for atomistically small crystal nuclei in HEN from vapours and
electrolytic solutions W* is a broken linear function of Ay. It is clear, however,
that this result is of more general validity. Indeed, from eq. (4.6) we find that
W* depends linearly on Ay according to the general formula
W* = -n*Afi + <J>* (4.44)
provided that eq. (4.43) is satisfied and that <P* = <P(n*) is A^i-independent.
For instance, for atomistically small nuclei of condensed phases eqs (3.44),
(3.77) and (4.2) lead to
56 Nucleation: Basic: Theory with Applications
W* = -n*A/j + (In* - £*) (4.45)
in the case of HON and to
W* = -n*&n + (hi* -E* - OtA*) (4.46)
in the case of HEN (for brevity, E* = £„« and A* = A„«). In the particular case
of 2D nuclei of monomolecular height A* - agn* and from eq. (4.46) (or
from (3.78), (3.79) and (4.2)) it follows that
W* = -n*&n + (hi* - E*-a0osn*) (4.47)
for 2D nucleation on foreign substrate and that
W* = -n*A/( + (hi* - E* - a0on*) (4.48)
for 2D nucleation on own substrate. Inspection of eqs (4.44)-(4.48) shows
that, in conformity with (3.89), for nucleation of condensed phases 0* can
be represented most generally by the formula
** = hi* - E* - a,A* (4.49)
in which crs = 0 and as > 0 correspond to HON and HEN, respectively.
For the atomistically small nuclei of gaseous phases W* from (4.44) may
also be a linear function, but of the pressure p rather than of the supersaturation
A/i. This is so because for such nuclei <P* is a function of Ap: according to
(2.1), (3.90) and (4.4) we have
** = n*A,u - (p* - p)V* + <j>* - O^A*. (4.50)
Combining (4.44) and (4.50) we find that, atomistically, for nucleation of
gaseous phases W* is given by
W* = V*p + (.$* -p*V* - o,A*) (4.51)
where again crs = 0 and crs > 0 correspond to HON and HEN, respectively. As
seen from this expression, if in a certain pressure range V* and the bracketed
term are /?-independent, W* will decrease linearly with lowering p.
Equations (4.45)-(4.48) and (4.51) are concrete forms of eq. (4.44). As
exemplified in Tables 4.1 and 4.2, with properly defined supersaturation Ap
(see Chapter 2), eqs (4.44)-(4.48) are the atomistic formulae for the A^i
dependence of W* for HON and HEN of condensed phases in vapours,
solutions, melts, etc. In these equations n* and <J>* (respectively, the bracketed
terms which are explicit expressions for <P* from (4.49)) are unknown constants
and, for this reason, the atomistic theory can predict values of W* only with
the help of model considerations for the nucleus size «*, the nucleus binding
energy E* and the nucleus/substrate contact area A* [Venables and Price
1975; Lewis and Anderson 1978; Stoyanov 1979]. Such considerations are
needed as well for the determination of A*, the nucleus volume V*, the
pressure p* inside the gaseous nucleus and the nucleus total surface energy
0*, since in (4.51) they are also unknown parameters. Moreover, from eqs
(4.44)-(4.48) and (4.51) it is hard to see whether the atomistic theory leads
to a W*(A(i) dependence satisfying the general thermodynamic requirement,
Nucleus size and nucleation work 57
eq. (4.13), for annulment of W* at the supersaturation A/js corresponding to
the spinodal.
Table 4.1 Work W* for atomistic HON and HEN in various cases.
Nucleation
in the case ol'
vapour condensation
boiling
solute condensation
melt crystallisation3
melting3
electrocrystallization
W*
(HON)
-n*kT In (pip,)
+ h* - E*
V*p + $* -p*V*
-n*kT\n(CICc)
+ Xn*-E*
n*AseT - E*
-n*\s,(T-2Tc)-E*
-
W
(HEN)
- n*kT In (plpe) + h* - E* - oyl*
-n*kT In (1/1,) + h* - E* - a.A*
V*p + $• -p*V*~ oyi»
- n*kJ In (CIQ) + h* - E* - Cy4*
n*&s,T - E* - cyl*
-n*As,(T- 2Te) - E* - Cy4*
-«*s^o((Pe- (p) + h* - E* - rjsA*
awhen Afj is given by eq. (2.23) in which Aj'e = XiTe
Table 4.2 Work W* for atomistic 2D HEN of monolayer nuclei on
foreign (o_< * G) or own (as = a) substrate in various cases.
Nucleation in the case of W*
vapour condensation -n*kT In {pipe) + (A - a0<7s) n* - £*
-n*kT In (///e) + (A - aaas) n* - £*
evaporation or sublimation -n*kT In ipjp) + (A - ao°s) n* ~ £*
-n*kT In (IJI) + (A - a0as) n* - £*
solute condensation -n*kT In {CICe) + (A - a0<Ts) n* - E*
dissolution ~n*k'F In {CJC) + (A - a0as) n* - E*
meli crystallization" n*AseT - a0osn* - £*
melting* -n*AsJ' + (2A - a0<ys) n* - E*
electrocrystallization -n*zten{(ps - <p) + (A - a0oj n* - E*
electrodissolution -n*z-,efj{p-<pc) + (A - aQOs) n* - E*
Vhen Aft is given by eq. (2.23) in wbich Ase = A/7"e
Chapter 5
Nucleation theorem
The considerations in the preceding chapter show that, depending on the
concrete model for the nucleus shape and size, in the various cases of nucleation
the nucleation work W* and the nucleus size n* are different functions of Ap.
Notably, however, the classical theory provides the rather general expressions
W* = (l/2>*A/i (5.1)
W* = n*&fj (5,2)
which relate W*, n* and A^/ in the considered cases of EDS-defined 3D and
2D nuclei of condensed phases regardless of the assumed nucleus shape.
This is a signal that there can exist a more general, model-independent
relationship between W*, n* and Ajj. Indeed, inspection of the pairs of
equations (4.7)-(4.8), (4.24)-(4.25), (4.30)-(4.31), (4.32)-(4.33), (4.35)-
(4.36) and (4.38)-(4.39) for W* and n* shows that not only the classical
formulae (5.1) and (5.2), but also eqs (4.43) and (4.44) of the atomistic
theory can be amalgamated into a single and in this respect more general
expression of the form
dW*/dAv = -n*. (5.3)
This relation was noted by Nielsen [1964] and Kostrovskii et al. [1982] in
analysing particular cases of nucleation with the aid of the classical theory,
and in another context, a similar formula was used by Gibbs [1928, p. 264].
Despite its great generality, as can be verified with the help of (2.10) and
the pairs of equations (4.16)-(4.17), (4.28)-(4.29) and (4.40)-(4.41), eq.
(5.3) is not applicable to bubble nucleation. Moreover, it is valid only for
EDS-defined nuclei and in this sense its generality is limited: a most general
formula must connect W* and A/j, which are invariants with respect to the
choice of the dividing surface, with a quantity which, in contrast to «*, does
not depend on this choice. One may then ask the important question: is there
any formula relating in a most general, universal way some thermodynamic
characteristics of the nucleation process which are invariants with regard to
the choice of the dividing surface? The answer is positive and the nucleation
theorem represents such a universal relationship between the nucleation work
and the nucleus size. In its most general, model-independent form the nucleation
theorem for one-component nucleation was proven first by Kashchiev [1982]
with the help of phenomenological considerations. Later, it was substantiated
statistically by Viisanen et al. [ 1993] for one-component HON and generalized
statistically by Strey and Viisanen 11993] and thermodynamically by Viisanen
et al. [1994| and Oxtoby and Kashchiev [1994] for multicomponent HON.
Nucleation theorem 59
Model-dependent results related to the nucleation theorem were obtained by
Nielsen [1964], Allen and Kassner [1969], Anisimov et al. [1978, 1980,
1982, 1987], Baidakov era/. [1980],Kostrovskiie/a/. [1982], Anisimov and
Cherevko [1982], Anisimov and Vershinin [1988] and Bedanov et al. [ 1988],
New findings have also been reported recently [Ford 1996, 1997a; McGraw
and Laaksonen 1996; Laaksonen 1997; Luijten 1998],
Let us now prove rigorously the nucleation theorem first phenomenologically
and then thermodynamically. In the phenomenological proof we shall use
eqs (3.86) and (4.6) with ^regarded as a phenomenological quantity defined
only to satisfy the energy balance W = -nA/i + <Z>. The thermodynamic proof
will be done with the help of eq. (4.5) with <£* determined from the very
general, but still concrete, formula (3.83).
5.1 Phenomenological proof
In its phenomenological formulation [Kashchiev 1982] the nucleation theorem
for one-component nucleation reads:
If the work for cluster formation is expressed in the form ofeq. (3.86) and
ifW* and <P* are diffe re ntiable functions of Aft, the relation
dW*/dAv = -n* + d&t/dAv (5.4)
holds true, <P* = <P{n*(A^), Ap] being the effective excess energy of the
nucleus.
To prove the nucleation theorem in the above most general form, following
the original analysis [Kashchiev 1982], we find first die Ay derivative of W*
from eq. (4.6):
dW*/dAv = ~n* - (Ap - d<P*fdn*)(dn*/dAv) + d^ldA^i. (5.5)
Then, considering n as a continuous variable, from (3.86) and the condition
for extremurn, eq. (4.1), we find that
A^ - (d<&dn)n=n* = 0. (5.6)
This equation is in fact a most general, phenomenological form of the Gibbs-
Thomson equation for one-component nucleation. Since d&*/dn* = (d<Df
dn)n=n*, owing to (5.6) the d«*/dA|i term in (5.5) vanishes. Thus, eqs (5.5)
reduces to eq. (5.4) which was to be proven.
Clearly, since throughout the derivation of eq. (5.4) no assumptions
whatsoever are made about the physical nature of #*, the nucleation theorem
in this phenomenological form is valid for any kind of one-component
nucleation (HON, HEN, 3D, 2D, involving atomistically small nuclei of a
few molecules only, occurring in external fields, etc.). While n* and <&*
depend separately on the choice of the dividing surface, the sum -n* + d<P*l
dAp in eq. (5.4) does not. The nucleation theorem thus relates two invariants
with respect to the choice of the dividing surface, W* and -n* + d<P*/dAn,
and that makes it of universal validity.
60 Nucleation: Basic Theory with Applications
Let us now apply the nucleation theorem (5.4) to the particular case of 0*
defined by the very general, but already concrete expression
0* = n*A» - (p* - p)V* + 0* + 0S* - 0sO (5.7)
where 0* = 0[n*(A/j), Ap:] and 0* = 0s[n*(A^i), Ap]. This expression is valid
for arbitrary choice of the dividing surface and follows from (3.88) upon
taking into account (2.1), (3.9), (4.4) and the annulment of the ns term in
(3.88) owing to the fact that for the nucleus jis = p:old [Toschev 1973a].
Naturally, 0* from (5.7) is obtainable also by setting equal the right-hand
sides of (4.5) and (4.6). Using 0* from this equation allows rationalizing the
<90*/(&Vi term in (5.4) not only in the cases considered hitherto, but also in
all possible cases of nucleation in which the nucleus effective excess energy
is of the above form. In addition, with the help of <P* from (5.7) it becomes
possible to answer the question: why does d<P*/dAn appear in (5.4), but does
not in (5.3)? Is the reason for its absence in eq. (5.3) due to the fact that this
equation relates W*, n* and Ap. only for condensed-phase nuclei which,
moreover, are defined by the EDS?
Our task is thus to find dcp*/dAfi with the aid of #* from (5.7). It is more
convenient, however, to calculate first the derivative <9#*A9ju0]d, since we
can profit from the three 'bulk' and three 'surface' Gibbs-Duhem relations
at constant T [Guggenheim 1957]
dp = Pold<Kld <5-8)
<IP = Pnew<Kew (5-9)
dp* = - p*w d^oid (5.10)
d0s,O =-'WKid <5-n)
d0s* = -pidpo,d (5.12)
d0* = -n*&nM. (5.13)
Here pold (m~3), pnew (m~3) and p*w (m~3) are, respectively, the number
densities of the molecules in the bulk old and new phases and in the nucleus,
n* is the Gibbs surface excess in the number of molecules (n* accounts for
the choice of the dividing surface defining the nucleus), and nads and n*ds are
the numbers of the old-phase molecules adsorbed on the substrate respectively
in the absence and presence of the nucleus on it. In (5.10) the chemical
potential ^newn* °f die molecules in the nucleus is replaced by (iM, since
according to (4.4), /inew n* = po]d. When the EDS is chosen as a dividing
surface, by definition,
n* = 0, (5.14)
and when there is no adsorption of molecules of the old phase on the substrate,
"a*ds = "neb = 0 (5.15)
so that then none of the total surface energies 0*, 0* and 0S 0 depends on
iU0jd. In the general case n*, n^,s and nads can also be positive or negative
Nuclealion theorem 61
depending on the choice of the dividing surface and on the character (positive
or negative) of the adsorption. We note that the case of HON is also describable
by (5.15), as then there is no substrate (and, hence, adsorption) in the system.
Using (5.8)-(5.13), from (5.7) we find that
9**%oid = n* dA/i/d/i„u - V*(p*w - poU) - n* - n*ds + n^. (5.16)
As by definition
p*w=n*/V* (5.17)
and since because of (2.1), (5.8) and (5.9)
from (5.16) we finally get
3**/9A,U= n*(pold/ft*ew) (1 - fttw/Pnew V(l " Pold/Pnew)
- (n* + n*ds - na(ls)/(l - fWp„ew)- (5.19)
This expression shows that when <P* is defined by (5.7), the role of the
d&*ld&n term in the nucleation theorem is two-fold. First, through the n*
summand this term allows for the departure of the number density p*w of
the molecules in the nucleus from their number density pnew in the bulk new
phase. Second, this term accounts explicitly for the choice of the dividing
surface and for the character of the adsorption (the second summand in
(5.19)). Both summands in (5.19) vanish under the following conditions: the
n* summand - when the nucleus and the bulk new phase have the same
density (then p^w = Pnew)'trie second summand - when the nucleus is EDS
defined either in HON or in HEN on non-adsorbing substrates (then (5.14)
and (5.15) are valid). This means that
df/dAp = 0 (5.20)
is a good approximation for EDS-defined nuclei of condensed phases (then,
typically, p*w = pnew) in the cases of both HON and HEN if in the latter
case the substrate is so weakly adsorbing that n^is - nads = 0. We thus see
that the reason for the absence of the d&*/dAp term in eq. (5.3) is that this
equation follows from formulae for W* and n* describing only EDS-defined
nuclei of condensed phases. The conclusion is [Kashchiev 1982] that eq.
(5.3) is a particular form of the nucleation theorem. This form is applicable
to nucleation of condensed phases when the nucleus effective excess energy
<J>* is determined by (5.7), the nucleus is defined by the EDS and the substrate
(in the case of HEN) is not covered more than, e.g. a few percent by adsorbed
molecules of the old phase.
Substituting 30*/dAfi from (5.19) in (5.4), we can now find what is the
form of the nucleation theorem in the particular case of <J>* specified by
(5.7). The result is
W^dAfi = - An*/( 1 - pM/PlKJ (5.21)
where An* is given by
62 Nucleation: Basic Theory with Applications
An* = n*(l - pold/p*w) + n* + na*ds - na(ls. (5.22)
Recalling (5.18), we see that eq. (5.21) can be represented equivalently and
even more simply as
dW*ldiioili = -An*. (5.23)
Equations (5.21) and (5.23) are two equivalent forms of the nucleation
theorem for isothermal one-component nucleation. Though less general than
the phenomenological form (5.4) of the nucleation theorem because of their
relying on <P* from (5.7), eqs (5.21) and (5.23) cover practically all commonly
considered cases of nucleation. For instance, it can be verified that n* and
W* given by the pairs of equations (4.16)-(4.17), (4.28)-(4.29) and (4.40)-
(4.41) of the classical theory of gas-phase nucleation satisfy (5.21) and
(5.23) under the condition that eqs (5.14) and (5.15) are obeyed. Recently,
eq. (5.23) was substantiated statistically by Viisanen etal. [1993] and derived
thermodynamically (see below) by Oxtoby and Kashchiev [ 1994] in the case
of HON for which n^s = nads = 0. Here, we have generalized eq. (5.23) by
proving that it holds true also for HEN, i.e. when n„A, *nads * 0.
We now return to the quantity An* which is most remarkable with its
independence of the choice of the dividing surface defining the nucleus. This
independence is mandatory, because W*, A/j, /i0|d, pold and pnew in (5.21)
and (5.23) are invariant with respect to this choice and so must, therefore, be
An*. The physical meaning of An* is very simple and can be perceived
easily by considering a spherical nucleus formed heterogeneously on a spherical
substrate, the two spheres having a common centre. In this case n*^ = 0
because after the nucleus formation the substrate is not any more in contact
with the old phase and adsorption is impossible. Curves CF and DF in Fig.
5.1 depict schematically the spatial variation of the molecular densities pi
and p* in this system, respectively before and after the appearance of the
nucleus on the substrate. As illustrated by the shaded area DFC in Fig. 5.1,
An* is the excess number of molecules in the spatial region occupied by the
nucleus over that present in the same region before the nucleus formed. This
is so, as in Fig. 5.1 the geometrical representation of the various summands
in (5.22) is as follows: n* = p*w V* is the area ADEG, n*pold/ p*w = poMV*
is the area ABFG, - ns* > 0 is the area DEF, n^ = 0 has no area, and nads
is the area BCF. In Fig. 5.1 the arbitrarily chosen dividing surface is positioned
sufficiently away from the EDS (for this reason nf < 0) in order to emphasize
that An* and all other quantities characterizing the nucleus (e.g. p„ew, V*,
0*, etc.) remain well defined even for atomistically small nuclei of a few
molecules only. Figure 5.1 shows clearly also that An* is independent of the
location of the dividing surface: for example, An* is again represented by
the hatched area DFC if the EDS is chosen as a dividing surface. As then n*
= 0, from (5.22) we find that for EDS-defined nuclei
An* = n*(l - ft,u/p*ew) + "ads - "ads <5'24)
in the case of HEN and
An* = n*(l- pMlpL) (5.25)
Nucleation theorem 63
Pre.'
>-
05
LU
Q
Pold
0
n
V.
e
A
An*
/... .
r
\
n'P<*/Pn«'
".'
E
F
0
■rMrury d'v4ng
surface
V*
VOLUME
Fig. 5.1 Spatial variation of the molecular density before (solid curve Pj) and after
(solid curve p*J the formation of a condensed-phase nucleus around a smaller-size
substrate. The step-like dashed line represents the molecular density corresponding to
an arbitrarily positioned dividing surface which defines the nucleus volume V*. The
positions of the substrate surface and the EDS are also indicated. The shaded area
represents the number An* of molecules which are additionally involved in the
building up of the nucleus.
in the case of HON. These formulae show that both in HEN on weakly
adsorbing substrates (then «*ds - nads ~ 0) and in HON An* is given by the
same formula, eq. (5.25). In both these cases An* becomes merely equal to
«*, i.e.
An* = it*. (5.26)
if the EDS-defined nucleus is much denser than the old phase (then p£w »
p0jd). To a very good approximation, eq. (5.26) is applicable, e.g„ to nucleation
of liquids or crystals in sufficiently dilute vapours or solutions. When pZw
~ pnewi (5.24) and (5.25) can be used for estimating An* with the help of
theoretically determined values of n* for EDS-defined nuclei.
The above considerations reinforce the view [Toschev 1973a] that the
EDS is the most convenient dividing surface for describing the energetics of
one-component nucleation. Indeed, according to (5.21), (5.23) and (5.25),
for EDS-defined nuclei the nucleation theorem takes the simpler equivalent
forms
dW*/dA/i = -**(1 - Po.d/Pn*w)/d - Potd'PneJ C5-27)
dW*/tKid = -«*(! - Poid/p*eW) (5.28)
both for HON and for HEN on weakly adsorbing substrates. In the large
64 Nucleation: Basic Theory with Applications
class of cases of nucleation of condensed phases, typically, p*ew ~ pnew so
that for EDS-defined nuclei of such phases eq. (5.27) leads to the nucleation
theorem in its simplest form (Kashchiev 1982]:
dW*/dA/u = -n*. (5.29)
It is worth noting once again that the classical formulae for W* and n* in
Sections 4.2 and 4.3 satisfy only the above simplified forms of the nucleation
theorem. The important implication is, therefore, that the classical nucleation
theory relies, tacitly or not, on the assumptions involved in the derivation of
eqs (5.27)-(5.29). Clearly, the assumption about the EDS as a dividing surface
which defines the nucleus is among the most significant ones.
5.2 Thermodynamic proof
The thermodynamic proof [Oxtoby and Kashchiev 1994] of the nucleation
theorem is analogous to the phenomenological one. It relies on eq. (4.5)
which is valid for arbitrary choice of the dividing surface, but gives W* in
terms of the nucleus effective surface energy &*, rather than in terms of the
nucleus effective excess energy 0* as does eq. (4.6).
Our task now is to rederive eqs (5.21) and (5.23). We first differentiate
W* from (4.5) with respect to /ioii:
dIV*/d^old = V*d(j> -/>*)/<K,d + (p-p*) dV*/dpM
+ (3<%73v*)(dV*/d,uold) + d<P'/diiM. (5.30)
Then, using (4.3) to cancel the second and the third summand and recalling
the Gibbs-Duhem equations (5.8) and (5.10) as well as eq. (5.17), we transform
this expression into
dW*/dtiaU = - n*(l - Aw/ft™) + 3*s*/3Mow (5.31)
and, accounting additionally for (5.18), into
dlV/dA/u = -«*(1 - A,id/p„*ew)/(l - pold/pnew) + d<p;/d&u. (5.32)
The last step is to find the derivatives of the nucleus effective surface
energy <P* which, according to (3.83), is given by
** = 0* + &*-ft.o- (5.33)
This is easily done with the help of the 'surface' Gibbs-Duhem equations
(5.11H5.13):
d0*/dnM = -nf - «1 + «.ds (5.34)
d4>*/dA,u= (-n* - «1 + nadsV(I - ftM/ftew)- (5.35)
Thus, using this result and the definition (5.22) for An*, we see that (5.31)
and (5.32) take the form of, respectively, eqs (5.23) and (5.21) which were
to be proven.
Nucleation theorem 65
Looking back at eqs (5.31) and (5.32), we note that their derivation involves
no assumptions whatsoever about the physical nature of <P*, This means
that, along with eq. (5.4), eqs (5.31) and (5.32) can be regarded as alternative
phenomenological forms of the nucleation theorem provided that in them
<PS* is a phenomenological quantity defined only to satisfy the energy balance
required by eq. (4.5).
5.3 Generalizations
The nucleation theorem in the form of eq. (5.23) was generalized by Strey
andViisanen [1993], Viisanen er a/. [1994] and Oxtoby andKashchiev [1994]
for multicomponent HON. The latter generalization covers also the case of
non-isothermal HON which can occur, for instance, in crystallization of
melts and in polymorphic transformation of solids.
Following Oxtoby and Kashchiev [ 1994], let us first generalize the nucleation
theorem for the case of non-isothermal one-component nucleation. For clarity,
it is necessary to note that in this section the term non-isothermal is used in
the sense that A/i can be given different values by choosing different temperature
T of the system. Despite the change of A^i by means of 7, however, the
nucleation process itself occurs at constant temperature - the same which is
chosen to determine the A^i value. If we repeat the derivation outlined in
Section 5.2 by allowing for the variation of T through appropriate sdT terms
added to the right-hand sides of the Gibbs-Duhem equations (5.8)-(5-13)
[Guggenheim 1957], we shall find that
iW*/dnM = -An* - AS*(dr/d>old). (5.36)
Here An* is given again by (5.22), and AS* is defined as
AS* = J*ewn*(l - 50idPoW/5„*wASw) + s*"* + 4dSn«*dS- ■s.d.n.ds (5.37)
where j*w and so[ti are, respectively, the entropies per molecule in the
nucleus and in the old phase, if is the surface excess entropy (per molecule)
of the nucleus, and s*ds and sads are the entropies per adsorbed molecule,
respectively in the presence and absence of the nucleus on the substrate.
Physically, the quantity AS* is the excess entropy of the molecules in the
spatial region occupied by the nucleus over the entropy of the old phase in
the same region before the nucleus formed. Similar to An*, AS* is invariant
with respect to the choice of the dividing surface and, as required, this makes
invariant also the whole r.h.s. of (5.36).
Equation (5.36) is the general form of the nucleation theorem for one-
component both HON and HEN under non-isothermal conditions. It holds
true for arbitrary choice of the dividing surface and passes into (5.23) when
T is constant, as then dT = 0. It is an extension of the result of Oxtoby and
Kashchiev [1994] which is obtained under the restriction 0,* = 0S 0 = 0 and
which is thus valid only for HON.
In the experimentally interesting case of changing ,t/0|d by varying T at
66 Nucleation: Basic Theory with Applications
constant pressure p of the old phase d/joJd = -so]ddT (see eq. (2.19)). Then,
for HON or HEN on weakly adsorbing substrates, again for EDS-defined
nuclei, with the help of (5.14) and (5.15) eq. (5.36) simplifies substantially
and takes the form
cW*/d/iold = -n*(l - 4w/J0ld)- <5-38)
This can be represented equivalently as
dW*/dAji = -«*(1 - .s*w/sold)/(l - Wsold). <5-39)
since according to (2.1) and (2.19) 4unew = -snewdTand dAjiAhjo]d = l-snew/
sold-
Comparison of eqs (5.38) and (5.39) with eqs (5.27) and (5.28) shows that
the role of the density factors in the case when ^0|d and A^i are changed by
changing p at constant T is played by the entropy factors when the ^old and
A^i changes are brought about by variation of T at constant p. It is worth
pointing out, however, that when the entropies of the molecules in the EDS-
defined nucleus and in the bulk new phase are nearly the same (i.e. when
■5*ew = -snew), e1- (5.39) leads again to the nucleation theorem in its simplest
form [Kashchiev 1982]
dW*/dAfi = -n*. (5.40)
It is this form of the nucleation theorem which is satisfied by the classical
theory of one-component nucleation in melt crystallization. This means that,
as already noted, this theory is limited by the assumptions involved in the
derivation of (5.40), an important one being that the nucleus is EDS defined.
Allowing now for the presence of more than one component in the old
phase and in the nucleus, we can follow the modus operandi leading to eq.
(5.36) in order to generalize the nucleation theorem to cover the case of
multicomponent non-isothermal nucleation. Using the Gibbs-Duhem relations
(5.8)-(5.13) with the extra terms in them necessary to account for the presence
of more than one component in the system [Guggenheim 1957], we find that
in this case eq. (5.36) retains its form:
3W*/3f,Mj = -An* - ASHdT/df,^). (5.41)
Here, analogously to (5.22) and (5.37),
An* = n* (1 - Poky/piTewj) + n* + n*dSii - nAi (5.42)
AS* = 5ueW|,-Hi (1 - soldjp0idj/snew/pncwj)
+ .ss*n* + .sa*SJn*d!J - Jadsjiads,;. <5-43)
the subscript i - 1, 2, . . ., k referring the respective quantity to component
i in the ^-component system. It should be noted that AS* is ^-independent and
can be calculated if the quantities on the right of (5.43) are known only for
one of the components, e.g. component 1.
Equation (5.41) is the general form of the nucleation theorem for
multicomponent non-isothermal both HON and HEN. It is an extension of
Nucleation theorem 67
the result of Oxtoby and Kashchiev [1994] obtained for HON, i.e. in the
particular case of «*dSi, = nads (- = 0. When nucleation is isothermal, dT = 0
and eq. (5.41) reduces to
with An,* given by (5.42). The nucleation theorem in this form applies to
both HON and HEN and is also an extension of the known result for
multicomponent isothermal HON [Strey and Viisanen 1993, Viisanen et al.
1994, Oxtoby and Kashchiev 1994]. It is important to note that, in contrast
to one-component nucleation, in the case of multicomponent nucleation the
EDS cannot be used for simplifying (5.41)-(5.44), since in this case it plays
no special role. Indeed, even when the EDS is chosen to make n* = 0 for,
say, component /=1, still in general n* * 0 for the other components (( *1)
in the system. For example, for binary HON of condensed phases in vapours
or solutions (then nadsi = n*dSj = 0 and pQ[^i « p*ew,;)» in accordance
with (5.42) such a choice gives Anf = n* for component 1, but
An* = n* + n*2 & n* for component 2.
5.4 Integral form
The nucleation theorem in its various forms given hitherto relates in a most
general way the infinitesimally small changes of W* and A/i or ^/0|d.
Mathematically, it is therefore a differential equation which can be solved to
yield in an equally general way the unknown nucleation work W* as a
function of Ap or^/0id [Kashchiev 1982]. The so-obtained solution is, in fact,
the integral form of the nucleation theorem. Restricting ourselves only to
one-component systems, from (5.23) and (5.36) we readily find that
An*d//old (5.45)
lUold
for isothermal nucleation and that
[A«*+ As*(dT/dMoM)]dM0,d (5.46)
for non-isothermal nucleation. Here W*' = W* (ii'M) is the value of W* at a
chosen reference value ii'M of the chemical potential of the molecules in the
old phase. For EDS-defined nuclei of condensed phases these expressions
can be simplified: according to (5.29) and (5.40), for both isothermal and
non-isothermal HON or HEN on weakly adsorbing substrates we have
[Kashchiev 1982]
n* dA^i (5.47)
An
where A// is a reference supersaturation, and W = W*(Afi').
68 Nucleation: Basic Theory with Applications
Equations (5.45)-(5.47) are integral forms of the nucleation theorem for
one-component nucleation which hold true even for atomistically small nuclei
of a few molecules only. The first two of them are most general, model-
independent formulae for W* which reveal that the nucleation work is done
solely to create the necessary excesses in the number of molecules and in the
entropy within the spatial region occupied by the nucleus. They show that
W* is obtainable without making any assumptions about the nucleus density,
shape and surface energy provided that An* and AS* are known from a
theory which is free of such assumptions. Even more, the whole concept of
surface and surface energy of the nucleus can be abandoned if the An*(/J0|d)
and AS*(p0]d) dependences can be found by using non-classical approaches.
The currently developing density-functional approach (see Chapter 8) is
certainly very promising in this respect and has already produced results
which are particularly valuable, because they do not rely on the concept of
Gibbs dividing surface.
The nucleation theorem in its integral forms, eqs (5.45)-(5.47), allows
determination of the absolute value of W* only when W*' is known
independently. Since thermodynamics requires that W* vanish at the old-
phase chemical potential /volds or supersaturation A^is which correspond to
the spinodal point (see eq. (4.13)), if we choose n'old = ^ioU s or A/<' = A/(s,
for W in (5.45)-(5.47) we shall have W*' = 0. For instance', eqs (5.45) and
(5.47) then result in the following integral forms of the nucleation theorem:
•Void
n*dA^/. (5.49)
Afj
In the absence of spinodal, /Jold s = A^/s = «> and these formulae become
W*0/Old)= [ tsn*ipaXi (5.50)
•'''old
W*(A/i)= f n*dA(i. (5.51)
It is instructive to exemplify the potential applicability of the integral
nucleation theorem by considering EDS-defined nuclei of condensed phases
for which eq. (5.49) is in force in the cases of HON and HEN on weakly
adsorbing substrates. We restrict the example to 3D nucleation and assume
that the classical Gibbs-Thomson equation (4.38) gives adequately the n*(A,u)
dependence for all supersaturations from 0 to A^is. Using (4.38) to perform
the integration in (5.49) leads easily to
1V*(4/i) = 4r3«JfJe3t/27 A/u2 - 4c3u;;CJ3f/27 A/i2. (5.52)
Comparing this equation with eq. (4.39) of the classical theory, we see
Nuclealiott theorem 69
that although both equations predict a linear dependence of W* on 1/A^i2,
their right-hand sides differ by an additive constant which equals zero in the
absence of spinodal limit, as then A/js = °°. This constant depends only on T
and is particularly important for A/v approaching A/vs, since by making W*
vanish at Ay = A^s rather than at Ay —> °°, it removes the fundamental
inconsistency of the classical theory. In this sense (5.52) can be regarded as
the thermodynamically consistent formula of the classical theory for the
nucleation work W* in the case of nucleation of condensed phases. However,
this formula remains hypothetical until it is clarified if the classical Gibbs-
Thomson equation (4.38) can be used in the whole supersaturation range
from 0 to A^is. It is worth noting that the thermodynamically consistent
W*(Afj) dependence (5.52) is in qualitative agreement with the numerical
W*(A[i) dependences obtained by Cahn and Hilliard [1959], Oxtoby and
Evans [1988], Nishioka et al. [1989] and Nishioka [1992] by employing
density-functional methods for studying HON. Also, the A^is term in (5.52)
corresponds to the D(T) term in the W*(Afi) formulae of McGraw and
Laaksonen [1996, 1997].
Chapter 6
Properties of clusters
We have seen in Chapter 3 that the main difficulty in the determination of the
Gibbs free energy G(n) and, thereby, of the work W(n) for formation of an n-
sized cluster arises from the uncertainty concerning the cluster excess energy
Gex(n). The reason for this uncertainty is that the cluster is, in fact, a phase
of finite size. The properties of such a phase differ from those of the
corresponding bulk (i.e. infinitely large) phase. That is why, in eq. (3.9), the
pressure pn inside the n-sized cluster and the chemical potential ^new„ of the
molecules in the cluster are not equal to the outside pressure p and the
chemical potential ^/new of the bulk new phase under otherwise the same
conditions. In addition, the presence of interface boundary between the cluster
and the old phase leads to the appearance of the cluster surface energy $«)
in the energy balance. Let us now consider the effect of the cluster size on pn,
^/ncwn and some other thermodynamic characteristics of homogeneously formed
clusters. Unless especially noted, the considerations will be confined to one-
component clusters defined by the EDS.
6.1 Inside pressure
For the pressure p„ inside a cluster of n molecules we have the generalized
Laplace equation (3.10) so that for a concrete dependence of pn on n we need
a model presentation of 0. In the scope of the capillarity approximation § is
given by eq. (3.17) which, upon being used in (3.10), yields the following
formula for EDS-defined clusters of both condensed and gaseous phases:
pn=p + 2cnGn/3 Vnm + (l/3)d(c„cv,)/d V„l/3. (6.1)
For clusters of condensed phases Vn and n are related in a simple way by eq.
(3.13) and this allows expressing/7„ from (6.1) as an explicit function of n:
P„ = p + 2cna„/3 uj'V3 + {l/3v^)d(cnon)/dnm. (6.2)
The representation of pn from (6.1) in explicit dependence of n for clusters
of gas phases is complicated because of the non-linear relation, eq. (3.14),
between V„ and n in this case.
For spherical clusters of both condensed and gas phases cn = (36?r)1/3 =
constant and Vn is conveniently expressed by eq. (3.22) as a function of the
cluster radius R. Equation (6.1) then becomes
pn = p + 2aJR + doJdR
(6.3)
Properties of clusters 71
which is the known formula [Ono and Kondo 1960; Toschev 1973a; Abraham
1974a] forp„ of EDS-defined spherical clusters.
In the classical nucleation theory cn and <yn are regarded as n-independent
quantities whose values c and <J characterize, respectively, the specified
cluster shape and the specific surface energy of the planar interface (<J is
independent of the choice of the dividing surface and is the limiting value of
on for n —> °°). In this approximation eqs (6.1) and (6.2) take the simple form
Pn = p + 2coB V„10 (6.4)
for clusters of both condensed and gas phases and
p„ = p + 2caB uJ'V3 (6.5)
for clusters of condensed phases. Accordingly, eq. (6.3) reduces to the familiar
Laplace equation (3.24).
Equations (6.4), (6.5) and (3.24) say that the molecules inside any n-sized
cluster are under pressure p„ > p and that p„—>p when n (and, hence, Vn or
R) tends to infinity. This is seen also in Fig. 6.1a which depicts the dependence
of pn on n from eq. (6.4) (with V„ related to n in accordance with (3.21))
and from eq. (6.5) for steam bubbles in water at T = 583 K and for water
droplets in vapours at T = 293 K under the atmospheric pressure p = 0.1
MPa. The calculation is done with c = (36;r)"3 (spheres) and the parameter
values listed in Tables 3.1 and 3.2. In the range of n < 100 molecules the
dependences shown should be regarded as more or less qualitative, since
employing the usual thermodynamic methods for the description of EDS-
defined clusters of such small size is questionable.
6.2 Chemical potential
The chemical potential [inewn of the molecules in an n-sized cluster is given
by eq. (3.11). It is different from the chemical potential ji„w(p) of the molecules
in the bulk new phase at pressure/>, because the pressure p„ inside the cluster
is higher than the pressure p outside it.
To find the dependence of ^n!.w„ on n we use again the capillarity
approximation and consider EDS-defined clusters first of condensed phases.
For them V„ can be approximated by (3.13) if they are treated as incompressible.
Performing the integration in eq. (3.11) with V„ from (3.13) and using p„
from (6.2) then results in
J'new.n = ^new(?) + 2c„a„ v%3/3n"3 + ( ujf/3)d(c„CJ„)/dn"3. (6.6)
For spherical clusters c„ = (36;r),/3, n = (ATtHv^R21 and (6.6) gives conveniently
ju„ as a function of the cluster radius R:
^new.n = ^new(P) + 2a„V0IR + VgdaJdR. (6.7)
Turning now to EDS-defined gaseous clusters, we can employ eq. (3.14)
to perform the integration in (3.11). Doing that and using (6.1) yields
72 Nucleation: Basic Theory with Applications
1500
,_1000
c
'■ 500
0
: 6
1.0
e> o.5
droplet
(a)
v bubble
i 7~i I ... 1
_i i i i_
bubble
(<=)
droplet
200
400
600
800
1000
n
Fig, 6,1 Size dependence of (a) pressure inside cluster, (b) cluster chemical
potential, unci (c) cluster specific surface energy for water droplets in vapours
at T = 293 K and steam bubbles in water at T = 583 K.
Ainew,,, = /i„e«r(P) + WIn [I + 2c„ajip V,,"3 + (l/3/»)d(c„a„)/d V„"3] (6.8)
where Vn is a complicated function of n obtainable from eq. (3.14) with/7,,
from (6.1). For spherical clusters, however, it is much simpler to express Vn
through the cluster radius R rather than through n. Using (3.22) in (6.8) and
recalling that for this cluster shape c = (36¾)10 then leads to
^new,n = //«„(?) + Win [1 + laJpR + (\lp)r\aJiK\. (6.9)
In the scope of the classical nucleation theory with n-independent c„ and
0"„, from eqs (6.6) and (6.8)
/'new.n
it follows that
--rt„ew(p) + 2cciv2„a/3nm
(6.10)
Properties of clusters 73
for clusters of condensed phases [Zettlemoyer 1969] and that
Ainew.,, = //«„(>) + kT In (1 + 2CCJ/3/7 V„"3) (6.11)
for gas-phase clusters. Accordingly, from (6.7) and (6.9) we find that if the
clusters are spherical,
AW.« = ^™w(P) + 2<yvo/R (6.12)
when they are liquid or solid [Zettlemoyer 1969] and
A'new.n^newW + WlnO + lolpR) (6.13)
when they are gaseous [Kaischew and Mutaftschiev 1962].
Equations (6.6)-(6.13) show that^new„(p„) —>^imw(p) when n (and, hence,
V„ or S) tends to infinity. This is understandable, since when the cluster is
sufficiently large, the pressure p„ inside it is practically equal to the outside
pressure p. The dependence of ^newn on n for spherical water droplets in
vapours at T = 293 K and for spherical steam bubbles in water at T = 583 K
and p = 0.1 MPa is depicted in Fig. 6.1b. The calculation is done with the
help of eqs (6.10) and (6.11) (in the latter V„ is expressed via n as required
by eq. (3.21)). The parameter values used are those in Tables 3.1 and 3.2. As
already noted, in the range of n < 100 molecules the dependences in Fig.
6.1b may have only a qualitative character because of the questionable
applicability of usual thermodynamics to EDS-defined clusters of such small size.
Equations (6.10) and (6.11) provide direct evidence that the chemical
potential ^new>„* of the molecules in the nucleus equals the chemical potential
^0id of the molecules in the ambient old phase. Indeed, setting n - n* in
(6.10) and (6.11), inserting n* from (4.7) in (6.10) and V* from (4.15) in
(6.11) and accounting for A^i from (2.1) and for p* and A^i from (4.14)
and (2.10) leads again to eq. (4.4). In the same way it is easy to verify that
iUnew.n > /*oid for n< n* and that /inew „ < ^01<i for n>n*. Physically, the former
inequality implies that the subnuclei cannot form spontaneously, but only by
chance: transfer of molecules from a spatial region (the old phase) where the
chemical potential is lower to a spatial region (the subnucleus) where it is
higher is thermodynamically unfavoured, i.e. it is not a 'natural process'
[Guggenheim 1957].
6.3 Vapour pressure
A bulk condensed phase can coexist with its vapours at a given temperature
T if the vapour pressure is equal to the equilibrium (or saturation) pressure
pc{T). For that reason pe is often called vapour pressure of the condensed
phase. Likewise, a vapour pressure pe„(T) is necessary for a given «-sized
cluster of the condensed phase to coexist with the vapours around it. Since
pe and/?e „ are related to [ie and/4new,n, and /ye and ^ncwfl are not equal because
of the finite size of the cluster, the vapour pressure p,, „ of the n-sized cluster
will also differ from pe.
74 Nucleation: Basic Theory with Applications
To find/>e „ we use the ideal-gas approximation for the vapours and assume
that the cluster is incompressible. Then, from eqs (2.5) and (6.10) we have
J<oW<Pe.„) = Ve + kT In (pcJPe) (6.14)
and
PvewJPJ = ^newteci.) + 2eO vg* ft/l1* (6.15)
for the chemical potentials of the molecules in the vapours around the n-
sized cluster and of the molecules inside the cluster, respectively. Equation
(6.15) expresses fincv/in in the scope of the classical nucleation theory and in
it fimv(pCt„) >s given by eq. (2.6). Coexistence between the cluster and its
vapours is possible only when fuwwn(pn) = fiQ\d(ptl„) s0 tnat fr°m (2-6), (6.14)
and (6.15) it follows that
/7en=/7eexp [2c<yv^/3kTnu3 + vQ(pen-pc)/kT\ (6.16)
or, to a good approximation,
pejl = pe exp {lea u$3 /3kTn m). (6.17)
For spherical clusters (6.17) takes the form of the well-known Thomson
(Lord Kelvin) equation [Thomson 1870, 1871]
p^ = pc exp (2ouo/kTR). (6.18)
The approximation of the classical theory for constant <r„ can be relaxed
by using eq. (6.6) in lieu of (6.10) throughout the above derivation. Equation
(6,17) is then replaced by
Pen = />e exP {( v™/3kT)[2cnon/nm + d{cnan)/dn1/3]}, (6.19)
and eq. (6,18) becomes [Ono and Kondo 1960; Abraham 1974a]
Pen = Pc exp [(v0/kT){2oJR + daJdR)]. (6.20)
The above formulae apply to EDS-defmed condensed-phase clusters
surrounded by their vapours. Similarly, it is possible to speak of the vapour
pressure pen of gas-phase clusters embedded into bulk condensed (liquid or
solid) phases. Physically,pen is again that pressure of the vapours within the
n-sized gaseous cluster, which ensures the coexistence (i.e. the chemical
equilibrium) between the cluster and the condensed phase around it. Hence,
/?en should not be confused with/?n which is the pressure inside the gaseous
cluster necessary only for the cluster's mechanical equilibrium. However,
despite their different physical nature, pcn and p„ can have the same value
for a cluster with particular size. This means that such a gaseous cluster has
the important property of being simultaneously in mechanical and chemical
(i.e. complete thermodynamic) equilibrium. According to eqs (4.3) and (4.4),
the nucleus is a cluster which has just this property. In the scope of the cn, on
= constant approximation, the other clusters (the sub- and the supernuclei)
are only in partial thermodynamic equilibrium: though usually assumed to
be in mechanical, they are not in chemical equilibrium, for they cannot
coexist with the old phase.
Properties of clusters 75
To determine the vapour pressure /?en of spherical gaseous clusters in the
cn,an = constant approximation of the classical theory we use eq. (6.11) in
the form
+ kTIn [i + 2co/3(pen - 2co/3 Vnm) VnI/3]. (6.21)
This expression takes into account that when the pressure inside the cluster
is /?e„, in conformity with (6.4), the outside pressure is pen - 2co/3 Vn,/3.
With this outside pressure, from (2.6) it follows that we can use the formula
Vo\M,n ~ 2c<7/3 Vn1/3) = ^ + v0(p^n - 2co/3 Vn3'3 - /7e) (6.22)
for the chemical potential of the molecules in the condensed phase around
the gaseous cluster (this phase is now the old one). Since coexistence between
the cluster and the old phase requires equality of the above two chemical
potentials, pcn can be found by setting equal the right-hand sides of (6.21)
and (6.22) noting that due to (2.5)
^new(pe,« - 2cO/3 V„l/3) = p.e + kT In [(pen - 2ca!3 Vnlf3)/pel
since now the gas phase is the new one. The result is
/>e.n = Pe exp [ - (vo/kT)(2coB V„l/3 + pe -peJ] (6.23)
or, to a good accuracy,
p^ = pe exp (- 2covo/3kT Vn1/3). (6.24)
In these formulae Vn is a complicated function of n obtainable from (3.21)
withp replaced by pQn - 2cal3 V„i/3. For spherical clusters it is more convenient
to express Vn through the cluster radius R with the help of (3.22). Equation
(6.24) then becomes [Ono and Kondo 1960]
Pen = /7e exp (- 2ov<JkTR). (6.25)
If we want to avoid the approximation of the classical nucleation theory for
rt-independent cn and an, we should repeat the above derivation, but with the
aid of eqs (6.1) and (6.8) instead of (6.4) and (6.11), respectively. Equations
(6.24) and (6.25) then take the form
Pe,n = pc exp {- (vo/3kT)[2cno„/V^ + d(cno„)/d V„1/3]} (6.26)
Pw = Pe exp [- (v(i/kT)(2oJR + don/dR)] (6.27)
for EDS-defined gaseous clusters with arbitrary and with spherical [Ono and
Kondo 1960] shape, respectively.
The dependence of /?e„ on n from eqs (6.17) and (6.24) is illustrated in
Fig. 6.2a for spherical water droplets at 7= 293 K and steam bubbles at 7 =
583 K (the parameter values used are those listed in Tables 3.1 and 3.2). For
the smallest clusters (e.g. of n < 100 molecules) these dependences are more
or less only qualitative, since usual thermodynamics may fail in this range of
cluster sizes. As seen,/7e^—->pc torn —»°°, but while pentpe for the droplets.
76 Nucleation: Basic Theory with Applications
a.
c
0
Q.
0
o
5
4
3
2
1
0
15
10
o
<d 5
0
1.0
0.5
- \
- \
■
Sv^^droplet
bubble
(a)
-
;
-
-
i . . . i . , . i
(b)
(c)
. . . 1 . . i
200
400 600
n
800 1000
Fig. 63. Size dependence of (a) vapour pressure of water droplets in vapours at
T = 293 K and of steam bubbles in water at T = 583 K, (b) solubility of crystallites
of sparingly soluble salts in aqueous solutions at room temperature, and (c) melting
point of ice crystallites in water at atmospheric pressure.
Pe,n - Pe f°r tne bubbles. The physical reason for this distinction is that the
molecules at the convex surface of the droplet and at the concave surface of
the bubble are bound, respectively, more weakly and more strongly than the
molecules at the planar liquid/gas interface. Evaporation of the molecules
is then, respectively, less and more difficult from the curved cluster
surface than from the planar surface and this is reflected by the above
inequalities betweenpt (which is the vapour pressure over the planar surface)
and peM.
Properties of clusters 77
6.4 Solubility
Let us consider a condensed-phase cluster of n molecules which is in a
solution. Clearly, the increased chemical potential of the molecules in the
cluster will affect the cluster solubility, i.e. the solute concentration at which
the n-sized cluster can coexist with the solution. In other words, if Ce „ is this
concentration, we should expect it to differ from the equilibrium concentration
(or the solubility) Ce of the bulk condensed phase of solute. The effect is
completely analogous to the effect of the cluster size on the cluster vapour
pressure />e „, because the chemical potential of the molecules in dilute vapours
and solutions depends logarithmically on pressure and solute concentration,
respectively (see eqs (2.5) and (2.11)). This means that for the solubility Ce „
of an n-sized cluster of condensed phase we can rewrite eqs (6.17) and (6.19)
with pe „ and pc replaced by Ce „ and Ce, respectively:
CCJ, = Ce exp (IcOv™BkTn"3) (6.28)
Ce,„ = Cc exp{(i>f BkT)[2cna„/nm + d(c„o-„)/dn"3]). (6.29)
For spherically shaped clusters these dependences become (cf. eqs (6.18)
and (6.20))
Cejl = Ce exp (2avo/kTR) (6.30)
Ce,„ = Ce exp [(vdkT)(2oJR + dcytt/?)]. (6.31)
Equations (6.29) and (6.31) account for the dependence of the cluster
specific surface energy <J„ on the size of the EDS-defined cluster and, in the
cn,Gn = constant approximation of the classical theory, simplify to eqs (6.28)
and (6.30), the latter being the known formula of Ostwald [1900] for the
solubility of small crystals in dilute solutions. According to (6.28) and (6.30),
the solubility of the condensed-phase clusters increases with decreasing cluster
size. This is seen in Fig. 6.2b which represents the dependence of Ce„ on n
for spherical clusters, resulting from eq. (6.28) with the parameter values
listed in Table 6.1 (these values are typical for crystallites of sparingly soluble
salts in aqueous solutions at room temperature). We note again that in the
small-size region (e.g. for n < 100 molecules) the application of eqs (6.28)-
(6.31) is questionable. Also, for more concentrated solutions, in these equations
Ce „ and Ce must be replaced by the corresponding activities (see Chapter 2).
Table 6.1 Values of various quantities used for calculation of
different dependences for nucleation of crystallites of sparingly
soluble salts in aqueous solutions at T = 293 K
(nm!)
0.05
(nm)"
0.46
(m-3)
10"
a
(mj/m2)
100
D
(um2/s)
1000
Jn
1
"calculated from da= (6iV^)''
78 Nucleation: Basic Theory with Applications
6.5 Melting point
The dependence of the chemical potential of the molecules in a given cluster
on the cluster size has an impact also on the melting point of the crystalline
clusters when they are small enough. The melting point is the absolute
temperature Te^ at which an n-sized crystalline cluster can coexist with the
melt around it. This quantity is analogous to the melting point Te of the
corresponding bulk crystal, i.e. to the temperature at which the bulk crystal
and the melt are in two-phase equilibrium. Similar to pen and Ce „, Te „ can
be found from the condition for equality of the chemical potentials jinew„ and
/<0id of the molecules in the n-sized cluster (the crystallite) and the old phase
(the melt) in which the cluster is formed.
According to eqs (6.10) and (2.19), for ^inew„ and [ioU we have the
expressions
/<„ew,„(7;.„) = fc(p) + s^,(.Te)(.Te - re,„) + lea vflln™ (6.32)
iUdd(^) = MP) + WnXn - n,„) (6.33)
which are restricted by the approximation of the classical theory for n-
independent c„ and an and by the approximations snew,„(7) = constant =
Sncw(T'e) and ■Soid(T) = constant = sM(Tt) for the molecular entropies Jnew-„ and
sM of the crystallite and the melt in the temperature range between Te „ and
re. Setting equal the right-hand sides of the above equations we thus find that
rM = re - lea v^/3Asjim (6.34)
where As,, s sM(Tc) - sn=w(7e) is the melting entropy per molecule. If the
crystalline cluster is regarded as spherically shaped, (6.34) takes the form of
the known formula of Thomson [1886]
re,„ = TC- 2avd&sj<. (6.35)
The approximation for constant cn and on can be relaxed by employing
eq. (6.6) instead of (6.10) for the above derivation. The resulting generalization
of (6.34) and (6.35) for EDS-defined clusters is
Te.„ = Te - ( v^/3 Ase)[2c„a„/n"3 + d(c„a„)/dn"3] (6.36)
when the clusters are arbitrarily shaped and
TtJl = TC- (vo/Ast)(2oJR + dcjJdR) (6.37)
when they are spherical.
Equations (6.34) and (6.35) show that the melting point of the crystalline
clusters is below that of the corresponding bulk crystal, the depression being
more pronounced for the smaller clusters. Physically, this is a consequence
of the weaker (on average) binding of the molecules at the finite-area surface
of the crystalline cluster compared with their binding at the infinitely large
crystal surface. The magnitude of the effect for spherical ice clusters in water
at atmospheric pressure is illustrated in Fig. 6.2c which depicts the dependence
of Te„ on n, calculated from eq. (6.34) with c = (36;r)"3 and the parameter
Properties of clusters 79
values given in Table 6.2. It is worth reiterating that the applicability of eqs
(6.34)-(6.37) to the smallest clusters (e.g. of n < 100 molecules) is questionable.
Table 6.2 Values of various quantities used for calculation of different dependences
for nucleation of ice in water under atmospheric pressure
"0
(nm3)
0.033
do
(nm)*
0.4
(K)
273.15
Asjk
2.6
a
(mj/m2)
26
77(230 K)
(mPas)b
43
7)(240 K)
(mPas)b
16
%
1
Calculated from dQ = {&vGi7t)hi
Calculated from eq. (13.47)
6.6 Specific surface energy
The specific surface energy is an important parameter in the physical chemistry
of surfaces in general and in the classical nucleation theory in particular. For
instance, eqs (6.1), (6,6), (6.8), (6.19), (6.26), (6.29) and (6.36) tell us that
Pn> Wm Pc,n> Ce,n ^d Tcn for a cluster of specified shape (c„ = constant)
depend on n not only directly, but also through the ^-dependence of the
cluster specific surface energy on. If the effect of the cluster size on an
manifests itself more strongly for smaller values of n, it might be expected
that the approximation an = constant = a of the classical theory may introduce
significant inaccuracy in the thermodynamic description of the smaller clusters
even in the size range in which such a description is unquestionable. Clearly,
finding the n-dependence of on is an important problem and this may explain
the interest in this problem shown first by Gibbs [1928] and then by many
others ITolman 1949; Kirkwood and Buff 1949; Defay and Prigogine 1966;
Nishioka 1977, 1987, 1992; Rasmussen 1982a; Rasmussen et al. 1983;
Schmelzer and Mahnke 1986; Larson and Garside 1986; Sohnel and Garside
1988; Nishioka et al. 1989; Dillmann and Meier 1989, 1991; Hadjiagapiou
1994; Laaksonen and McGraw 1996; McGraw and Laaksonen 1997].Despite
the great effort put in solving this problem, there is no generally agreed
opinion on the dependence of on on n. We shall now outline briefly only the
best known solutions of the problem. For further reading we refer to the
detailed considerations of Ono and Kondo [1960], Rusanov [1967, 1978]
and Baidakov [1994] as well as to a recent paper by Schmelzer et al. [1996].
When solving the problem of the dependence of an on n the choice of the
dividing surface is of prime importance. Gibbs [1928] considered clusters
defined by the so-called surface of tension (ST) which is chosen in such a
way that the value of an is minimal with respect to the values resulting from
any other choice of the dividing surface. He derived a differential equation
for the dependence of the specific surface energy on the curvature of the ST,
which for ST-defined spherical clusters reads [Gibbs 1928]
d(ln <rST„)/d(ln R$T) = (260//^)/(1 + 26^^). (6.38)
80 Nucleation: Basic Theory with Applications
Here the subscript 'ST' indicates that 0"„ is referred to the ST, and 50 is
defined by
S0=R-RST (6.39)
where R and SST are the radii of the EDS- and the ST-defined clusters,
respectively. The Gibbs parameter 8(, is thus the distance between the EDS
and the ST. Gibbs 11928] solved eq. (6.38) by assuming (i) that S0 is constant
and (ii) that RST » 2¾. He thereby found that
ctst„ = a exp (- 2<y«sT) (6-40)
where o (independent of the choice of the dividing surface) is the specific
surface energy of the planar interface (i.e. the value of <7STn at R$i = °°). We
may note, however, that Gibbs could have solved eq. (6.38) without using
the second of the above assumptions. Indeed, the solution of the full eq.
(6.38) with constant So is represented as
<%,„ =0/(.1 + 2So/RST) (6.41)
which is the formula of Tolman [1949] and Kirkwood and Buff [1949].
Equation (6.38) was derived by Gibbs for the specific surface energy
corresponding to the ST. However, it can be shown [Kondo 1956; Ono and
Kondo I960] that
d(ln cVr„)/d(ln Rsr) = d(ln cr„)/d(ln R)
so that eq. (6.38) remains in force also when the specific surface energy is
referred to the EDS (we keep the notation 0"„ in this case). This means that
with the help of (6.39) eq. (6.38) can be rewritten as
d(ln o-„)/d(ln R) = (2fyR)/(l + <%«)■ (6-42)
If we now adopt Gibbs's assumptions, the second in the less restrictive
form R » S0, we arrive again at Gibbs's equation (6.40), but for EDS-
defined clusters:
cr„ = a exp (- 2So/R). (6.43)
Similar to the derivation of (6.41), we can integrate the full equation
(6.42), i.e. without utilizing the condition R » 80. The result is
a„= a/0 + So/R)2 (6.44)
which is the analogue of (6.41) in the case of EDS-defined clusters. This
equation, which does not seem to have been reported hitherto, can be
generalized to hold for arbitrarily shaped clusters of both condensed and
gaseous phases if with the aid of (3.22) R is replaced by the volume Vn of the
EDS-defined cluster:
a„=o/(l+cdb/3V„ia)2. (6.45)
Here c is the cluster shape factor equal to (36;r)"3 for spheres, 6 for cubes,
etc.
Properties of clusters 81
In accordance with (3.13), for condensed-phase clusters the explicit n-
dependence of a„ from (6.45) is given by
a„ = al(\ + cdotfi'i'V'3)2. (6.46)
For gaseous clusters, however, the dependence of a„ on n remains only
implicitly expressed by (6.45). Indeed, in this case V„ and n are related in a
complicated way by the equality
n = (plkT)V„ + (2ca/3kT)V^[l + (c,5„/3V„"3)/(l + c5„/3V„"3)3] (6-47)
which follows from (3.14) upon using pn from (6.1) with cn - constant = c
and dCT„/d V„"3 calculated from (6.45).
The above results show that the effect of the cluster size on the cluster
specific surface energy is controlled by the Gibbs parameter 8^: &Q > 0 leads
to a lower specific surface energy for the smaller clusters, and S0 < 0 has the
opposite effect (see, e.g. eqs (6.44)-(6.46)). It is, therefore, very important to
know the sign and the absolute value of Sq. Unfortunately, such a knowledge
cannot be obtained from experiment, since neither the ST nor the EDS are
real physical objects and it is thus impossible in principle to measure the
distance (¾ between them. Under such circumstances it remains only to
resort to theoretical arguments and/or to determinations of 1¾ with the aid of
numerical calculations. On such grounds it is believed that, typically, J Sq | is
of the order of the molecular diameter d0, but the question about the sign of
Sq is still open [Ono and Kondo 1960; Defay and Prigogine 1966; Nishioka
1977, 1992; Hadjiagapiou 1994; Baidakov 1994, 1995], The argument
concerning the magnitude of | Sq | is that since the EDS is positioned inside
the surface layer (the spatial zone with varying molecular density), the ST
can be expected to be close to the EDS and, hence, also inside this layer.
Ergo, So should not exceed the thickness of the surface layer which at low
enough temperatures comprises one to a few molecular layers. Another unclear
point is whether it is possible to treat (¾ as an R, /?ST-independent quantity
which may be approximated [Ono and Kondo 1960] by the distance between
the EDS and the ST for the planar interface, i.e. by the limiting value of (¾
for both /?—>«> and R$T —> °°. Recently, Hadjiagapiou [1994] has shown that
Sq may decrease linearly with R, and Schmelzer et al. [ 1996] have demonstrated
that different 5a(R,RST) functions can affect essentially the size dependence
of the cluster specific surface energy.
Figure 6.1c illustrates the n-dependence of a„ from eqs (6.45) and (6.46)
for spherical EDS-defined steam bubbles and water droplets, respectively.
For the bubbles T= 583 K and/> = 0.1 MPa, and n is determined with the
help of (6.47). The parameter values used are c = (36;r)"3, ^ = 0.1 nm and
those listed in Tables 3.1 and 3.2. As seen, 0"„ does not differ more than 10%
from its limiting value a for n = «> (i.e. for the planar interface) when n >
1000 molecules. In the range of n < 100 molecules the change of d„ with
respect to cris already greater. However, in this range of n values the predicted
effect cannot be regarded without reservation, because then the Sq(R, Sst) =
constant approximation is uncertain and, more importantly, the very
82 Nucleation: Basic Theory with Applications
applicability of usual thermodynamics is questionable. From this point of
view, the numerous attempts in the nucleation literature to take account of
the ^-dependence of o„, e.g. by means of eq. (6.41), appear more or less
speculative. In the absence of a firm knowledge about the actual change of
on with n we shall confine all further considerations within the approximation
on - constant = g of the classical nucleation theory. In Chapter 8 we shall see
that an alternative determination of the cluster specific surface energy is
possible in the scope of quasi-thermodynamics.
Chapter 7
Equilibrium cluster size distribution
In the theory of nucleation the work W(n) to form a cluster (more generally,
a density fluctuation) of n molecules is just a convenient conceptual device.
This will become evident in Part 2 where we shall see that the nucleation
theory can be formulated on entirely kinetic grounds and, hence, without
having the slightest idea of W(n). Nonetheless, W{n) is of prime importance
for the theory, because it can be used for determination of the concentration
C(n) (m~3 or m~2) of n-sized clusters in the old phase at truly stable or
metastable equilibrium. The actual physical objects that carry out the nucleation
process are the clusters and it is C(n) which is the immediate participant in
the theoretical description of the kinetics of the process.
In Chapter 3 we have seen that the formation of the smallest clusters is
associated with elevation of the Gibbs free energy of the system. This is
related to the fact that the chemical potential of the molecules inside these
clusters is higher than that of the molecules in the ambient old phase (see
Section 6.2). From a thermodynamic point of view, therefore, the appearance
of these clusters is an 'unnatural process' [Guggenheim 1957], i.e. they
should never form in the old phase. Yet, they do form, but only by chance,
i.e. randomly in both time and space, for this is the sole way of circumventing
the thermodynamic law. An important implication of this probabilistic nature
of the cluster formation is that, even at equilibrium, in the old phase there
exists a temporally fluctuating and locally different number of clusters of
various sizes. This number can be averaged over long enough observation
time and over the system volume and then divided by this time and volume
(or the respective area of the substrate surface in the case of HEN on a
substrate). The result is the time-independent, spatially uniform function
C(n) introduced above which is called equilibrium cluster size distribution,
because it characterizes the cluster population in the old phase under conditions
of truly stable or metastable equilibrium. It must be kept in mind, however,
that though well defined, in the latter case C(n) is just a theoretical abstraction,
since a supersaturated old phase can stay only temporarily in the corresponding
metastable state.
The nucleation literature is abundant with solutions of the problem for
C(n) (e.g. Frenkel [1939, 1955]; Lothe and Pound [1962]; Walton [1962,
1969b]; Feder et al. [1966]; Reiss and Katz [1967]; Reiss et al [1968,
1997]; Zettlemoyer [1969,1977]; Abraham [1974a]; Rasmussen etal. [1983];
Suck Salk and Lutrus [1988]; Mutaftschiev [ 1993]; Ford [ 1997a]). Nevertheless,
so far there is no generally agreed formula for C(«), particularly in the case
of HON in vapours. In this case, Lothe and Pound [1962] have argued that
84 Nucleation: Basic Theory with Applications
cluster rotation and translation in the vapours may have a significant effect
on W(n) and, thereby, C(ri). In Chapter 3 this effect has not been considered
and, hereafter, we shall also refrain from dealing with it, because the issue is
still unresolved: there are counter-arguments [Reiss and Katz 1967; Reiss et
al 1968, 1997; Kikuchi 1969; Blander and Katz 1972; Radoev et al. 1986;
Ford 1997a] that this effect can have much smaller impact than originally
suggested. A clarifying analysis of existing expressions for C(«) has recently
been presented by Wilemski [1995]. In principle, the problem of finding
C(«) is sufficiently hard both physically and mathematically so that we shall
confine our considerations only to a derivation of a formula for C(«) which
satisfies the Law of Mass Action. Satisfying this law is an important condition
for self-consistency and, hence, reliability of C{n), for it is hard to conceive
a reason for which C{n) could be in violation of this fundamental law of
nature.
7.1 Equilibrium concentration of clusters
To find the equilibrium cluster concentration C(«), following Frenkel [1955],
we can consider the n-sized cluster as a product of a 'reaction' of aggregation
of single molecules. In other words, the cluster is a new 'chemical' species,
an «-mer, formed of n monomers (i.e. single molecules) in accordance with
the reversible 'reaction' (« = 1, 2, . . .)
«[C,]^[CJ (7.1)
where [CJ and [Cn] are the 'chemical' formulae of the monomer and the n-
sized cluster, respectively. It is important to recall that whereas in HON the
monomers are only formally, in HEN they are really distinguishable from
the molecules of the old phase. Since we consider mutual equilibrium between
the n-mers of all possible sizes n = 1, 2, . . ., there is no need to know if in
reality the formation of the n-sized cluster occurs as specified by (7.1).
Indeed, for maintenance of equilibrium the actual mechanism of the process
is of no importance and we are free to adopt even an imaginary one by
requiring only that it be sufficiently simple for theoretical handling. In this
respect eq. (7.1) is quite suitable, because in accordance with the
thermodynamic condition for chemical equilibrium [Guggenheim 1957; Landau
and Lifshitz 1976] we can write
Wo\<i = Vn, (7.2)
for any «=1,2,.... Here fj„ is the chemical potential of the n-mer
considered as a separate 'macromolecule', and it has been accounted that in
equilibrium, by definition, ^, = ,u0]d. The problem is, therefore, to express nn
in terms of the concentration C(«) of the w-mers, i.e. of the n-sized clusters.
To do that we consider first HON and denote by C0 (m~3 or m1) the
concentration of sites in the system on which the clusters of the new phase
can form. Typically, the clusters are scores of molecular diameters away
Equilibrium cluster site distribution 85
from each other so that, as is customary in nucleation theory, we can neglect
the cluster-cluster interactions and treat the cluster population as an ideal
multicomponent mixture of cluster-free sites and n-mers of all sizes. For
such a mixture, the chemical potential itn of the n-mers (which are now the
nth component in the mixture) is related to their chemical potential G(n) in
a reference one-component phase of these n-mers (then C(n) = C0) by the
formula [Guggenheim 1957]
H, = Gin) + kT In lC(n)/C0]. (7.3)
The important point with this formula is that in it G(n) is just the quantity
already introduced in Section 3.1. This is so, because while there the n-sized
cluster was considered as a separate thermodynamic system with corresponding
Gibbs free energy, now it is scaled down to a single 'macromolecule' in an
ensemble of other 'macromolecules'. Hence, if the other 'macromolecules'
in the old phase were just clusters of the same size n, we would have had
H„ = G(n). In reality, however, the cluster population is a mixture of
'macromolecules' of all sizes and the logarithmic term in (7.3) allows for the
entropy of mixing in the absence of cluster-cluster interactions.
Now, substituting n„ from (7.3) in (7.2) and using eqs (2.1), (3.3) and
(3.5), we find that the equilibrium cluster size distribution is of the form (n
= 1,2, ...)
C(n) = C0 exp [ - W(n)lkT\. (7.4)
It follows from here that the monomer concentration C(l) is related to C0 by
C, = C0 exp ( - WJkT) (7.5)
so that C(n) from (7.4) can be expressed alternatively as (n = 1, 2, . . .)
C(n) = C, exp (- [Win) - W,]lkT) (7.6)
where, for brevity, C\ = C(l) and W] = W(l). We note one more equivalent
presentation of C(n) from eqs (7.4) and (7.6), which shows explicitly that the
equilibrium cluster size distribution satisfies the Law of Mass Action.
Multiplying the r.h.s. of eq. (7.6) by the quantity (C|/C0)""'exp [(n - l)W1/
kT], which equals unity by virtue of (7.5), changes (7.6) into (n = 1, 2, . . .)
C(n) = Co(Ci/C0)" exp (- [W(n) - nW{\lkT\. (7.7)
This equation expresses the Law of Mass Action with the exponential factor
playing the role of the equilibrium constant of the 'reaction' represented by
eq. (7.1).
Turning now to HEN, we see that the above derivations for C(n) remain
entirely in force after replacement of G(n) in (7.3) with the quantity G(n) +
(|)s(rt) - 0S q. The difference 0s(n) - (j)s 0 is necessary in (7.3) to account that the
total surface energy of the substrate is not the same before and after the
occurrence of the heterogeneous 'reaction' (7.1) of clustering of n old-phase
molecules into an n-mer on the substrate. With this modification of eq. (7.3)
and with the aid of (2.1), (3.3) and (3.47) we arrive again at eqs (7.4)-(7.7).
86 Nucleation: Basic Theory with Applications
The conclusion is, therefore, that these equations apply to the cases of both
HON and HEN and are thus of most general validity provided that the
cluster-cluster interactions are negligible. Allowing for these interactions is
also possible (see, e.g. Abraham [1974a]), but the problem becomes very
complicated mathematically. We note as well that the equivalence of eqs
(7.4), (7.6) and (7.7) for C(n) is ensured by eq. (7.5) which relates C0, Cx and
Wh Equation (7.5) is thus a general condition which must be obeyed by the
limiting value W{ of any physically sound model for the W(n) dependence.
This equation can be used for determination of anyone of the above three
quantities if the other two are known.
Equations (7.4), (7.6) and (7.7) show that the so-obtained equilibrium
cluster size distribution C(n) has several important properties. First, it is of
Boltzmann type: in accordance with first principles [Landau and Lifshitz
1976], the probability C(n)/C0 to find an n-sized cluster on a given nucleation
site is exponentially proportional to the work W(n) to form the cluster (eq.
(7.4)). Second, C(n) is self-consistent in the sense [Girshick 1991; Wilemski
1995] that it satisfies the Law of Mass Action (eq. (7.7)) and returns the
identity C(l) = C, at n = 1 (eqs (7.6) and (7.7)). Third, C(n) from eq. (7.6)
is the thermodynamic counterpart of the equilibrium cluster size distribution,
eq. (12.3), derived by purely kinetic considerations in Chapter 12. It must be
pointed out that in the classical nucleation theory of HON in vapours C(n) is
usually represented by eq. (7.6), but with W, = 0. It is clear, however, that
neglecting Wl is inadmissible not only because it upsets the above-mentioned
self-consistency of C(n), but also because it can have a considerable quantitative
effect on C(n) [Barnard 1953; Girshick and Chiu 1990; Wilemski 1995].
The concentration C0 of the nucleation sites on which the clusters of size
n = 1, 2,.. . can form is a parameter which has a specific presentation in the
various cases of nucleation. For example, in the most frequently encountered
cases of HEN C0 has the following form:
C„ = NJAS (m"2) (7.8)
for 3D or 2D HEN on a substrate without nucleation-active centres on it (Ns
is the number of adsorption sites on the substrate surface of area As; hence,
for 2D nucleation on own substrate /Vs = AJoq and C0 = lla0 = 1019 m"2),
C0 = NJA, (nf2) (7.9)
for 3D or 2D HEN on a substrate with /Va active centres (e.g. impurity
molecules, structural point defects, etc.) which are always less than Ns,
C„ = NJV (m"3) (7.10)
for 3D HEN in the volume V of an old phase in which there are /V, active
centres (e.g. impurity molecules or ions),
C0 = /VaWs/V(m-3) (7.11)
for 3D HEN in an old phase containing Ms seeds each of them with N, active
centres on its surface (the seeds are, e.g. microparticles whose surfaces act
as microsubstrates for HEN).
Equilibrium cluster size distribution 87
For HON the determination of C0 is also easy when the old phase is a
condensed one. Then each of the available M molecules in the old phase
plays, in fact, the role of an active centre for nucieation and, analogously to
(7.10),
C0 = M/V= l/v0 = 1028 to 1029 m"3. (7.12)
For HON in gaseous phases, for reasons of symmetry between the
condensed-to-gaseous and gaseous-to-condensed phase transitions, to a certain
approximation it can be considered that C0 is again given by (7.12). The
accuracy of this approximation can be checked with the help of eq. (7.5) in
the typical case of HON in dilute vapours which can be treated as ideal gas
containing only monomers. Then
C,=plkT (7.13)
and also, according to (2.8) and (3.44),
W, = X - kT In (p/pe), (7.14)
because Et = 0. For the equilibrium (or saturation) pressure pc we have the
integrated Clapeyron-Clausius formula [Glasstone 1956]
pc = p0e-Mr (7.15)
where />o(Pa) is a pre-exponential factor. With C\ from (7.13) and W{ from
(7.14) eq. (7.5) yields
Ca = (pJkT)e)JkT = p0lkT. (7.16)
This formula gives C0 in the case of HON in vapours. It is indeed close to eq.
(7.12), since often p0 can be approximated as/>0 = kT/v0 [Moelwyn-Hughes
1961],
It must be emphasized that, in fact, knowing C0 for HON in gaseous
phases is not necessary (provided, of course, W{ is known), since in this case
the convenient formula for C(«) is eq. (7.6) with Ct from (7.13). For instance,
in the scope of the classical theory for W(n), from (7.6) with the help of
(2.8), (3.39) and (7.13) we find that (n = 1, 2 )
C(«) = CUc exp [n In S - (aolkT)(nm - 1)]. (7.17)
Here Cle, given in ideal-gas approximation by
Cle = PclkT, (7.18)
is the concentration of the molecules in the gas phase at the equilibrium (or
saturation) pressure pe, and
S = plpc = CxICif (7.19)
is the so-called supersaturation ratio. Equation (7.17) is the self-consistent
classical equilibrium cluster size distribution discussed by Wilemski [1995]
and named so because it both satisfies the Law of Mass Action and at
n = 1 returns the identity C(l) = C{ with C, from (7.13). This equation thus
88 Nucleation: Basic Theory with Applications
corrects the familiar formula of the classical nucleation theory [Frenkel
1939, 1955]
C(n) = C, exp [n In S - (aolkT)nlri] (7.20)
which, as seen, follows from (3.39) and (7.6) after neglecting W\ in respect
to W(n). The inconsistency of eq. (7.20) was noted by many authors [Courtney
1961; Dufour and Defay 1963; Feder et al. 1966; Blander and Katz 1972;
Goodrich 1964; Ziabicki and Jarecki 1984; Ziabicki 1986; Shizgal and Barrett
1989; Girshick andChiu 1990; Girshick 1991; Katz 1992] and its further use
in the theory of nucleation should be avoided, because it is only a rather
unsatisfactory approximation to (7.17).
Though we are free to use any of eqs (7.4), (7.6) or (7.7) thanks to their
equivalence, the convenient formula for C(«) appears to be (7.4) when C0 is
known independently, and (7.6) when we have such a knowledge of C^. If
both C0 and Cj are known independently as, e.g. in HEN on a substrate
whose surface is free of nucleation-active centres, eq. (7.7) may also be
convenient for usage (this is the equation which is widely employed in the
atomistic theory of thin film nucleation from vapours [Walton 1962, 1969b;
Venables 1973; Venables and Price 1975; Lewis and Anderson 1978; Venables
et al. 1984]). Equation (7.4) tells us also that the dependence of C(n) on Ay
and the material parameters of the system is determined entirely by the work
W(n) for cluster formation (it is this one-to-one correspondence between
W(n) and C(n) that makes W(n) a quantity of key importance in the theory of
nucleation). Therefore, using eqs (3.39), (3.42), (3.60), (3.62), (3.66), (3.71)
and (3.75), from (7.4), (7.6) or (7.7) we can obtain general expressions for
C(n) in the cases of both HON and HEN considered in Chapter 3 in the scope
of the classical nucleation theory. Similarly, the respective general atomistic
formulae for C(n) in HON or HEN follow from (7.4), (7.6) or (7.7) upon
employing eqs (3.44), (3.77)-(3.79).
We can thus write down the following often needed classical formulae (n
= 1,2,...):
C(n) = C0 exp [(Afi/kT)n - (aaJkT)nm] (7.21)
C(n) = C] exp [(Ai//,<ir)(n -1)- (aaJkT)(nm - 1)] (7.22)
for either HON or 3D HEN of condensed phases (r% = a for HON, and <%
= frr < a for HEN of caps, lenses, etc. - see eq. (4.42)),
C(n) = C0 exp {[(Ay - aefAa)/A;7> - (bK/kT)nm) (7.23)
for 2D HEN of condensed phases on foreign (Aa # 0) or own (Ac = 0, aa =
a0) substrate and
C(n) = C„ (p/p„f exp[(A^7> - [cac!l3pnm(kDu,]nm) (7.24)
for either HON (r% = oj or HEN (ocf = f cr< a) of gas phases (the dependence
of p„ on n is given by (3.63) for spherical (1//= 1) or cap-shaped (V < 1)
gaseous clusters).
Equilibrium cluster size distribution 89
The respective atomistic formulae for clusters of condensed phases are (n
= 1,2, ...)
C(«) = C0 exp ([(A/( - l)n + En]lkT) (7.25)
C(n) = C, exp (l(&/u - X)(n -1) + E„]lkT) (7.26)
in the case of HON (then £, = 0),
C(n) = C0 exp [[(Aft - X)n + E„ + cylJrtT) (7.27)
C(n) = C„ (CJCof exp ([ESJI + as(An - na0)]/kT} (7.28)
in the general case of HEN (then E] * 0) of arbitrarily shaped clusters (for
them An * naG) on a foreign substrate and
C(n) = C0 exp [[(Aft - X + a0os)n + E„]/kT] (7.29)
C(n) = C0 (CJCo)" exp (EJkT) (7.30)
in the particular case of 2D HEN of monolayer clusters (for them An = na0)
on a foreign (crs* (T) or own (crs = a) substrate.
In writing eqs (7.28) and (7.30) it is taken into account thatAi equals the
molecular area a0, and the quantity Es„ 2 0 is defined by (n = 1, 2, . . .)
Es,„ = E„ - nE, = nu, - U„. (7.31)
Physically, this quantity is the 'substrate' binding energy of the n-sized
cluster, i.e. the work for dissociating the cluster present on the substrate into
n single molecules (monomers) also on the substrate. The second part of eq.
(7.31) is obtained with the aid of En from (3.32) and shows that Es„ differs
from E„ only by the chosen zero of the potential energy: «ol(t for £„ and u i for
EBlr We see also that Es] = 0 (a cluster of size n = 1, i.e. a single molecule
adsorbed on the substrate, cannot be dissociated into more such molecules).
With appropriately defined C0, C,, aet and A,u eqs (7.21)-(7.30) give the
equilibrium cluster size distribution C(n) in various concrete cases of HON
or HEN in vapours, solutions, melts, etc. For example, with A^i from (2.8),
aef = <J and CJCic = p/pc, eq. (7.22) turns into the self-consistent formula
(7.17) of the classical theory for HON of liquids or solids in vapours. Also,
eqs (7.21) and (7.23) are widely used for describing HON or HEN in vapours,
solutions and melts with Aji from (2.8), (2.9), (2.13), (2.14), (2.16) and
(2.23) [Hirth and Pound 1963; Nielsen 1964; Zettlemoyer 1969; Lewis and
Anderson 1978; Christian 1975; Kelton 1991; Sohnel and Garside 1992].
With A/u from (2.27) eq. (7.29) is in agreement with the results of the atomistic
theory of electrochemical nucleation [Milchev et al. 1974; Milchev 1991],
and when C\ is identified with the concentration of adsorbed molecules, eq.
(7.30) is the Walton atomistic formula for HEN of thin films by molecular
beam condensation [Walton 1962, 1969b]. We note as well the atomistic
formula for C(n) for HON in dilute vapours (n - 1, 2, . . .),
C(n) = Cu S" exp (- [(n - \)l - E„]/kT), (7.32)
which follows from (7.26) with Ai/ from (2.8) and S from (7.19). Recalling
90 Nucleation: Basic Theory with Applications
that in HON E{ - 0, we observe that eq. (7.32) is a complete analogue of the
self-consistent classical equation (7.17), because nX - En is just the atomistic
representation of the classical total surface energy aon of the n-siztd
liquid or solid cluster (cf. eqs (3.20) and (3.33)).
We can now amalgamate all eqs (7.17), (7.21)-(7.30) and (7.32) into
three equivalent most general formulae which show explicitly the role of the
super saturation A^/ and of the effective excess energy O(n) of the n-sized
clusters in determining their equilibrium concentration C(n). This is achieved
by insertion of W(n) from (3.86) into eqs (7.4), (7.6) and (7.7) (n = 1,2,.. .):
C(n) = C0 exp [- 0{n)/kT] en&fJ,kT (7.33)
C(n) = C, exp {- Wn) - <P{\lkT) ein'l)^lkT (7.34)
C(n) = C0 (CyC0)K exp {[n0l - <P(n)]/kT}. (7.35)
In these most general formulae for C(«), in which <P{ = ¢(1), $(«) is
given thermodynamic ally by eq. (3.87) or (3.88) depending on the choice of
the Gibbs dividing surface and atomistically by eq. (3.89) or (3.90), but it
can also be considered as a phenomenological quantity. Since in many cases
<&{n) is practically Ay -independent (Table 3.3 shows that an exception is,
e.g., gas-phase nucleation), eqs (7.33)-(7.35) tell us (i) that the concentration
C(n) of the clusters of a given size n increases exponentially with increasing
Afj and decreasing <&{ri) and (ii) that the equilibrium constant of the clustering
'reaction' (7.1) (this is the exponential factor in eq. (7.35) which expresses
explicitly the Law of Mass Action) is determined by the net gain n<P{ - #(n)
> 0 in effective excess energy on assembling n monomers into an n-sized
cluster. It is important to note that as the derivation of eqs (7.4)-(7.7) is not
restricted by any conditions concerning A/i, all formulae for C(n) resulting
from them are applicable regardless of whether the old phase is supersaturated
(Ap > 0), saturated (A^/ = 0) or undersaturated (A^/ < 0). From eq. (7.33) we
thus see that since typically 'Pin) is a positive quantity which increases more
or less steadily with increasing n, when Aju < 0, most generally, C(ri) diminishes
with n and vanishes in the n —»<*> limit (in 2D HEN this is so only in the
Aju < ae(Ao range). On the contrary, when Aju > 0 and the old phase is
supersaturated, C(«) -> «= in the limit of n —» *». This behaviour of C(«)
reflects the fact that when A/i 5 0, the old phase is in truly stable thermodynamic
equilibrium, i.e. in a state in which occurrence of first-order phase transition
and, hence, nucleation is impossible. Then C(n) is a real physical quantity
describing the actual equilibrium population of clusters in the old phase.
Similarly, the divergence of C(ri) with increasing n when A/i > 0, i.e. when
the old phase is in metastable equilibrium, implies that C(ri), though
mathematically well-defined, is only a theoretical abstraction which represents
an imaginary cluster size distribution. In this case the actual population of
clusters cannot be found in the scope of equilibrium thermodynamics - in
the next Part we shall see that kinetic considerations are necessary for its
determination.
Figure 7.1 is an illustration of the typical size dependence of the equilibrium
Equilibrium cluster size distribution 91
100 120 140
Fig. 7.1 Equilibrium size distribution of water droplets in vapours at T = 293 K and
In S = - 1.5, 0 and 1.5 (as indicated): solid curves - eq. (7.17); dotted curve - eq.
(7.37). The double arrow indicates the width of the nucleus region.
cluster concentration C(n) when n is considered as a continuous variable.
The solid curves represent the classical equation (7.17) for HON of spherical
water droplets in vapours at T = 293 K and Afi/kT = In S = - 1.5, 0 and 1.5
(as indicated). The values of v0, a and pc are those listed in Table 3.1. We
recall that the quantitative relevance of the curves in Fig. 7.1 is increasingly
questionable for the smaller clusters. As seen, C(n) decreases steeply with n
when the vapours are undersaturated (Aft < 0) or saturated (A/( = 0). As then
C(n) is a really existing cluster size distribution, we read from Fig. 7.1 that
the classical theory predicts the presence of 1 and 10s droplets of, say, 12
water molecules in vapours with volume of 1 m3 at AplkT =-1.5 and 0,
respectively. Increasing A/( thus has a strongly stimulating effect on the
droplet formation: A^i is indeed the driving force of the process. When the
vapours are supersaturated, the whole C(n) curve is further shifted upwards
- in particular, we see from Fig. 7.1 that already 106 droplets of 12 water
molecules can be found in 1 m3 at Afi/kT= 1.5. However, at this Afi value the
course of the C(ri) curve is fundamentally different: the curve exhibits a
minimum at n = n* and diverges for n —> «>. The position n* of the minimum
of C(n) coincides precisely with that of the maximum of the W(n) function
(this is obvious from eq. (7.4)), so that n* in Fig. 7.1 is nothing else but the
number of water molecules in the nucleus droplet at Afi/kT = 1.5. Since the
water vapours are in metastable equilibrium when A^i > 0, the cluster population
represented by the uppermost solid curve in Fig. 7.1 does not exist in reality.
Nonetheless, as will be seen in Section 13.1, the descending branch of the
C(n) curve at Aft > 0 can be used for a reasonable estimation of the actual
92 Nucleation: Basic Theory with Applications
concentration of the droplets of subnucleus size n < n* and even of the
concentration of the nuclei themselves. In this sense it can be said that even
for supersaturated systems C(n) is physically relevant as long as n < n*. In
the n > n* range, however, C(n) differs already qualitatively from the actual
cluster size distribution (see Section 13.1). The conclusion is, therefore, that
it is only the ascending branch of C(n) which is totally irrelevant for physical
considerations. The appearance of this branch of C(n) when A/j > 0 is just a
mathematical manifestation of the fact that a supersaturated old phase is in
metastable, and not truly stable, equilibrium.
Equations (7.4), (7.6), (7.7), (7.33)-(7.35) are equivalent general expressions
for the equilibrium cluster concentration. However, they do not give C(n) as
an explicit function of n. Since in theoretical analyses it is often necessary to
use the explicit C(n) dependence, let us now see how such an approximate
dependence can be obtained from eq. (7.4) in the A/j > 0 case without any
loss of generality. Recalling the general property of W(n) to pass through a
maximum at n - n* when the system is supersaturated, we can approximate
the W(n) dependence in the vicinity of the nucleus size «* by the truncated
Taylor series
W(n) = W* + (\l2)(i2Wlin2)„,A" - «*)2- (7.36)
Using this expression for W(n) in (7.4), we find that for supersaturated
systems, to a good approximation [Zeldovich 1942],
C(n) = C* exp [P2(n - n*)2] (7.37)
provided \n -n*\ < 1//3. Here C*sC(n*) = C0 exp (- W*/kT) is the equilibrium
concentration of nuclei, and the numerical factor fi > 0 is given by
/3 = [(- d2W/dn\,„,/2kT]"2. (7.38)
Geometrically, /3 characterizes the curvature of the W(n) curve at n = «*: a
greater /3 value corresponds to a sharper maximum of W(n) at the nucleus
size. To estimate /3 we can use the classical formulae for W(n). For example,
in the cases of both HON ((% = a) and 3D HEN (ocl < a) of condensed
phases, from (3.39), (3.60) or (3.66) and from (4.38) and (4.39) it follows
that (A/j > 0)
fi = aoJ9kTn*in = 9Afi4/16lcTa1 als= WBkTn*2 = Aii/6kTn*. (7.39)
Similarly, for 2D HEN of condensed phases on foreign (Aa * 0) or own (Ac*
= 0) substrates, with the aid of (3.71), (4.32) and (4.33) we find that, classically,
(A/j > aefAo-)
02 = bMkTn*3'2 = (A/j - a^Aaf/kTb2!^ = W*/4kTn*2
= (A/j - acfAo)/4kTn*. (7.40)
These relations tell us that j} is an increasing function of A/j and that, typically,
f}= 0.01 to 1, since in most cases of nucleation AfilkT - 0.1 to 5 and n* = 1
to 100.
Equation (7.37) is the desired explicit and at the same time general
Equilibrium cluster size distribution 93
dependence of C{ri) on n for supersaturated systems. It should be kept in
mind, however, that it is an approximate formula which gives sufficiently
accurately the equilibrium concentration only of those clusters whose size is
not too different from the nucleus size n*. The dotted curve in Fig, 7.1
illustrates the accuracy of eq. (7.37) in the considered case of HON of water
droplets at Ap/kT = 1.5 (in this case n* = 53 and hence, according to (7.39),
/3 = 0.07). As seen, eq. (7.37) approximates well the corresponding C(«)
dependence from (7.17) solely in the nucleus region (known also as critical
region) around n = «*. This region extends from n = ri\ to n = «2 which are
defined by
n} = n* - ^fip (7.41)
n2 = n* + ^/2/2{3 (7.42)
so that its width A* = n2 - rt] is given by
A* = 7Tv2/p. (7.43)
This formula reveals the physical significance of /3: its reciprocal is about
half the width A* of the nucleus region which is illustrated by the double
arrow in Fig. 7.1. As shown by Zeldovich [1942], the nucleus region itself
has a simple physical meaning: all clusters of size n from this region are
energetically equivalent, because the difference between the nucleation work
W* and the work W(n) for their formation is less than the thermal energy kT.
For that reason the equilibrium concentration of the clusters from the nucleus
region is nearly equal to the equilibrium concentration C* of the nuclei (see
Fig. 7.1).
7.2 Equilibrium concentration of nuclei
Let us now use the general formulae for C{ri) derived above in order to
determine the equilibrium concentration C* (m~3 or rrf2) of nuclei. This
quantity plays an important role in the theory of nucleation and also, to a
first approximation, gives the actual concentration of nuclei under conditions
of stationary nucleation (see Section 13.1). Setting n = n* in eqs (7.4), (7.6)
and (7.7), we readily obtain the following most general, equivalent expressions
(n* = 1,2 ):
C* = C0 exp (- W*/kT) (7.44)
C* = Q exp [- (W* - W{)lkT\ (7.45)
C* = CdCt/Coy* exp [- (W* - ^W^/kT] (7.46)
where W* is the nucleation work. Despite the equivalence of these expressions,
eq. (7.44) is the one that is most frequently seen in the nucleation literature
[Hirth and Pound 1963; Zettlemoyer 1969]. With W] neglected in respect to
W*, eq. (7.45) is the Frenkel formula for C* [Frenkel 1939, 1955] which, as
already noted, is not a particularly good one because of its self-inconsistency
94 Nucleation: Basic Theory with Applications
and quantitative inaccuracy. Equation (7.46) expresses explicitly the Law of
Mass Action: the n*-sized nucleus is constituted of n* single molecules.
With the help of W* from (4.6), keeping their generality and equivalence, we
can represent eqs (7.44)-(7.46) in the form (n* = 1, 2, . . .)
C* = C0 exp (- <f*/kT) e"'&>"tT (7.47)
C* = C, exp [- (** - <Pi)lkT\ e'"*-"^'^ (7.48)
C* = Co(C,/Co)"* exp [(«**, - <t>*)lkT] (7.49)
where 0* is the nucleus effective excess energy. We note that these formulae
can be obtained directly from eqs (7.33)-(7.35) by setting n = n* and that in
them, in general, both n* and 0* are functions of Aft.
With concrete expressions for C0 (or C]), W* (or **), Wi (or #,) and «*,
eqs (7.44)-(7.49) apply to all cases of nucleation. The general thermodynamic
formulae for W* and $*, valid for arbitrarily chosen Gibbs dividing surface,
are eq. (4.5) (with <PS* from (5.33)) and eq. (5.7). In the scope of the classical
nucleation theory these formulae result in specific Aft dependences of W*,
n* and #*. For instance, substituting W* from (4.33), (4.39) and (4.41) in
eq. (7.44), we find that [Hirth and Pound 1963; Zettlemoyer 1969] (A^i > 0)
C* = C0exp (-4c3vlalmkT^n1) (7.50)
in the cases of both HON (¾ = a) and 3D HEN ((% < a) of condensed
phases, that [Hirth and Pound 1963; Zettlemoyer 1969] (Aji > aefAa)
C* = C0 exp [- fcV/4J;T(A/( - ac!Ao)] (7.51)
for 2D HEN of condensed phases on foreign (A<7 * 0) or own (Act = 0, aef =
a0) substrate and that [Hirth and Pound 1963; Blander 1979] (0 < p < pe)
C* = C0 exp [- 4c3al,mkT(p* -pf] (7.52)
for either HON ((% = a) or 3D HEN (oc[< a) of gaseous phases, p* - p
being related to A^i via (2.10) and (4.14). It is worth noting that for HON of
condensed phases in vapours, in conformity with (7.5), (7.13), (7.18) and
(7.19), the classical theory gives C0 in (7.50) in the form
C0 = tjpJkT) exp (c ujf olkT), (7.53)
because, classically, W, = - A/j + c v™ a (see eq. (3.39)). This expression for
C0 is only an approximation to (7.16), for it results from using cVq a instead
of X (cf. eqs (3.20) and (3.33)) in the exact formula Wj = - A/i + X which
follows from the atomistic equation (3.44) with £, = 0.
In addition to the classical formulae for W* and 0* we have general
atomistic expressions for these quantities, eqs (4.44)-(4.51). Employing these
equations in eqs (7.44)-(7.49) readily yields C* in the scope of the atomistic
theory of nucleation. For example, with W* from Tables 4.1 and 4.2 substituted
in (7.44)-(7.46) we have concrete atomistic formulae for C* in various cases
of nucleation. Particular atomistic formulae for C* follow also from eqs
(7.25)-(7.30) with n set equal to «*, e.g.
Equilibrium cluster size distribution 95
C* = C0 exp {[(Afi - X)n* + E*]/kT] (7.54)
in the case of HON of condensed phases and
C* = C0 (C,/C0)"* exp (E*/kT) (7.55)
in the case of 2D HEN of monolayers of condensed phases (E* = £„» and g*
= Esll* are the respective binding energies of the nucleus). Clearly, these
equations are more detailed forms of (7.47) and (7.49) in these two cases,
and (7.55) is the known formula of Walton [1962,1969b], In fact, eqs (7.47)-
(7.49) are already the most concisely written general atomistic formulae for
C*, because in them <J>* contains the information about the kind of nucleation
(see eqs (4.49) and (4.50)). Since in (7.47)-(7.49) n* and ** are unknown
functions of Aft, these equations become practically working formulae in the
n* —» 1 limit when in certain Aft ranges both n* and #* remain constant with
respect to A^i (see Section 4.4). As this can be the case with n* and <P* for
condensed-phase nuclei, eq. (7.47) thus reveals that at high enough
supersaturations (then n* —> 1) the concentration C* of such nuclei is a
simple exponentially increasing function of Aft controlled by two unknown
parameters: n* and C0 exp (- <P*/kT).
The C*(Afi) dependence is quite different, however, at sufficiently low
supersaturations when n* » 1: eqs (7.50)-(7.52) show that, classically, this
dependence is controlled by ocf, K and Ao. Figure 7.2 depicts C* from eq.
(7.50) as a function of AfilkT = In S in the cases of HON of spherical (1// =
1, aef = a) and HEN of hemispherical (y = 1/2, c% = (l/2)"3rj) water
droplets in vapours at 7"= 293 K. The numbers at the symbols on the C*(Aft)
curves indicate the number n* of water molecules in the nucleus droplet
(according to eq. (4.38)) at the corresponding supersaturation. The values of
v0, o and ps used in the calculations are those given in Table 3.1, for HON
C0 = 2.6 x 1027 m~3 is evaluated from (7.53), and for HEN it is assumed that
C0 = 1019 irr2 which corresponds to a substrate surface free of active centres.
As seen from Fig. 7.2, C* in both HON and HEN increases strongly with
increasing Aft. Indeed, if in HON the volume of the vapours is 1 m3, we
observe that while at AfilkT = 1.2 the number of nucleus droplets in the
vapours is 5, at AfilkT = 1.4 it is already 5 x 107. Similarly, in the case of
HEN on a substrate with surface area of 1 m2, at the same AfilkT values the
respective number of nucleus droplets on the substrate is 5 x 105 and 109.
The physical reason for this strong impact of Aft on C* is the diminishing of
both n* and W* with increasing Afi (in HON n* = 103 and 53 molecules and
WlkT = 62 and 40 at the above AfilkT values, and in HEN both n* and W*
are twice as small, since the nucleus droplet is hemispherical). We note again
that, being classical, the curves in Fig. 7.2 are more or less qualitative because
of the rather small size n* of the nucleus.
Figure 7.2 gives information also about the relative role of HON and
HEN at different supersaturations. Suppose that HON and HEN occur
simultaneously, the former in the bulk of a container of water vapour with
volume of 1 m3, and the latter on a substrate with surface area of 1 m2, the
substrate being, e.g. a wall of the container. As already noted above, at
96 Nucleation: Basic Theory with Applications
1 i , , , , i r i
0 12 3 4
Au7kT
Fig. 1.2 Supersaturation dependence of the equilibrium concentration of nuclei:
curves HON and HEN - eq. (7.50)for, respectively, spherical and hemispherical water
droplets in vapours at T = 293 K. The numbers at the circles and triangles indicate
the nucleus size at the corresponding supersaturation.
Afj/kT= 1.2 there will be 5 x 105 nucleus droplets on the substrate and only
5 nucleus droplets in the bulk of the vapours. When Afi/kT = 2, however, the
number (= 1018) of nucleus droplets in the bulk is already by far greater than
their number (= 1014) on the substrate. We are thus led to a conclusion of
great practical importance: in systems in which HEN is possible to occur
alongside HON (as this is almost always the case), while HEN is predominant
at lower supersaturations, in the range of higher A^i values HON takes over.
This conclusion clarifies further the stimulating role of various foreign bodies
(substrates, seeds, impurity molecules, ions, etc.) on nucleation-they affect
both C0 and W* and thus change C* as required by the general equation
(7.44).
Chapter 8
Density-functional approach
The classical nucleation theory developed in the scope of the cluster approach
predicts that under typical conditions n* < 100 molecules. This statistically
small number of molecules in the nucleus raises a number of important
questions which still remain unanswered by the classical theory. First, what
actually is a cluster, i.e. is there a physically objective criterion for positioning
the Gibbs dividing surface? Second, what is the shape, structure and density
of small clusters? Third, to what accuracy can the specific surface energy o"
be treated as independent on the cluster size nl Fourth, what in fact is the
physical meaning of a for small clusters when, geometrically, they may have
fewer 'bulk' than 'surface' molecules? And, of principle importance, why is
W* ^ 0 at Afi = A^is, in contradiction with the requirement of eq. (4.13)? As
already noted in Chapter 3, to avoid at least some of the above difficulties,
the atomistic theory of nucleation was developed. This theory operates with
the cluster binding energy En rather than with the cluster surface energy and
is thus not concerned with the third and the fourth of the above questions.
However, it is still unable to answer the remaining ones. Clearly, a radical
way of dealing with these and other related questions is to follow an approach
which is entirely different from the cluster approach used in both the classical
and the atomistic theories.
Such an approach is the density-functional approach employed first by
Cahn and Hilliard [1959] and more recently, e.g. by Abraham [1974b, 1979],
Hanowell and Oxtoby [1984], Oxtoby and Evans [1988], Nishioka et al.
[1989], Hoyt [1990], Tomino etal. [1991], Zeng and Oxtoby [1991], Nishioka
[1992], Granasy [1993a, b, 1996a, b], Granasy etal. [1994], Baidakov [1994,
1995], Laaksonen and Oxtoby [1995], Laaksonen [1997], Shen and Oxtoby
[1996] and Talanquer and Oxtoby [1994, 1995, 1996, 1997] with the aim of
avoiding the use of dividing surface. Limiting the considerations to one-
component HON under isobaric-isothermal conditions, we shall now outline
the density-functional theory of nucleation which is based on the density-
functional approach. This theory was considered also in recent review articles
by Kelton [1991], Laaksonen et al. [1995] and Oxtoby [1992a, b, 1998].
8.1 General considerations
We have seen that when the system is in state 2 (Fig. 3.2), in reality there is
no dividing surface between the molecules of the old and new phases, but a
continuous change of the molecular number density p(m~3) through a transition
98 Nucleation: Basic Theory with Applications
zone along a spatial axis (Fig. 3.1). This transition zone is called surface
layer [Guggenheim 1957]: 'surface'-because this zone is what we perceive
experimentally as a surface, 'layer'- because this surface has a finite rather
than zero thickness. Let p(r) be the density of the molecules at point (x, y, z)
with position vector r(x, y, 7). The existence of spatial inhomogeneity of the
molecular density p is the reason for the appearance of such inhomogeneity
also in a number of other quantities characterizing the system. In particular,
the spatially constant pressure p of the old phase becomes an r-dependent
tensor with components />,j,(r) (i, k = x, y, z), and the Helmholtz free energy
/ per molecule becomes locally different and a function of r, p and its
derivatives, which we shall denote for short as /(r). That is why usual
thermodynamics which operates with r-independent quantities cannot be
employed for determination of the nucleation work W*. For example, the
Helmholtz free energy F2 of the system in state 2 will be given by
Filp}= [ /Wp(r)dr, (8.1)
since p(r)dr is the number of molecules in the differentially small volume
dr = Ax dy 6z around point r (the integration is over the whole volume V of
the system). In this way F2 and, thereby, the Gibbs free energy
G2 = F2 + pV (8.2)
of the system in state 2 become functions of the function p(r), i.e. Junctionals
of the molecular density. We shall denote these functionals as F2[p) and
G2[p} in distinction with the usual notations F2(p) and G2(p) for F2 and G2
when these are functions of the variable (and not of the function) p. The
density-functional theory operates with the functionals F2[p) and G2[p)
which thus give the name of this theory. These functionals are non-local,
because / accounts for the interaction of the molecules at point r with the
other molecules in the system.
Let us now see how one can determine the work W[p] to form a density
fluctuation (not a cluster) having a density profile p(r). Rather than a function
of the cluster size n, now Wis afunctional of p, since it is again given by eq.
(3.4) in which, according to (8.1) and (8.2),
G2[p}= f l/(r)p(r)+/7]dr. (8.3)
Jv
Since eq. (3.1) for the Gibbs free energy G, of the system in state 1 remains
unchanged, using this equation and (8.3) in eq. (3.4) leads to
W[p)= [ d/(r)-//old]p(r) + p])dr (8.4)
Jv
where it is taken into account that the constant total number M of molecules
in the system is equal to p(r)dr, the integration being over the whole
volume V of the system.
Equation (8.4) is the general formula for the work to form a density
fluctuation of arbitrary density profile p(r) in the case of one-component
Density-functional approach 99
HON under isothermal-isobaric conditions. It is valid for any shape of the
density fluctuation even when p is different from the new-phase density pnew
at every point r in the fluctuation. Equation (8.4) does not rely on the concept
of dividing surface between the density fluctuation and the old phase and
makes no distinction between 'bulk' and 'surface' molecules in the fluctuation.
It is thus rid of the limitations (physical meaning, size dependence, etc.) of
the classical theory with respect to the cluster specific surface energy o.
Equation (8.4) shows that W is a functional of the molecular density not
only because p appears explicitly in the integral, but also because / is a
function of p and, possibly, its derivatives. Since now the nucleation work
W* is the work done to form that particular density fluctuation (to be called
nucleus fluctuation or nucleus, for brevity) whose density profile p*(r)
corresponds to a saddle point in functional space, the problem of finding W*
is a standard variational problem. Namely, one has to find the quantity
W* = W{p*} (8.5)
where p*(r) is the solution of the variational equation (SWis the variation of
W with respect to p)
(SW{p})p=P* = 0. (8.6)
Equations (8.5) and (8.6) are the analogues of eqs (4,2) and (4.1),
respectively. In order to solve (8.6) it is necessary to know the character of
the/(r) function. Without losing the essential physics, the simplest to assume
is that besides of r,/is a function only of p and its first derivatives pf = dpi
di(i = x, y, z). In such a case, the solution p*(r) of eq. (8.6) with W defined
by (8.4) can be found with the help of the Euler equation [Korn and Korn
1961]
^- (fp - Paid P + p) - ? -&- {fp - Mo.d P + p)] = 0- (8.7)
r ' Pi Jp = p*
Upon performing the differentiation this equation takes the form (p£ = d2p/
didk with i, k - x, v, z)
Equation (8.8) plays the role of the Gibbs-Thomson equation in the density-
functional theory and corresponds to eq. (4.4). It is a differential equation of
second order in the unknown function p*(r) which, in order to be physically
acceptable, must satisfy given boundary conditions on the surface of the
volume V of the system. Finding p*(r) (which is the analogue of «*) upon
100 Nucieation: Basic Theory with Applications
solving (8.8) under the concrete boundary conditions, substituting it into
(8.4) and performing the integration completes the determination of W*
from (8.5) in the scope of the density-functional theory.
Equations (8.4) and (8.8) show clearly that the results of the density-
functional theory depend crucially on the modelling of the/(r) function.
Most important in this modelling is to account for non-local effects, i.e. for
the interaction between the molecules in the volume dr with those in another
volume dr' around an arbitrary point (x\ y, z) with position vector r'. The
comprehensive way of doing that is to express/as a function not only of the
unary (i.e. p), but also of the other molecular distribution functions - the
binary, ternary, etc. ones. As this procedure is accompanied with formidable
mathematical difficulties, in practice it is necessary to use various
approximations. Such approximations are the gradient approximation of Cahn and
Hilliard 11958, 1959] which accounts for the dependence of/on p and its
first derivatives p' and the hard-sphere approximation of Oxtoby and Evans
[1988] which represents/as a sum of Helmholtz free energies due to harsh
repulsion and weak attraction between the molecules. Other approximations
are also possible and an obvious alternative is the quasi-thermodynamic
approximation described by Ono and Kondo [I960]. Let us now consider
briefly the gradient and the hard-sphere approximations. The quasi-
thermodynamic approximation has not been used so far in the theory of
nucieation and will be described in more detail in Sections 8.4 and 8.5.
8.2 Gradient approximation
The gradient approximation is the pioneering one in the density-functional
theory of nucieation. It was introduced by Cahn and Hilliard first for
determining the interfacial energy of a system with non-uniform density
[Cahn and Hilliard 1958] and then for analysing nucieation in two-component
incompressible fluids [Cahn and Hilliard 1959]. The assumption is that/
is a function only of p and its first derivatives p' and can be approximated
as
/(r) =/u[p(r)] + K[p(r)][Vp(r)]2. (8-9)
Here/U is the Helmholtz free energy (per molecule) which the system would
have had if it were not only locally (at point r), but everywhere with the
same density p, K > 0 is a p-dependent coefficient, and (Vp)2 = (dp/ck)2 +
(dpfdy)2 + (dp/dz)2 is the squared gradient of p. Equation (8.9) is a truncated
expansion of/in gradients of p, and the gradient term describes the departure
of the actual energy / at point r from the energy /u of a uniformly dense
system with density p. Physically, this term accounts for the interaction of
the molecules at point r with the other molecules in the system and thus
makes/(r) a non-local function of r. The contribution of the gradient term
vanishes when (Vp)2 —» 0 and since K is positive, this term favours the
levelling-off of the density inhomogeneity in the system.
Density-functional approach 101
Using /from (8.9) in (8.8) leads to the equation of Cahn and Hilliard
[1959]
d(fZp*)ldp* - [d(K*p*ydp*](Vp*)2 - 2K*p*(V2p*) = pold (8.10)
where /* =fn(p*), K* s K(p*) and V2p* s d2p*ldx2 + d2p*ldy2 + d2p*/dz2.
For a spherically symmetrical nucleus fluctuation this equation simplifies
essentially, as then p* is a function of one variable only - the radial distance
r from the nucleus centre (Fig. 8.1). Indeed, in spherical coordinates (Vp*)2
= (dp*/dr)2 and V2p* = (2/r)(dp*/dr) + cfp'/dr2 [Korn and Korn 1961] so
that in this case (8.10) reads
2K*p*(d2p*/d^) + (4K*p*/r)(dp*/r\r) + [d(AT*p*)/dp*](dp*/dr)2
= d(/*p*)/dp*-/J„ld, (8.11)
t
P"
Pnew*
Pold - -
0 r
Fig. 8.1 Spatial variation of the molecular density of a condensed-phase nucleus
(the shaded area corresponds to the number An* of molecules which are additionally
involved in the building up of the nucleus).
the boundary conditions for an old phase with infinitely large volume being
[Cahn and Hilliard 1959]
p*(«) = Poid, (dp*/dr)r=0 = 0. (8.12)
When the solution p*(r) of eqs (8.11) and (8.12) is found, the nucleation
work W* can be determined from (8.5) with W from (8.4) in which f is
expressed by (8.9), and dr = 47a2 dr for the considered spherical symmetry.
If one is interested in calculating W* for isothermal nucleation in a system
with constant volume rather than pressure, the term ^i0idp-P m (8-4) should
be replaced by^oidPoid (/^d is the Helmholtz free energy per molecule of the
102 Nucleation: Basic Theory with Applications
old phase) - the resulting formula for W* then coincides with the original
formula of Cahn and HiUiard [1959J.
Cahn and Hilliard [1959] analysed the properties of p* and IV* which
follow from eqs (8.4), (8.5), (8.11) and (8.12) both most generally and in a
model representation of the dependences of f* and K* on p* with the help
of the regular solution theory of mixtures. They found that at sufficiently low
supersaturation A^/ the nucleus fluctuation resembles the classical nucleus
cluster in that (i) the density pn*w at its centre approaches the density pnew of
the bulk new phase, (ii) the specific surface energy a associated with the
nucleus surface layer is close to that of a planar surface layer, and (iii) the
appropriately defined mean radius of the nucleus is determined by an equation
which is analogous to the Gibbs-Thomson equation (4.9). Accordingly, W*
increases to infinity with Ap -» 0 in the way predicted by the classical
formula (4.11). However, p* and W* have a markedly different behaviour
for Ap approaching the spinodal supersaturation A^/s: the density p*w at the
nucleus centre tends to the spinodal density poit^s of the old phase, the nucleus
becomes infinitely large and its surface layer infinitely diffuse and, most
importantly, W* decreases gradually until vanishing at Aju = A^/s. The latter
is exactly what is required by eq. (4.13) and implies that the density-functional
theory is free of the fundamental inconsistency of the classical theory which
predicts that W* = 0 only at Afj = «>. More recently, the gradient approximation
was used by Nishioka et al [1989], Hoyt [1990], Tomino et al. [1991] and
Nishioka [1992] in the case of multicomponent nucleation and by Baidakov
[1994, 1995] in the case of bubble nucleation. In an extension of the gradient
approximation, Unger and Klein [1984] also analysed nucleation near the
spinodal and found that if the range of molecular interaction is long enough,
the nucleus can obtain a fractal structure. This prediction was supported later
by Monte Carlo simulation results [Monnette et al. 1988].
8.3 Hard-sphere approximation
This approximation was introduced by Oxtoby and Evans [1988] with the
aim of avoiding the gradient one which is valid for a relatively smooth
variation of p with r. Following Tarazona and Evans [1983] and Evans etal.
[1986], Oxtoby and Evans [1988] expressed/analogously to the molecular
Helmholtz free energy /vnw of a VDW fluid with uniform density pu = M/V.
As from thermodynamics [Guggenheim 1957] F-~ P(V)dV, with the help
of eq. (1.1) wefindthat/vDw = f^is given by [Landau and Lifshitz 1976]
/vDw=/«f + Win \b'pj{\ -b'pj] -a'pu (8.13)
where/rcf is a p0-independent reference energy. The b' term in this formula
accounts for the harsh repulsion between the molecules at distances smaller
than the molecular diameter d0. This term becomes important when the
density pu of the VDW fluid approaches its upper limit 1 lb' and the molecules
are so close to each other that they repel themselves like hard spheres. The
Density-functional approach 103
a' term allows for the relatively weak molecular interaction at distances
greater than da and is determined by the expression [Landau and Lifshitz
1976]
a'=-(1/2) f u(r) dr. (8.14)
Here «(r) < 0 is the pairwise interaction potential, i.e. the energy of interaction
between two molecules at a distance r from each other, and dr = An?- dr.
In the spirit of the VDW formula (8.13), Oxtoby and Evans [ 1988] assumed
that for a system with non-uniform density p(r),/can be represented as
m = A WD] + (1/2) J p(r>.(|r - r'|) dr' (8.15)
where the integration is over the volume of the system. In the above
approximation/h is the hard-sphere Helmholtz free energy per molecule and
is a function of r only through p. It corresponds to the first two terms on the
right of eq. (8.13) and can be determined either by direct identification with
these two terms or, as done by Oxtoby and Evans [1988], by employing
another more accurate /h(p) formula. In accordance with the last term in
(8.13) and a' from (8.14), the integral in (8.15) accounts for the attraction
between the considered molecule at point r and the p(r') dr' molecules in the
volume dr' around another point r' anywhere in the system. This is so,
because ua is only the attractive part of the pairwise potential u between two
molecules at a distance |r - r'| from each other. We thus see from (8.15) that
in the hard-sphere approximation of Oxtoby and Evans [1988] /does not
depend on the derivatives of p. Owing to this difference from the gradient
approximation, the hard-sphere one has the important advantage that it can
be used for analysing systems with arbitrarily steep spatial variation of p.
However, the hard-sphere approximation has limitations of its own, because
it is of mean-field type.
Wither) from (8.15) we can now use (8.8) to find the equation whose
solution p*(r) represents the nucleus density profile. Since now fix) is
independent of the first derivatives of p, <XJp)ld pf = 0 for each i = x,y,z and
(8.8) leads to the equation of Oxtoby and Evans [1988]
Ph[p*(r)l + j" P*(r')««(|r - r'|) dr' = ^ (8.16)
in which ^h = <XJ^p)ldp is the hard-sphere chemical potential. The analogy
between this equation and eq. (4.4) is obvious, since in the scope of the hard-
sphere approximation the sum of the two terms on the left of (8.16) is the
chemical potential of the molecules at point r in the inhomogeneously dense
system. Equation (8.16) is an integral equation in the unknown density profile
p*(r) of the nucleus. In view of (8.5), solving this equation and using the so-
obtained p*(r) dependence in (8.4) and (8.15) gives the nucleation work W*.
Oxtoby and Evans [1988] and Zeng and Oxtoby [1991] solved (8.16)
104 Nuc.leation: Basic Theory with Applications
numerically for nuclei of spherical symmetry by employing a concrete ^/h(p)
dependence and Yukawa or Lennard-Jones attractive potential «a. Oxtoby
and Evans [1988] found that although the radius R* of the classical nucleus
cluster may correspond to the radius of the surface layer of the nucleus
fluctuation, the density p*w at the centre of the fluctuation can depart
significantly from the density pnew of the bulk new phase. The nucleation
work W* determined by them differs from the classical one in the way
already noted in Section 8.2: with increasing A^/, W* decreases faster than
predicted by the classical equation (4.11) until vanishing at A/j = A^/s as
required by eq. (4.13). This finding is of principal importance, since it shows
that in the hard-sphere approximation the density-functional theory is also
free from the thermodynamic inconsistency of the classical theory with respect
to the finite value of W* at the spinodal.
8.4 Quasi-thermodynamics
The basic difficulty in the analysis of the thermodynamic state of the old
phase in the presence of a density fluctuation in it arises from the inability of
usual thermodynamics to operate with r-dependent quantities. Here and in
Section 8.5 we shall see that for such an analysis it is appropriate to use the
so-called quasi-thermodynamics [Ono and Kondo 1960] which describes the
energy properties namely of systems with locally different molecular density.
Quasi-thermodynamics is a generalization of the usual thermodynamics in
the sense that it postulates the possibility for determination of all intensive
thermodynamic quantities via a local energy balance in the vicinity of point
r where the density is p(r). The so-determined molecular Helmholtz free
energy/becomes a function of r both explicitly and/or implicitly through p
and its derivatives. The density-functional theory operates namely wifh/(r)
and that makes quasi-thermodynamics its natural basis. Following Ono and
Kondo [1960], we shall now present some basic quasi-thermodynamic
dependences which are valid for arbitrary symmetry of the density
inhomogeneity in one-component systems at constant absolute temperature
T. These dependences are generalizations of those given by Ono and Kondo
[1960] for planar and spherical surface layers.
In quasi-thermodynamics the number density p of the molecules varies in
space and is defined as
p(r) = l/i;(r) (8.17)
where v (m3) is the volume occupied by a molecule at point rU, y, z).
Accordingly, the total number M of molecules in the system is given by
W= f p(r)dr (8.18)
Jv
where dr = dx dy 6z is a differentially small volume around point r, and the
integration is over the whole volume V of the system.
Density-functional approach 105
The intensive thermodynamic quantities depend on the molecular density
and for that reason in quasi-thermodynamics they become locally defined
quantities. Analogously to (8.1), the total Helmholtz free energy F of the
system is given by
F{p}= f/(r)p(r) dr, (8.19)
since/(r) is the Helmholtz free energy per molecule at point r. In this way
F is a functional of p just as it is in the density-functional theory. In the
considerations to follow we shall restrict ourselves by the assumption [Ono
and Kondo 1960] that/is independent of the derivative of p.
Since in a system with non-uniform density the pressure is a tensor with
components />,fc(r) where i, k = x, y, z, in quasi-thermodynamics the role of
the r-independent scalar thermodynamic pressure p is played by the various
components of the pressure tensor. We shall now see how one can define an
effective spatially varying pressure p^.r) which is a scalar and can be used
instead of the pressure tensor for a convenient presentation of the basic
quasi-thermodynamic relations in a general form.
Let us consider a differentially small volume dr around an arbitrary point
r and increase it by the elementary volume <5dr. As known from the theory
of elasticity [Landau and Lifshitz 1965], the change of the volume from dr
to dr + SAt is related to the change Selk(r) of the components e,*(r) of the
deformation tensor by the formula ((, k = x, y, z)
5dr = [Z&„(r)]dr. (8.20)
Accordingly, the elementary work 5w done by the system for the increase of
the volume dr will be [Landau and Lifshitz 1965]
Sw(r) = [ I Z/7ft(r)&,t(r)] dr. (8.21)
i k
On the other hand, due to energy conservation, the work done by the
system leads to a decrease of the free energy fl.r)p(r) dr in the volume dr. We
shall therefore have
Sw(r) = -p(r)dr <§f(r) (8.22)
under the condition that the number p(r)dr of the molecules in the volume
dr is fixed. This condition means that the variation d(p dr) equals zero, i.e.
dr dp<r) + p(r) Sir = 0. (8.23)
Setting equal the right-hand sides of eqs (8.21) and (8.22) and accounting
for (8.20) and (8.23), we arrive at the formula
df(T)ldp(T)=pe((T)lp1{T) (8.24)
in which the variation sign is replaced by the sign for partial differentiation
(to indicate that T is constant), and pef is an effective pressure at point r,
defined as
Pcf (r) = [S Zpft(r) «5^(r)]/[S &i7(r)]. (8.25)
' k i
106 Nucleation: Basic Theory with Applications
Equation (8.24) is analogous to the known thermodynamic relation
[Guggenheim 1957] dfidp - pip2 for systems with uniform density, pressure
and Helmholtz free energy per molecule. Thus, the introduction of the effective
pressure /?ef is very convenient, for it gives the possibility for many of the
usual thermodynamic formulae to be directly used in quasi-thermodynamics,
but only locally, i.e. for every differentially small volume dr, since the
p(r)dr molecules in this small volume are formally with uniform number
density and under uniform (and thus scalar rather than tensorial) pressure
/?ef(r). For example, with the help of pet eq. (8.21) takes the familiar form Sw
= Pefddr, and the chemical potential fi(r) at point r can be defined by
^(r)=/(r) + Pef(r)/p(r). (8.26)
This expression follows from the thermodynamic formula [Guggenheim 1957]
(cf. eq. (8.2))
^/pdr=/pdr+ /7efdr
for the Gibbs free energy up dr of a uniformly dense phase of p dr molecules
with Helmholtz free energy fp dr in a volume dr under pressure /?ef. Also, the
quasi-thermodynamic Gibbs-Duhem relation (at constant T)
dPd(r) = p(r) d^(r) (8.27)
is obtained directly from the familiar thermodynamic formula (cf. eqs (5.8)
and (5.9)) upon replacing the uniform pressure p with pef. Naturally, eq.
(8.27) can be derived by differentiating (8.26) and accounting for (8.24). We
note as well that eq. (8.26) can be obtained rigorously by minimizing F from
(8.19) under the condition of constant M from (8.18) and the assumption that
/is independent of the derivatives of p(r) (see Ono and Kondo [I960]).
Equations (8.24)-(8.27) are basic relations in quasi-thermodynamics. They
are generalizations of the known quasi-thermodynamic formulae for the
cases of one-component two-phase systems with planar or spherical surface
layers [Ono and Kondo 1960], since then/?ef equals the tangential component
Pj of the pressure tensor (see below). Also, in the case of uniformly dense
phases pef ~ p and these equations pass into the usual thermodynamic relations.
Indeed, in this case the pressure tensor is with components pxx - pvy = pzz
- p and pxy - pxz = pyx -pvz = pa = pv = 0 [Ono and Kondo I960]. Hence,
It'LpikSeik=p'Ldeii
i k i
and from (8.25) it follows that pef = p.
When the system is with r-dependent density p across a planar surface
layer, Pxx = pyy = pT(z), pzz = Pn = P and Pxy = pxz = pyx = pyz = pzx = pzy = 0
if the z axis is chosen to be normal to the surface layer (see Ono and Kondo
[1960]) (for this reason pN andpT are called, respectively, normal and tangential
components of the pressure tensor). Also,
SdA = dA(Sexx + SeY}) (8.28)
dr = dAdz, ddr/SdA = dz (8.29)
Density-functional approach 107
where dA = dxdy is a differentially small area, and SdA is the small change
of dA leading to the change <Sdr of the volume dr. Then, />ct from (8.25) takes
the form
Pa= [Pt(*« + Se„) + pSeJ/(Se„ + *» + &n)-
With the help of (8.20) and (8.28) this expression becomes
Pef = Upt - p)(SdA/6A) + p(Sdrldr)](dr/Sdr)
which, owing to (8.29), leads to />ef = pT(z).
The determination of pet for a system with a spherical surface layer is
analogous. In this case p„ s Ptf(r), pm = p„ = /)T(r) and pre = pr<p = per=Pe?
= P<pr = P<pe-Qm spherical coordinate system r,8,<p with origin at the centre
of the sphere representing the surface layer [Ono and Kondo I960]. In addition,
SdA = dA(Seee+Se„) (8.30)
dr = d4dr, Sdr/SdA = dr (8.31)
where dA = ? sin 9 d9 dtp. With the aid of (8.20), (8.30) and (8.31) from
(8.25) it follows again that />ef = />T(r).
8.5 Quasi-thermodynamic formulation
We are now in position to give a quasi-thermodynamic formulation of the
density-functional theory of nucleation. As in Sections 8.1-8.3, we shall
restrict the considerations to one-component HON in an old phase of constant
total number M of molecules at fixed temperature T and pressure p. The
extension of the results to HEN is not difficult.
The work W{p} to form a density fluctuation (not a cluster) having arbitrary
density profile p(r) is the functional given by eq. (8.4). This is so, because
in conformity with (8.18) and (8.19), eqs (8.1), (8.3) and (8.4) are in fact
quasi-thermodynamic relations. What we can do now is to use eq. (8.26) for
eliminating J{r) in (8.4) and expressing W{ p) in terms of ^i(r) and/>ef(r):
W[p) = [ (Mr) -A!„id]p(r) + [p -/7ef(r)]} dr. (8.32)
Jv
This equation is the general quasi-thermodynamic formula for the work to
form an arbitrary density fluctuation in the case of one-component HON. It
is valid for any shape of the density fluctuation and for varying molecular
density everywhere in the fluctuation. Since eq. (8.32) is derived without
using the Gibbs dividing surface, it is not concerned with the difficulties of
the classical nucleation theory with respect to the definition of the cluster
size, density and surface energy. Equation (8.32) tells us that the work to
form a density fluctuation is done for changing the chemical potential and
the pressure everywhere in the system from ^old and p to their local values
p(r) and />ef(r), respectively. Typically, these changes are concentrated in the
spatial region occupied by the density fluctuation and decay relatively quickly
108 Nucleation: Basic Theory with Applications
outside this region. This is so because the range of the molecular interactions
determining the density profile p(r) and, thereby, the ^/(r) and pCf(r)
dependences is rather limited. For that reason, W{p) from eq. (8.32) is
virtually independent of the volume V of the system as long as V is large
enough. In the particular case of HON of spherically shaped density fluctuations
/7ef(r) = pT(r) and dr = 47ZT2 dr so that eq. (8.32) takes the form
Msys
W{p)=4i:\ {Mr)-fiM]p(r) + \P~P-r(rmrdr (8.33)
Jo
in a spherical coordinate system with origin at the centre of both the fluctuation
and the system which is considered as a sphere of radius Rsys (in practically
all calculations it is possible to use /?sys = «, since p(r) ~ ^oi(i and /?T(r) = p
not too far away from the fluctuation).
Using W{p) from (8.32), we can now find the nucleation work W* which
is again determined by (8.5). This means that we must find the solution p*(r)
of eq. (8.8) and substitute it in (8.32). In our quasi-thermodynamic formulation
the/(r) function which has to be used in (8.8) is given by eq. (8.26) under the
assumption that it is independent of the derivatives of p(r). Thus, the summation
terms in eq. (8.8) vanish and thanks to the Gibbs-Duhem relation (8.27), eq.
(8.8) reduces to
H *(r) = nM (8.34)
where /j*(r) = Ui(r)]p=p«,r).
Equation (8.34) is the analogue of eq. (4.4) and plays the role of the
Gibbs-Thomson equation of the classical theory, since the p(r) function
which satisfies (8.34) is merely the density profile p*(r) of the nucleus
fluctuation. This profile (illustrated in Fig. 8.1) is really special: as seen from
(8.34), it is the only one that makes the chemical potential ^i(r) have the
same value jiold everywhere in the system. This reflects the fact that the
nucleus is that particular density fluctuation which is in chemical (albeit
labile) equilibrium with the old phase. We note as well that eq. (8.16) of
Oxtoby and Evans [1988] appears as a particular case of eq. (8.34).
Combining eqs (8.5), (8.32) and (8.34) yields the nucleation work W* as
W*= f [p-/>cf*(r)]dr (8.35)
Jv
where />ef*(i") s [pc!(r)]p-p»m is the effective pressure in the system
corresponding to the density profile p*(r) of the nucleus. In the particular
case of a spherically shaped nucleus fluctuation, />ef*(r) = pr*(r) s [px('')]p=p»(r)
and dr = Am2 dr so that this formula takes the form
[p - Pi*^)]!2 dr (8.36)
o
which follows also from (8.33) and which was used by McGraw and Laaksonen
[1996] with R - ~.
Density-functional approach 109
Equations (8.34)-(8.36) are basic results of the density-functional theory
of nucleation in its quasi-thermodynamic formulation. The first of them
defines the density profile p*(r) of the nucleus (rather than its size n*, as it
is not a cluster any more), and the other two allow the calculation of the
nucleation work W* through the effectively averaged value p*t of the
components (respectively, through the value /?* of the tangential component)
of the pressure tensor generated in the old phase by the appearance of the
nucleus. The /u(r) dependence which is necessary as input in eq. (8.34) when
solving it for the unknown p*(r) function can be determined by model
considerations or by the methods of statistical thermodynamics. An example
of model fj(r) dependence with statistical mechanical justification is the
expression used by Oxtoby and Evans [1988], Zeng and Oxtoby [1991J and
Hadjiagapiou [1994] in the l.h.s. of (8.16) and the similar formula described
by Ono and Kondo [ 1960, eqs (38.12) and (38.21)]. When the solution p*(r)
of eq. (8.34) is found, the dependence of /?*f (or pj) on r can be obtained
by integrating the Gibbs-Duhem relation
d/4(r)-p*(r)d^ld (8.37)
which is eq. (8.27) applied to the nucleus and in which, by virtue of (8.34),
fi0id stands for^*(r). Since in principle for a uniformly dense phase/?ef(r) is
r-in dependent, most generally, /?*f (r) ~ p = constant for r sufficiently far
from the nucleus and /?*f (r) ~ p* = constant when r is close enough to the
centre of the nucleus provided that the nucleus has a uniform density pn*w
with corresponding pressure p* in its core (the same is true, of course, for
p*{r) in the case of spherical nucleus). We note as well that eq. (8.36)
contains information about the asymptotic dependence of /?* on r, in
order for W* to be finite when Rsys = *>, p* must tend to p faster than r"3 for
r —» °°.
Equation (8.35) is remarkable with its physical simplicity. It shows that
for the formation of the nucleus, work is done solely for changing the initially
uniform pressure/? in the system to the spatially varying effective pressure
/j*f which is an appropriate average of the components of the pressure
tensor characterizing the system in the presence of the nucleus. This is quite
understandable: since according to (8.34) the chemical potential ju* is the
same at any point r both inside and outside the nucleus, the only change
occurring in the system because of the rearrangement of the molecular density
from pold to p*(r) is the change in pressure. It is thus equally correct to state
that the nucleus is a pressure rather than a density fluctuation. Actually, these
two fluctuations go hand in hand, since pressure and density are coupled by
the equation of state of the nucleating phase.
At different values of ^/old the nucleus surface layer is differently distanced
from the centre of the density fluctuation and is with different diffuseness.
This introduces a dependence of p% on ^/old which, along with the p(ji0\d)
one, makes W* from (8.35) a function of ^0,dand, hence, Afj. It is mandatory
for this function to obey the nucleation theorem in the form of eq. (5.21) or
(5.23), since this theorem is a consistency test for any theoretical formula for
110 Nucleation: Basic Theory with Applications
W* derived for one-component isothermal nucleation. We can now check if
W* from (8.35) satisfies eq. (5.23). Differentiation of both sides of (8.35)
with respect to fjo]d and employment of the Gibbs-Duhem relations (5.8) and
(8.37) results in
dW*fdvold = - f [p*(r) - pold] dr. (8.38)
Jv
According to (8.18), the integral of p*(r) in this equation represents the total
number A/ of molecules in the system (this is seen in Fig. 8.1 in which M
corresponds to the area under the solid curve). The integral of pold is the
number pQ\$y of the molecules which would have been in the system if it
were a uniformly dense old phase (in Fig. 8.1 this number corresponds to the
area under the pold line). Hence, the whole integral in (8.38) (visualized by
the shaded area in Fig. 8.1) is exactly the excess number An* of molecules
in the nucleus, as required by the nucleation theorem in the form (5.23). This
means that, in fact, eq. (8.38) is the nucleation theorem in the framework of
the density-functional theory. In the particular case of spherically shaped
nucleus (Fig. 8.1), in spherical coordinate system with origin at the nucleus
centre, eq. (8.38) simplifies to the expression
ftfsys
dW*/dvou = ^x [p*(r) - Pm]? dr (8.39)
Jo
which follows also from (8.36) and in which the approximation Rsys = «> can
practically always be used with sufficient accuracy.
The fact that W* from (8.35) obeys the nucleation theorem has an important
practical implication. Namely, even without knowing the /?*f (r) dependence
in (8.35) we can determine the dependence of W* on fjQ]d (or A^/) with the
help of the integral nucleation theorem (5.45) if only the nucleus density
profile p*(r), i.e. the solution of eq. (8.34), is known. In conformity with eq.
(5.45) or eqs (8.38) and (8.39) we shall have
W*(Moid) = W*'+ J"^ j£[p*(r) - pold] drj dMold (8.40)
for arbitrarily shaped nuclei and
lP*(r) - ftu] r2dr \ dfiM (8.41)
Mold IVo J
for spherical nuclei (W*' = W* (^„id) aI>d, a* already noted, Rsys = °° in most
cases of interest). If the metastability of the old phase is limited by a spinodal,
in these integral forms of the nucleation theorem it is convenient to set n'0\$
- A'oid.s. as *nen H™' = 0 (see eq. (5.48)). In the absence of spinodal (then
A'oid s = °°). an appropriate choice for Jt„id and W*' m me above formulae is
/iJul = =» and W*' = 0 (see eq. (5.50)). Equations (8.35) and (8.40) thus offer
two alternative ways of finding the W*(pol^) dependence which plays a key
Density-functional approach 111
role in nucleation theory. Mathematically, they merely express the equivalence
between the familiar view that the nucleus is a density, fluctuation (p*(r) in
eq. (8.40)) and the somewhat unusual one that it is a pressure fluctuation
(/>*f(r) in eq. (8.35)).
Now, it is important to establish the correspondence between eqs (4.5)
and (8.35) which give W* in the scope of the cluster and the density-functional
approach, respectively (note that since we consider HON, in (4.5) we have
0S = 0*). Suppose that we want to perform the integration in (8.35) without
knowing the concrete p*f (r) dependence. From a quasi-thermodynamic point
of view, this is what Gibbs [1928] wanted when introducing a dividing
surface between the nucleus and the old phase. Sticking to his idea, we
choose a dividing surface which is arbitrarily positioned around the nucleus
and defines its volume V*. Next, we postulate (as Gibbs [1928] did) that
both the nucleus and the old phase are under uniform pressures /?* and /?,
respectively. This means that />cf(r) in eq. (8.35) is a step function of r:
P*f (r) = P* inside the nucleus and />et (r) = p outside it. The integration is
then reduced to integration of the r-independent pressure difference p - p*
only over the nucleus volume V* and the result is eq. (4.5) (with <P* = 0*)
in which, mathematically, 0* compensates for the error introduced by the
stepwise approximation of />er(r). Physically, 0* is the nucleus total surface
energy and it must obey the equation
#* = (p*-p)V*+ f [p-p*f(r)]dr (8.42)
Jv
in order for W* from (4.5) and (8.35) to be equal to each other-
Equation (8.42) is nothing else but a quasi-thermodynamic definition of
the total surface energy 0* of a homogeneously formed one-component
nucleus separated from the old phase by an arbitrarily shaped and positioned
Gibbs dividing surface. This important equation makes it possible to calculate
0* in the scope of the density-functional theory: to this end the nucleus
density profile p*(r) must first be found as a solution of (8.34) and then used
in (8.37) for obtaining />ef(r) after integration. In the particular case of
spherical nuclei 0* = 4/Efi*2cr, V* = (47t/3)fi*3, p*f (r) = p* (r), dr = Ato2 dr
and (8.42) leads to the known formula (eq. (15.5) of Ono and Kondo [1960],
eq. (1.26)ofRusanov [1967])
a= (l/3)(p* -p)R* + [p- p*(r)](r/R*fdr (8.43)
Jo
for the specific surface energy o of arbitrarily positioned spherical dividing
surface (SSJ,S = «> for a system with sufficiently large volume). It is worth
noting that eq. (8.42) defines a also for a planar interface with area A;: then
«f = A,a, p* = p and, if the z axis is perpendicular to the interface plane,
/>cf(r) = pt(z) and dr = Atdz- Equation (8.42) then passes into the
Backer formula (eq. (13.5) of Ono and Kondo [1960], eq. (1.12) of Rusanov
[1967])
112 Nucleation: Basic Theory with Applications
<*=J ip-Priz)]dz (8.44)
where the limits of integration are shifted to infinity with virtually no loss of
accuracy.
Finally, looking back at the questions at the beginning of this chapter, we
see that since the density-functional theory does not describe the nucleus as
a cluster with surface energy, it is not concerned with the first four of these
questions. Moreover, the nucleation work W* calculated with the help of this
theory for model systems [Cahn andHilliard 1959;Oxtoby and Evans 1988;
Nishioka et al. 1989; Nishioka 1992] satisfies the thermodynamic requirement
for annulment at the spinodal, eq. (4.13). The density-functional theory of
nucleation is thus successful also in answering the last of these questions
and, thereby, in overcoming the fundamental inconsistency of the classical
theory. Not surprisingly, however, this theory has problems of its own such
as employing realistic interaction potentials «(jr - r'j) between the molecules,
using good enough approximations for/(r) and fj(r) and obtaining analytical
results for the p*(r) and W*(&p) dependences. It might be expected that the
above quasi-thermodynamic formulation of the density-functional theory
will be helpful in getting such analytical results.
Part 2
Kinetics of nudeation
This Page Intentionally Left Blank
Chapter 9
Master equation
In the thermodynamic analysis of a cluster of given size we are interested
neither in the history of its appearance nor in its future evolution, i.e. we
consider the cluster as a static formation which is in internal thermodynamic
equilibrium. In addition, we treat the cluster as being in partial or complete
thermodynamic equilibrium with the ambient old phase (and the substrate in
the case of HEN). As a matter of fact, however, a given cluster is only a
transient formation with a certain lifetime which depends on the cluster size
and is often as short as is a nano- and even a picosecond. Apart from the non-
trivial question about the applicability of usual thermodynamics to such
short-living formations, it is clear that a more comprehensive description of
the nucleation process requires kinetic considerations based on a concrete
mechanism by which the clusters grow and decay and thus change their size.
The kinetic treatment of nucleation in the scope of the cluster approach
was initiated by Farkas [1927] who materialized the idea of L. Szilard for
cluster formation as a result of a series of consecutive attachments and
detachments of single molecules. Later, the Szilard model of nucleation was
used by Kaischew and Stranski [1934a], Becker and Doring [1935] and
Zeldovich [1942] and is still the basis of the modern theory of nucleation.
Following more recent developments [Andres and Boudart 1965; Katz et at.
1966; Kashchiev 1971, 1974, 1984a], we shall now formulate the master
equation of nucleation, i.e. the basic kinetic equation which describes the
evolution of the process with the help of a generalization of the Szilard
model. This generalization is essential, as it makes it possible for the so-
formulated master equation to describe kinetically all stages of the overall
process of first-order phase transition, and not only its initial, nucleation
stage. Throughout this part our considerations will be restricted to one-
component nucleation which, however, can be either homogeneous or
heterogeneous. Results concerning the master equation of multicomponent
nucleation and its solutions can be found elsewhere (e.g. Reiss[l950] ;Sigsbee
[ 1969]; Wilemski [ 1975]; Binder and Stauffer [ 1976]; Stauffer [ 1976]; Temkin
and Shevelev [1984]; Greer etal. [1990]; Shi and Seinfeld [1990a]; Kozisek
and Demo [1993b]; Wu [1993]; Nishioka and Fujita [1994]; Wilemski and
Wyslouzil [1995]; Wyslouzil and Wilemski [1995]).
9.1 General formulation
When describing nucleation kinetically in the framework of the cluster approach
116 Nucleation: Basic Theory with Applications
various assumptions are made about the system and the processes taking
place in it at the molecular level, but only the following two are indispensable
[Kashchiev 1974, 1984a]:
(i) There exist clusters in the old phase which consist of different number
n of molecules (or atoms) (n = 1, 2, . . .),
(ii) Transformations of n-sized clusters into m-sized ones at time t occur
with certain, in general, time-dependent frequencies fnm(t) (s_1) (n, m =
1,2,...).
These are the basic assumptions of the kinetic theory of nucleation, which
allow a mathematical formalism to be developed for a detailed description of
the evolution of the process. Assumption (i) reflects one of the most outstanding
features of the nucleation process: the initial localization of the new phase in
nanoscopically small spatial regions. This assumption is thus in line with the
fact that first-order phase transitions occur along a path of non-uniform
transformation of the density of the old phase into the density of the new one
(see Chapter 3). Assumption (ii) warrants the evolution of the nucleation
process under time-dependent conditions, because the transition frequencies
fnm are not confined by requirement for their independence of t. Since these
frequencies are regarded as known, they remain in the final theoretical results.
These results are, therefore, very general: they can be used in various particular
cases of nucleation provided in each case the transition frequencies fnm are
independently determined by concrete model considerations. Such
considerations are needed to reveal how the frequencies fnm depend on time t and the
conditions (e.g. the supersaturation Aji) imposed on the old phase and to
enable the final results of the theory to be expressed in terms of experimentally
controllable parameters. It is worth noting that assumptions (i) and (ii) make
nucleation congenial with other processes involving clusters, for instance,
association in vapours, micellization in solutions, metal-ligand complexing,
coagulation in colloid suspensions, etc.
The mathematical formalism of the kinetic description of nucleation is
largely predetermined by assumption (i): the unknown function that is sought
as a solution of the kinetic master equation is the time-dependent concentration
of the clusters of a given size n. The master equation itself is easily written
on the basis of assumption (ii), but is very complicated mathematically
because of the fact that equally sized clusters can have different shape (and,
possibly, structure) and have to be distinguished by both size and shape. A
great mathematical simplification results if it is additionally postulated that
the clusters of given size have only one shape: the cluster size n then becomes
the sole parameter to characterize the clusters. Any physically meaningful
shape can be assigned to the clusters, but it is the equilibrium one [Kaischew
1950, 1951; Zettlemoyer 1969; Toschev 1973b; Mutaftschiev 1993] that is
commonly used. Binder [1977] showed that it is possible to think also in
terms of appropriately averaged cluster shapes and transition frequencies,
and Ziabicki [1968] demonstrated how the restriction for a fixed cluster
Master equation 117
shape can be relaxed with respect to the master equation. Neglecting everywhere
in the following the differences in the shape of the equally sized clusters, let
us now write down the master equation governing the concentration Z„(t)
(m~3 or m"2) of clusters of size n at time t. We note immediately that representing
the cluster concentration in this way means that, like the equilibrium cluster
size distribution considered in Chapter 7, Zn(t) is treated as independent of
the space coordinates. This spatial uniformity of Zn(t) implies that Zn(t) is a
quantity of mean-field type: it represents the actual cluster size distribution
averaged over a unit volume of the system (or a unit area of the substrate
surface in the case of HEN on a substrate). Accounting for the spatial non-
uniformity of the cluster concentration is also possible (see, e.g. Voloshchuk
and Sedunov [1975]; Williams [1984a, b|), but is beyond the scope of our
considerations.
Figure 9.1 shows schematically how a cluster of n molecules can increase
or decrease its size. The arrow beginning from size n and ending at size m on
the size axis symbolizes the quantity fnmZn which gives the number of n —>
m transitions undergone by the n-sized clusters per unit time and volume (or
area in the case of HEN on a substrate). It is seen that a change of the cluster
size may occur as a result of attachment and detachment of other clusters
of 1, 2, 3, , . . molecules to and from the considered rc-sized cluster (as in
Part 1, we keep treating the single molecules as clusters of size n = 1).
! I ! ! *
\ l< II*
: i i !"-
*—i 4—*—i
[ ; rnn^m •
; | nm n j
\ ! *i
i * 4 i -
n-2 n-1 n n+1 n+2 .
CLUSTER SIZE
Fig. 9.1 Schematic presentation of the possible changes in the size of a cluster ofn
molecules. The concentration ofn-sized clusters diminishes because ofn —> m
transitions (the arrows leaving size n) and increases thanks to m —*n transitions (the
arrows ending at size n).
118 Nucleation: Basic Theory with Applications
This model of cluster growth and decay [Andres and Boudart 1965; Katz et
al. 1966; Kashchiev 1971,1974,1984a] is thus a generalization of the classical
Szilard model [Farkas 1927] which allows for cluster gaining and losing
single molecules (i.e. monomers) only.
The frequencies/^(() of the transitions between the various cluster sizes
determine the kinetics of the nucleation process in particular and of the
overall process of first-order phase transition in general. In Fig. 9.1 the
arrows leaving size n show that due to the n —» m transitions the concentration
Zn{t) of n-sized clusters will be diminished per unit time by the quantity
mi)
£ /™(')Z„(0
m = \
where M(i) is the total number of molecules in the old phase at time t.
Conversely, the arrows ending at size n illustrate the role of the reverse, i.e.
the m —» n, transitions: owing to them Zn(t) will increase per unit time by the
quantity
MO
£ U(»zm(t).
m=l
In addition to the above changes in the concentration Z„(t) of the n-sized
clusters, it is possible that such clusters appear and vanish as a result of non-
aggregative processes occurring at certain rates which we shall denote by
K„(f) and £„(r), respectively.
On the other hand, the change of Zn(i) per unit time is expressed
mathematically by the derivative dZ„(r)/dr. The balance between the above
quantities thus leads to the sought master equation of first-order phase transitions
in one-component systems [Kashchiev 1984a] (n = 1, 2, . . ., M)
mo
dT Z"(0 = Bi VW)ZM -fnm(t)Z„(t)] + km - £„(*). (9. l)
Equation (9.1) is a set of ordinary differential equations of first order. In
general, these equations are non-linear because of the dependence of the
transition frequencies on the unknown cluster concentration Z„(r). The solution
Z„(t) of the master equation is unambiguous and physically acceptable when
it satisfies the initial condition (n = 1, 2, . . . , M)
Z„(0) = Z„,0 (9.2)
where Z„0 is the (a priori known) cluster size distribution at the initial
moment t - 0. Clearly, due to mass conservation, Z„(r) and M(t) are connected
by the relation
mo
£ nZM = M0)IV (9.3)
in which the volume V of the system must be replaced by the area As of the
Master equation 119
substrate surface in the case of HEN on a substrate (then M(t) is the total
number of molecules on the substrate at time t).
Equation (9.1) makes it possible in principle to follow with time the entire
transition of the considered one-component old phase from an arbitrary initial
(e.g. thermodynamically stable or tnetastable) state into any other time-
independent (e.g. stable, metastable or stationary) state. In particular, eq.
(9.1) describes the kinetics of one-component HON or HEN at variable
supersaturation Ap(t) in a system which is open for mass exchange, i.e. for
which K„(t) * 0, L„(t) * 0 and M = M(t). Nucleation of thin films [Zinsmeister
1966, 1968, 1969, 1970, 1971; Venables 1973; Venables and Price 1975;
Stoyanov and Kashchiev 1981; Lewis and Anderson 1978; Venables et al.
1984; Zinke-Allmang et al. 1992] is an example of a process occurring in
such a system; then usually K„(t) = L„(t) = 0 for n > 2, and K, and L, are
given by the rates of monomer impingement onto and desorption from the
substrate, respectively. For a closed system, K„(t) = L„(t) = 0 for any n so that
M = constant. Equation (9.1) then becomes [Kashchiev 1971] (n = 1, 2,...,
M)
ft Z„ (/) = Z_ [fm„(0Z„W ~L„W„(t)l (9.4)
eq. (9.2) remains the same and eq. (9.3) changes to
M
X «Z„(() = MIV (or MIAJ = constant. (9.5)
A great simplification of the master equation (9.4) for closed systems is
achieved by assuming that the transition frequencies f„m are independent of
both the time t and the cluster concentration Z„. However, this assumption
narrows the application of eq. (9.4) only to nucleation at time-independent
supersaturation A/v, since only then can fnm be constant with respect to t
provided that the clusters gain and lose only monomers whose concentration
Z, is also invariable with time. Under this assumption eq. (9.4) becomes a set
of linear ordinary differential equations and passes into the master equation
used by Andres and Boudart [1965] when treating nucleation as a multistate
process. It is instructive to note that eq. (9.4) is completely analogous to the
Pauli master equation in quantum mechanics [Huang 1963]. This is not
surprising upon realizing that n and/wf?I can be juxtaposed, respectively, to
the number of the quantum state of the system and to the probability for
quantum transition from state n to state m. A master equation similar to eq.
(9.1) was used by Ree et al. [1962] in studying the kinetics of random walks
and related physical problems again under the condition of t- and Z„-
independent transition frequencies^. The application of eq. (9.4) in connection
with computer experiments was considered by Binder [1977]. Mathematically,
eqs (9.1) and (9.4) fall into the class of equations describing Markoff processes
(see, e.g. Graham and Haken [1971]).
For the considerations to follow it is convenient to introduce the quantity
120 Nucleation: Basic Theory with Applications
j„(t) (with dimension m
for HEN on a substrate) defined
1,2, ...,M)
by [Kashchiev 1971, 1974, 1984a] (n
2 lfmm(')Zn(t)-fm^(t)Zm(t))
(9.6)
Physically, jn(t) is the rate (per unit volume or unit area in HEN on a
substrate) of appearance at time t of clusters of size greater than n, i.e. the n,
/-dependent net flux of clusters along the size axis. Figure 9.2 illustrates this
perception: the arrows connect all sizes tri <n with an arbitrary size m>n,
which means that the sum of all of them symbolizes the partial net flux
;„.„(») = I [/™-»,WZ„W-/™WZ„W]
i—i * i—<
! mn m !
" '< '•
►—4 * +.
1 2 .
. .n-1 n n+1
CLUSTER SIZE
Fig. 9,2 Schematic presentation of the net flux jlim through size n. This flux is the
sum of the forward and backward fluxes (the arrows) due to m' —> m and m —* m'
transitions with m' values from 1 to n.
through size n. This partial flux is the mth summand in the first sum in eq.
(9.6) and represents the net concentration of m-sized clusters formed per unit
time from clusters of size m' < n. Summation of jnm over all sizes m> n
yields the total flux/rt(r) from eq. (9.6) which parallels the formula used by
Katz et al. [1966] with time-independent transition frequencies fnm.
With the help of jn from (9.6) the master equation (9.4) can be represented
in a simple and physically transparent form. Indeed, using eq. (9.6) to determine
the difference jn_\ -j„, after some algebra we find that (/n = 0)
Master equation 121
M
jn -1 (t)-jj.t)= I( \fmii{t)Zm{t)-fmn{t)Zn(t)]
so that setting equal the left-hand sides of this equation and eq. (9.4) results
in [Kashchiev 1974, 1984a] (n = 1, 2, . . ., M)
ftUt)=in~x(t)-Ut)- (9.7)
Written down in this way, the master equation for closed systems is of
most general validity, as it corresponds to the known continuity equation in
hydrodynamics [Landau and Lifshitz 1988]. It shows that in the absence of
non-aggregative generation and/or annihilation of clusters the change in the
cluster concentration is due solely to the cluster 'motion' along the size axis.
Through the flux difference jn_x - j„, this 'motion' is controlled by the
frequencies f„m of monomer, dimer, trimer, etc. attachment and detachment
to and from the clusters of the considered size n.
So far, we have treated the number n of molecules in an n-sized cluster as
a discrete variable assuming only integer values. This approach is quite
natural from a physical point of view and originates from the pioneering
papers on nucleation [Farkas 1927; Kaischew and Stranski 1934a; Becker
and Doring 1935]. However, it is rather cumbersome as far as mathematics
is concerned, for it involves summations and finite differences. To avoid this
inconvenience the pioneers of the nucleation theory and most notably Zeldovich
[ 19421 worked with integrals and derivatives upon adopting the other possible
approach - consideration of n as a continuous variable. At present both
approaches are widely used in the theory of nucleation IZettlemoyer 1969,
1977, 1979; Abraham 1974a] (see also Part 1) so that it is necessary to
represent the above results in terms of continuous n. This is readily done by
replacing the sums with integrals in the corresponding equations. For an
open system the master equation (9.1) takes the form (1 < n < M(ij)
jr Z(n, t) = [/(m, n, t)Z{m, t) -j{n, m, t)Z(n, t)] dm
+ K{n, t) - L{n, t), (9.8)
the initial condition (9.2) remains unchanged, and eq. (9.3) becomes
.Mil)
nZ(n, t) d« = M{t)IV (or M(t)/As) (9.9)
where Z{n, t) = Zn{t),f(n, m, 0 =fnm(t), K(n, t) = K„{t) and L{n, t) = Ln{t).
Accordingly, for a closed system the master equation (9.4) reads [Kashchiev
1971] (1 <n<M)
-j- Z(n, t) = \f(m, n, t)Z(m, t) -f(n, m, r)Z(«, t)] dm, (9.10)
122 Nucleation: Basic Theory with Applications
and in eq. (9.9) MlV (or M/As) is constant with respect to r. Again for a closed
system the fluxj'fn, r) =jn(t) from (9.6) becomes [Kashchiev 1971]
j(n, t) = \ \\ [/("«'. "i. OZ(m', r) -/(m, m', r)Z("», 01 dm' [ dm
(9.11)
and the master equation (9.7) takes the standard form of continuity equation
[Kashchiev 1974] (1 < n < M)
^Z(«,r) =-■£;(«, 0 (9.12)
upon using the truncated Taylor expansion
j(n - 1, t) =;'(«, r) + \j^j(n, /)][(« - 1) - n]
of j(n - 1, r) around point n. It is easily verified that substitution of j(n, t)
from (9.11) in (9.12) leads to the master equation (9.10). This means that, as
it should be, eqs (9.11) and (9.12) are merely an equivalent representation of
the master equation (9.10).
The great mathematical difference between eqs (9.8) and (9.10) and eqs
(9.1) and (9.4) is that while the latter are sets of many ordinary differential
equations, the former are single integro-differential equations. Though working
with a single equation may be advantageous, the complicated character of
eqs (9.8) and (9.10) allows solving or examining them analytically only in a
limited number of simple cases corresponding to rather restrictive assumptions
about the transition frequencies. It should be noted that treating n as a continuous
variable and using integrals and derivatives instead of sums and finite
differences can be expected to yield mathematically reasonable results when
1 « n « 2/„„.min/ (9.13)
where ^, min denotes the smallest among the transition frequencies/^. This
condition is based on the rigorous formula of Ree etal. [1962, eq. (123)] and
shows that the concentration of clusters already of more than a few molecules
can be calculated sufficiently accurately in the framework of the continuous
approach provided that long enough time has elapsed from the onset of the
process. In particular, the requirement concerning t is always met in solving
stationary problems, for then t = °°.
As already noted, the master equation (9.1) (and, hence, (9.8)) makes it
possible in principle to follow the entire evolution of the process of first-
order phase transition: from the onset of the process - the moment of
supersaturating the system, i.e. putting it in metastable thermodynamic
equilibrium, to the end of the process - the moment of reaching a stable
thermodynamic equilibrium. Clearly, the master equation (9.1) or (9.8) can
describe also the process of relaxation of the system from a given thermo-
dynamically stable or metastable state to another such state when this process
Muster equation 123
does not involve formation of a microscopically large new phase, but rather
a size redistribution of the nanoscopically small clusters existing in the
system. Though without sharp boundaries between themselves, the following
three main stages of the overall process of first-order phase transition might
be distinguished:
(1) early (nucleation) stage,
(2) advanced (coalescence) stage,
(3) late (ageing) stage.
At the early stage, the dominating process is nucleation, i.e. creation of a
population of clusters of various, typically nanoscopically small sizes. The
cluster concentration is relatively low and the clusters grow and decay by
gaining and losing monomers only. The consumption of monomers by
the growing clusters is negligible and, initially, the supersaturation can
remain nearly unchanged even when it is determined by the monomer
concentration.
The advanced stage is characterized by the onset of coalescence, i.e. the
merger of two or more smaller clusters into a new bigger one. The contacts
between the clusters of various sizes begin to play an increasingly important
role in cluster growth, because the cluster concentration is already relatively
high. Many of the initially present monomers are already captured by the
growing clusters and this may cause a significant decrease in the supersaturation
and, hence, a virtual halt of nucleation.
Throughout the late stage, ageing of the newly formed disperse phase
occurs either by continuing coalescence between the already formed clusters
(coalescence regime) or by decay of the smaller clusters by loss of monomers
which feed the growth of the larger clusters (Ostwald-ripening regime). The
process terminates when the clusters reach a concentration and mean size
ensuring the thermodynamic equilibrium of the system. Clearly, this can
occur only if the system is closed for mass exchange.
Solutions of the master equation (9.1) or (9.8) are hard to find either
analytically or numerically if we want to describe the evolution of the entire
process of new-phase formation. However, the master equation can be
simplified considerably when we are interested in solutions covering only
one of the above stages. These simplified forms of the master equation
appear as particular cases corresponding to specific assumptions about the
mechanism of cluster growth and decay. For instance, eq. (9.4) was shown
[Kashchiev 1971] to be able to describe nucleation and coalescence from a
unified point of view. We shall now consider three particular forms of eq.
(9.1), which are widely used for separate description of the above-mentioned
three stages of the overall process of first-order phase transition. These
considerations should demonstrate the ability of the master equation (9.1) or
(9.8) to provide a unified description not only of nucleation and coalescence,
but also of ageing in one-component systems.
124 Nucleation: Basic Theory with Applications
9.2 Nucleation stage
The theory of nucleation is concerned with the processes occurring at the
early, nucleation stage of the overall process of first-order phase transition.
Then, as shown by Nielsen [1964], it is highly unlikely for the clusters of n
= 2, 3,... molecules to come into mutual contact, because their concentration
is still rather low. The chance that the clusters lose dimers, trimers, etc. is
also rather low. That is why, as an acceptable approximation, for the nucleation
stage we can set (n, m - 1, 2, . . .)
/™W = 0 (9.14)
for \n - m\ > 1. This means postulating that cluster growth and decay take
place only by monomer attachment to and detachment from the clusters, and
this is simply the Szilard model of nucleation [Farkas 1927]. As now/„„,(/)
* 0 only for \n - m\ = 1, the clusters can change size only by nearest-size
transitions. This is illustrated in Fig. 9.3 in which the black arrows symbolize
the number of forward (n —> n + 1) and backward (n —> n - 1) transitions per
unit time and volume (or area in HEN on a substrate). It is seen that the
appearance of the clusters is a result of a series of 'bimolecular reactions'
between monomers and n-mers [Zettlemoyer 1969] (n = 1, 2, . . .)
—i
Jn-1
. rN
>
IS
9nZn
fn-1Zn-1
» (
in
IS
>
1^
9n+1Zn+1
fnzn
> i
( »-
n-1 n n+1
CLUSTER SIZE
Fig. 9.3 Scheme of [he Szilard model of nucleation, according to which a cluster of n
molecules changes size only as a result of monomer attachment and detachment. The
net flux (the white arrow) through a given size is the difference between the respective
forward and backward fluxes (the black arrows).
Master equation 125
[Z,] + [Z„] ^ [Z„ +l] (9.15)
where [Z,] and [ZM] are the 'chemical' formulae of the monomers and the «-
sized clusters, respectively. Equation (9.15) is of great physical significance,
for it is an expression of our concept concerning the molecular mechanism
of cluster formation. The important question about this mechanism was not
(and could not be) answered by the thermodynamic theory of nucleation,
considered in Part 1. In this respect the kinetic approach to nucleation is, of
course, superior to the thermodynamic one.
Accounting for (9.14) reduces the master equation (9.1) to (n = 1, 2,..., M)
ft Ut) =/„ - i(')Z„ _,(()- g„(')ZM -fMZ„(t)
+ g„ + l (t)Z„ + , (/) + K„(t) - L„(t) (9.16)
where/„(/) =f„,„+i(t) and g„(t) =f„M.\(t) ait, respectively, the frequencies of
monomer attachment to and detachment from an n-sized cluster (by definition,
here and everywhere below/0 = 0, g[ = 0 and ZM+1 = 0). Equation (9.16) is
the master equation of nucleation in open systems. For closed systems Kn(t)
= L„(t) = 0 and it becomes (« = 1, 2, . . ., M)
ftW) =/„-iWZ„.,(r) -g„(t)Zr(t) -/„(/)Z„(/) + S„+1(0Z„+,W. (9.17)
This master equation is the basis of the theoretical description of the
kinetics of one-component nucleation in mass-conservative systems for which
M = constant with respect to time t. In this form, i.e. with time-dependent
nearest-size transition frequencies/„ and g„, eq. (9.17) was derived by Kashchiev
[1969b] in an analysis of nucleation at variable supersaturation. It is a
generalization of the master equation (n = 1, 2, . . ., M)
±Z„(t) =/„ _ ,Z„ _,(<)- SAM -/„Z„(r) + gn + ,Z„ + ,(0 (9.18)
formulated first by Tunitskii [1941] with time-independent/„ and gn. As
already noted, this independence of the transition frequencies on time implies
applicability of eq. (9.18) only to isothermal nucleation at constant super-
saturation. The generalized master equation (9.17) is not restricted in this
respect and has recently found application in studies on nucleation at variable
supersaturation [Kozisek 1988,1990,1991; Kozisek and Demo 1993a; Ludwig
and Schmelzer 1995].
It is important to note that the master equation (9.17) (and, hence, (9.18))
can be given the form of the continuity equation (9.7), since owing to (9.14)
the fluxj„(r) from (9.6) contains only two summands (see the white arrow in
Fig. 9.3), i.e. (n= 1,2, . . ., M)
jn(t) =f„(t)Z„(t) - g„+1(r)Z„+,(0. (9.19)
We now turn to the question about the analogues of eqs (9.16)-(9.19)
when n is considered as a continuous variable. There are at least two ways
of answering this question which were used by Tunitskii [1941] andZeldovich
126 Nucleation: Basic Theory with Applications
[19421 to transform eq. (9.18) into a partial differential equation of Fokker-
Planck and Fick-Kramers type, respectively. Mathematically, these two ways
are not completely equivalent, and physically, the resulting equations are of
different practical value [Goodrich 1964; Zinsmeister 1970; Dadyburjor and
Ruckenstein 1977; Shizgal and Barrett 1989; Wu 1992b; Slezov and Schmelzer
1994]. Since the equation obtained by Zeldovich [1942] does not require
knowledge of the detachment frequency g„, it has found a much wider use in
the theory of nucleation.
Denoting f„(t) by f(n, t) and g„(t) by g(n, t), let us follow Tunitskii [1941]
and approximate f„-i(t)Z„_l(t) and j„+i(J)Z,l+|(r) by the truncated Taylor
expansions about point n:
f(n- \,t)Z(n- l,/)=/(n,/)Z(n,r)
d
-d-[f(n,t)Z(n,t)]\[(n-l)-n]
+ \\~^{f(n,t)Z(n,t)}\\(n-\)-
g(n + l,r)Z(n + \,t) = g(n,t)Z(n,t)
d
-[g(n,t)Z(n,t)]\[(n-
-) 16 V"' ' /*-l"» */J (IA" + 0 — ni
on
+ 2 -£tUC".')Z(«,<)] [(«+ l)-«]2.
Replacement of the first and the fourth summands in (9.16) by the right-hand
sides of these expressions leads readily to (1 < n ^ M)
-sH
^ Z(«, t) = -^| [/(K, t) -g{n, /)]Z(«, ,)
~ 2 Jn~|[/("' ° + S("' 'm"-')]) + K(n' ° ~ L(n' °'
(9.20)
This master equation describes nucleation in systems open to mass exchange
and is the analogue of (9.16). In the particular case of K(n, t) = L(n, t) = 0
(mass-conservative systems) and time-independent transition frequencies/^,/)
=f(n) and g(n, t) = g(«) it passes into the equation of Tunitskii [1941 ]. In this
case, in mathematics eq. (9.20) is known as the forward Kolmogorov equation
for stochastic processes, and in physics as the Fokker-Planck equation for
diffusion in a force field. The / - g term in the large parentheses in (9.20)
accounts for the 'drift' of the clusters along the size axis, the 'drift velocity'
being
Master equation 127
v(n,t)=f(n,t)-g(n,t). (9.21)
In real terms, u(n,t) is merely the rate of deterministic (non-random) increment
of the cluster size n, i.e. the growth rate dn/dt of the n-sized cluster. Similarly,
the/+ g term in (9.20) describes the 'diffusion' of the clusters along the size
axis, i.e. the fluctuative (random) changes in the cluster size n. The
corresponding 'diffusion coefficient' (l/2)[/(n, t) + g(n, 1)] depends on both
n and t.
The problem with the practical use of eq. (9.20) is that we must know the
two transition frequencies f(n, t) and g(n, t) in order to be able to solve it
(naturally, we have the same problem also with eqs (9.16)-(9.18)). This is
really a problem, since whereas finding the n, t dependence of the attachment
frequency / is quite straightforward, the determination of the detachment
frequency g as a function of n and t is by far not so easy (see Sections 10.1
and 10.2). It is therefore advantageous to eliminate the latter frequency from
the master equations (9.16)-(9.18) and in 1942 Zeldovich found a way to do
that with respect to eq. (9.18).
Generalizing the approach of Zeldovich [1942], let us define a function
C„(t) with the help of the equality [Kashchiev 1969b] (n = 1, 2,. .., M - 1)
A(')C„(r)-g„+1WC„+1W = 0 (9.22)
and employ this function to exclude g„(t) and g„ + ,(/) from eq. (9.16). The
resulting equivalent master equation contains C„(/) instead of gn(t) and reads
[Kashchiev 1969b] (n = 1, 2, . . ., M)
j-tZ„(t) ^-fcMRiiyUD-WW)]
-/„(/)C„(r)[Z„(/)/C„(0 - Z„+1(/)/C„+,(<)] + K„(t) - £.„(/). (9.23)
Denoting C„(t) by C(n, /), we can use twice the truncated Taylor expansion
Z(n+ l,r)/C(fl + 1,/) =Z(n, t)/C(n, t)
+ 1-^ [Z(«, t)/C(n, /)]|[(n + 1) - n] (9.24)
to approximate the first and the second bracketed finite differences in (9.23)
by the partial derivatives - d[Z(n - 1, t)/C(n - 1, t)]/dn and - d[Z(n, t)/C(n,
t)]/dn, respectively. Then, the so-obtained new finite difference can again be
represented as a derivative with the aid of the analogous Taylor expansion
fin - 1, t)C(n - 1, t) -r- [z(n - 1. t)/C(n -1,0]
on
= f(n, t)C(n, /)-^ [Z(n, t)/C(n, f)]
+ ^ /(«, OCfn, f) -^ [Z(n, t)IC(n, /)]![(« - 1) - «].
128 Nucleation: Basic Theory with Applications
We thus arrive at the master equation of nucleation (1 <n < M)
(9.25)
-f Z(„, 0 = / {/(«, t)C(n,t)j- [Z(n, t)IC(n, t)\)
dt dn [ dn J
+ K(n, l)-L(n,t)
which is the analogue of (9.16) or (9.23) and which describes the kinetics of
the process in open systems when n is treated as a continuous variable. In the
case of nucleation in closed systems, K(n, t) = L'n, t) = 0 and (9.25) reduces
to (1 <n <M)
/Z(n, 0 = ■£ {/(«■ t)C(n, t) -^ [Z(n, t)IC(n, t)]\. (9.26)
dt dn \ dn
This is the master equation proposed by Kashchiev [ 1969b] for description
of nucleation at variable supersaturation. It is the counterpart of eq. (9.17)
and is able to provide such a description, because in it the transition frequencies
fin, t) and gin, t) (the latter through C(n, 1)) are allowed to change with time
t. This ability of eq. (9.26) was used by PUschl and Aubauer [1980] in an
investigation of phase transformation under conditions resulting in the alteration
of fin, t) and C(n, t) during the process. In the particular case of time-
independent f(n, t) =f(n) and C(n, t) = C(«), which corresponds to isothermal
nucleation at constant supersaturation, eq. (9.26) takes the form of the known
equation of Zeldovich [1942] (1 < n < M)
j-t Z(n, t) = -£ \f(n)C(n) -^ [Z(n, t)IC(n)}\. (9.27)
It must be noted that eqs (9.26) and (9.27) can be readily written down in
the form of the continuity equation (9.12) whenj„(0 from (9.19) is represented
as [Kashchiev 1969b]
j(n, t) = -f(n, t)C(n, t) / [Z(n, t)IC(n, /)]. (9.28)
dn
This result fory(n, t) is obtained from (9.19) with the aid of (9.22) and (9.24)
and is a generalization of the formula given by Frenkel [1955] for the particular
case of time-independent f(n, t) = f(n) and C'n, t) = C(n).
Physically, eqs (9.25)-(9.27) are analogous to eq. (9.20), as they also
describe diffusion in a force field. This is easily seen if, following Frenkel
[1955], we first rewrite j(n, t) from (9.28) in the equivalent form
j(n, t) = v(n, t)Z(n, t) -fin, t) -^ Z(n, t) (9.29)
where
v(n,t) = [f(n,t)lkT] j-/ [kTln C(n, t)\ . (9.30)
Master equation 129
Comparing (9.20) and (9.25), we then note that7(11. t) from (9.28) or (9.29)
corresponds to and has the structure of the expression in the large parentheses
in eq. (9.20). Hence, the second term in (9.29) is the 'diffusion' flux of
clusters, arising from their random 'motion1 along the size axis. Indeed, the
clusters of various sizes are not of equal number and the existing concentration
gradient dZfdn causes cluster 'diffusion' with a 'diffusion coefficient'/(«, i)
which is a function of n and t. The first term in (9.29) is the 'drift' flux of
clusters, v(n, t) being the n, /-dependent 'velocity' of directed (and not
random) cluster 'motion' along the size axis. In accordance with the usual
definitions of particle mobility, field force and chemical potential, v{n, t) is
the product of the cluster 'mobility' f{n, t)/kT on the n axis and the 'force'
- dfj/dn acting on the cluster because of the non-uniformity of the cluster
'chemical potential' p = kTIn C(n, t) along the size axis. As already pointed
out, in real terms u(«, i) is nothing else but the growth rate d«/d/ of the «-sized
clusters. We note, however, the difference between eqs (9.30) and (9.21)
which define this quantity. This difference and the difference in the respective
'diffusion coefficients' (\/2)[f(n, t) + g(n, t)] and j{n,t) in eqs (9.20) and
(9-25) are an expression of the non-equivalence of these two equations.
The above considerations show that Zeldovich's idea to introduce the
function C„ is really remarkable: this function allows eliminating the detachment
frequency gn from the master equations (9.16)-(9.18) in a mathematically
elegant and, as will be seen in Chapters 10 and 12, physically extremely
fruitful way. In his analysis, Zeldovich [1942] invoked physical arguments
when writing eq. (9.22) in the particular case of time-independent flv gA and
C„. To emphasize that such a priori physical argumentation is completely
unnecessary for the definition of Cn{t) with the help of eq. (9.22), so far we
did not address the question about the physical meaning of Cn{t) and will not
do that until Chapters 10 and 12. Drawing attention to the fact that C„(t) is
free of whatever physical assumptions concerning the nucleation process is
worthwhile because of the widespread incorrect opinion that the introduction
of Cn{i) in the master equations (9,16)-(9.18) corresponds to the imposition
of artificial constraints to the nucleating system. Nothing of the kind has
been done hitherto: Cn{i) is merely the solution of (9.22), given by [Kashchiev
1974, 1984a] (n = 2, 3, . . ., M)
C„{t) = C,(/)lfi(ry2(0 . . .fn-iWgiiOgtf) . .. g„(t)l (9.31)
and is mathematically well defined as long as its value C((r) at n = 1 and the
transition frequencies /„(/) and gn(t) are well defined, too. This important
relation shows explicitly how Cn(t) can be found as a function of n and t
when we know from kinetic considerations the «, t dependence of/,(/) and
g„(t). In the particular case of time-independent C\ and transition frequencies,
eq. (9.31) becomes the formula noted by Zinsmeister [1970] and Lewis and
Halpern [1976]. The point to remember is that gn(t) and Cn{t) ait completely
equivalent in respect to their usage in the master equation of nucleation: they
are just substitutes for each other so that knowing gn(t) allows obtaining
C„(t) and vice versa (when, of course, /„(/) is also known). As we shall see
130 Nucleation: Basic Theory with Applications
in Section 10.2 and Chapter 12, Cn(i) has the physical significance of cluster
concentration. The most notable with respect to Cn{t) is, however, that it is
that particular cluster concentration which enables the establishment of a
fundamental link between the nucleation kinetics and thermodynamics and,
thereby, the introduction of thermodynamic quantities as parameters in the
master equation of nucleation.
Looking back at eqs (9.20), (9.25)-(9.27), we see that they are single
partial differential equations of second order in contrast to their counterparts,
eqs (9.16)-(9.18) and (9.23) which are sets of a great number of ordinary
differential equations of first order. When solving eqs (9.20), (9.25)-(9.27)
we therefore need one initial (at t = 0) and two boundary (at n = 1 and n =
M) conditions. The initial condition is the one already given by eq. (9.2). The
boundary conditions may differ in dependence of the problem considered,
but the most often used ones read:
Z(\,t) = Z[{i) (9.32)
Z{M, t) = 0. (9.33)
Equation (9.32) requires knowledge of the actual concentration Z\ (r) of clusters
of size n = 1 as a function of time, and eq. (9.33) implies that the largest
possible cluster comprising all M available molecules can never appear in
the system. In Sections 15.1 and 24.1 we shall find the solution of the master
equation of nucleation at time-independent boundary condition (9.32) and,
respectively, zero and non-zero initial condition (9.2). The case of nucleation
at zero initial condition, but time-dependent boundary condition (9.32) was
considered elsewhere [Kashchiev 1985]. Other important solutions of the
master equation of nucleation will be obtained in Chapter 12 and Sections
13.1 and 17.1.
9.3 Coalescence stage
With the advancement of the phase-transition process in time, the cluster
concentration increases and so does the probability for mutual contacts between
the clusters of different size. The process is already at its coalescence stage:
an n-sized cluster can become larger not only by attaching a monomer to
itself, but also by coalescence with one or more clusters each of them containing
2, 3, .. . molecules. Clearly, the leading role in the coalescence process will
be played by the binary contacts, i.e. those which result in the appearance of
an m-sized cluster via merger of two smaller clusters - one of n molecules
and another of m - n molecules. To a certain approximation, we can therefore
neglect the multiple contacts between clusters and represent the frequency
fam of forward cluster-size transition, from size n to a larger size m, as
proportional to the concentration Z,„_n(t) of the clusters of m - n molecules
(m > «):
ftl„{t) = (0,un.nit)Zm_,lt). (9.34)
Here the frequency factor (o„ m_„ (m3/s or m2/s) is the number of coalescence
Master equation 131
events that can be experienced per unit time by one n-sized cluster and one
(m - n)-sized cluster in a system of unit volume (or unit area in the case of
HEN on a substrate). The physical meaning of (0„m_„ can be revealed by
model kinetic considerations (see Section 10.3). As either of the coalescing
clusters can be considered as the one which increases its size, it is obvious
that <a„ m_„ = com_„„.
For the frequency of the backward transition, from size m to a smaller size
n, we can use again eq. (9.14), since the detachment of monomers, dimers,
etc. from the clusters is virtually unaffected by the cluster concentration. In
other words, without much loss of accuracy, at the coalescence stage we
have
/„„(') = 0, (9.35)
but only for m - n > 1.
Substituting/„„ and/„„ from (9.34) and (9.35) in eq. (9.1) thus results in
(n = 1, 2, . . .,Afl
m(t)Z„(0
(9.36)
(9.37)
and gn =/„,„_! are the already introduced frequencies of monomer attachment
and detachment, respectively, the 1/2 stands to compensate for the double
summing and the second sum is an equivalent representation of
Mil)
Z «w„(/)Z„,_„(/).
As before, by definition/0 s 0, g, = 0 and ZM+, = 0.
Equation (9.36) is the master equation describing (along with the initial
condition (9.2)) the coalescence stage of the overall process of first-order
phase transition iu open systems. For closed systems K„(t) = L„(t) = 0 and
with M = <»> eq. (9.36) takes the form of the equation used by Shi and
Seinfeld [1990b] in a study of the effect of pre-existing particles on nucleation.
Compared with the master equation (9.16) of nucleation, it has two extra
terms (the sums) which take account of the formation of larger clusters in the
system as a result of coalescence events due to binary contacts between the
existing clusters. Consequently, eq. (9.36) in fact describes simultaneous
nucleation and coalescence, i.e. not just the second, but both the first and the
second stages of the phase-transition process. For that reason, in various
forms it was used for analysing, e.g. the kinetics of nucleation in the presence
d
it
Here/„,
■Z„(0
given
= /„-!
4
+ K,
by
i(')Z„-,W-
n-2
m = 2
,(<) - L„(t).
/„«) =i
gn(t)Z„(t) -/„(0Z„(f) +
,(t)Zn_m(t)Zm(t)-Zn(t)
;,.«+i(') = <%.iMZiw.
■g»+l
M(0-
z
132 Nitcleation: Basic Theory with Applications
of pre-existing particles [Shi and Seinfeld 1990b] and the kinetics of thin
film nucleation and growth [Zinsmeister 1966, 1968, 1969, 1970, 1971;
Venables 1973; Venables and Price 1975; Lewis and Anderson 1978; Venables
etal. 1984],
In many cases the solution of eq. (9.36), which is a set of ordinary differential
equations, is more easily obtained when considering n as a continuous variable.
We have seen in Section 9.2 that the four/„,gM terms in (9.36) can be represented
as - dj(n, f)ldn with;'(n, t) from (9.29). Using this result and replacing the
sums by integrals, we transform (9.36) into the following equation (1 < n
<M)
7, Z("' t) = lt\ f(n- ° ^ Z("' ° " V{n',)Z(n' °
f
n, t) 0){n, m, t)Z{m, t) dm + K(n, t) - L(n, t)
-~ | (Dim, n - m, t)Z{n - m, i)Z{m, t) dm
(9.38)
in which 0)(n, m, t) = 0)nil,(t), and u(n, t) = dn/dt is the monomer-mediated
growth rate of the n-sized clusters.
Equation (9.38) describes simultaneous nucleation (the dldn term) and
coalescence (the integral terms) in open systems and for that reason ii is in
fact applicable to the first two stages of the overall phase-transition process.
It is a single integro-differential equation and its solution Z(n, 0 must satisfy
the initial condition (9.2). In addition, it requires two boundary conditions
such as, for instance, eqs (9.32) and (9.33). Finding solutions of either eq.
(9.36) or eq. (9.38) is a hard mathematical problem and was attempted, e.g.
by Sutugin and Fuchs [1970], Sutugin et al. [ 1971 ], Vincent [ 1971 ], Pratsinis
[1988] and Shi and Seinfeld [1990b].
9.4 Ageing stage
Ageing of the cluster population nucleated and developed during the first
two stages of the phase-transition process can take place when the system is
closed for mass exchange. Ageing occurs in the absence of nucleation by
two main mechanisms.
The first mechanism is the coalescence mechanism already considered in
Section 9.3. It remains operative also at the ageing stage provided that the
clusters in the closed system are sufficiently mobile to make mutual contacts
and stable enough to lose neither monomers nor multimers. The gn terms in
(9.36) then vanish and as K^t) = L„(t) = 0, in this purely coalescence regime
of ageing this equation takes the form of the von Smoluchowski equation
lOverbeek 1952; Voloshchuk and Sedunov 1975] (n = 1, 2, . . ., M)
Master equation 133
/T-l
jj Z„ (t) = i jE fi>,„,„_„(0 Z„_„(f)Zm(0
-Z„(0 Z a)„„(t)Z„(t). (9.39)
This set of ordinary differential equations was derived by von Smoluchowski
[1916, 1917] with time-independent co„m and M - » to describe the kinetics
of coagulation in colloid suspensions. When n is considered as a continuous
variable, it becomes a single integro-differential equation (see also Williams
[1984a, b])(l <n<M):
4p Z(n, /) = \ (0(m, n-m, t)Z(n - m, t)Z(m, t) dm
at 2- J,
- Z(n, 0 &>(«, m, ()Z(m, /) dm. (9.40)
With time-independent frequency ftXn, m),M = °° and lower limit of integration
shifted to zero, eq. (9.40) takes the form which is most often seen in the
literature on coagulation (e.g. [Voloshchuk and Sedunov 1975]).
Equations (9.39) and (9.40) are the master equations of ageing by the
coalescence mechanism and their solutions are subject to the initial condition
(9.2). Solving either (9.39) or (9.40) is not an easy problem and was attempted
in numerous theoretical studies on coagulation-mediated ageing of colloid
suspensions [von Smoluchowski 1916, 1917; Todes 1949a; Friedlander 1961;
Swift and Friedlander 1964; Baroody 1967; Voloshchuk and Sedunov 1975;
Tambour and Seinfeld 1980; Williams 1984a, b, 1985; Hendricks and Ernst
1984; Sampson and Ramkrishna 1985; Pilinis and Seinfeld 1987; Ratke
1987; Muralidhar and Ramkrishna 1986, 1989], Equations (9.39) and (9.40)
were used also to describe the ageing of thin discontinuous films when it
occurs by coalescence due to diffusion motion of the separate clusters on the
substrate surface [Ruckenstein and Pulvermacher 1973; Me'tois et al. 1974;
Kashchiev 1976a]. This problem which is a 2D analogue of the classical
problem of coagulation in colloid suspensions was reviewed, e.g. by Kashchiev
[1979a], Kern et al. [1979] and Stoyanov and Kashchiev [1981].
The second mechanism recognized as responsible for the ageing process
is the so-called Ostwald ripening effective when the smaller clusters lose
monomers and decay, and the larger clusters capture these monomers and
grow [Ostwald 1901]. This mechanism operates, e.g. when the clusters in the
closed system are immobile so that they cannot contact each other. The
coalescence terms (the sums) in eq. (9.36) thus vanish in this purely Ostwald-
ripening regime and, recalling that Kn(t) = L„{t) = 0, we obtain [Kashchiev
1969b] (n= 1,2, . . ., M)
-rrZ„(<) =f„ - i(')4 - i(<) - Ut)W) -/„(')Z„(0 + g„ t ,(/)Z„ „ ,(0-(9.41)
134 Nucleation: Basic Theory with Applications
This is the master equation of ageing by the mechanism of Ostwald ripening
and it should be solved under the initial condition (9.2). The coincidence of
(9.41) with the master equation (9.17) of nucleation at variable supersaturation
implies that eq. (9.17) is applicable throughout all stages of the overall
process of first-order phase transition if this process takes place in a closed
system in the absence of coalescence between the clusters of n = 2, 3, . . .
molecules. In fact, when coalescence does not occur in the system, the phase
transition does not pass through a coalescence stage: after termination of the
nucleation stage the process enters immediately the ageing stage. An example
of such coalescence-free process is the crystallization in solid solutions closed
for mass exchange. The crystalline clusters are then immobile and if the total
concentration of solute is sufficiently low, coalescence does not take place,
because the clusters are always so far away from each other that they cannot
make mutual contacts as a result of their growth.
The analogue of eq. (9.41) in terms of continuous n is eq. (9.38) with
K(n, t) = L(n, t) = 0 and annulled coalescence terms (1 < n < M):
^ Z(n, 0 = -JH /(«, 0 ^ An, t) - v(n, t)Z(n, t) . (9.42)
This is the master equation of ageing in the regime of pure Ostwald ripening
and its solution Z(n, t) is subject to the initial condition (9.2) and two boundary
conditions such as, e.g. (9.32) and (9.33). Since eq. (9.42) is an equivalent
presentation of (9.26) with j{n, t) from (9.28) or (9.29), like (9.17), this
equation or eq. (9.26) describes the kinetics of both the nucleation and the
ageing stages, i.e. the entire evolution of the phase-transition process in the
absence of coalescence. This important property of eq. (9.26) was used by
Piischl and Aubauer [1980] to study the overall kinetics of transition of
thermally treated alloys to different thermodynamic ally stable states. It must
be pointed out, however, that when eq. (9.26) or (9.42) is employed lor
analysing solely the stage of ageing, it can be simplified essentially by neglecting
the 'diffusion' flux -fdZldn with respect to the 'drift' flux uZ. Hence, provided
that \fdZ!dn\« \uZ\, eq. (9.42) becomes [Todes 1946, 1949a]
4- Z(n, t) =--0- M", 0Z(n, t)]. (9.43)
at on
This approximate master equation of ageing by the mechanism of Ostwald
ripening has the form of the known continuity equation [Landau and Lifshitz
1988]. It is a partial differential equation of first order and has to be solved
under the initial condition (9.2) and only one boundary condition. We emphasize
that, strictly speaking, it is not justified to use eq. (9.43) in the whole range
of cluster sizes from n = 1 to n = M if growth of some of the clusters is
fluctuative rather than deterministic. For such clusters the 'diffusion' flux
dominates the 'drift' flux and the size interval corresponding to them has to
be excluded from the size interval in which (9.43) is solved.
It seems that Todes [1946] was the first to employ eq. (9.43) for analysing
the kinetics of ageing in the Ostwald-ripening regime. Subsequently, the
Master equation 135
problem was studied theoretically by many authors [Todes and Khrushchev
1947; Todes 1949a; Lifshitz and Slezov 1958,1961; Wagner 1961;Chakraverty
1967; Markworth 1973,1986; Flynn and Wanke 1974; Kahlweit 1975; Wynblatt
and Ahn 1975; Ahn and Tien 1976; Ahn et al. 1976; Wynblatt and Gjostein
1976; Wynblatt 1976; Dadyburjor and Ruckenstein 1977; Ruckenstein and
Dadyburjor 1977; Brailsford and Wynblatt 1979; Gelbard and Seinfeld 1979;
Coutsias and Neu 1984;Ratke 1987; Kukushkin and Slezov 1996; Kukushkin
and Osipov 1996]. The effect of neglecting the 'diffusion' term in (9.42) and
working with the approximate master equation (9.43) when determining the
cluster size distribution in ageing by the Ostwald-ripening mechanism was
investigated by Dadyburjor and Ruckenstein [1977].
Summarizing, we see that the general master equation (9.1) (or (9.8))
describes all stages of the overall process of first-order phase transition.
Without losing the essential physics, it does the same in its simplified form
given by eq. (9.36) (or (9.38)). This simplified master equation contains as
particular cases the master equations (9.16) (or (9.25)), (9.39) (or (9.40)) and
(9.41) (or (9.42) and (9.43)) which describe, respectively, nucleation, ageing
by coalescence and ageing by Ostwald ripening. Hereafter, the subject of our
considerations will be the master equation of nucleation in the forms given
by eqs (9.18), (9.26) and (9.27).
Chapter 10
Transition frequencies
In nucleation kinetics the transition frequencies fnm play the same central
role as that of the nucleation work W{n) in the thermodynamics of nucleation.
In principle, knowing/,,,, allows determination of all important characteristics
of the nucleation process without resort to thermodynamic quantities. Finding
the attachment (or forward) and the detachment (or backward) frequencies,
however, is not equally easy. This is because the attachment, e.g. of a molecule
to a given condensed-phase cluster depends above all on the state of the old
phase, which is not so difficult to describe, as this phase is a bulk one. On the
contrary, the detachment of a molecule from the cluster is largely determined
by the cluster properties which are poorly known, since the cluster is a new
phase of finite size. That is why, in many cases it is more convenient to
eliminate the detachment frequencies from the theoretical description. As in
Section 9.2, this can be done with the aid of the function C(n,l) which can be
obtained in a very general form in the framework of thermodynamics, since
this function is merely the equilibrium cluster size distribution which would
exist at the momentary value of the supersaturation in the system (see Section
10.2 and Chapter 12). The attachment frequencies thus appear as indispensable
kinetic parameters in the theory of nucleation. In this chapter, we shall
consider in some detail the frequency/,, of monomer attachment and relatively
briefly the frequency g„ of monomer detachment. As we know from Section
9.2, the frequencies/„,„ of multimer attachment and detachment are only of
limited significance for the nucleation process. Yet, for completeness, these
frequencies will also be considered at the end of this section, since they are
important for the post-nucleation stages of the phase-transition process.
10.1 Monomer attachment frequency
In practically all cases of interest, monomer attachment to the clusters is
controlled by mass and/or heat transfer. Heat transfer may be important
when the heat required for or released during attachment must be conducted
sufficiently quickly to or away from the cluster. Mass transport is always
needed and it occurs usually by three mechanisms: direct impingement of
molecules upon the cluster surface, diffusion of molecules towards the cluster
(either in the volume of the old phase or along the surface of a substrate) and
transfer of molecules across the cluster/old phase interface. In the following,
we shall consider attachment controlled only by mass transport, since this
kind of control is most often encountered in practice.
Transition frequencies 137
(1) Direct-impingement control
Direct impingement of molecules upon the cluster surface can control the
monomer attachment frequency/,,, for instance, when the old phase is gaseous.
Figure 10.1 illustrates this mechanism of monomer attachment to condensed-
phase clusters in the cases of HON and HEN on a substrate. In the case of
HON (Fig. 10.1a), according to the kinetic theory of gases, the number #„ of
collisions per unit time between Zn spherical n-sized clusters and Z, monomers
can be represented as [Moelwyn-Hughes 1961] (n = 1, 2,. . .)
&= 4jt(/J + RQ)\(kTI2K){\fmn + Mm^\mZnZx (10-1)
(b)
(a)
<c> (d)
Fig. 10.1 Direct-impingement mechanism of attachment of molecules (the circles) to
(a) spherical duster in HON, (b) and (c) cap-shaped cluster in HEN on a substrate,
and (d) disk-shaped cluster in HEN on a substrate.
where R0 = (3vt-J4rf) "3 and mo are the molecular radius and mass, respectively.
As by definition/,, = Y„xJZ„, taking into account that the radius R and the
mass m„ of the n-sized cluster are given by R - Rf/im and m„ = nnj0, from
(10.1) it follows that (n = 1, 2,. ..)
f,M) = y,fvll3(kTI2nm0)m[(n + l)/n]"2(n"3 + l)2Z,(f). (10.2)
Here c = (36.T)"3, and %, a number between 0 and 1, is the socking coefficient
which takes into account that some of the impinging monomers may not be
attached to the cluster. When Zt does not depend on time (, /„ is constant
with respect to time (provided that y„ is time-independent) and (10.2)
138 Nucleation: Basic Theory with Applications
becomes the formula for the monomer attachment frequency given by Andres
[1969].
The important point now is the determination of the actual monomer
concentration Z,. As already noted in Section 7.1, when the old phase is in
truly stable equilibrium, Zj is nothing else but the equilibrium monomer
concentration C, given by eq. (7.5). A basic assumption in the kinetic theory
of nucleation is that this result remains valid also when the old phase is in
metastable equilibrium: it is postulated that if the whole system is well
equilibrated, Zi is again equal to C\ from (7.5). As discussed by McDonald
[1963], this assumption is based on the fact that Zx and C{ are very large
numbers. They can thus coincide even though for a supersaturated system
the actual and the equilibrium cluster size distributions Z0 and C(n) are
increasingly different from each other with increasing n. Hence, identifying
Z] with C\ and using eq. (7.13) which is a particular form of (7.5) for HON
in dilute vapours, from (10.2) we find that for large enough clusters in
vapours behaving as ideal gas/„ reads (n » 1)
fn(t) = ynCv^lpOyilTtmok-DV. (10.3)
With constant pressure p of the vapours, this is the familiar expression [Volmer
1939; Frenkel 1955; Hirth and Pound 1963; Zettlemoyer 1969] for the time-
independent attachment frequency/, represented as the product of the monomer
sticking coefficient /„, the Hertz-Knudsen impingement rate p/(27tm0kT)1'2
of monomers and the area AkR2 = cuQl3n2n of the surface of the n-sized
cluster. This means that eq. (10.3) can be used also for large enough clusters
of non-spherical shape if the factor c is regarded as the shape factor introduced
in Chapter 3 (e.g. c = (36;r)i/3 for spheres, c = 6 for cubes, etc.).
Turning now to HEN of condensed-phase clusters on a substrate, we see
that eq. (10.3) is directly applicable to this case. For instance, for cap-shaped
clusters (Fig. 10.1b) we have
fn(t) = %[(1 - cos ew)/2iffy\ew)]O6Kvi)i,3[p(M27tm0kT)m]nm,(lQA)
since the lateral area of the cap is 2^R2(1 - cos 0W) = (36;n>2)1/3rt2/3(l - cos
0w)/2y/2/3(0w), 1//(0W) being given by eq. (3.56). As required, eq. (10.4) passes
into (10.3) at 0W = ^(complete non-wetting). Evidently, eq. (10.4) is relevant
when the substrate is immersed into the vapours and the molecules can strike
the cap-shaped cluster from all directions (Fig. 10.lb). It needs some
modification in the important case of nucleation of thin films during molecular
beam condensation. When the beam is perpendicular to the substrate surface
(Fig. 10.1c), the molecules impinge on the projected area KR1 sin2 6W =
06nvl)mnm sin2 ^/4^(^) of the cap and (10.4) changes to (0W < 7tl2)
f„(t) = /„[sin2 0w/4v/2/3(0w)](36^o2)1/3/(/)«2/3. (10.5)
With time-independent pressure p and impingement rate I (m~2 s_l), eqs
(10.4) and (10.5) are the known formulae for/„ in condensation of vapours
via 3D HEN of caps [Hirth and Pound 1963; Zettlemoyer 1969].
Following the reasoning behind eqs (10.4) and (10.5), it is a simple matter
Transition frequencies 139
to determine /„ also in the case of 2D HEN of condensed-phase clusters on
a substrate when the clusters attach only molecules striking their periphery
(Fig. 10.Id). Considering clusters of monolayer height, for the attachment
area at the periphery of an n-sized 2D cluster we have dohn112 so that the
analogues of (10.4) and (10.5) read (n = 1, 2, . . .)
/„(/) = 7„Mob(')/(2^m0*r)"2]«"2 (10.6)
/„(r) = y„bd*Kt)nm- (10.7)
As in Section 3.2, here bn]/2 is the length of the cluster periphery, bis a shape
factor (e.g. b = 2(7ta0)112 for disks, b = ia1^1 for squares, etc.) and it is
assumed that the width of the attachment area is equal to the molecular
diameter d0.
Equations (10.3)-( 10.7) show that if the sticking coefficient % is considered
as n-independent, the monomer attachment frequency increases with cluster
size according to a simple power law with exponent which depends on the
cluster dimensionality. This increase is seen in Fig. 10.2 in which the solid
curves display the dependence of/„ on n calculated from eq. (10.3) for HON
of spherical water droplets in vapours at T = 293 K under constant pressure
p - 2/7e (curve 2) and 4/>e (curve 4). The parameter values used are those
listed in Table 3.1. Figure 10.2 illustrates also the linear increase of/„ with
p or /, which is equivalent to an exponential increase with Ap, since p = pe
exp (AplkT) and I = /e exp (AplkT) (see eqs (2.8) and (2.9)).
8x109
6x109
'(0
2x109
0
1 100 200 300 400 500 600
n
Fig. 10.2 Dependence of the frequencies of monomer attachment and detachment on
the cluster size in HON of water droplets in vapours at T = 293 K: solid curves -
attachment frequency fn under direct-impingement control according to eq. {10.3) at
p/pe = 2 and 4 (as indicated); dashed curve - detachment frequency gn under
evaporation control according to eq. (10,72).
140 Nucleaiion: Basic Theory with Applications
It must be noted that the dependences seen in Fig. 10.2 may differ both
quantitatively and qualitatively from the real ones because of the unknown
dependence of the sticking coefficient yn on n. The problem of finding yn as
a function of n is an important problem per .ye, but is beyond the scope of our
considerations. In principle, yn is likely to decrease with diminishing n especially
in the case of HON, since thermal accommodation of the impinging monomers
is more difficult when the clusters are smaller. In the limit of two monomers
colliding to form a dimer in HON, even the mediation of a third body might
be necessary for ensuring the dissipation of the kinetic energy of the sticking
monomers and for making /„ essentially non-zero. Apart from allowing for
the energetic impediments accompanying the sticking of a monomer to a
suitable site on the cluster surface, yn takes account also of the number of
such .sticking sites on this surface. While, for example, for a liquid cluster
practically each site on its surface is equally suitable for sticking because of
the molecular roughness of the liquid surface, for a faceted crystalline cluster
the molecular smoothness of its surface creates a deficit of sticking sites and
is a substantial obstacle to monomer attachment [Stranski and Kaischew
1934; Kaischew and Stranski 1934a; Volmer 1939]. Also, adsorption of foreign
molecules on the cluster surface can alter appreciably the number of sticking
sites and thereby the value of yn [Bliznakow and Kirkova 1957; Bliznakow
1958; Hirth and Pound 1963]. Most generally, yn incorporates in itself the
effects of the thermal accommodation of the arriving monomers and the
molecular status (structure, adsorption coverage, etc.) of the cluster surface.
In some cases it may be convenient to represent yfl as a product of the
coefficients yt„ and % „, also numbers between 0 and I, accounting separately
for the effects of thermal accommodation and surface status, respectively:
The above considerations concern the frequency fr] of attachment of
monomers to clusters of condensed phases in their vapours. In the reverse
situation, i.e. when the cluster is gaseous, the enlargement of its size n by I
occurs through detachment of a molecule from its surface and its passing
into the volume of the cluster. This means that for gaseous clusters in their
own condensed phase /„ is physically equivalent to the frequency g„ of
monomer detachment from a condensed-phase cluster in vapours. For that
reason, analogously to gn from (10.68), in the case of evaporation control/,,
in HON of a gas phase is given by eq. (10.3) with u0 replaced by the volume
kTlpn occupied by a molecule in the «-sized gaseous cluster and with p
replaced by pe„:
/«(') = y«c\p,J(2KmQkT)m}[kTipn{t)]mnm. (10.9)
Classically, the pressure pn inside the cluster and the cluster vapour pressure
pe„ are represented by eqs (6.4) and (6.24). It must be noted, however, that
other factors (e.g. heat conductivity and inertial or viscous forces in the
condensed old phase) can also control fn for gaseous clusters [Kagan 1960;
Skripov 1972; Blander and Katz 1975; Blander 1979; Baidakov 1995].
Transition frequencies 141
(2) Volume-diffusion control
Diffusion of molecules from the volume of the old phase towards the cluster
surface can control monomer attachment to the clusters, e.g. in nucleation in
liquid or solid solutions. Figure 10.3 illustrates this mechanism of monomer
transport in the cases of HON and HEN on a substrate.
(b) (c)
Fig. 10.3 Volume-diffusion mechanism of attachment of molecules (the circles) to
(a) spherical cluster in HON, (b) cap-shaped cluster in HEN on a substrate, and
(c) disk-shaped cluster in HEN on a substrate.
Confining ourselves only to nucleation of condensed phases, let us first
consider HON of spherical clusters. Analogously to eq. (10.3), for sufficiently
large clusters/,, is given by
/„(<) = 7,Jd,„W4tffi2 (10.10)
where jin (m~2 s-1) is the incoming diffusion flux of molecules to the surface
of the n-sized cluster of radius R and surface area 4jtR2 = c D2/3 n2'3, c being
equal to (36s)"3. The diffusion flux is calculated by the Fick formula
U.„ = DtfZrldr)^ (10.11)
in which D (m2s_1) is the coefficient of monomer diffusion in the solution.
Equation (10.11) shows that finding jdJI requires knowing the concentration
Zr(r) (m " 3) of monomer solute as a function of time f and the distance r from
the cluster centre (the coordinate system is spherical with origin in the cluster
142 Nucleation: Basic Theory with Applications
centre). This concentration is the solution of the diffusion equation in spherical
coordinates [Carslaw and Jaeger 1959; Crank 1967]
-dT = ^Tr{rS7) (,0',2)
under initial and boundary conditions
Zr(i) = Z, at t = 0, Z£f) = 0 at r = R, ZJf) = Z, at r = ■». (10.13)
The so-formulated problem is analogous to that solved by K. Neumann
(see Volmer [1939]; Nielsen [1964]) for diffusion-controlled growth of an
isolated crystallite in a solution. The distinction is in the boundary condition
at r = R, which in crystal growth accounts for the presence of non-vanishing
concentration of monomers at the crystallite surface. In the case considered
here we are interested solely in the incoming (and not the net) diffusion flux
to the cluster and that is why the monomer concentration at the cluster
surface is set equal to zero. When the monomer concentration Z, (m~3) in the
bulk of the solution (i.e. at r = °°) is constant with time, the solution of eqs
(10.12) and (10.13) is of the form (e.g. Overbeek [1952]; Nielsen [1964])
Zr(t) = Z,(l -(«//•)( 1 -erf [(r - R)l(ADtfn\\) (10.14)
where
erf (x) = (2/7Cm) \ exp (-jc/2) djr' (10.15)
Jo
is the error function [KomandKorn 1961], From eqs (10.11) and (10.14) we
thus find that
;d,„ = (DZ,/fi)[l + (R2/7rDt)"2l (10.16)
This expression shows that after time t > R2/D the diffusion flux becomes
time-independent and takes the stationary value
7d)„ = DZ,« (10.17)
which we shall use in the considerations to follow. We can do that without
much loss of accuracy, because with R = 1 nm and a typical value of 1000
;/m2/s for the diffusion coefficient D in liquid solutions we find that after
time t > 1 ns the volume-diffusion supply of monomers to the cluster proceeds
already in stationary regime. In solid solutions D is much smaller than in
liquid ones, but then nucleation takes hours and days so that the stationary
approximation (10.17) for jdjI may also be acceptable. Hence, introducing
jd„ from (10.17) in (10.10) leads to
f„ = y„iKRDZh (10.18)
This relation can be generalized to cover the case of nucleation under
conditions of variable concentration Zi(r) of the monomers in the bulk of the
solution. If the variation of Zx with time is sufficiently slow, in quasi-stationary
Transition frequencies 143
approximation^,, is again given by (10.17), but with Z, = Zt(t). As to Z,, as
already discussed, we must set it equal to the equilibrium monomer
concentration Cj and calculate it from eq. (7.5) in which according to (7.12)
C0 = l/v0, and due to (2.14) and (3.44) W, = I-kTIn (OCc) for sufficiently
dilute solutions. Recalling that to a good approximation the solubility Ce and
the molecular heat X of dissolution are related by [Moelwyn-Hughes 1961]
Ce=C0e-MT, (10.19)
we find that Z\ = C. This is an expected result, because in the case considered
Z| is just another notation for the concentration C of dissolved molecules.
Thus, since for condensed-phase clusters R = (c/47r)"2u„'3fl"3, in quasi-
stationary approximation /, from (10.18) can be represented as (n » 1)
f„(t) = K(4^c)"2Di/3DC(/)n"3 (10.20)
where c = (36s)"3.
In deriving eqs (10.18) and (10.20) we have treated implicitly the«-sized
cluster as immobile, and the diffusing molecules as material points. In fact,
the cluster can also perform a diffusion (Brownian) motion in the solution
when the solution viscosity t] is sufficiently low, since the diffusion coefficient
D„ of the cluster is determined by the equation [Einstein 1905]
D„=kT/6icriR. (10.21)
That is why a more accurate formula for /„ requires accounting for the
diffusion motion both of the cluster and of the monomers whose diffusion
coefficient is D = kTI67U]R0. As shown by von Smoluchowski [1916, 1917]
(see also Overbeek [1952]), for that purpose it is necessary that D and R in
(10.18) be replaced, respectively, by the effective diffusion coefficient D„ +
D and the effective radius R + R0. The latter change allows for the non-zero
radius Rr, of the diffusing molecules. Hence,/, from (18) becomes
fn = yA'C(R + Ro)(D„ + D)Z]. (10.22)
With y„ = 1, this is the von Smoluchowski formula for/, when monomer
attachment is controlled by volume diffusion (e.g. Overbeek [1952]).
To express/,, from (10.22) in terms of n for condensed-phase clusters we
can use (10.21) in the form D„ = DR0/R. Recalling that R = Ron"3, R0 = (3v0/
4tf)"3 and Z, = C, we thus find (n = 1, 2, . . .)
AC) = rn(4"c)"2ui'3DC(/)(n~"3 + l)(n"3+ 1) (10.23)
where c ~ (36;r)1/3. This equation is the analogue of eq. (10.2) and for n » 1
it passes into (10.20). Comparing (10.20) and (10.23), we see that accounting
for the cluster mobility and the non-vanishing molecular radius results in an
increase of/, by not more than a factor of 4.
Equation (10.20) is easily obtainable because of the spherical symmetry
of the concentration field around the spherical cluster. If the cluster has
another (for example, polyhedral) shape, this symmetry is broken and the
diffusion problem becomes very complicated [Wilcox 1977]. For that reason,
only to a certain approximation, eq. (10.20) can be viewed as a general
144 Nucleation: Basic Theory with Applications
formula for arbitrarily shaped clusters with corresponding shape factor c
(e.g. c - (36;r)l/3 for spheres, c = 6 for cubes, etc.). It should be noted as well
that eqs (10.20) and (10.23) do not take into account the influence of the
neighbouring clusters on the concentration field around the considered n-
sized cluster. As these clusters compete to capture diffusing monomers, /„
diminishes when the concentration fields around the clusters overlap
appreciably. This effect is negligible provided the distance between the clusters
in the solution is greater than about 10 times their diameter [Ham 1958].
Equation (10.20) tells us that under volume-diffusion control /„ increases
with increasing cluster size n and concentration C of monomers in the bulk
of the solution. If yn = constant, the dependence of/„ on n follows a power
low with exponent 1/3 (instead of 2/3 under direct-impingement control),
and the increase with C is linear. Like under direct-impingement control, this
linearity is equivalent to an exponential dependence on A^/, since due to
(2.14) C = Ce exp (Ay/kT). The solid curves in Fig. 10.4 illustrate /„ from
(10.20) as a function of n at two fixed concentrations C = 5Ce (curve 5) and
10Ce (curve 10). The parameter values used are given in Table 6.1 and are
typical for crystallites of sparingly soluble salts in aqueous solutions at room
temperature. Concerning y„, it may be noted that in nucleation in solutions
thermal accommodation of the monomers is hardly a problem because of the
presence of the solvent molecules so that in (10.8) yin = 1 is a reasonable
approximation.
3x107
2x107 -
1x10'
'.
-
~ /
^-^^
^ n*
I
. . I . . . . 1
^^
~~~~i^^*^
'
n*
1
1<L—^
5
100
200
300
400
500
n
Fig. 10.4 Dependence of the frequencies of monomer attachment and detachment on
the cluster size in HON of crystallites of sparingly soluble salts in aqueous solutions
at room temperature: solid curves - attachment frequency f„ under volume-diffusion
control according to eq. (10.20) at CICe = 5 and 10 (as indicated); dashed curve -
detachment frequency gn under the same control according to eq. (10.72).
Transition frequencies 145
We now turn to the case of HEN on a substrate. Unlike under direct-
impingement control, if the condensed-phase clusters are cap or disk shaped
(Fig. 10.3b and c),/„ cannot be determined with the aid of eq. (10.20) by
accounting only for the difference between the attachment area of the sphere
and of the cap or the disk. This is so, for the spherical symmetry of the
concentration field around the cluster is distorted by the presence of the
substrate. The corresponding diffusion problem is highly complicated,
especially for a disk-shaped cluster which can attach molecules only to its
circular edge. Yet, as a rather crude approximation, for cap-shaped clusters
of condensed phases with large enough wetting angle 6W (e.g. 6W > TtiA) eq.
(10.20) can be used in the form
/«« = r«[(i - cos ew)/v/1/3(ew)](6A0)t/3£»C(0n1/? (10.24¾
which accounts for the fact that the lateral area of the cap is 2nR2(\ - cos 0W)
and that R = [3v0f47ty{6v,)}mnm. This equation is the analogue of (10.4). As
required, at 0W = it (complete non-wetting) it passes into eq. (10.20).
(3) Surface-diffusion control
Surface diffusion can be the transport mechanism controlling^ in HEN on
a substrate. In this case the clusters are cap or disk shaped and the monomers
are the adsorbed molecules of the old phase which diffuse along the substrate
surface towards the cluster periphery with surface-diffusion coefficient Ds
(m~2 s-1) (Fig. 10.5). Surface diffusion is the dominant mechanism of transport
of monomers to the clusters on the substrate first of all in HEN of liquids or
solids from vapours [Hirth and Pound 1963; Sigsbee 1969; Lewis and Anderson
1978]. When HEN occurs on a substrate in contact with a condensed phase
(e.g. liquid or solid solution), the surface mobility of the adsorbed molecules
is relatively low and surface diffusion is often negligible with respect to the
other transport mechanisms. Surface diffusion is also very sensitive to the
substrate temperature, since Ds depends on T according to the Frenkel-type
formula [Lewis and Anderson 1978]
Ds = di vs exp (- EJkT) (10.25)
(a) lb)
Fig. 10.5 Surface-diffusion mechanism of attachment of molecules (the circles) to
(a) cap-shaped, and (b) disk-shaped cluster in HEN on a substrate.
in which vs (= 1013 s"1) is the vibration frequency of an adsorbed molecule,
ds (= 0.5 run) is the length of a molecular jump along the substrate surface,
and £"sd is the activation energy for surface diffusion.
We consider again clusters of condensed phases. To find the frequency f„
of monomer attachment controlled by surface diffusion we must know the
146 Nucleation: Basic Theory with Applications
incoming diffusion fluxjdn (m l s ') of monomers to the cluster periphery.
Indeed, analogously to (10.10), now/„ is given by
/„(') = 7Jd,n(')27r/r (10.26)
where R' is the radius of the cluster base (R' = (iV(JA%)in [sin 0W/
i//3(0w)]«l/3 for cap-shaped clusters (Fig. 10.5a) and R' = R = (ao/^'V2
for disk-shaped clusters of monomolecular height (Fig. 10.5b)).
The incoming diffusion flux can be calculated from the Fick formula
h„ = DJdZr/dr)r = R. (10.27)
which is the 2D analogue of (10.11). Here Zr{t) (m~2) is the time-dependent
radially symmetrical concentration of adsorbed monomers at a distance r
from the centre of the cluster base and is the solution of the 2D diffusion
equation of Burton et al. [1951]
^=AJLfr^ + /_|, (I0.28)
under initial and boundary conditions (10.13) with R' instead of R (the
coordinate system is cylindrical with origin in the centre of the cluster base
and z-axis perpendicular to the substrate). In this equation I is the impingement
rate (i.e. the flux of adsorbing molecules), and Td is the desorption time (i.e.
the mean residence time of a molecule on the substrate), so that Zr/Td is the
local desorption flux of molecules. This flux is greater at higher substrate
temperature, because Td decreases with increasing 7 according to [Lewis and
Anderson 1978]
Td=(l/v,)exp (Ede8/*:r) (10.29)
where Edes is the activation energy for desorption.
Like (10.12), eq. (10.28) also has a stationary solution satisfying the
initial and boundary conditions (10.13) with R replaced by /?'. As in the
volume-diffusion case, this solution is obtained from (10.28) upon setting
dZJdt = 0 and may be expected to hold for t > R'2/Ds. This gives t > 1 /is
with R' = 1 nm and a rather low surface-diffusion coefficient Ds = 1 pm2/s.
The analysis of Sigsbee [1971] also shows that for typical experimental
times the stationary solution of eq. (10.28) describes adequately the
concentration of adsorbed molecules around the cluster. This solution is of
the form [Burton et al. 1951; Vetter 1967; Sigsbee 1971, 1972]
Zr = Z,[l - Kofr/^yKof/m,)] (10.30)
where Ko is the zeroth-order modified Bessel function of second kind. The
quantity A,, defined by the Einstein-type formula
1,= (¾¾1°. (10.31)
is the mean diffusion distance travelled by an adsorbed molecule before
desorption (typically A, = 1 nm to 1 ,i/m).
Transitionfreqitencies 147
Employing Zr from (10.30) for the calculation of the derivative in (10.27)
leads to (e.g. Vetter [1967]; Sigsbee [1971])
jd,„ = (D^AHK.yrao/KoCRV/O] (10.32)
where K} is the first-order modified Bessel function of second kind. In
quasi-stationary approximation this formula for7dn can be used with time-
dependent concentration Z,(r) (nr2) of the adsorbed molecules far away
from the cluster (i.e. at r = °°). In this approximation, from (10.26) we thus
find that for both caps and disks
/„(/) = Y„2KDsZl(t)(R'as)lKl(R'l^yK0(R-lls)l (10.33)
This equation reveals that the frequency of monomer attachment controlled
by surface diffusion is a complicated function of the cluster size. This function
is relatively simple only for sufficiently small or large clusters. As done, for
example, by Vetter [ 1969], Sigsbee [ 1971,1972] and Stowell [ 1972a], recalling
the asymptotics of the Ko and K, functions, we can simplify (10.33) to
f„(t) = Y„27cDsZt(t)/ln (Xs/R') (10.34)
for small clusters (R' « As) and to
/„(r) = 7„2jrDsZ,(/)(/ms) (10.35)
for large clusters (/?'» A,). However, as shown by Kashchiev [1978], we
can use the single approximate formula
27r(«'M,) [K,(/ra,)/K„(R7;t,)] = 1.3 + 2xR'll.
for clusters of any size as long as /?7AS > 10-4. Since typically only clusters
of sizes satisfying the condition 10-4</?7As< 1 are involved in the nucleation
process, the above result means that for both caps and disks/, from (10.33)
can be approximated by the simple expression (n - 1, 2, . . .)
fn(t) = y#*Dfr(t) (10.36)
where c* ~ 1 to 5 is the so-called capture number due to surface diffusion.
The possibility to use the n-independent capture number c* in (10.36) was
discussed by a number of authors [Lewis 1970; Venables 1973; Stowell
1974a; Lewis and Rees 1974; Venables and Price 1975; Lewis and Halpern
1976; Lewis and Fujiwara 1976; Lewis and Anderson 1978; Stoyanov and
Kashchiev 1981]. A Monte Carlo simulation of Pocker and Hruska [1971]
and rigorous calculations of Temkin [1977] showed that, indeed, c* is virtually
n-independent and has a value between 1 and 5. It must be noted that in
deriving eqs (10.33)-(10.36) we did not take account of the influence (see,
e.g. Bartelt et al. [1999]) that neighbouring clusters can have on the attachment
of adsorbed molecules to the considered n-sized cluster. Nonetheless, these
equations are applicable also to the case of other clusters present around the
cluster under consideration as long as the neighbouring clusters are away
from the considered one at a distance greater than about 1 OR' [Sigsbee 1971,
1972; Stowell 1972a; Kashchiev 1978, 1981; Lewis and Anderson 1978].
148 Nucleation: Basic Theory with Applications
Clearly, eqs (10.33)-(10.36) apply also to clusters of non-circular base, since
c* can be regarded as accounting also for the effect that the different shapes
of the cluster periphery may have on/„ [Lewis and Rees 1974; Lewis and
Anderson 1978],
Equations (10.33)-(10.36) do not allow for the possible motion of the
considered n-sized cluster along the substrate surface. At present, there exist
experimental and theoretical studies which provide evidence that under
favourable conditions dimers, trimers and even clusters comprising scores of
molecules can migrate on the substrate surface as entities (for reviews see,
e.g. Masson et al. [1971]; Geguzin and Kaganovski [1978]; Kern et al.
[1979]; Kashchiev [1979a]; Stoyanov andKashchiev [1981]; Jensen [1999]).
Zinsmeister [ 1969] and Lewis [ 1970] were the first to consider the effect that
cluster mobility might exert on the kinetics of HEN on a substrate. If we
want to allow for this effect on/„, just as in the case of volume-diffusion
control, we must replace D3 and R' in eqs (10.33)-(10.36) by the effective
coefficient Ds„ + Ds of surface diffusion and the effective radius R' + R0 of
the cluster base (Ds „ is the surface-diffusion coefficient of a cluster of n
molecules). As a result, since typically R' + R0 « As and Ds„ < Ds, in
accordance with (10.34) and (10.36)/„ will be increased not more than
twice. Obviously, this correction can be incorporated into the capture number
c* so that eq. (10.36) may be regarded as a formula which gives^, also when
the clusters on the substrate are mobile (in this case/„ is given more accurately
by eq. (10.95) at m-n= 1).
It remains now to express Z,, the concentration of adsorbed monomers far
away from the cluster, in terms of experimentally controllable parameters
such as the impingement rate / in molecular-beam condensation on the substrate,
the pressure p of the vapours in contact with the substrate, the solute
concentration C in the solution in which the substrate is immersed, etc.
When the system is in equilibrium, as already noted, Zj equals the equilibrium
monomer concentration C1 determined by eq. (7.5) so that, using (3.86) and
(3.89) with A, = a0, we have
Z, = C, = C0 exp [(A(i - X + E, + osa0)/kT\. (10.37)
This formula, in which C0 = l/a0 in accordance with (7.8), shows that Z\ is
proportional to exp (A/j/kT), i.e. to />, / or C in the cases of A^i defined by eqs
(2.8), (2.9) and (2.14).
In principle, however, nucleation may begin before the establishment of
the equilibrium concentration C, of monomers on the substrate. This means
that, depending on the time scales of the adsorption and nucleation processes,
Z1 may be either varying or constant (and equal to Cj) during nucleation.
The increase of Z\ from zero at the moment t - 0 of putting the bare substrate
in contact with the old phase to the time-independent concentration C,
corresponding to adsorption equilibrium at t = °»is governed by the differential
equation (e.g. Lewis and Anderson [1978])
dZ,(t)/d/ = /- Z,(t)/Td.
(10.38)
Transition frequencies 149
This equation quantifies the balance between the adsorbing (/) and the desorbing
(Zj/Tjj) molecular fluxes and is valid for sufficiently low adsorption coverage
of the substrate. It has a simple solution [Korn and Korn 1961] which in
view of the initial condition Zi(0) = 0 reads:
Z](0 = /Td[l-exp(-;/Td)]. (10.39)
This formula tells us that if HEN on the substrate occurs for t « Td, Z|
increases linearly with time according to (e.g. Lewis and Anderson [1978])
Zx(t) = It. (10.40)
Conversely, for t» Td, the concentration of adsorbed monomer is constant
and given by (e.g. Lewis and Anderson [1978])
Z, = C, = /rd = (Vvs) exp [(4i + E^ikT}. (10.41)
This is the concentration of monomers on the substrate at adsorption equilibrium
and it corresponds to that from (10.37) with C0 from (7.8) and Ap from (2.9).
This correspondence leads to the conclusion of general validity that, to a first
approximation, the energy E\ + as«o is merely the monomer desorption
energy Edcs. Indeed, since/e-C0vs exp (- K/kT) [Lewis and Anderson 1978],
comparison of the r.h.s. of (10.37) with that of (10.41) yields Ex + osa0 =
In many cases nucleation is observed experimentally over periods of time
much longer than Td. Combining eqs (10.31), (10.36) and (10.41), we find
that in these cases/, is given by the formula [Stowell 1974a; Lewis and
Halpern 1976; Lewis and Anderson 1978] (n = 1, 2, . . .)
/„=7nc*DsTd/=^c*As2/ (10.42)
which applies to clusters of arbitrary shape. Physically, this formula means
that if yn = 1, nearly all molecules impinging upon the diffusion zone
around the cluster (the area of this zone for caps or disks is about %Xl
when R' « As) are attached to the cluster independently of its size n.
Equations (10.39)-(10.42) describe Z[ and/, in HEN on a substrate during
condensation of molecular beams. When the substrate is in contact with
vapours with pressure p, in all these equations I should be replaced by the
Hertz-Knudsen impingement rate pi(2Km^kT)m. For example, in quasi-
stationary approximation, eq. (10.42) then becomes (n = 1, 2, . . .)
f„(t) = ync*DsC0KpP(t) (10.43)
where
KP = Td/C0(27tm0kT)m (10.44)
is the Langmuir constant of adsorption [de Boer 1953]. Naturally, eq. (10.43)
is readily obtainable from (10.36) withZ| determined from the low-coverage
form of the Langmuir adsorption isotherm (e.g. de Boer [1953])
ZJCq = C^Cq = Kvp
(10.45)
150 Nucleation: Basic Theory with Applications
where Ct/C0 is the coverage of the substrate by monomers in adsorption
equilibrium, and C0 is given by (7.8).
Similarly, when the substrate is immersed in a solution with concentration
C of monomer solute, after time t» Td, in quasi-stationary approximation
/„ is determined by
/„(») = y„c*D,C0KcC(t) (10.46)
where Kc is the constant in the low-coverage Langmuir adsorption isotherm
[deBoer 1953]
Zj/C0=Cl/C0=KcC. (10.47)
Since C0i>o=4), eq. (10.47) shows that when Kc = u0, the volume concentrations
CJda and C of solute in the adsorbed monolayer and in the bulk of the
solution are equal.
The solid and the dotted lines in Fig. 10.6 depict, respectively, the exact
and the approximate dependences (10.33) and (10.43) (with c* = 1.9) of
/„ on n at two constant pressures p ^ 2/?e (lines 2) and 4/?e (lines 4).
The calculation is done for hemispherical water droplets (for them R' =
(ivolln)"3n"3) in vapours at T= 293 K. The parameter values used are listed
in Table 3.1 and it is assumed that the substrate is so chosen that A, = 50 nm.
n* n*
J I , I 1
1 100 200 300
n
Fig. 10.6 Dependence of the frequencies of monomer attachment and detachment on
the cluster size in HEN of hemispherical water droplets on a substrate in vapours at
T = 293 K: solid curves and dotted lines - attachment frequency f„ under surface-
diffusion control according to eqs (10.33) and (10.43), respectively, at p/pe = 2 and 4
(as indicated); dashed and dot-dashed curves - detachment frequency gn under the
same control according to eq. (10.89) with /e-n from the exact and the approximate
formulae (10.33) and (10.43), respectively.
Transition frequencies 151
In the calculation Zx in (10.33) is expressed with the help of (10.45) and it
is taken into account that DSC0KV = X2 /(2 7tm0kT)m. As seen from Fig. 10.6,
the dependence of/„ on n is indeed rather weak and is practically negligible
in the n < 100 size range which is of interest in the theory of nueleation. The
linear increase of/„ with increasing p is also manifested in the figure. As in
the cases of direct-impingement and volume-diffusion control, this is equivalent
to a proportionality of f„ to exp (A^i/kT).
It is important to keep in mind that monomer attachment by surface diffusion
generally occurs in parallel with that by direct impingement or volume diffusion.
That is why it is of interest to know whether the contribution of surface
diffusion to /„ is dominant or negligible in comparison to that of each of
these two volume-transport processes. Let us first consider attachment of
monomers by simultaneous direct impingement and surface diffusion. For
hemispherical clusters (then 0W = nil), from (10.4) and (10.43) we find that
the ratio rD1/So between/, by direct impingement and/„ by surface diffusion
is given by
rDVsD=(27t/c*)(R/^)2. (10.48)
Since 27tfc* & 1, this formula shows that surface diffusion dominates over
direct impingement as long as R < A, [Sigsbee 1971, 1972], This condition
is usually fulfilled, for example, in HEN during condensation of vapours on
a substrate (then very often ^ > 10 nm). It is worth noting that in 2D HEN
of monolayer disks the role of surface diffusion is even greater [Kashchiev
1978, 1981], because direct attachment of molecules occurs only at the
cluster periphery and, hence, on an area proportional to R (see eqs (10.6) and
(10.7)) rather than to R2 as in the case of 3D HEN of caps.
Let us now consider monomer attachment by simultaneous volume and
surface diffusion. Again for hemispherical clusters, according to (10.24) and
(10.46), for the ratio >Vd/sd between/„ by volume diffusion and/n by surface
diffusion we find
rvD/SD= {27t/c*){RD/CQKcDs). (10.49)
We can, therefore, conclude that surface rather than volume diffusion controls
monomer attachment in 3D HEN on a substrate when the clusters have radii
R < CqK^DJD. As in most cases only clusters of R < 10 nm are of interest for
nueleation, with C0 = \/a0= 10l9m~2andD = 10~9m2 s_1, this condition will
be satisfied for substrate/solution systems for which KCDS > 10~36 m5 s~'.
When Kc and/or Ds are sufficiently small (i.e. when the adsorption coverage
and/or the monomer surface diffusivity are low), KCD^ < 10~36 m5 s"1 and
volume diffusion is the transport mechanism supplying monomers for
attachment to the cap-shaped clusters in HEN on a substrate.
(4) Interface-transfer control
Transfer of molecules across the cluster/old phase interface can control their
attachment to the clusters mostly in nueleation of liquids or solids in condensed
phases (e.g. melts, solutions* etc.). In attachment controlled by interface
152 Nuclealion: Basic Theory with Applications
transfer, the molecule to be attached is in immediate contact with the condensed-
phase cluster (Fig. 10.7) and can join the cluster by making a random jump
over a distance comparable with its diameter d0. The probability for such a
jump is characterized by the jump frequency [Frenkel 1955]
V = Vo,„ exp (- EitJkT) (10.50)
•(- n -•
(a)
(b) (c)
Fig. 10.7 Interface-transfer mechanism of attachment of molecules {the circles) to
(a) spherical cluster in HON, (b) cap-shaped cluster in HEN on a substrate, and
(c) disk-shaped cluster in HEN on a substrate.
where v0.« ( = 1013 s*1) is the vibration frequency of a molecule of the old
phase at the surface of an n-sized cluster, and Ettn > 0 is the activation energy
for interface transfer. If the area with which a cluster of n molecules is in
contact with the monomers is Acn, in the volume d^icn around the cluster
there are Z(<ioACi„ molecules that can be attached to the cluster (in interface-
transfer control, by definition, the monomer concentration 7,x is uniform
right up to the cluster surface). The attachment frequency /„ is then merely
the product of Vj „, Zyd^iQ „ and the sticking coefficient % so that we have
(B=1.2,...)
fn = %v0m exp C- E^kDZidoA^. (10.51)
Equation (10.51) is the general formula for/„ in the case of interface-
transfer control. For each particular cluster shape it is possible to determine
the contact area Acn as a function of n and thus find the dependence of/„ on
the cluster size n if %,. v0„ and EAn are known. It is worth noting that now,
as in the cases of volume- and surface-diffusion control, for clusters in a
condensed old phase yXlJ in (10.8) can also be set equal to unity, since such
a phase can be a good mediator in the monomer thermal accommodation
during attachment.
The practical use of eq. (10.51) is rendered difficult not only by our poor
knowledge of yn, but also of v0 „ and £it.n- Indeed, the motion of the monomers
Transition frequencies 153
adjacent to the cluster surface can be affected considerably by this surface.
This means that both v0>„ and Eit„ in (10.51) may vary with n (especially in
the small-size range) and differ substantially from the vibration frequency vd
and activation energy Ed for the diffusion-type jumps of the molecules in the
bulk of the old phase. In the absence of sufficient knowledge about v0jl and
Ens,, the common approach [Volmer 1939; Frenkel 1955; Zettlemoyer 1969]
is to accept that, approximately, v0„ = constant = vd and Eit „ = constant = Ed.
Then, as the monomer diffusion coefficient is given by [Frenkel 1955]
D = dl vd exp (- EJkT), (10.52)
from (10.51) it follows that (n = 1, 2, . . .)
f„ = yn(DZl/d0)A^. (10.53)
This expression is useful for nucleation in solutions, since then D is often
known independently and, as already discussed, Z\ is related to the solute
concentration C which is an experimentally controllable parameter.
Alternatively, employing the Stokes-Einstein formula (cf. eq. (10.21))
D = kTlind0i} (10.54)
allows/,, to be written down in terms of the viscosity 7) of the phase (solution,
melt, etc.) around the cluster (n = 1, 2, . . .):
/„ = Ukmitdl r))Z,Ac,„. (10.55)
This equation is particularly convenient for calculating /„ for nucleation
of crystals in melts, as then r/ may be known independently, and Z] = 1/1¾.
With this approximation for Zj, eq. (10.55) was used in a number of papers
considering this case of nucleation IToschev and Gutzow 1967a; Toschev
1973a; Rowlands and James 1979; Gutzow et al. 1985]. The possibility of
introducing D or 7] as a parameter determining /„ for nucleation of crystals
in melts was utilized also, e.g. by Kelton etal. [1983], Kozisek [ 1988, 1990,
1991], Toner et al. [1990] and Kelton [1991]. The temperature dependence
of 7] is usually taken into account by the Vogel-Fulcher formula [Vogel
1921; Fulcher 1925] (see also Tammann and Hesse [1926])
J)(7) = f)0 exp [EJk(T- Td)] (10.56)
where 7)0 is a practically /"-independent factor, Ev is the activation energy for
viscous flow, and Td is the absolute temperature at which 7) diverges.
Before proceeding further it must be noted that for nucleation of crystals
in melts the use of the approximation Zj = l/u0 leads to the qualitatively
incorrect result that/„ from (10.51), (10.53) or (10.55) does not depend
explicitly on A^i. Indeed, if we want to be consistent with the postulate that
Z, equals the equilibrium monomer concentration C,, according to (3.86),
(3.89), (7.5) and (7.12), in the case of HON of condensed phases (Fig. 10.7a)
the correct expression for Z, reads
Z, = C, = (l/i>o) exp [(Afi - X)lkT\ (10.57)
154 Nucleation: Basic Theory with Applications
since then E{ = 0 and 0"s = 0. This expression applies also to 3D HEN on a
substrate when practically only monomers that are not in contact with the
substrate (Fig. 10.7b) are attached to the clusters (this is so, e.g. for cap-
shaped clusters with such a large wetting angle 6W that attachment of monomers
to their periphery can be ignored). In the case of 2D HEN of condensed-
phase clusters of monomolecular height the monomers joining the clusters
are at the cluster periphery and, hence, in contact with the substrate (Fig.
10.7c). Then, Zx in (10.51), (10.53) or (10.55) is equal to CJdr, where C,,
given by (10.37), is the equilibrium concentration of monomers on the substrate.
Hence,
Z[ = C,/d0 = (Co/do) exp [(A/i - X + E, + o&,)/kT\ (10.58)
where 0/(¾ = ^latAi ~ l'"o- From the above two equations we see that the
correction to the commonly used approximation Zi = llv0 for nucleation of
crystals in melts is the exponential factor which introduces in/„ from (10.51),
(10.53) or (10.55) the same dependence on Aft as that of /„ in the cases
already considered of attachment controlled by direct impingement, volume
diffusion and surface diffusion. In eq. (10.57) this factor is practically just a
number: with A^i from (2.23) and with typical values of Asjk = 1 to 5 we
find that exp [(Aji - X)lkT\ = exp (- Asjk) = 0.4 to 0.007. In eq. (10.58),
however, this factor is a function of T and depends on the energy E, + <7sa0
which, as noted above, is approximately equal to the monomer desorption
energy Edes. Comparison of the last two equations shows that when E, + O"s0o
= 0, the concentrations (in nrT3) of the monomers in the adsorbed monolayer
and in the bulk of the melt are equal.
It is important to bear in mind that in some cases eqs (10.53) and (10.55)
may not describe adequately /„, because in them D and r/ are quantities
characterizing the molecular motion in the bulk of the old phase rather than
in the vicinity of the cluster surface. This means that, more generally, D and
7) in (10.53) and (10.55) should be treated as unknown theoretical parameters.
In practice, this is equivalent to using eq. (10.51), since in it it is possible to
reduce the two unknown parameters v0n and Eitn to one, Eit. This can be
done by ignoring the ^-dependence of Eit„, i.e. by assuming that £itn =
constant = Eit, and by employing the approximation Von - constant = kT/hp
of Turnbull and Fisher [1949] which follows from the theory of reaction
rates [Glasstone etal. 1941] (hp is the Planck constant). Then (10.51) transforms
into the expression (« = 1,2,.. .)
/„ = y„(kT/hp) exp (- EyJkT)Z\d<^„ (10.59)
which, with yn = 1, Z, - l/u0 and size-dependent EIX, is the formula for/,
proposed by Turnbull and Fisher [1949] for nucleation in condensed phases.
The unknown activation energy Eit for interface transfer is expected to be
close to the activation energy determining the temperature dependence of D
or r/ when the molecular jumps across the cluster/old phase interface are not
too different from those in the bulk of the old phase.
We can now obtain concrete expressions fovf„ for nucleation of condensed
Trunsitionfrequencies 155
phases in solutions and melts when monomer attachment is controlled by
interface transfer. For nucleation in solutions, expressing the cluster lateral
area Ac„ in terms of n and setting Z\ = C in the cases of HON and 3D HEN
illustrated in Fig. 10.7a,b, from (10.53) we find in quasi-stationary
approximation that (n = 1, 2, . . .)
/„(') = %(c v™ lck)DC(t)nm (10.60)
for HON of spheres (c = (367t)"3), cubes (c = 6), etc. and that (n = 1, 2,. . .)
/«(') = %[(i - cos ejny/'^eMcvf/dojDC^n113 (io.6i)
for 3D HEN of caps (c = (367t)"3). As already noted, the use of Z, = C for
3D HEN implies that practically none of the molecules that join the cluster is
in contact with the substrate. In the case of 2D HEN of clusters of monolayer
height (Fig. 10.7c) mainly molecules adsorbed on the substrate are likely to
join the cluster and then Z, = CJd^ where Cl is specified by eq. (10.47). Hence,
Z, = C0KcC/d0 = KcClv„ (10.62)
in the concrete case of low-coverage Langmuir adsorption (this equation for
Zi is a particular case of the general equation (10.58)). With Ac n = dgbn"2,
from (10.53) it thus follows that (n = 1, 2, . . .)
f„(t) = Y„(blv0)DKcC(t)n"2 (10.63)
for disks (b = 2(7ta0)l/2), square prisms (b = 4«^), etc. of monolayer height.
Whenever necessary, in eqs (10.60), (10.61) and (10.63) D can be replaced
by the solution viscosity T) with the aid of the Stokes-Einstein formula
(10.54). It is worth keeping in mind as well that in these equations D (or r\)
is, in fact, an effective quantity characterizing the random jumps of the
molecules across the cluster/old phase interface and having values which
may differ appreciably from those of the diffusion coefficient (or the viscosity)
in the bulk of the solution. We also note that at 8W = K (complete non-
wetting), as required, eq. (10.61) passes into eq. (10.60).
For nucleation of solids in melts, again in the cases of HON and HEN
illustrated in Fig. 10,7, with the help of eq. (10.57) for Z,, from (10.55) we
find that (n = 1, 2, . . .)
/„ = Jaicv^n itdl)(kTlvarf) exp [(A/j - X)lkT]nm (10.64)
for HON (c is the shape factor appearing in (10.60)) and that (n = 1, 2, . . .)
/„ = 7,,1(.1 -cos ejl2\f'\ew)Kcv™/i7cd2aXkT/v0ri)
exp [(A// - X)/kT]n2B (10.65)
for 3D HEN of caps (c = (36;r)"3). Similarly, using Z, from (10.58) in eq.
(10.55) yields (n = 1,2, . . .)
/„ = y„(.b/5Xd0)(kTlu0ri) exp [(Aij - X + £, + osa0)/kT\n"2 (10.66)
for 2D HEN of clusters with monolayer height (b is the shape factor entering
eq. (10.63)). Comparison of eqs (10.64) and (10.65) shows that under the
156 Nucleation: Basic Theory with Applications
same conditions fn for caps in HEN is smaller than/n for spheres in HON.
According to (10.65),/, diminishes with the decrease of the wetting angle
0W. When 0W —» K (complete non-wetting), as required,/„ from (10.65)
passes into/n from (10.64) with c - (36;r),/3 (HON of spheres). We note as
well that without the exponential factors in them, eqs (10.64)-(10.66)
correspond to the expression for/, used by Gutzow ex al. [1985] in the scope
of the Z] = l/u0 approximation.
Equations (10.60), (10.61), (10.63)-(10.66) tell us that when/„ is controlled
by interface transfer, similar to the cases of control by direct impingement
and diffusion, it increases as a power function of the cluster size n (provided
the variation of yn with n is negligible). As under diffusion control, in the
case of nucleation in solutions /„ increases linearly with increasing solute
concentration C, i.e. it is again proportional to txp(A^i/kT). The same explicit
exponential variation of/„ with A/i is in force also for nucleation of crystals
in melts. The dependence of/n on the cluster size n is illustrated by the solid
curves in Fig. 10.8 for monomer attachment to spherical clusters in HON of
ice in water. The curves are drawn for T = 230 and 240 K (as indicated)
according to eq. (10.64), Aju is calculated from (2.23), and the parameter
values used are listed in Table 6.2. As seen, attachment of monomers to the
clusters in nucleation of solids in melts is quite sensitive to changes in the
temperature, the effect being due mainly to rj rather than to A^. This sensitivity
is particularly high when nucleation occurs in a temperature range in which
12x10:
n
Fig. 10.8 Dependence of the frequencies of monomer attachment and detachment on
the cluster size in HON of ice in water at T = 230 and 240 K (as indicated): solid
curves - attachment frequency fn under interface-transfer control according to eq.
(10.64); dashed curves - detachment frequency g„ under the same control according
to eq. (10.78).
Transition.frequencies 157
the melt viscosity varies strongly with T (e.g. near the glass-transition
temperature Ts of the melt).
As monomer transfer across the cluster/old phase interface occurs in series
with any of the three transport processes considered hitherto, it is of interest
to have a criterion about the conditions under which it controls the attachment
of the monomers to a given cluster. Such a criterion is needed above all for
nucleation in solutions, for then interface transfer is preceded by the relatively
slow process of volume diffusion of the monomers towards the cluster surface.
Considering successive volume diffusion and interface transfer, from (10.20)
and (10.60) (used with v0 = (7d6)dl) we find that for HON of spheres the
ratio rVD/iT between /„ controlled by volume diffusion and by interface transfer
is given by
rVMT = 2«-"3. (10.67)
As seen, rvD/rr < 1 for h > 8. This means that volume diffusion can be
expected to be the attachment-controlling transport process for the larger
clusters. However, this conclusion holds only when the diffusion coefficient
in eq. (10.60) can be approximated by that in eq. (10.20). As already noted,
this approximation may not always be realistic, since the diffusion jumps of
the molecules across the cluster/solution interface and in the bulk of the
solution may have quite different activation energies EAn and E&.
10.2 Monomer detachment frequency
In contrast to attachment which is largely determined by parameters specific
for the bulk old phase (e.g. p or Q, monomer detachment depends on parameters
characterizing the cluster rather than the bulk new phase. This is so because
being a phase of finite size, an n-sized cluster has properties which may
differ appreciably from those of the bulk new phase (see Chapter 6). As these
properties are not easy to determine especially for the smallest clusters,
finding the frequency gn of monomer detachment is more difficult than
finding the monomer attachment frequency/„. For that reason, in obtaining
gn it is convenient to use indirect methods based on a priori knowledge of/„.
One such method profits the fact that the detachment of molecules from
the clusters is the process opposite to their attachment. In other words, if
attachment of a molecule to an (n - 1 )-sized cluster is the forward 'reaction'
occurring with frequency/„ _ 1? detachment of the same molecule from the
resulting n-sized cluster is the backward 'reaction' which takes place with
frequency gn. Consequently, if we know the conditions under which gn and
/„ _ j are equal, we can obtain gn simply by identifying it with/n _, corresponding
to these conditions.
This method is easy to apply to the case of HON of liquids or solids in
vapours. In this case we know from thermodynamics that an n-sized cluster
neither grows nor decays when the pressure p of the vapours around the
cluster equals the cluster vapour pressure ptlt (see Section 6.3). The cluster/
158 Nucleation: Basic Theory with Applications
vapour coexistence has a dynamic character and is maintained thanks to the
equality in the number of attachments,/, _ ,, and detachments, g„, per unit
time. In view of (10.3), this balance between gn and/„_ , at p = pen yields
(n» 1)
gn = 7„^cv^\pJ(2Km0kT)m](n - l)2'3 (10.68)
which is the formula used already in the pioneering works on nucleation
[Farkas 1927; Kaischew and Stranski 1934a; Becker and Doling 1935].
Since this formula is obtained with/, corresponding to direct-impingement
control, it gives gn for HON of condensed phases when monomer detachment
is controlled by evaporation [Hirth and Pound 1963]. As noted in Section
10.1, for a gas-phase cluster in its own condensed phase the detachment of
monomers from the cluster surface is the process by which the cluster gains
molecules. That is why the monomer attachment frequency/, in HON of a
gas phase from its own condensed phase (see eq. (10.9)) corresponds to g„
from (10.68) with v0 replaced by kTlp„.
With appropriate modification, eq. (10.68) determines g„ also in HON of
a gaseous phase in its own condensed phase. When the molecules of the gas
inside an n-sized gaseous cluster are lost by direct impingement upon the
cluster surface, in conformity with eq. (10.3), in this case g„ is governed by
the pressure /?„ inside the cluster (this is so because g„ for gaseous clusters
corresponds to/ for condensed-phase clusters in vapours). For that reason,
with Pc changed to p„ and u0 replaced by the volume kT/p„ _ ( occupied by
a molecule in the (n - 1 )-sized gaseous cluster which results from the n-sized
one after the detachment event, eq. (10.68) gives g„ in HON of a gas phase
from its own condensed phase:
g„ = 7» -,c(kr/p„, 0m<J>e.„ - ,//>*)[/>„/(2;rmoW)"2](n - 1 )2'3.(10.69)
Classically, />„ is determined by the Laplace-type equation (6.4),pcn is given
by (6.24), p* is specified by (4.14), and the factor pcn _ ,//>* is necessary to
ensure the equivalence of eq. (10.69) to eq. (10.90) when in the latter/ from
(10.9) and H'(n)from (3.42) are used for the determination of g„ for gaseous
clusters in the case of direct-impingement control.
The above method of finding gn for nucleation in vapours is easily
applicable also to HON of condensed phases in solutions. The role of the
vapour pressure pen is then played by the solubility Ce„ of the n-sized
cluster (see Section 6.4). Hence, using (10.20) and (10.60) in the balance
equation gn = / _ !(Ce„), we find that (n » 1)
g„ = In - i(4nc)"2ui'3DCe.„(n - l)"3 (10.70)
for volume-diffusion control and that (n = 2, 3, . . .)
g„ = Yn-Acvf ld0)DC^(n- l)m (10.71)
for interface-transfer control.
Equations (10.68), (10.69), (10.70) and (10.71) show that knowledge of
such thermodynamic characteristics of the clusters as pcn , p„ and Ce „ is
Transition frequencies 159
indispensable for the explicit determination of the dependence of gn on the
cluster size n. However, the smallness of the clusters is a serious obstacle in
their thermodynamic description and that makes the problem of finding the
size-dependence of g„ a very hard one. As discussed in Part 1, the classical
theory of nucleation overcomes the difficulties in the thermodynamic treatment
of the clusters by employing the capillarity approximation. In the scope of
this approximation, for clusters of condensed phases, pc „ and Ce „ are given
by eqs (6.17) and (6.28), respectively. Consequently, we can unite eqs (10.68),
(10.70) and (10.71) into the single formula (c = (36;r)"3 for spheres, 6 for
cubes, etc.)
Sn =/e.»- l exP (2cvj?3o/3kTnul). (10.72)
Here/e „ _ , is the value of the attachment frequency/„ _ 1 at the equilibrium
pressure />„ or concentration Ce (i.e. at Aft = 0):
An- i = In- ^vf-\pJ(2mn^T)"\n - If1 (10.73)
for HON in vapours, and
An- i = In- \{Anc)xavfDCJ,n - l)1'3 (10.74)
or
An-1 = %- i("f/4)OQ» -1)2'3 (10.75)
for HON in solutions depending on the attachment mechanism (volume
diffusion or interface transfer, respectively). Naturally, eqs (10.73)-(10.75)
are merely eqs (10.3), (10.20) and (10.60) with p = pe or C = Ce.
Equation (10.72) shows that, in contrast to/„, gn is insensitive to changes
in the actual monomer pressure p or concentration C as long as yn and a are
also constant with respect to p or C. This means that unlike/„, gn does not
depend on the supersaturation Ay when this is varied isothermally. With
increasing n, g„ first diminishes strongly because of the exponential factor
and then increases weakly owing to the pre-exponential factor which takes
into account that larger clusters have larger surface area and, hence, a higher
chance to lose molecules. This dependence of gn on n is depicted by the
dashed curves in Figs 10.2 and 10.4 for the case of HON at T = 293 K of,
respectively, water droplets in vapours (evaporation control) and crystals in
solutions (volume-diffusion control). The curves are drawn according to eq.
(10.72) with the parameter values given in Tables 3.1 and 6.1. As seen, while
for the smaller clusters detachment dominates attachment, for the larger
clusters attachment takes over. At a particular cluster size n* which depends
on p or C (i.e. on A/y) the attachment and detachment frequencies are equal.
As will be shown in Chapter 12, n* is precisely the nucleus size appearing
in nucleation thermodynamics (see Chapters 3 and 4) so that, kinetically, the
nucleus is merely that particular cluster which has the same chance to gain
or lose a molecule per unit time [Volmer 1939]. It must be emphasized that
since eq. (10.72) relies on the classical formulae for/>e „ and Ce „, its applicability
to the smallest clusters is questionable. For that reason, in the size range of,
160 Nucleatian: Basic Theory with Applications
say, n < 100 molecules the g„ curves in Figs 10.2 and 10.4 are more or less
qualitative.
Another method of finding gn is the one based on the theory of reaction
rates and already used in Section 10.1 for determination off, when attachment
is controlled by interface transfer. Following Turnbull and Fisher [1949], in
analogy with (10.51), we can represent g„ for liquid or solid clusters in
condensed phases as (n = 2, 3, . . .)
g„ = Y«- iv0.„ exp ( - Eaa.JkT) Z,4A„„ ,. (10.76)
This relation implies reversibility between the forward and backward
'reactions' of attachment and detachment: the same number y„_ lZld0Ac„_l
of molecules that can join the (n - l)-sized cluster to make it a cluster of n
molecules can also leave the so-formed n-sized cluster when this is transformed
back into a cluster of n - 1 molecules. According to the reaction-rate theory
[Glasstone et al. 1941], the molecular vibration frequency v0n is size-
independent and the same for both attachment and detachment: vB„ = kTlhf.
Thus, it is the activation energy Edet „ for detachment from an n-sized cluster
that makes g„ from (10.76) essentially different from the corresponding
attachment frequency/„ from (10.51). Clearly, in order to be able to determine
gn we need a knowledge of E^t «- 1° tne absence of such a knowledge it is
convenient to relate Efeln to the activation energy Eit„ for interface transfer
(i.e. for attachment) and the work W(n) for cluster formation. This is easily
done upon noting that, by definition, for isobaric-isothermal attachment and
detachment
E,v-i = Ga,„-G2(«- 1)
Edey, = Ga,„ - G2(n).
Here G2(n - 1) and G2(n) are the Gibbs free energies of the system in its
initial and final states when the cluster in it is of size n - 1 and «, respectively,
and Ga „ is the Gibbs free energy of the system in the activated state when the
cluster size is somewhere between n - 1 and n. Since G2(n) is related to W(n)
by eq. (3.4), it follows from the above two equations that
Ede.,„ = Eit,„-l + ^(1-1)-^)-
In combination with (10.51), this formula allows representing g„ from (10.76)
as (n = 2, 3, . . .)
g„ =/„_ , exp [[W(n) - W(n - l)]lkT). (10.77)
Equation (10.77) shows that, as in (10.72), thermodynamics enters again
the description of the kinetic quantity g„. When concrete dependences of/„
and W(n) on n are introduced in it, eq. (10.77) gives g„ in the cases of both
HON and HEN of liquids or solids in condensed phases. In fact, as we shall
see below, eq. (10.77) is a very general formula which is valid for whatever
case of nucleation of condensed or gaseous phases and for any mechanism
of monomer attachment and detachment. It parallels the formula of Volmer
Transition frequencies 161
[1939] for the g„/f„ _ , ratio in various cases of nucleation and was used
recently in numerical studies on the kinetics of HON of crystals in melts
[Kelton etal. 1983; Kozisek 1988,1990,1991; Kelton 1991]. In this case of
HON, with the aid of/„ from (10.64) and the classical formula (3.39) for
W(ri), from (10.77) we find that
Sn = [%- t(cv™IZKdlXkTIvvT]) exp (- UkT)(n - l)m)
exp (2c vf allkTn'") (10.78)
since, as noted by McDonald [1963], nm - (n - l)2'3 = 2/3n"3. We used this
approximation to show that in the scope of the classical nucleation theory eq.
(10.77) leads again to a dependence of gn on n, which is similar to that given
by (10.72). Despite this similarity, the bracketed pre-exponential factor in
(10.78) does not have the physical significance of attachment frequency
/e „ _ i at equilibrium (i.e. at Ap = 0) when Ap in (10.64) is controlled by the
melt temperature T according to (2.20). If, however, Ap in (10.64) is controlled
by the melt pressure p, this pre-exponential factor is again the attachment
frequency/e„ _ t resulting from (10.64) at Ap = 0. This is so, since in this case
Ap is given by [Volmer 1939]
Afl = Ave(p-pc) (10.79)
when compressibility is negligible (pc is the equilibrium pressure, and Aue =
Vom - vQc is the difference between the volumes u0m and u0c occupied at
equlibrium by a molecule in the melt and in the nucleating crystal phase).
Verification that eq. (10.77) is indeed a generalization of (10.72) is easily
done with the help of W(n) from (3.39), Ap from (2.8) and (2.14), />e.„ and
Ce„ from (6.17) and (6.28), the McDonald approximation given above and
/„ from (10.3), (10.20) and (10.60): after some algebra eq. (10.77) leads to g„
from (10.68), (10.70) and (10.71) in the respective cases.
The dashed curves in Fig. 10.8 display the dependence of g„ on n for
spherical ice clusters in water at T = 230 and 240 K (as indicated). The
calculation is done according to eq. (10.78) with X = TeAse and the parameter
values listed in Table 6.2. As seen, despite that g„, in contrast to/„, does not
depend explicitly on Ap = Asc(Te - T), it changes significantly with 7 mainly
as a result of the 1}(T) dependence. Again, g„ and/„ are equal at the nucleus
size n* which is smaller at the lower T value (i.e. at the higher supersaturation
Ap). We note also that since eq. (10.78) is derived with the aid of the classical
W(n) dependence (3.39), the reliability of the g„ curve in Fig. 10.8 is
questionable for the smallest clusters of, e.g. n < 100 molecules,
A third method of finding the monomer detachment frequency is based on
model probabilistic considerations and was used for the determination of gn
in the atomistic theory of HEN of condensed phases on a substrate [Zinsmeister
1970; Stoyanov 1973; Lewis and Halpern 1976; Lewis andAnderson 1978].
Let us consider 2D clusters of monolayer height on an atomically smooth
substrate with adsorption sites of concentration C0 given by eq. (7.8). For
monomer attachment by the surface-diffusion mechanism/„ is determined
by eq. (10.36) where Zb the concentration of adsorbed monomers, is given
162 Nucleation: Basic Theory with Applications
by (10.37) in general or by (10.41), (10.45) or (10.47) in particular under
conditions of adsorption equilibrium. Divided by C0, this concentration may
be interpreted as the probability that a surface site at the cluster periphery is
occupied by a monomer which can make a surface-diffusion jump and attach
to the cluster. Analogously, monomer detachment is determined by the
concentration Q of molecules inside the monolayer cluster (the molecules
are assumed to be close-packed). Dividing this concentration by C0 gives
unity for the probability that a peripheral site inside the cluster is occupied
by a molecule ready to leave the cluster. Before making a diffusion jump
away from an n-sized cluster, however, the molecule must break its bonds
with the cluster. From Boltzmann statistics, the probability for this to happen
is equal to exp [- (Esn - £SM_ \)/kT], because the cluster must change its
'surface' binding energy from Esw to Es„ _ ] (Esn is defined by eq. (7.31)).
Thus, in analogy with (10.36), when detachment is controlled by surface
diffusion, in 2D HEN of condensed-phase clusters of monolayer height gn is
given by [Zinsmeister 1970; Stoyanov 1973; Lewis and Halpem 1976; Lewis
and Anderson 1978] (« = 2, 3, . . .)
g«=7n- ic*DsC0 exp [- (£s,, - E,fl_ x)lkT\ (10.80)
where y„ _ , appears because of the reversibility between attachment and
detachment.
Comparing this equation with eqs (10.42), (10.43) and (10.46), we observe
that under surface-diffusion control, in contrast to/„, g„ is also A^/-independent
when the supersaturation is varied isothermally. As in the other considered
cases of transport control, g„ depends essentially on the energetic status of
the cluster - the exponential factor in (10.80) is the atomistic analogue of the
exponential factors in (10.72) and (10.78). This is easily seen in the case of
HEN on the own substrate which is characterized by os = (7 and o, = 0. In this
case, with the help of (7.31) and the classical approximation (3.81) for E„,
from (10.80) it follows that
S„ =/eiB-i «p (bK/2kTnm). (10.81)
Here b = 2(7ra0),/2 for disks, 4aq2 for square prisms, etc. and, in conformity
with (10.36) and (10.37),
/c„-i = 7n-ic*DsCu (10.82)
is the attachment frequency at equilibrium (i.e. at A/u - 0) when the
concentration of monomers on the substrate is C] e (in general, C] c is given
by Z[ from (10.37) at Ay = 0, and in particular, by Cu = /eTd, C{ e = C0KpPe
and CXfi = C0KcCe as required by eqs (10.41), (10.45) and (10.47)). The
exponential factor in (10.81) results from using the McDonald-type
approximation nm - (n - 1)1/2 ~ l/2«l/2. Physically, this factor accounts for the
effect of the cluster size on the vapour pressure or the solubility of the 2D
cluster on the substrate (it is the 2D analogue of the exponential factor in eqs
(6.17) and (6.28)).
Concerning eq. (10.80), we also note that it is obtainable as a particular
Transition frequencies 163
case of eq. (10.77) in the scope of the atomistic theory of nucleation of
condensed phases. Indeed, for 2D clusters of monolayer height W(n) is given
by (3.78) so that
W(n) - W(n - 1) = kT In (Q/C,) + Es„ . , - E5,„ (10.83)
by virtue of (7.31) and (10.37). When W(n) - W(n - 1) from this formula and
/„ _ i from (10.36) with Z, = C\ are introduced in (10.77), we arrive again at
eq. (10.80).
Looking back at the particular results for gn obtained by the above three
methods, we observe that eq. (10.77) appears as a general formula giving g„
in various cases of nucleation in the scope of both the classical and the
atomistic nucleation theory. We are thus led to the question: is this generality
relatively limited (after all, eq. (10.77) is a model-dependent relation derived
within the caveats of the reaction-rate theory) or, on the contrary, is it an
expression of some fundamental physical principle? The answer to this question
is in favour of the latter anticipation: eq. (10.77) is a manifestation of the
basic principle of microscopic reversibility or detailed balance.
To come to this answer we must change diametrically our line of thinking.
Hitherto, we determined g„ by purely kinetic arguments and only at a later
stage used knowledge from thermodynamics to reveal the size dependence
of ptJl, Ce„, W(n) and £„, in eqs (10.68), (10.70), (10.71), (10.77) and
(10.80). It is possible, however, to conceive gn as a quantity defined from the
very beginning with the help of thermodynamic parameters. This method of
finding g„, proposed first by Zeldovich [1942], is thus conceptually different
from the three methods described so far. Though allowing the determination
of gn in a most general way, it deprives this quantity from being a strictly
kinetic one and explicitly introduces thermodynamics in the kinetic description
of nucleation.
The idea behind the Zeldovich method is that when the system is in
thermodynamic equilibrium, kinetically, the equilibrium is maintained by
the detailed balance (i.e. the balance holding at any cluster size n) between
the number of attachments to all n-sized clusters in the system and the
number of detachments from all (n + l)-sized clusters in it. Clearly, the
cluster population which is needed kinetically to ensure this microscopic
reversibility or detailed balance is represented by the equilibrium cluster size
distribution C(n) given thermodynamically by eq. (7.4). The condition for
detailed balance thus reads [Zeldovich 1942] (n = 1, 2, . . .)
/„C(n) = g„+1C(n+l). (10.84)
From this expression, owing to (7.4) or (7.6), we obtain again eq. (10.77).
The conclusion is, therefore, that (10.77) is indeed a much more general
result for gn than just a model-dependent formula in the scope of the reaction-
rate theory. Regarded from this point of view, however, g„ from (10.77) is
not any more a purely kinetic quantity: it is determined by both kinetics
(f„ _,) and thermodynamics (W(n)).
Equation (10.84) and, hence, eq. (10.77) are applicable only to nucleation
164 Nucleation: Basic Theory with Applications
occurring under time-independent conditions. In practice, it is important to
know gn also when the process takes place at variable supersaturation,
temperature, etc. In this case the role of C(n) is played by the quasi-equilibrium
cluster size distribution C(n, t) corresponding to the momentary values of
A/(, T, etc. and defined as [Kashchiev 1969b] (n = 1, 2, . . .)
C(n, t) = C|(I) exp (- [W(n, t) - W^t)]/kT(t)} (10.85)
where C^t) = C(l, t) and W\(t) = W(i, t). This equation is a generalization
of (7.6) and in it, according to (3.86),
W(n, t) = ~nAfi(t) + <P(n, t) (10.86)
is the work to form an «-sized cluster at the supersaturation A,u existing at
time t. The cluster effective excess energy <P is given in general by (3.87)
(for EDS-defined clusters) or by (3.89) or (3.90) and may also be a function
of time. As C(n, t) is the cluster size distribution which would ensure equilibrium
in the system at variable T, A/u, etc., it satisfies the condition for detailed
balance in the form [Kashchiev 1969b] (n = 1, 2, . . .)
/„(0C(n, t) = g„ + , (t)C(n + 1, t). (10.87)
This equation is a generalization of (10.84) and with C{n, t) from (10.85)
leads to the sought general formula for gn(t) under time-dependent conditions
(« = 2, 3, . . .)
SM =/,-.(0 exp {[W(n, t)-W(n-l, t)]/kT(t)}. (10.88)
Clearly, this formula is equally applicable to clusters of both condensed and
gaseous phases in whatever case of nucleation (HON, HEN, 3D, 2D, classical,
atomistic, etc.).
At this point, let us recall eq. (9.22) and compare it with (10.87). We
employed (9.22) to define a new function C„(t) in the master equation (9.23)
of nucleation by considering fn(t) and g„(r) as independently known kinetic
quantities. With respect to eq. (10.87) the situation is reversed: we presumed
that we knew the function C(«, t) and used this function to define the unknown
detachment frequency g„(r). The fact that eqs (9.22) and (10.87) coincide
upon setting C„(t) s C(n,t) means that C„(t) in the master equation of nucleation
has the physical significance of the quasi-equilibrium cluster size distribution
given by (10.85). As to eqs (9.22) and (10.87), they represent a universal
relation between/,, gn and C„ and can be used for obtaining anyone of these
three quantities provided the other two are independently known. Equation
(9.22) or (10.87) thus establishes the link between kinetics (/„, g„) and
thermodynamics (C„) of nucleation (see also Chapter 12) and, for this reason,
is a central formula in nucleation theory.
Let us now illustrate the applicability of eq. (10.88) to the case of isothermal
3D HEN of cap-shaped condensed-phase clusters on a substrate when monomer
attachment is controlled by surface diffusion. In this case, classically, W(n, t)
from (10.86) is given by (3.60) with
A/j(t)/kT = In [/(()/41, In [pit)/pe] or In [C(f)/Ce],
Transition frequencies 165
and for/„ we have eq. (10.33) and its respective approximation (10.42),
(10.43) or (10.46). Thus, with the aid of the McDonald approximation nm -
(n - \)m = 2/3n"3, from (10.88) we find that
gn =/0.-1 e*P \2y"hew)cv™o/3kTnml (10.89)
where/e „ _,, the frequency of monomer attachment at equilibrium (i.e. at A^t
= 0), is given exactly by (10.33) at Z, = Cx e and approximately by (10.82)
with C|... = /eTd, C, e = C0^ppe or C, e = C^KJO^, respectively. We note that at
y/ = 1 (HON) eq. (10.89) takes the form of eq. (10.72) and that it predicts a
constancy of g„ with time despite the changing supersaturation in the system
(this constancy is due to the general property of g„ for condensed-phase
clusters not to depend explicitly on A^i).
The dashed and the dot-dashed curves in Fig. 10.6 display, respectively,
the exact and approximate size dependence of gn from (10.89) for hemispherical
water droplets in vapours at T = 293 K. The calculation is done with 1// =
1/2, c* = 1.9 and Xg = 50 nm, and the values of the other parameters are those
given in Table 3.1. Again, the nucleus is that particular cluster (its size is «*)
for which attachment and detachment occur at equal rates.
It remains now to see how the gn formulae obtained above have to be
modified when the cluster size n is treated as a continuous variable. The
necessary changes are very simple indeed: in eqs (10.68)-(10.76), (10.78),
(10.81), (10.82) and (10.89) n - 1 must be changed to n, in eqs (10.77) and
(10.88)/„ _ , and W(n) - W(n - 1) must be replaced by/(n, t) and 3W(n, t)l
dn, respectively, and in eq. (10.80) % and d£,„/dn should be used instead of
yn _ i and E^n - Es „ _,. The same replacement of the finite differences by the
dW/dn and d£s„/drt derivatives is necessary in eq. (10.83). For instance, eqs
(10.88) and (10.89) take the form
g(n, t) =/(n, t) exp {[dW(n, t)ldn]lkT(t)} (10.90)
g(") =/e(n) exp [2y/"3(ew)c(jfo-/3it7>i"3], (10.91)
respectively, wherefc(n) s/e„. Like (10.88), eq. (10.90) is a most general
formula applicable to clusters of both condensed and gaseous phases in
whatever case of nucleation. In the case of HON (then 1//= 1) of condensed
phases in vapours at constant T, for example, it is easily verified that eq.
(10.91) follows from (10.90) with/(n, r) from (10.3), W(n, t) from (3.39) and
Aji(r) = kT In [p(t)lpe]. Also, with/(n) from (10.9), W(n) from (3.42), Au
from (2.10) and/j* from (4.14), eq. (10.90) leads again to g(n) from (10.69)
in the case of HON of gaseous clusters in their own condensed phase.
10.3 Multimer attachment frequency
As already noted in Chapter 9, the multimer attachment frequency/^, (m >
n) is not of immediate interest for the theory of nucleation, because at the
nucleation stage of the process of first-order phase transition it is negligibly
small in comparison with the monomer attachment frequency fn. For
166 Nucleation: Basic Theory with Applications
completeness, however, we shall briefly consider fnm in the two different
cases of attachment controlled by mobility or growth coalescence of condensed-
phase clusters of size n and m-n which contact to make a new clu ster of size
m > n.
(1) Mobility coalescence
Mobility coalescence is operative when an (m - n)-sized cluster can travel a
certain distance before attaching itself to the considered rc-sized cluster which
may also be mobile. Attachment controlled by mobility coalescence occurs
mainly in fluid old phases such as vapours, liquid solutions and melts. For
instance, in HON of liquids or solids in vapours, in the direct-impingement
regime the number of collisions between spherical clusters of size n and
m - n is given by eq. (10.1), but with m0, R0 and Z\ replaced by the mass
mm _ n, radius R,„ _ „ and concentration Zm_„ of the (m - «)-sized clusters
[Moelwyn-Hughes 1961]. For that reason, since mm_n = m0{m - n) and
Rm „ „ = Ro(tn - n)]/3, in the same way as /„ from (10.2), fnm can be written
down as (m > n)
fnmit) = yajH - nc uf {kT/27Cm0)m[m/(m - n)n\m
[nm + {m - n)l,3]2Zm .n(t) (10.92)
where y/um _ n is the sticking coefficient. As seen, in the m - n = 1 case
(monomer attachment), (10.92) passes into (10.2) in which yn = ynj.
As another example, let us consider HON of condensed phases in liquid
solutions when multimer attachment is due to mobility coalescence controlled
by quasi-stationary volume diffusion. In this case, for spherical clusters,
fnm is given by eq. (10.22) with D, R0 and Z, replaced by Dm _ n> R^ _ n
and Zm _ n. Since the diffusion coefficient Dm_n and the radius Rm _ n of the
(m - n)-sized cluster are related by eq. (10.21), fnm can be represented by the
von Smoluchowski-type formula (e.g. Overbeek [1952]) (m > n)
fnJt) = yn,m-n{toc)xav]?D[n-m + {m- n) " l»]
[nm +{m- n)m]Zm _„(t) (10.93)
where c = (367t)m. Again, in the m - n = 1 limit fnm from this equation turns
into the monomer attachment frequency/,, from (10.23). For analytical work
with eq. (10.93), as done by von Smoluchowski (see, e.g. Overbeek [1952]),
it is possible to make use of the approximation [n " l/3 + (m - n)~ I/3][«l/3
+ (m-«)1/3] = 4.
The last example to consider is multimer attachment in the regime of
mobility coalescence controlled by surface diffusion of cap- or disk-shaped
clusters in, respectively, 3D or 2D HEN on a substrate. In this case/n0I can
be found in the same way as in the case of volume-diffusion control. The
only difference is that the diffusion problem of von Smoluchowski [1916,
1917] has to be solved in the 2D space of the substrate surface, since the
(m - «)-sized clusters diffuse solely along this surface with a coefficient
Ds m_n of surface diffusion [Ruckenstein and Pulvermacher 1973; Kashchiev
1976a]. The relative motion of the (m - n)-sized clusters and a given n-sized
Transition frequencies 167
cluster to which they could be attached can be accounted for by regarding
the real process as a 2D diffusion of material points of concentration Zr(r)
and diffusion coefficient Ds„ + Dsm _ „ towards the periphery of an immobile
cluster with radius R'+ R'm_n of its base. The unknown radially symmetrical
concentration Zr{t) (m~2) of (m - n)-sized clusters at a distance r from the
centre of the n-sized cluster at time t is the solution of the 2D diffusion
equation in a cylindrical coordinate system with origin in the cluster centre.
This equation is eq. (10.28) without the / and Td terms in it, because the (m
- n)-sized clusters are assumed neither to arrive at nor to desorb from the
substrate surface. The initial and boundary conditions are again given by
(10.13), but with Z] and R replaced by Z„_„ and R' + R'm_n, respectively.
The solution of this 2D diffusion problem is known [Carslaw and Jaeger
1959; Crank 1967] and the most essential in it is that, in contrast to the 3D
diffusion towards a sphere (see eq. (10.14)), a steady-state concentration of
(m - n)-mers cannot be established around the considered n-sized cluster on
the substrate. That is why, the incoming surface-diffusion flux jd „ w _ n of
(m - n)-mers is time-dependent even asymptotically: for t» (R' + R'm-„ )2/
4(Z\„ + £>,,„_„) it is given by [Carslaw and Jaeger 1959; Crank 1967]
h.„.m -„(') = 2(Z\„ + Ds „ _ „)Z„ _ „/(# +/?;_„)
In [4(Z\„ + Ds„_„)tl(R' + k;_„)2]. (10.94)
As pointed out by Kashchiev [1976a], however, to a good approximation it
is possible to consider the logarithmic factor in (10.94) as time-independent
and to represent it as
l/ln[4(Ds„ + DSJn_n)tl(R' + R'm.„)2] = e"
where e' is a number between 0.1 and 0.4. The reason to do so is the fact that
over a wide range of x » 1 values the 1/ln x function varies within the limits
indicated (e.g. 1/ln x = 0.43 to 0.11 for* = 10 to 104). From the definition
/nra(') - Yn,in-nJii.n.m-i,(')2rtR' + R'm-n), which is the analogue of (10.26), we
thus find that in quasi-stationary approximation/„m is of the form (m > n)
/™W = 7„,m-„c*(Os.„ + Os,m_ „)Z„_„(t) (10.95)
as long as t > (/?' + R'm-„)2/(Daj, + DSJn_n). For instance, for attachment of
the smallest multimers of, say, m-n< 10 molecules to much larger and thus
much less mobile n-sized clusters, an estimate with (R' + R'm_„) = R' = 1 nm
and (Dsrt + Dnm_n) = Ds„_n = 100 nm2/s shows that the above formula for
fnm is a working one for t > 0.01 s, a condition which is often met in practice.
As in eq. (10.36), in (10.95) the numerical factor c* s iitg = l to 5 is the
respective capture number. An expression for fnm, which parallels (10.95),
was proposed by Venables [1973] and Venables and Price [1975], We note as
well that in the m - n = 1 limit, since yn ] = yn and Dsl = Ds, eq. (10.95) can
be used for determination of the frequency/„ of surface-diffusion-controlled
monomer attachment to mobile 3D or 2D clusters on a substrate (cf. eq.
(10.36) which is derived under the assumption thatDs„s 0).
168 Nucleatioti: Basic Theory with Applications
Equation (10.95) tells us that the dependence of/nn, on n at fixed m - n is
governed by the size dependence of the diffusion coefficient D%n (provided
Yn,,n - n 1S treated as constant with respect to n). In surface diffusion of
clusters on a substrate, however, this dependence is not necessarily expressed
by an inverse proportionality to the cluster radius, as it might be expected in
analogy with eq. (10.21) [Lewis 1970; Massone/a/. 1971; Kern et al. 1971,
1979; Metois et al. 1972, 1974; van der Eerden et al. 1977; Kashchiev
1979a; Jensen 1999]. A rather general formula for this dependence, which
unites many of the known ones, reads [Ruckenstein and Pulvermacher 1973;
Kashchiev 1976a]
D,n = D'n~d (10.96)
where D' > 0 and a' > 0 are free parameters obtainable on the basis of
concrete model considerations concerning the mechanism of surface migration
of the clusters. When Dsn from this formula is substituted into eq. (10.95),
we find that the frequency of surface-diffusion-controlled multimer attachment
in 3D or 2D HEN on a substrate is given by [Kashchiev 1976a] (m > n)
fnmit) = %,m - n^TX [« " ^ + (/»-«)" *] Z„, _ „(t). (1 0.97)
(2) Growth coalescence
An (m - rc)-sized cluster can be attached to a cluster of size n even when the
two clusters are immobile in HON in condensed phases such as solid solutions,
vitrified melts, etc. or in HEN on a substrate. Attachment of this kind is
of interest mainly for supersaturated systems if at least one of the n- and
(m - rc)-sized clusters is of supernucleus size so that it can grow irreversibly
in radial direction, contact the other cluster after some time and thus make
possible the attachment event.
Let us confine the analysis to spherical clusters in HON and to cap- or
disk-shaped clusters in 3D or 2D HEN on a substrate. To determine the
multimer attachment frequency fnm we assume that the clusters are randomly
positioned and consider first HON. Two clusters of n and m - n molecules
and respective radii R and Rm __ n will coalesce when the distance between
their centres becomes equal to R + Rm _n. As in the case of mobility coalescence,
it is convenient to think again of an effective cluster of radius R + Rm _ „,
whose centre coincides with the centre of the n-sized cluster, and of Zm _ „
mathematical points representing the centres of all the (m - «)-mers in unit
volume of the system. These points move radially towards the centre of the
effective cluster with velocity d(R + Rm _ „)Idt which is the sum of the
velocities dR/dt and dRm . Jdt of the n- and (m - «)-sized clusters. An
attachment event can occur only when one of the points reaches the surface
of the effective cluster. Hence, since the attachment frequency fnm is merely
equal to the product of the number 4n(R + Rm _ n)2[d(R + R,n _ „)/dr]Z„, _ „ of
points reaching the cluster surface per unit time and the sticking coefficient
Yn,m - m we find that for condensed-phase clusters (m > n)
/™W = Y,,m-n^W13 + (m-n)m]2[vn(t)n-213
+ vm.H(t)(m-n)-2a]Zm.H(t) (10.98)
Transition frequencies 169
where vn = dn/dt is the cluster growth rate and it is taken into account that R
= (47tlZv0fnnil1 and Rm.„ = (47t/3v0)m(m - n)"3.
Clearly, eq. (10.98) is applicable only to clusters of supernucleus size,
since v„ > 0 only when n > n* (the subnuclei tend to decay so that,
deterministically, vn < 0 for n < n*). However, the applicability of (10.98)
can be extended over the subnucleus size range with the help of the
approximation vn = 0 for n < n*. This results in/nOT = 0 when both n < n* and
m - n < «*, i.e. in ignoring growth-coalescence-controlled contacts between
subnucleus clusters. We note also that as the cluster growth rate v„ is largely
determined by monomer attachment and detachment, in conformity with
(9.21) we have
vn(')=M)-gM- (10.99)
Hence, v„ ~fn for large enough supernucleus clusters, because /„ » gn for
n » «*. In the scope of these approximations, from (10.98) it follows, for
instance, that (m > n)
fnmU) = %.„ _ „"ofn(t)Zm _ „ « (10.100)
for supernucleus clusters of size n » n* (for them v„ - f„) which attach
subnucleus multimers of size m - n < n* (for them vm _„ = 0). From (10.98)
we also find that (m = In)
/™(0 = 7nJn-n^riV„(t)Zm_n(t) (10.101)
for clusters and multimers of nearly equal supernucleus size n~m-n>n*.
From eqs (10.98)-(10.101) we thus see that using concrete expressions for
the frequencies/, and gn of monomer attachment and detachment at different
transport mechanisms (see Sections 10.1 and 10.2), we can obtain also the
multimer attachment frequency /„m controlled by growth coalescence of
immobile clusters. For example, for attachment of large enough supernucleus
multimers (m-n» n*, vm_„ = /m_„) to large enough supernucleus clusters
(n > > «*, u„ =/„) in solid solutions, with the help of c = (36tc)"3 and v0 =
(7tl6)dl, from (10.60) and (10.98) we find that (m > n)
/™W = %.m-*(% + 7m-n)(^/6)d*DC(t)[nln + (m - n)ln?Zm _„(t)
(10.102)
when monomer attachment and, hence, growth of both the multimers and the
clusters is controlled by interface transfer.
Let us now consider randomly positioned caps of wetting angle 6W ^ Jt/2
or disks of monolayer height in HEN of condensed phases on a substrate.
To find the multimer attachment frequency/„m in this case we may repeat the
above analysis by taking into account, however, that now attachment can
occur when one of the moving mathematical points representing the multimers
reaches the periphery (and not the surface) of the effective cluster. The
length of this periphery is 2n(R' + R'm_„) where, in conformity with (3.13)
and (3.52), R' = (3uo/4jr)1,3[sin ew/y/'/3(ew)]n"3 and R'm_„= (3tV47r)"3[sin
ew/y/"3(ew)](m - n)"3 for the caps and R' = R = (ao/7t)"2nm and R'„_„ =
170 Nucleatiott: Basic Theory with Applications
Rm-n = (a^K)v2(m - n)"2 for the disks. Consequently, since now the number
of points reaching the cluster periphery per unit time is determined as 2it(R'
+ R'm-n )[d(/?' + R'm-,t )/dt]Zm_n, the multimer attachment frequency is given
by (m > n)
fnmC) = 7,,,,,, - „(7rY6)"3[sin 9Jyl,\9„)]2u2/3[nm + (m - „)"3J
x [v„(t)n ~ m + v„ _ „(»)(« - n) - m)Zm _ „(r)
(10.103)
for caps with 8W < nl2 in 3D HEN on a substrate and by (m > n)
/„M = 7,,,,,. - M"ln + (m - n)m][v„(t)n ' "2
+ "„,-„«(« -n)-"2]Z„,_„(/) (10.104)
for disks of monolayer height in 2D HEN on a substrate. Like (10.98), eqs
(10.103) and (10.104) are valid only for supernucleus clusters. When both
n < n* and m-n<n*, approximately,/™ = 0. We note that eq. (10.103) is
essentially the formula found by Vincent [1971] in an analysis of growth
coalescence of cap-shaped supernucleus clusters on a substrate. Also, for
clusters and multimers of almost equal supernucleus size n = m - n > n*
eq. (10.104) simplifies to (m = In)
/»,(') = %.„, - MpnW,, -,,(') (10.105)
which, with y/Jm_„ = 1 and n-independent cluster growth rate v„, leads to the
expression used by Venables [1973] and Venables and Price [1975] for the
rate of growth coalescence in thin film formation. Again, in (10.103)-( 10.105)
vn ~fn for supernucleus clusters of size n » n* so that when the growth of
such disk-shaped clusters of monolayer height is controlled, e.g. by monomer
surface diffusion, from (10.36) and (10.104) we find that (m > n » «*)
/™,W = r^-fC'aoDsZMln"2 + (m - «)"2][y„«- "2
+ 7m - „(»» - n) - "21Z„ _ „(/) (10.106)
provided that the monomer surface-diffusion length X^ is greater than the
radii of the coalescing clusters. We recall that under conditions of adsorption
equilibrium Z, in this formula is time-independent and given by (10.37) in
general and by (10.41), (10.45) or (10.47) in particular. It is worth remarking
as well that setting % = %,_„ = 1 and using the approximation [n"2 + (m -
n),l2][n ~ m + (m - n)" "2] = 4 (which parallels that of von Smoluchowski
[1916, 1917]) may be helpful for analytical work with/„„ from (10.106).
When the product of the two bracketed factors in it is approximated by 4, eq.
(10.106) follows directly from (10.105) with v„ replaced by/„ from (10.36).
Summarizing the above results, we see that in all cases considered, in
accordance with (9.34), the multimer attachment frequency/„„ is proportional
to the concentration Zm _ „ (m ~ 3 or m ~2) of (m - n)-mers in the volume of
the old phase or on the substrate. This results from neglecting the multiple
contacts between coalescing clusters and limits the applicability of the derived
Transition frequencies 171
formulae to such periods in the evolution of the process of first-order phase
transition during which the total number of the clusters in the system is
sufficiently small. Comparison of eq. (9.34) with eqs (10.92), (10.93), (10.95),
(10.97), (10.98), (10.100)-(10.106) reveals the physical meaning of the
frequency factor 0)n>m_n in the cases considered above.
10.4 Multimer detachment frequency
Similar to gn, the multimer detachment frequency fm„ (with m> n) can be
determined either independently by model kinetic considerations or with the
help of thermodynamic quantities by invoking the principle of detailed balance.
We shall consider only the latter possibility [Kashchiev 1971, 1974].
The cluster population in a system in equilibrium is described by the
equilibrium cluster size distribution C(n) given by eq. (7.4), (7-6) or (7.7).
The principle of detailed balance requires equality of the total number of
forward and backward transitions in the system per unit time not only between
a given cluster size n and its nearest size m = n + 1, but also between the
same size n and any other size m = n + 2, n + 3, etc. Under time-dependent
conditions, the mathematical expression of this requirement is thus the relation
[Kashchiev 1971, 1974]
fnm(t)C(n, t) =fmn(t)C(m, t) (10.107)
whtrtfnM and/mw are, respectively, the frequencies of attachment of a multimer
of m - n molecules to an n-sized cluster and of detachment of such a multimer
from an m-sized cluster, and C(n, t) is the quasi-equilibrium cluster size
distribution given by eq. (10.85). Equation (10.107) is a generalization of
(10.87) and passes into it in the particular case of nearest-size transitions
when m = n + 1. Graphically, eq. (10.107) implies equality of the lengths of
each two opposing arrows between any two sizes n and m in Fig. 9.2. With
time-independent transition frequencies and equilibrium cluster size
distribution, eq. (10.107) was noted by Andres and Boudart [1965] and used
by Katz et al. [1966] in an analysis of droplet nucleation in associated
vapours.
Now, substituting C(n, t) from (10.85) in (10.107), we find that
fmntt) =fnm(t) exp {[Mm, 0 - W(n, t)VkT{t)} (10.108)
where W(n, t), given by (10.86), is the work to form an n-sized cluster at the
supersaturation existing at time t. When tn> n, this equation is the general
formula for the frequency fmn of detachment of a multimer of m - n molecules
from an m-sized cluster. It shows that fmn is readily obtainable once the
multimer attachment frequency/Mm and the work W for cluster formation are
separately known. Using concrete expressions for/nm from Section 10.3 and
for W from Chapter 3, from (10.108) we can calculate the multimer detachment
frequency in various cases of interest. This was done, e.g. in a study of the
effect of dimer attachment and detachment on the kinetics of droplet nucleation
172 Nucleation: Basic Theory with Applications
in vapours IKashchiev 1976b], Equation (10.108) is a generalization of (10.88)
and turns into it in the case of monomer detachment when m = n + 1.
10.5 General formulae
We have seen hitherto that in each particular case of nucleation we have
concrete expressions for the attachment and detachment frequencies as functions
of material parameters (pe, Ce, D, r/, etc.) and experimentally controllable
variables (p, C, T, etc.). In some cases, however, especially in theoretical
work, we need general formulae for the frequencies /„ and gn of monomer
attachment and detachment, as these are of greatest interest for the theory of
nucleation. We shall now summarize the results obtained in Sections 10.1
and 10.2 by presenting them in a few general formulae applicable to a large
class of cases of one-component nucleation of condensed phases.
We begin with/„. As noted in Section 10.1, it contains always the factor
exp (A/j/kT). In all considered cases in which A/( is not defined by eq. (2.20),
with the help of Aft from (2.8), (2.9), (2.14) and (10.79) used with time-
dependent p, I, C and T, all eqs (10.3)-(10.7), (10.20), (10.23), (10.24),
(10.42), (10.43), (10.46), (10.60), (10.61), (10.63)-(10.66) can be represented
by the single general formula (« = 1, 2, . . .)
AW =/«.„(0 exp [Au(/)ftrW] (10.109)
where/e„ is the monomer attachment frequency at equilibrium, i.e. at A^i -
0. For instance, in the particular cases of nucleation considered in Section
10.2/e„ is given by eqs (10.73)-(10.75) and (10.82) with a concrete 7\r)
dependence introduced in the temperature-dependent parameters in them. It
must be especially emphasized and kept in mind that eq. (10.109) is not
applicable to the important case of Aft controlled by T in accordance with eq.
(2.20): in this case eqs (10.64)-(10.66) suggest a general presentation of/„
in the form (n - 1, 2, . . .)
/„(<) = /o.„M exp {An(t)lkT(t)]. (10.110)
This equation is similar to (10.109), as in it/0„ is also the value off„ at A/j
= 0. However,^) „ does not have the physical meaning of monomer attachment
frequency at equilibrium, since it is determined at the actual temperature T
rather than at the equilibrium temperature 7e. Thus,/U„ is an implicit function
of Afi mainly because of the temperature dependence of the parameters (e.g.
1} from (10.56)) taking part in its determination.
Let us now consider g„. We already have the most general formula for g„:
it is represented by eq. (10.88) or (10.90) each of which is an expression of
the principle of detailed balance. We can use it in conjunction with eq.
(10.109) in order to find a less general formula for g„ holding only in the
scope of the validity of (10.109). Namely, insertion of/„ from (10.109) in
(10.88) and (10.90) and accounting for W(n, r) from (10.86) yields (« = 2, 3,
...)
Transition frequencies 173
f„W=/e,„-iWexp ([*(n, t)-<P(n-l,rWT(0) (10.111)
or, if n is considered as a continuous variable,
g(n, t) = /e(n, /) exp {[d0(n, t)ldn]lkT(t)}. (10.112)
With concrete expressions for the monomer attachment frequency /e „ at
equilibrium and the cluster effective excess energy 0 from (3.87) or (3.89),
eqs (10.111) and (10.112) describe the monomer detachment frequency in all
cases of nucleation of condensed phases when, as emphasized above, Afi is
not defined by eq. (2.20). These equations show that, in contrast to/„, gn
does not depend on the supersaturation in the system as long as A/j is changed
isothermally and <P is A^i-independent. This is not so, however, when Afi is
varied by means of Tas required by (2.20). Indeed, from (10.86), (10.88),
(10.90) and (10.110) we then find that (n = 2, 3, . . .)
g,,(t) =/o.„- i(') exp ([*(„, r) - <P(n - 1, t)]lkT(t)\ (10.113)
when n takes only integer values and that
g(n, t) =f0(n, t) exp [[d<D(n, t)ldn]lkT(t)} (10.114)
when n varies continuously (/0(n, t) =f0,n)- As seen from these equations, g„
is an implicit function of Afj through the temperature T. In the scope of the
capillarity approximation, with 0 from (3.87), eqs (10.112) and (10.114)
lead to eqs (10.72), (10.78), (10.81), (10.89) and (10.91) determining g„ in
particular cases of nucleation. Similarly, when the atomistic equation (3.89)
for 0 is used in (10.111), g„ from (10.80) can be obtained with the aid of
(7.31), (10.37) and (10.82).
Finally, we recall that the general formulae for the frequencies /„,„ of
multimer attachment and detachment are eqs (9.34) and (10.108), respectively.
Chapter 11
11 Nucleation rate
The central problem in the theory of nucleation is the determination of the
nucleation rate J(t) (rrf3 s"1 or m~2 s_1) which is the frequency of appearance
at time t of supemuclei per unit volume or area of the system under
consideration. Accordingly, the general definition of J reads
where £ (m~3 or m~2), given by
MO
?(/)= I Z„(f), (11.2)
» = »•(0+1
is the concentration of supemuclei in the system. In the literature £is often
called concentration of nuclei, but to avoid confusion we shall not use this
term here, because in our terminology nuclei are only the n*-sized clusters.
When the cluster size n is regarded as a continuous variable, eq. (11.2)
becomes
?(») = Tin, t) An. (11.3)
J nit)
As seen, in order to be able to calculate the nucleation rate J we must
know the actual concentration of the differently large clusters of supemucleus
size, i.e. the solution Z„(t) = Z(n, t) of the master equation (9.1) or (9.8).
Alternatively, however, J can be expressed with the help of the flux j*(t) =
J»*(i)(r) =j[n*(t), t] through the nucleus size «* and of the actual concentration
Z*(t) = Z„,w(r) = Z[n*(t), t] of nuclei. Indeed, differentiating Cfj) from (11.3)
with respect to t, accounting that Z[M(t), r] = 0 (cf. eq. (9.33)) and substituting
the result in eq. (11.1) leads to
#)=^-.-^)^. (...4)
J„.„, 3t
it
For a closed system or for an open system in which the supemuclei do not
appear or vanish as a result of non-clustering processes, in eq. (9.1) or (9.8)
we have K„(t) = Ln(t) = 0 for n > n*. This means that we can use the master
equation in the form of eq. (9.12) in order to replace the time derivative dZ/
dt in the above integral by the size derivative - dj/dn and carry out the
integration. Taking also into account that j[M(t), i] = 0, we can thus represent
(11.4) as [Kahlweit 1970; Kashchiev 1974, 1984a]
Nucleation rule 175
dn*{t)
]{t)=j*V)-Z*{t)—^. (11.5)
Equation (11.5) is a general expression for J and is valid for whatever
mechanism of cluster formation provided the supernuclei in the system come
into being or disappear as a result of clustering processes only. Physically, it
shows that the nucleation rate equals the flux;* through the nucleus size n*
solely in the case when n* does not change with time. This case can be
realized when nucleation occurs at time-independent supersaturation A/i: for
instance, from eqs (4.32), (4.35) and (4.38) we see that, classically, dn*/dt<^
- dAfj/dt for either 3D or 2D nucleation of condensed phases so that Ay =
constant is a necessary condition for dn*/dt = 0. At variable A/;, due to the
motion of n* along the size axis, J is thus greater or less than/'* when Ay
increases or decreases with time, respectively. In general, the flux j* is given
by the equations
Mil) ri*(t)
j*(t) = Z £ lC(/)ZM(o -fnm{t)Ut)] (11.6)
n=«*(/) + l m = \
■r\r
Jinn I Ji
j*(')= | \ | [f(m,n,t)Z(m,t)-f(n,m,t)Z(n,t)]dm\dn
(11.7)
which result from (9.6) and (9.11). In the important particular case of cluster
formation by attachment and detachment of monomers only, i.e. according
to the Szilard model, from (9.19) and (9.28) it follows that;* is of the form
j*(t) =f*(t)2*(t) - g„.m+ ,Z„.„) + , (11.8)
j*(0 = -f*(t)C\t) \^-\Z(n, t)l(Xn, t)A (11.9)
Here f*U) =/„.(,,W = f[n*(t), t] and C*(t) = C„.m(t) s C[n*(t), t\ are,
respectively, the frequency of monomer attachment to the nucleus and the
quasi-equilibrium concentration of nuclei.
It must be pointed out that eq. (11.5) is of great practical value, as it shows
that if we know n* and ;'*, we can determine the nucleation rate without
having any information about the concentration f of the supernuclei in the
system. Hence, after obtaining J by means of (11.5), we can use it in eq.
(11.1) in order to find f with the aid of this equation rather than of eq. (11.2)
or (11.3). Indeed, integration of (11.1) under the initial condition f(0) = £,
results in
«») = £,+ [ J(t') it'
Jo
(11.10)
where f0 (m 3 or m 2) is the concentration of all supernuclei at the initial
moment t = 0.
176 Nucleation: Basic Theory with Applications
Going back to eqs (11.1) and (11.5), we note that the so-defined nucleation
rate J relies on the notion of nucleus and on a priori knowledge of the
nucleus size n*. This looks like a self-inconsistency in the definition of J,
since whereas J is a purely kinetic quantity, «* is obtainable by thermodynamic
considerations (see Chapter 4). This self-inconsistency, however, is only a
seeming one: in Section 10.2 it was noted and in Chapter 12 it will be shown
rigorously that the nucleus can be defined and its size n* determined by
entirely kinetic arguments. The real difficulty with using J is of practical
character: experimentally, it is hard to determine n* and thus check reliably
any theoretical prediction based on J from (11.1) or (11.5). The problem with
the experimental determination of the nucleus size is twofold. On the one
hand, morphologically, the nucleus is indistinguishable from any other of the
clusters populating the system. On the other hand, even if we could use non-
morphological methods of distinguishing the nucleus, usually it is built up of
such a small number n* of molecules (typically n* < 100) that the resolution
of most of the experimental techniques used nowadays does not allow
determination of n*. For that reason, what we often need in experiments is
to know also the rate J'(t) (rrf3 s"1 orm~2 s_1)of formation of clusters of size
n > ri, n' being a fixed, previously specified cluster size. For concreteness,
hereafter we shall consider n' as chosen to correspond to the resolution limit
of the particular experimental technique used for detecting the clusters in the
system. This choice of n is convenient, because then all clusters of size n >
n will be detectable and f will be the rate of formation of detectable clusters.
We can therefore call J' detectable nucleation rate (or detectable rate, for
brevity) and, in line with (11.1), define it by
^-(0 = ¾^. (ll.ll)
Here £' (m~3 or m"2) is the concentration of all detectable clusters in the
system and is given by
MiJ)
?'(f)= I Z„(») (11.12)
or, when n is treated as a continuous variable, by
*Mr)
f'(r)= Z(n, t)dn. (11.13)
To express / with the help of the flux/(r) =j„{t) =j(ri, t) through the size
n we can differentiate (11.13) with respect to t and use again the master
equation (9.12) to perform the resulting integration. Substituting the so-
obtained d£7dr in (11.11) and recalling that Z[M(t), t\ = 0 and j[M(t), t] = 0
yields
m =/(')■ (11.14)
This equation is equivalent to (11.11) and tells us that for whatever
Nucleation rate 177
mechanism of cluster formation the detectable nucleation rate / is always
equal to the flux/ through the size ri provided the detectable clusters appear
only as a result of clustering (i.e. if Kn(t) = L„(t) = 0 for n > ri). This is in
contrast with the relation between the nucleation rate J and the flux j* (cf.
eq. (11.5)) and is so because ri is time-independent by definition. As it
should be, when ri* does not change with time, in the particular case of ri =
ri* eqs (11.11)-(11.14) pass into eqs (11.1)-(11.3) and (11.5), respectively.
As to the flux/, it is given by eqs (9.6), (9.11), (9.19), (9.28) or (9.29) upon
setting n = ri. For example, from (9.28) and (9.29) we have
j'(t) = -f'(t)C(ri, t) 1-^ [Z(n, f)/q«. t)]j
= v(ri, t)Z(ri, t) -f'(t) M^z(«.') C'1-15)
where/' =/„■(!) =J{ri, t) is the frequency of monomer attachment to the ri-
sized cluster, and v{ri, t) is the growth rate of this cluster. This formula gives
/ when nucleation occurs according to the Szilard model of cluster formation
solely by monomer attachment and detachment.
It has to be noted that the role of eq. (11.14) is analogous to that of eq.
(11.5). Namely, after obtaining J'with the help of (11.14) we can use it in the
equation
?'(») = ?o+ f J'U')At' (11.16)
Jo
in order to find the concentration of all detectable clusters without resorting
to (11.12) or (11.13). This equation follows from (11.11) and in it fo(nr3 or
m"2) is the concentration of all detectable clusters at the initial moment t =
0. When ri* is time-independent and ri = «*, (11.16) is identical with (11.10).
The above considerations reveal how we can find both the nucleation rate
J and the detectable nucleation rate f when we knew the cluster size distribution
function Zn and its time evolution. Most generally, this function is the solution
of the master equation (9.1) or (9.8). In the theory of nucleation, which is
concerned only with the nucleation stage of the process of first-order phase
transition, Z„(r) or Z(n, t) is obtained by solving the simplified master equation
(9.16) or (9.25) under given initial and boundary conditions and at various
dependences of A^i on t. In the following sections we shall see what are the
solutions of (9.16) or (9.25) when the old phase is in three important states:
the equilibrium, the stationary and the non-stationary ones.
Chapter 12
Equilibrium
The state that is most easily described in the framework of the kinetic approach
to nucleation is the equilibrium one. The system can be in this state when it
is closed for mass exchange (then Kn- L„ =■ 0) provided that the transition
frequencies f„m are time-independent. The equilibrium cluster size distribution
C(«) is easily obtained from the condition
A(') = 0 (12.1)
expressing the principle of detailed balance or microscopic reversibility,
which forbids the flow of local fluxes along the size axis. Finding C{ri) in
this way is particularly important theoretically, because this quantity is
obtainable also by thermodynamic considerations (see Section 7.1). Comparison
of the kinetic and thermodynamic results for C(n) thus makes it possible to
establish the connection between nucleation kinetics and thermodynamics.
In general, the flux jn(t) in (12.1) is given by eq. (9.6). However, since the
concrete mechanism of cluster formation is immaterial for maintaining the
equilibrium,^ in (12.1) can be expressed with the help of the much simpler
equation (9.19) which gives it in the scope of the Szilard model of cluster
formation by attachment and detachment of monomers only. We observe as
well that substitution of jn(t) from (12.1) into the master equation (9.7)
results in dZlt{t)/dt = 0. This means that at equilibrium, as it should be, the
concentration of the clusters of various size does not change with time.
Let us now find kinetically the equilibrium cluster size distribution C(«).
Setting Z„(r) = C(n) and introducing j„ from (9.19) into eq. (12.1) yields
[Farkas 1927] (n = 1,2, . . .)
fnC(n)-gn + {C(n +1) = 0. (12.2)
As can be verified by direct substitution, the solution of this equation reads
[Zinsmeister 1970; Lewis and Halpern 1976] (n = 2, 3, . . .)
C(«) = C,(/,/2. . ./„_ i/8283 ...gn) (12.3)
where C( = C(l) is the equilibrium concentration of monomers.
Equation (12.3) is the sought kinetic formula for the cluster size distribution
at equilibrium. For an undersaturated or saturated old phase, i.e. when Ap
< 0 and the system is in truly stable equilibrium, C(«) from (12.3) represents
the really existing population of clusters in the system. When A/j > 0, however,
as then the old phase is supersaturated and thus in metastable equilibrium,
C(«) from (12.3) is only an imaginary cluster size distribution describing the
cluster population which would set up in the system if truly stable equilibrium
were possible.
Equilibrium 179
From eq. (12.3) we see that, as already noted, C(n) can be time-independent
and hence an equilibrium quantity only when /„ and g„ are not functions of
t, i.e. when T, A/(, etc. are kept constant. If, e.g. T and/or A^i are varied, the
detachment and/or attachment frequencies change with time and C(«) from
(12.3) becomes the quasi-equilibrium cluster size distribution C(n,t) defined
thermodynamically by eq. (10.85) and corresponding to the momentary values
of T and A/j. Indeed, since in this case the above derivation can be repeated,
but with Z„(t) set equal to C(n,r), from (12.1) and (9.19) it follows that the
quasi-equilibrium cluster size distribution is given by the equation [Kashchiev
1974, 1984a] (n = 2, 3, . . .)
On, t) = CmMViU) ■ ■ ./„-iM/S2(')g3(0 • ■ ■ g„(')l (12-4)
This equation is a generalization of eq. (12.3) and its comparison with eq.
(9.31) reveals the physical meaning of the function C„(t) introduced in the
master equation (9.23) of nucleation. The fact that the right-hand sides of
(12.4) and (9.31) coincide means that C„(t) = C(n, t), i.e. that C„(t) in the
master equation (9.23) has the physical significance of quasi-equilibrium
cluster size distribution (see also Section 10.2). The same is true for the
functions C(n, t) and C(n) in the master equations (9.26) and (9.27): these
functions are, respectively, the quasi-equilibrium and equilibrium cluster
size distributions. However, C(n) and C(n, t) from (12.3) and (12.4) cannot
be utilized in eqs (9.27) and (9.26), respectively, because in (12.3) and (12.4)
the cluster size n changes discretely. It is, therefore, necessary to have the
analogues of (12.3) and (12.4) when n is treated as a continuous variable.
Taking the logarithm of both sides of (12.3) and (12.4) and replacing the
sums with integrals, we find that these analogues are of the form [Kashchiev
1969b]
C(n) = C1exp|J In [f(m)lg(m))dm{ (12.5)
C(n,() = C,(r)exp j f ln[/(m,0/g(m,OJdml. (12.6)
Equations (12.3)-(12.6) are the kinetic formulae for the cluster size
distribution at equilibrium or quasi-equilibrium, for they give this distribution
entirely in terms of the transition frequencies /„ and g„ which are kinetic
quantities. On the other hand, C(n) and C(n, t) are given by the statistical
thermodynamic formulae (7.6) and (10.85) in which the work W for cluster
formation is a thermodynamic quantity. We can, therefore, compare C(n)
from (7.6) and (12.3) and C(n,t) from (10.85) and (12.4) in order to establish
the desired connection between nucleation kinetics and thermodynamics.
The result is an expression relating W to the monomer attachment and
detachment frequencies/„ and g„ (n = 2, 3, . . .):
W(n) = Wl+kT I, i«(gjfm.0 (12.7)
180 Nucleation.' Basic Theory with Applications
at equilibrium and
W(n, t) = IV, (/) + kT{t) i^ In [gm(t)/fm_ ,(0] (12.8)
at quasi-equilibrium. When n is regarded as a continuous variable, the above
sums can be replaced by integrals so that (12.7) and (12.8) transform into the
equations
W(n) = Wx + kT f In [g(m)/j{m)] dm (12.9)
W(n, t) = Wft) + k7\t) [ \n [g(m, t)ffim, t)] dm (12.10)
which can be obtained also by comparing (7.6) with (12.5) and (10.85) with
(12.6). We note that eqs (12.7) and (12.9) are similar to those proposed by
Kaischew [1937] and Zinsmeister [1970], and eqs (12.8) and (12.10) are
their generalizations for quasi-equilibrium [Kashchiev 1969b, 1974, 1984a].
As follows from (3.86) and (3.89), in eqs (12.7) and (12.9) the work IV,
= W(\) to form a cluster of size n = 1 is given by
IV, =-^u +A-E, -o,a0 (12.11)
both for HON (with £,=0 and as = 0) and for HEN on a substrate (then E,
*0and C7S;£0). Accordingly, the quasi-equilibrium quantity W^t) is expressed
as
Wx{t) = -A/i (0 + A- ^ - osa0 (12.12)
where the last three summands can also be time-dependent. Equations (12.11)
and (12.12) apply to nucleation of condensed phases, but they can be used
also in the case of gas-phase nucleation if in them X - E, - osa0 is replaced
by its counterpart <Z>( resulting from (3.90) at n = 1. It is worth noting as well
that E\ + osaQz= Edes (see Section 10.1) and that W] is connected with C] and
C0 by eq. (7.5) and by the generalized equation
W&) = kT{t) In [C0(0/C,(0] (12.13)
in the cases of equilibrium and quasi-equilibrium, respectively. This means
that if the concentration C0 of the sites on which clusters can form and the
concentration C, of the clusters of size n = 1 are known independently, the
work W\ to form these smallest clusters of the new phase can be calculated
from (7.5) or (12.13) without making use of (12.11) or (12.12). For instance,
in HEN on a substrate the C]/C0 ratio is merely the coverage of the substrate
by adsorbed molecules. This ratio can therefore be obtained with the help of
adsorption theories (see eqs (10.41), (10.45) and (10.47)) and used in (7.5)
or (12.13) for determination of IV,.
Physically, eqs (12.7)-( 12.10) are kinetic definitions of the work for cluster
formation. They thus make W a meaningful quantity even when nucleation
Equilibrium IS!
is described entirely on kinetic grounds, i.e. without using thermodynamics,
and allow finding this quantity provided the monomer attachment and
detachment frequencies f„ and gn are known independently. From a kinetic
point of view, therefore, the system is in truly stable equilibrium when/„ and
g„ are such that the sum or the integral in (12.7)-(12.10) does not have a
maximum with respect to n. When the transition frequencies result in such a
maximum, the system is in metastable equilibrium, i.e. the old phase is
supersaturated. In this case, as in thermodynamics, we can also speak of
nucleus, meaning the cluster which requires maximum work for its formation.
The nucleus size «*, however, is now determined entirely kinetically by
means of the transition frequencies. Indeed, differentiating W from (12.9) or
(12.10) with respect to n and using the result in eq. (4.1), we find that the
nucleus size n* is the solution of the equation IVolmer 1939; Zinsmeister
1970;Toschev 1973a]
f = g* (12.14)
or its generalized version [Kashchiev 1974, 1984a]
r(t) = g*w (12.15)
where g* = g(n*) or g*(t) = g[n*(t), t] is the frequency of monomer detachment
from the nucleus.
Physically, eq. (12.14) or (12.15) is the kinetic definition of the nucleus
and corresponds to the thermodynamic Gibbs-Thomson equation. It is the
third definition of «* (the other two are eqs (4.1) and (4.4)) and according to
it, kinetically, the nucleus is that particular cluster which gains and loses
monomers with equal frequencies [Volmer and Weber 1926; Kaischew and
Stranski 1934a; Volmer 1939] (see Figs 10.2, 10.4, 10.6 and 10.8). As to the
nucleation work W*, it is expressed kinetically by eqs (12.7)-(12.10) with n
set equal to n*. For instance, for isothermal nucleation at constant
supersaturation
W* = W, + kT Z In (g„/f„_ ,) (12.16)
or
W* = W, + kT f In [gMfltn)] dn (12.17)
when the cluster size is considered as a discrete or continuous variable,
respectively.
It is instructive to see how in two particular cases of nucleation the kinetic
formulae (12.3) and (12.7) yield results known from thermodynamic
considerations. Consider first HON of condensed phases in vapours, for which,
classically, the monomer attachment and detachment frequencies are given
by (10.3) and (10.72). Substituting/„ and gn from these equations in (12.7)
and recalling that in this case Aft is determined by (2.8), we get (a = cujf3)
182 Nucleation: Basic Theory with Applications
W{n) - WY = Z [kT In ipjp) + 2ao/3mi/3]
« -(n - l)Ap + (2aO/3) | m" 1/3 dm
= (- nA^/ + aan2B) - (- Aft + aG).
Thus, W(«) is given by the first bracketed summand in the last equality, i.e.
by eq. (3.39) which we already know from thermodynamics. We note that
the above kinetic result for W(ri) can be obtained by using eq. (12.9) in
combination with/(rc) and g(ri) from (10.3) and (10.91). Also, if we employ
j(n) and g(n) from (10.3) and (10.91) in eq. (12.14) in order to calculate
kinetically the nucleus size «*, we arrive again at the thermodynamic Gibbs-
Thomson equation (4.7). Similarly, as n replaces n - 1 when treated as a
continuous variable, with/(«) and g(n) from (10.9) and (10.69), eq. (12.14)
leads to the Gibbs-Thomson equation (4.16) for gaseous nuclei.
As a second case, let us consider 2D HEN of monolayer condensed-phase
clusters on a substrate which is in contact with vapours. The transition
frequencies/„ and gn are then determined by eqs (10.36) and (10.80) so that
using them in eq. (12.3) and recalling that at equilibrium Z} = C\, we obtain
C(n) = Ci(Ci/Co)n"'exp[(£Si2-£s,i + £S,3-Es2+. .. + ERJl- £SilI_ \)/kT\.
However, since Es] = 0 (see eq. (7.31)), this formula is in fact the Walton
atomistic formula (7.30) derived by statistical thermodynamic methods. Hence,
kinetics and thermodynamics yield again identical results for the equilibrium
cluster size distribution C{ri).
The ability of the above kinetic formulae to reproduce known
thermodynamic relations is a conditio sine qua non for the compatibility of the
kinetic and thermodynamic descriptions of nucleation. What is very important
with these kinetic formulae is also the possibility which they offer for
determination of C{n), W(n), n* and W* without resorting to the specific
surface energy a and other macroscopic parameters employed for characterizing
the smallest clusters of, say, n < 100 molecules. This possibility was used by
Stoyanov et al. [1970] in studying electron-stimulated nucleation in solid
solutions and its impact, e.g. on the formation of the latent image during
the photographic process. The same possibility is exemplified well by the
analysis of Yang and Qiu [1986] devoted to the application of the kinetic
approach to HON of condensed phases in vapours. Following them, let us
take the view that, due to the uncertainty in the validity of eq. (6.17) for
n —> 1, it is more realistic to express gn not by (10.72), but by the formula
gn^An-itxp(^QX/kTnx) (12.18)
where/e,w_i is given by (10.73), and X is the molecular heat of evaporation
or sublimation of the bulk new phase. The disadvantage now is that we must
work with two free parameters, x > 0 and /Jyq > 0, rather than with only one,
Equilibrium 183
the product cu„n(T(the value of this product is practically unknown, because
we do not have independent information about the shape factor c, the molecular
volume v0 and the specific surface energy oof the smallest clusters). Clearly,
eq. (12.18) is a generalization of (10.72): it passes into it when x = 1/3 and
/3Yq = 2 cv™ a/3X. As shown by Yang and Qiu [1986], physically, x and /Jyq
characterize the dependence of the evaporation or sublimation heat of the
cluster on the cluster size n. In this respect, it is worth noting that /JyQ is
analogous to the factor ftST = 0.2 to 0.6 in the Stefan-Skapski-Turnbull
formula
(J= PSSTXlv™ (12.19)
which relates a to the molecular heat A of evaporation, sublimation or fusion
[Stefan 1886; Skapski 1948, 1956; Turnbull 1950] (see also Walton [1969a];
Kern et al. [1979]; Kelton [1991]). A similar relation between a and X is
known to hold also for crystals in solutions [Nielsen and Sohnel 1971; Sangwal
1989; Mersmann 1990; Christoffersen ex al. 19911. With rjfrom (12.19) we
thus find that, classically, /Jyq = (2c/3)ftST where c = (36.T)"3 for spheres,
c - 6 for cubes, etc.
Considering n as a continuous variable (then/e„_ i in (12.18) is replaced
by/e(n)), substituting/, andg„ from (10.3) and (12.18) in (12.14) and accounting
for (2.8) yields [Yang and Qiu 1986]
n* = (/3yQA/A/, )"' (12.20)
which is a generalization of the Gibbs-Thomson equation (4.7). The work W
for cluster formation is also easily found from (12.9) with the help of (2.8),
(10.3) and (12.18):
W(n) = -nA/< + [jSyqA/U -*)]"1-*- (12.21)
This formula is physically relevant for x < 1, since when x> 1, the resulting
W(n) values are negative. Introducing n* from (12.20) into (12.21) leads to
the following expression for the nucleation work W* (0 < x < 1):
W* = (PYQX)[l'l(\lx - l)A/< "*-'. (12.22)
As seen, eqs (12.20)-(12.22) turn into the classical equations (3.39), (4.7)
and (4.8) when x = 1/3 and /Jyq = 2c v™ oBX. We note as well that n* and
W* from (12.20) and (12.22) satisfy the nucleation theorem in the form of
eq. (5.29). Due to this fact and the analogy between /Jyq and /3SST we can
conclude that with appropriately defined A^i (see Chapters 2) and a replaced
by C7ef from (4.42), eqs (12.18), (12.20)-(12.22) are applicable also to one-
component HON or 3D HEN of condensed phases in solutions and melts.
Analysing experimental data, Yang and Qiu [1986] found that x = 1/3 to
0.9 and /3yq = 0.33 to 1.14 are characteristic for HON of water, benzene and
ethanol droplets in vapours at T = 205 to 340 K. For comparison, we note
that, classically, x = 1/3 and ft-Q = 0.33 for spherical water droplets at 293
K (this ft.Q value is calculated from /3vq = 2c u„'3 oBX with X = 7 x 10"20 J
and with v0 and a from Table 3,1),
Chapter 13
Stationary nucleation
A necessary condition for the occurrence of stationary nucleation is the time
independence of the transition frequencies, i.e. the constancy of the temperature
and the supersaturation imposed on the system. Stationary nucleation is the
simplest case of nucleation, as then, by definition,
j„(t) = constant = Js (13.1)
for any «=1,2,.... Accordingly, from the master equation (9.7) and from
eq. (11.5) it then follows that dZJdt = 0 and that
J(t) = J» (13.2)
since d«*/dr = 0. This means that as in the case of equilibrium, the cluster
size distribution during stationary nucleation is also time-independent. In
contrast to equilibrium, however (cf. eqs (12.1) and (13.1)), this cluster size
distribution gives rise to a steady flux 7S > 0 along the size axis. According
to (13.2), this time-independent flux is the rate of nucleation in stationary
regime and finding it in the simplest case of cluster formation by the Szilard
mechanism of successive attachment and detachment of monomers only is
the problem that challenged the pioneers of the nucleation theory [Farkas
1927; Kaischew and Stranski 1934a; Becker and Dbring 1935]. We shall
now determine the stationary cluster size distribution Xn (rrf3 or m~2) and
the stationary nucleation rate Js (m~3 s"1 or m"2 s_1) corresponding to this
simplest mechanism of nucleation.
13.1 Stationary cluster size distribution
The size distribution Xn of the clusters during stationary nucleation which
occurs according to the Szilard model is the solution of the Tunitskii equation
(9.18). Setting Zn(t) = Xn in (9.18) and (9.19), recalling that dX„/dt = 0 and
using (13.1) leads to [Farkas 1927] (n = 1, 2, . . ., M- 1)
A=/A-^ + iXn + 1. (13.3)
In conformity with the assumption that in a system in metastable equilibrium
the actual monomer concentration is equal to the equilibrium one (see Section
10.1), for the stationary monomer concentration X] in (13.3) we can write
X, = C, (13.4)
where Cx is given by (7.5). Also, as the formation of cluster comprising all
M molecules of the old phase is ruled out, we have
Stationary nucleation 185
xM = o.
(13.5)
These two equations play the role of boundary conditions for the unknown
stationary cluster size distribution X„ and allow rewriting eq. (13.3) in the
form
A + 82*2=/, C,
Vs-/A + ^ + ,X„ + 1=0, (n =
^s~/tf- 1-¾ - 1 - 0.
2, 3,..., M- 2) (13.6)
This is a set of A/ - 1 linear algebraic equations in the M - 1 unknowns 7S,
X2, X3, .. ., XM_ i. It shows that Js and X„ can be time-independent and thus
stationary quantities only when the transition frequencies/, and #„ are constant
and when, in addition, the monomer concentration C, does not change with
time. This means that in a closed system stationary nucleation can occur
only until the moment when the concentration of the monomers begins to
diminish appreciably as a result of their 'consumption' by the clusters which
appear and grow in the system. We note as well that the constancy of the
monomer concentration with time is a necessary condition for stationarity of
the nucleation process, because C, enters (13.6) not only explicitly, but also
implicitly through the monomer attachment frequencies /,, /2, . . . (see
Section 10.1).
The algebraic set (13.6) can be solved exactly by various procedures
some of which are quite simple [Becker and Doring 1935; Zinsmeister 1970;
Abraham 1974a]. We shall now solve (13.6) in a standard way by invoking
the known Cramer rule in linear algebra. According to this rule, Xn can be
represented as [Korn and Korn 1961] (n = 2, 3, . . ., M - 1)
X„ = A„/A2. (13.7)
Here A„ and A 2 are the following determinants of order M - 1:
1*2
1-/2
1 0
1 0
1 0
1 0
1 0
1 0
1 0
0 ■
St ■
-/3-
0 ■
0 •
0 ■
0 •
0 •
0 ■
■ 0
• 0
• 0
-/,-
■ 0
■ 0
• 0
■ 0
• 0
/, c,
0
0
0
0
0
0
0
0
0 ■
0
0 ■
0 ■
«11 + 1 ■
-A +1 •
0 ■
0
0
0
• 0
■• 0
■■ 0
•■ 0
0
Jm -
■■ 0
•• 0
0 0
0 0
0 0
0 0
0 0
0 0
iSn-2 0
~/m-2 8m -
0 -Sm
(13.8)
186 Nucleatlon: Basic Theory with Applications
a; =
1 S2
1-/2
1 0
1 0
1 0
1 0
1 0
1 0
1 0
0 ■
gl ■
-/3-
0 ■
0 ■
0 •
0 ■
0 •
0 •
• 0
■ 0
0
•-/„-,
• 0
■ 0
• 0
• 0
■ 0
0
0
0
g„
-/„
0
0
0
0
0
0 ■
0 ■
0
g«11 •
-/„♦!•
0
0
0
0
0
0
0
0
0
'~Jm -
0
0
0 0
0 0
0 0
0 0
0 0
0 0
gM-2 °
~JM - 2 8 M -
0 -/„
(13.9)
In order to evaluate the above determinants we first expand A„ by the
minor of/,C, and then this minor by its minor of -/2. Continuing the latter
kind of expansion until the minor of -fn _ ,, we find that A„ can be written
down as (n = 2, 3, . . . , M - 1)
A„ = (-1)-^,/,/2.../...,4^,
Here the determinant A^ is given by
(13.10)
1 g„
1 -fn
1 0
1 0
1 0
1 0
0
£» + 1 '
-/,-1 ■
0
0
0
0
0
0
■ ' -fu - 3
0
0
0
0
0
gM-2
-/m-2
0
0
0
0
0
gM-l
~/m-
(13.11)
forn = 2, 3,..., M- 1 and, by definition, A'M - 1. This determinant satisfies
the recursion formula (n = 2, 3, . . ., M - 1)
= (-D*
g„A;,+i
(13.12)
"/«/n + l •••/m-
which follows from its expansion by the minors of 1 and gn. We note also
that AJ from (13.9) equals A'„ from (13.11) at n = 2. This means that the
evaluation of A^, with the help of (13.12) is necessary for obtaining both A„
and A2. This evaluation can be done by mathematical induction and, as it
can be verified by substitution in (13.12), the resulting expression for A^
reads (n = 2, 3, . . ., M- 1):
Stationary nucleation 187
a; = H)"-"/»/„+i •■•/«-■
t„ fJ^.-.L
(13.13)
Thus, using this expression for A;, first in (13.10) and then at n = 2, we find
that
A„=H)M(C,//„)/,/2.../m-1
for n = 2, 3, . . ., M- 1 and that
A'2 = (-l)M/2/3.../*-, 1
g«gnt\ ■■•g,r,
"» /»/,H
g2g3 ■
■/,„
«~ hh-f„
(13.14)
(13.15)
Finally, substitution of A„ and A2 from these formulae into (13.7) and some
rearrangement leads to (e.g. Zinsmeister [1970]) (n - 2, 3, . . ., M - 1)
/1/2 ---/.-1
' glgi ■■■£»
1+ z
g2g3 ■ ■ ■ g,„
2 /2/1 ■■•/,»
- hh---L
(13.16)
This equation represents the sought stationary cluster size distribution
when nucleation occurs only by attachment and detachment of monomers to
and from the clusters. It is the exact solution of the master equation (9.18)
under conditions ensuring stationarity of the process and gives X„ in a purely
kinetic form through the transition frequencies f„ and g„. It can be cast into
the following simpler form with the aid of the equilibrium cluster size
distribution C„ = C(n) from (12.3) (e.g. Andres [1969); Abraham [1974a])
(n = 1, 2, . . ., M- 1):
z
1
LC,„
1
/,„c„,
(13.17)
This equation is completely equivalent to (13.16) if in it C„ is determined
kinetically from (12.3) with the help of independently known /„ and g„.
Alternatively, however, if C„ is unknown kinetically because of lack of
information about the detachment frequency gn, we can use (13.17) with C„
expressed thermodynamically in accordance with (7.4). Equation (13.17)
then represents the kinetic quantity X„ as a 'mixture' of kinetics (f„) and
thermodynamics (C„).
As shown by Abraham [1974a], if C„ is introduced in eq. (13.3) from the
very beginning of the calculations, X„ from (13.17) can be obtained in a way
much simpler than that used above. In this respect it is instructive to see how
easily we can arrive at the same result for Xn when we treat n as a continuous
variable and employ the Zeldovich equation (9.27). This equation already
contains C„ and has to be solved in conjunction with the boundary conditions
ISS Nucleation: Basic Theory with Applications
(13.4) and (13.5) which are in fact (9.32) with Zt = C, and (9.33). Setting
Z(n, t) = X(n) results in dX(n)Idt = 0 so that (9.27) simplifies to an ordinary
differential equation of second order. As can be easily verified by direct
substitution, the general solution of this equation reads:
X(n) = C{n)
dm
: 1, c"
I f(m)C(m)
:a
J, f(m)C(m)
The integration constants c' and c" are readily determined from the boundary
conditions (13.4) and (13.5) as
dm
so that [Zeldovich 1942] (1 < n < M)
X(n) = C(.
"'[J, /(m)C(m)J J„
dm
f(m)C(m)'
(13.18
This equation coincides with eq. (13.17) when in the latter the sums are
replaced by integrals and M - 1 is approximated by M. The conclusion is,
therefore, that the Zeldovich equation (9.27) provides an adequate description
of the nucleation kinetics when n is considered as a continuous variable.
In deriving eqs (13.16) and (13.17) or (13.18) we did not make any
presumptions about the physical nature of the transition frequencies/,, and g„
or the equilibrium cluster size distribution C„. This means that these equations
are applicable to any kind of one-component stationary nucleation - HON,
HEN, 3D, 2D, atomistic, etc. With properly determined f„ and gn or C„ (see
Sections 7.1, 10.1, 10.2 and Chapter 12) they can be used for determination
of the stationary cluster size distribution in nucleation of condensed or gaseous
phases in vapours, solutions, melts, etc. However, eqs (13.16) and (13.17) or
(13.18) do not reveal explicitly the dependence of Xn on n. As this dependence
is also of interest, we shall now find it from eq. (13.18) at the expense of
some approximations concerning fin) and C(«), but without any loss of
generality. To do that we note that, due to the exponentially sharp maximum
of \IC{n) at n = n*, in the integrals of (13.18) we can (i) express C{n) by the
approximate formula (7.37), (ii) approximate fin) by its value/* at n = n*,
and (iii) replace M by *». The integration is then readily performed in terms
of the error function defined by (10.15), the result being
dm
f(m)C(m) f
j_r
*c* }„
^"■'"r> dm =
ji"2{1-erf[f)(n -«*)]}
(13.19)
Hence, from (13.18) we find that in all cases of nucleation the stationary
cluster size distribution is the following explicit function of the cluster size
(1 <«<Af):
Stationary nucleation IH9
X{n) = C(n)\-rt\fHX-n*)Y (1320)
We note that, to a good accuracy, this equation can be approximated by
[Nielsen 1964] (1 <n<M)
X(n) = 1 C(«){1 - erf [#« - n*)]) (13.21)
provided that
#n*-l)>l. (13.22)
This is so because when /3 and «* satisfy (13.22), erf [/3( 1 -n*)3 -erf (-<»)
= - 1 with an error of less than 15%. The important point concerning eq.
(13.21) is that it can be used for a reliable determination of X(n) in nearly all
cases of interest, since under typical experimental conditions /3 and n* obey
(13.22). Indeed, with the help of /3 from (7.39) and the approximation n* -
1 ~ «*, we see that (13.22) is equivalent to the requirement W*/kT> 3 which
is almost always met in practice. In any case, even when n* has its limiting
value of 1 and does not satisfy (13.22), since then erf [/3(1 - n*)\ - erf (0)
= 0, X(n) from (13.21) is in error with respect to X(n) from (13.20) by not
more than 50%.
Equation (13.20) or (13.21) unveils a remarkable property of the cluster
size distribution in stationary nucleation: this distribution is not sensitive to
the particular mechanism of monomer attachment and detachment and is a
universal function of C(n), n* and /3 which are quantities obtainable entirely
by thermodynamic considerations (see Chapter 4 and Section 7.1). We could
not expect that a priori, for X{ri) is an essentially kinetic quantity - in
contrast to C(n) which is a thermodynamic characteristic of the imaginary
equilibrium of the metastable system, X(n) describes the actual nucleation
process occurring in the system. The point is, however, that the independence
of X{n) of the transition frequencies is a result of the approximation /(«) ~
f* used to transform eq. (13.18) into (13.20) or (13.21). Hence, eqs (13.20)
and (13.21) merely reveal that the concrete kinetics of cluster growth and
decay by monomer attachment and detachment has virtually no effect on the
cluster size distribution under conditions of stationary nucleation. These
equations show that in stationary regime the cluster concentration X(n) decreases
steadily and strongly with increasing cluster size n and obeys the inequality
X(n) < C(n) for all n > 1. According to (13.21), at n = n*
X* =^C*. (13.23)
This means that the stationary concentration X* = X(n*) of the nuclei is just
half their equilibrium concentration C* [Zeldovich 1942]. Also, from (13.21)
it follows that the concentration of the subnuclei in the system is nearly the
same when the system is under conditions of stationary nucleation and when
it is in the thought state of metastable equilibrium: the X(n)/C(n) ratio is
190 Nucleatum: Basic Theory with Applications
between 1 and 1/2 for 1 < n < n*. This result is of practical value, as it allows
using the equilibrium cluster size distribution C(n) for an approximate
calculation of the actual concentration X(n) of the clusters of subnucleus size
when nucleation proceeds in stationary regime. In other words, with an error
of less than 50% for the subnuclei, X(n) from (13.21) can be approximated
by the following stepwise function:
X(n)IC(n) -.
(\<n <«*)
0, (n* < n < M).
(13.24)
Figure 13.1 depicts the size distribution of water droplets during stationary
HON in vapours at T = 293 K and plpc = 4.5. The solid and dashed curves
represent, respectively, the exact and approximate X(n) dependences (13.17)
and (13.21). For comparison, the corresponding equilibrium cluster size
distribution C(n) from eq. (7.17) is also shown by the dotted curve. The
calculations are done with the help of «*, C(n), /3and/„ from (4.10), (7.17),
(7.39) and (10.3). The y„, u0, o and pe values used are those listed in Table
3.1. Under these conditions we have Afi/kT= 1.5, n* = 53 and ji = 0.069. The
sums in eq. (13.17) are computed with M replaced by the effective value Mef
= 2n* = 106, since the summands with m>n* vanish rapidly with increasing
m. As seen in Fig. 13.1, eq. (13.21) approximates well the exact X(n) dependence
(13.17). We see also that, as n increases, X(n) starts departing appreciably
from C(«) only at the left end n, ~ 40 of the nucleus region whose width A*
~ 26 is illustrated by the double arrow (these values of n{ and A* are obtained
6x101'
O
2x1011
-
-—c*
X*
?1
I
A*
1 '■-•■■'
\ n*
1^
n,
1 10 20 30 40 50
n
70 80 90 100
Fig. 13.1 Stationary size distribution of water droplets during HON in vapours at
T = 293 K and p/pE=4.5: solid curve - eq. (13.17); dashed curve - eq. (13.21). The
dotted curve represents the corresponding equilibrium droplet size distribution,
eq. (7.17), and the double arrow indicates the width of the nucleus region.
Stationary nucleation 191
from eqs (7.41) and (7.43)). At the right end n2 « 66 of the nucleus region,
X(n) is already about 10 times less than C(n) and goes on diminishing for n
> «2 m contrast to the steep rise of C(n).
The variation of the X{ri)IC{n) ratio with n is displayed in Fig. 13.2. The
solid and the dashed curves represent eqs (13.17) and (13.21), respectively,
and correspond to the exact and the approximate X(ri) dependences shown in
Fig. 13.1. We see that the occurrence of the nucleation process in stationary
regime does not disturb the metastable equilibrium of the subnuclei outside
the nucleus region (X(n)/C(n) = 1 for n < nj) and that the X(n)/C(n) ratio is
a function of n practically only in this region. Also, the stationary concentration
of the supernuclei outside the nucleus region is vanishingly small in comparison
with their equilibrium concentration. This means that the nucleus region
appears as the 'bottleneck' of the nucleation process. When the clusters are
subnuclei of size n<nx, they preserve their equilibrium concentration, because
they are in a 'bottle' lying on the size axis with bottom at point n = 1 and
neck between points n = nx and n = n2. Due to fluctuative forward motion
(i.e. growth), some of these clusters can pass through the 'bottleneck' of
length A* = n2 - n i and become supernuclei of size n2. These supernuclei are
then so quickly driven off the 'bottle' as a result of fast deterministic growth
that, instead of assuming its equilibrium value C(n), the actual concentration
X(ri) of all supernuclei outside the 'bottle' remains practically equal to zero.
Hence, as an approximation, forX(n) it is possible to use the simple formula
[Kashchiev 1969a]
c
O
X
1.0
0.8
0.6
0.4
0.2
-
-
A'
n1
I
A*
I
\"2
1 10 20 30 40 50 60 70 80 90 100
n
Fig. 13.2 Size dependence of the X{n)/Qn) ratio for HON of water droplets in
vapours at T = 293 K and p/pe = 4.5: solid curve - eq. (13.17): dashed curve - eq.
(13.21): dash-dotted line - eq. (13.25); dotted line - eq. (13.24).
192 Nucleation: Basic Theory with Applications
X(n)IC(n)= j 1/2-(/3/^
0,
(1 <«<«,)
)(«-«*), (n,<n<n2) (13.25)
(n2 <n<M)
which follows from (13.21) after expanding erf [f}(n - «*)] in a truncated
Taylor series about n - n* and in which «i and «2 are given by (7.41) and
(7.42). Comparing (13.24) and (13.25) we note that the stationary concentration
of the subnuclei is approximated more accurately by the latter equation. The
dotted and the dash-dotted lines in Fig. 13.2 illustrate X(n)IC(n) from (13.24)
and (13.25), respectively.
13.2 Stationary rate of nucleation
The stationary nucleation rate 7S can be obtained from (13.6) without knowing
the stationary cluster size distribution X„. This possibility was used first by
Farkas [1927] and then by Kaischew and Stranski [1934a] and Becker and
Doring [ 1935]. As already noted in Section 13.1, 7S is just one of the M - 1
unknowns in (13.6) which is a set of linear algebraic equations and thus can
be solved exactly. Farkas [1927] and Kaischew and Stranski [1934a] missed
this opportunity and it were only Becker and Doring [1935] who arrived at
the exact formula for 7S. According to the Cramer rule [Korn and Korn
1961], J, from (13.6) is given by
Js = Aj/A'2 (13.26)
where A 2 is specified by (13.9), and Ay is the following determinant of order
M- 1:
(13.27)
/, c,
0
0
0
0
0
gl
-h
0
0
0
0
0 ■
gj ■
-h ■
0
0
0 ■■
0
0
0
" ~/m-3
0
0
0
0
0
gM-2
-JM-2
0
0
0
0
0
gM-t
~/m-
Obviously, for Ay we have
Ay=(-l)MC//2.../M_,
(13.28)
From (13.15), (13.26) and (13.28) we thus find easily that the sought exact
solution of (13.6) for 7S reads
Stationary nucleation 193
A=/,c,
1 + z
S2gj ■■■ Sn
2 /2/3 ■■•/»
(13.29)
This is the known formula of Becker and Doring [1935] which gives the
stationary nucleation rate in a purely kinetic form. What is important to note
is that no thermodynamic notions such as nucleus, nucleation work, etc. are
needed to use (13.29) if the transition frequencies/, and g„ are known from
model kinetic considerations. Owing to eq. (12.3), however, Js from (13.29)
can be written down also as (e.g. Andres [1969]; Abraham [1974a])
1
/„C„
(13.30)
This equation represents Js as a 'mixture' of kinetics (f„) and thermodynamics
(C„) if in it the equilibrium cluster concentration is considered as given by
the thermodynamic formula (7.4). Naturally, eq. (13.30) is eutirely equivalent
to (13.29) when Cn is determined kinetically from (12.3) by means of
independently known/„ and g„.
As shown first by Zeldovich [1942], it is easy to find Js also when n is
considered as a continuous variable. In this case, in accordance with (13.1)
and (11.9), /s is given by the formula
Js~-
-f*C* j£ [X(«)/C(«)]|
(13.31)
which tells us that in order to find /s we have to know the stationary solution
X(n) of the Zeldovich equation (9.27). This solution is represented exactly
by eq. (13.18) so that evaluation of the derivative of the X(n)IC(n) ratio with
respect to n at n - «* and substitution of the result in (13.31) leads to the
exact expression [Zeldovich 1942]
(M _dn__
J, f(n)C(n
(13.32)
Comparison of (13.30) and (13.32) shows that, like X(n), Js is also described
adequately when n is treated as a continuous variable.
Equations (13.29) and (13.30) or (13.32) are the general formulae for the
stationary rate of nucleation when the process occurs by the Szilard mechanism
which involves only nearest-size transitions of the clusters. With concrete n
dependences of/„ and g„ or C„ (see Sections 7.1, 10.1, 10.2, and Chapter 12)
they apply to any kind of stationary nucleation (HON, HEN, 3D, 2D, classical,
atomistic, etc.) of one-component condensed or gaseous phases in vapours-,
solutions, melts, etc. The numerical computation of Js from (13.29) or (13.30)
is not hampered by the typically large value of the total number M of the
molecules in the supersaturated old phase, since the result is not sensitive to
the choice of M provided that M obeys the condition M - Ma »n* (usually,
194 Nucleation: Basic Theory with Applications
it suffices to set Mef = In* or even Mcf = n2 = "* + ?rl/2/2j8) [McDonald 1963;
White 1969; Yang and Qiu 1986]. This reduction of the value of M to the
effective value Mef is possible because of the sharp maximum of the 1//„C„
function at n = n*, which makes negligible the contribution of all summands
with n outside the nucleus region of width A* = i£alfi.
Equations (13.29) and (13.30) or (13.32) are exact and easy to handle
numerically, but they say almost nothing about the properties of the stationary
nucleation rate. The question is, therefore, whether it is possible to represent
them without loss of generality in an approximate, but simpler and physically
more revealing form. We can do that by calculating either the integral in
(13.32) or the derivative in (13.31). In both cases, using eq. (13.19) at n = 1
or eq. (13.20), respectively, we find that
J^lfC* (13.33)
where C* is given by (7.44)-(7.49), and
s = 2j8/7r1/2{l - erf [j8( 1 -«*)!}■ (13-34)
The numerical factor z in which /3 is specified by (7.38) is the so-called
Zeldovich factor. When n* and /3 satisfy (13.22), as is usually the case, erf
[/3(1 - «*)] = - 1 and the Zeldovich factor takes its known form [Zeldovich
1942; Zettlemoyer 1969]
z = p/Ku2 = [(-d2WI<ln%,nJ2itkT]m (13.35)
(actually, Zeldovich [1942] obtained this expression for z, without 7f1/2 in the
denominator). Equation (13.34) is applicable even when the nucleus is so
small (e.g. when n* < 10) that the condition (13.22) is not fulfilled. The
difference between the z values calculated from (13.34) and (13.35), however,
is relatively small: as seen, z from (13.34) is always greater than z from
(13.35), but not more than by a factor of 2. Recalling that ji is a number
between 0.01 and 1 (see Section 7.1), we conclude that, typically, for the
Zeldovich factor we also have 0.01 < z < 1. We note also that, according to
(7.43), z from (13.35) is just the reciprocal of the width A* of the nucleus
region [Andres 1969].
Now, with the help of ji from (7.39) we find that, classically, in the case
of 3D nucleation (either HON or HEN) of condensed phases z from (13.35)
is given by (A^i > 0)
z = (W*/37CkTn*2)m = 3Au2/4(^jtra3CJe3t)"2. (13.36)
This particular form of the Zeldovich factor was found first by Becker and
Doring [1935]. In the case of 2D HEN of condensed phases on foreign (Act
* 0) or own (Acr= 0) substrates, the classical formula for z from (13.35) is
slightly different: with ji from (7.40) it follows that (e.g. Sigsbee [1969]) (A^i
> aaA&)
z = (W*l4KkTn*2)m = (A/< - aelAo)m/(kT)mbK. (13.37)
As seen, z from (13.37) is insignificantly less than z from (13.36) for the
Stationary nuclealion 195
same W*/kT and n* values. This is so because the classical W(n) curve for
2D nucleation is only a little less curved at its maximum than it is in the case
of 3D nucleation. It is worth noting as well that, to a good approximation, eq.
(13.36) can be used for calculation of the Zeldovich factor also in the case
of HON or 3D HEN of gaseous phases. It follows from eqs (3.62) and
(13.35) that in this case the exact expression for z is again of the form of the
first equality in (13.36), but with the divisor 3 replaced by the quantity 2(1
+ p/2p*) in which p* is given by (4.14). As seen, this quantity has values
between 2 and 3 when p is in the range of 0 < p < pc.
The dotted curve in Fig. 13.3 illustrates the A^i dependence of z from
(13.36) for HON of spherical water droplets in vapours at T= 293 K. The
calculation is done with A^i from (2.8), a = (36ffu^)"3, (¾ = a and the v0
and a values from Table 3.1 For comparison, thez(A^i) dependence (13.34)
and the exact variation of z with A;/ are also shown in Fig. 13.3 by the dashed
and solid curves, respectively (in accordance with (13.30) and (13.33), the
exact z(Afj) values are obtained from (13.30) upon dividing its r.h.s. by fC*
and replacing M by Met = 2n*). The numbers at the points on the solid curve
indicate the number «* of water molecules in the nucleus droplet at the
corresponding supersaturation (this number is calculated from (4.38)). We
see that the difference between the exact and the approximate z(A/y)
dependences is negligible in the supersaturation range of Afi/kT < 4 which is
typical for experiments. The good accuracy of the approximation (13.35) for
Au./kT
Fig. 13.3 Supersaturation dependence of the Zeldovich factor z in HON of water
droplets in vapours at T - 293 K: solid curve - exact dependence according to eqs
(13.30) and 113.33); dashed curve-eq. (13.34); dotted curve - eq. (13.36). The
numbers at the circles indicate the nucleus size at the corresponding supersaturation.
196 Nucleation: Basic Theory with Applications
z was noted also, e.g. by McDonald [1963], White [1969] and Kelton et al.
[1983]. Moreover, the weak variation of z with A^/ leads to the important
conclusion that z can be considered as A^/-independent for a given experiment
as long as this is carried out in a sufficiently narrow supersaturation range.
Equation (13.33) is one of the most important results in the theory of
nucleation and with z from (13.35) is well known in the literature [Zettlemoyer
1969]. It is worth noting that with z from (13.35), eq. (13.33) is obtainable
most easily from (13.31) upon using (13.25) for calculating the derivative of
the X{n)/C(n) ratio at n = n*. The physical meaning of eq. (13.33) becomes
clear when with the help of (7.43), (13.20) and (13.34) it is represented in the
equivalent form
J* = J*f*x*- (13.38)
This relation tells us that in the scope of the Szilard model of nucleation
the stationary nucleation rate is equal to an effective forward flux /*Xel
corresponding to the bimolecular 'reaction' of aggregation of a nucleus and
a monomer into a smallest, i.e. (n* + l)-sized, supcrnucleus. The effective
concentration Xe( of nuclei is equal to X* divided by the half-width A*/2 of
the nucleus region and is thus lower than the actual stationary concentration
X* of the nuclei. This reduction of X* is due to the fact that during their
growth some of the nuclei cannot reach the right end «2 - n* + A*/2 of the
nucleus region without shrinking back to subnuclei. This is so, because,
energetically, the supernuclei of size between n* and «2 <ux different by less
than kT from the nuclei (see Section 7.1) and are, therefore, subject to a
thermally activated random 'walk' in the nucleus region. The greater the
length A*/2 of the 'dangerous' distance that has to be 'walked' by the nuclei
before reaching the 'safe' size n2 guaranteeing further irreversible growth,
the smaller the number of the nuclei which become supernuclei of
macroscopically large size n » n2. Thus, effectively, only X*/(A*/2) nuclei
take part in the nucleation process and the nucleation rate Js is merely the
product of/* and X*/(A*/2). Similarly, the Zeldovich factor z in (13.33)
takes account of the loss of the nuclei during their Brownian 'motion' in the
nucleus region: it shows that only 1 out of \lz ~ 1 to 100 nuclei of thought
equilibrium concentration C* actually escapes from the nucleus region and
becomes an irreversibly growing supernucleus. The Zeldovich factor z in
(13.33) thus plays the role of the transmission coefficient in the reaction rate
theory [Gladstone et al. 1941 ]. Equation (13.33) itself is completely analogous
to the formula for the stationary reaction rate in this theory when the nucleus
and the monomer attachment frequency /* are regarded as corresponding,
respectively, to the activated complex of the nucleation 'reaction' and to the
frequency factor kT/hp (hp is the Planck constant).
Now, with the help of C* from (7.44)-(7.49) eq. (13.33) can be given
various equivalent and equally general forms which, physically, emphasize
different aspects of Js. For example, substituting C* from (7.44) in (13.33)
leads to the most often used formula of the nucleation theory (e.g. Hirth and
Pound [1963]; Zettlemoyer [1969])
Stationary nucleation 197
Js = A exp (- W*lkT) (13.39)
which is applicable to any kind of nucleation and in which A (nT3 s_l or
nr2 s~l) is defined by
A = zf*C0. (13.40)
Equation (13.39) is historically the first expression for the nucleation rate.
With undetermined pre-exponential factor A, it was proposed in the seminal
work of Volmer and Weber in 1926. Equation (13.40) unveils the physical
meaning of this factor: it is the product of z, f* and Ca and is thus an
essentially kinetic quantity accounting for the concrete kinetic (/*) and
spatial (C0) peculiarities in each particular case of nucleation. Recalling that,
typically, 0.01 < z < 1,/* = 1 to 1012 s_1 (see Section 10.1) and C0 = 1015 to
1029 m~3 or 10i0 to 1019 m~2 for nucleation in the volume of the old phase or
on a substrate, respectively (see Section 7.1), from eq. (13.40) we find that
in most cases A = 1013 to 104' nrV1 or A = 108 to 1031 nrV. Besides, the
smaller values are indicative for the presence of active centres, seeds, etc. in
the system and/or for a lower frequency of monomer attachment to the
nuclei.
We can now employ the general equations (10.109) and (10.110) to obtain
equally general formulae for A from (13.40). Using (10.109) and (10.110)
with /-independent A;/ and T, we find that in all cases of nucleation of
condensed phases
A=A'eA>"kT. (13.41)
Here the factor A' (m-3 s"1 or m~2 s"1) is given by
A'=zf*Ca (13.42)
when A^i is not controlled by T according to (2.20) and by
A'=zf*C„ (13.43)
when T is the parameter used to change A^i in conformity with (2.20). In
these expressions f* =/e„* and /0 =/0(1, are the values of/* at A^i = 0.
While /e* is physically the frequency with which the nucleus would attach
monomers at equilibrium, /0* does not have this meaning, because it is
determined at the actual temperature Trainer than at the equilibrium temperature
Tc. Thus, f* is a function of A^i through T.
Equation (13.41) shows that, to a good approximation, for nucleation of
condensed phases the kinetic factor A is an exponentially increasing function
of A/y if the supersaturation is varied without changing the temperature T.
This is so, since A' from (13.42) is nearly A^-independent. Indeed, /e is a
function of A,u only because of its proportionality to n*a (provided that the
sticking coefficient is size-independent). The exponent a lies between 0 and
2/3 depending on the concrete mechanism of monomer attachment (see Section
10.1). Hence, the variation of /c with A^ is rather weak. In addition, eqs
(13.36) and (13.37) show that, classically, theZeldovich factor z is also only
weakly changing with A/(. All this means that for practical purposes A' can
/9S Nucleation: Basic Theory with Applications
be treated as a constant with respect to Ap, especially in view of the more
strongly varying exponential factor in (13.41). In some cases A' is even
strictly A^i-independent. This is so, e.g. when nucleation is 3D and f* is
proportional to n*2'3 (cf. eqs (10.3)-(10.5), (10.60) and (10.61)). Then,
according to (4.38) and (13.36), n*m « A^T2andj°< Ap1 so that the product
zf* and thus A' from (13.42) is not a function of A^i if the sticking coefficient
is size-independent. For example, for direct-impingement-controlled HON
of condensed phases in vapours, combining (4.7), (4.8), (10.3) at p = pc,
(13.36) and (13.42) yields
A' = /Va/lS^mo^fPeUo/iWo (13.44)
where y* = %., C0 is given by (7.16) or (7.53), and c = (36^)"3 for spheres,
c = 6 for cubes, etc. We note that, owing to (7.15), {p^VtJkT) = (pqVqIIcT) exp
(-MkT) ~ exp (-MkT), A being the molecular heat of evaporation or sublimation.
We shall keep in mind also that with C0 determined from (7.8), (7.9) or
(7.11), eq. (13.44) can be applied to HEN of cap-shaped droplets or crystals
if in it c is replaced by 367T/(2 + cos8w) (this follows from (13.42) upon
using (10.4) instead of (10.3) for expressing f* and upon taking into account
that, due to (3.56), 4i/(6w)/(l - cos e„)2 = (2 + cos t%)).
The above conclusion about the simple exponential increase of A with Ap
is not valid for the cases of nucleation of condensed phases when Ap is
controlled by means of T. For instance, when Ap is related to T by (2.20), A'
is given by eq. (13.43) in which /0 is a strong function of Ap through T. For
that reason, the overall Ap dependence of A from (13.41) is actually governed
by the dependence of A' on Ap. This can be seen, e.g. in the case of HON of
crystals in melts when Ap is determined from (2.20) and monomer attachment
is controlled by interface transfer. From (4.7), (4.8), (10.64), (13.36) and
(13.43) we find that in this case
A'= 7*[(c3o-A:7")"2/9 ;r3/2d2r/] C0e-MT (13.45)
where X is the molecular heat of crystallization, c = (36;r)"3 for spheres, c =
6 for cubes, etc., and C0 is given by (7.12). Similarly, using (10.65) instead
of (10.64) shows that with c3 = 367T/(2 + cos 8W) and C0 determined from
(7.8), (7.9) or (7.11), eq. (13.45) gives A' in the case of HEN of cap-shaped
nuclei attaching monomers which are not in contact with the substrate. Since
the melt viscosity t) is an exponential function of T (see eq. (10.56)) and
thus, implicitly, of Ap, A' from (13.45) also depends exponentially on Ap.
Recalling that according to (2.20) higher supersaturation requires lower
temperature and that r/ increases with lowering T, from (13.41) and (13.45)
we conclude that A decreases strongly with increasing Ap provided yisAp-
independent. This conclusion has a general validity for nucleation of crystals
in condensed phases when Ap is determined by (2.20) and is in contrast with
the conclusion, based on eqs (13.41) and (13.42), about the exponential
increase of A with Ap in nucleation at isothermally varied supersaturation.
Turning now to nucleation of gaseous phases in their own condensed
phases, we note that the Ap dependence of the kinetic factor A from (13.40)
Stationary nucleatian 199
is found [Blander and Katz 1975; Blander 1979] to be usually negligible
when the supersaturation is changed at constant T. For example, classically,
A is practically A^i-independent (provided y* is constant) when monomer
detachment from the gaseous nucleus is evaporation controlled. Indeed, then
f* is given by eq. (10.9) at n = n* so that, since the nucleus vapour pressure
/>„„. is equal to the pressurep* inside the nucleus, from (4.16), (4.17), (13.36)
and (13.40) we find that [Blander and Katz 1975; Blander 1979]
A = y^allS^mo)"2^ (13.46)
where C0 is specified by (7.12), and c = (36.T)"3 for spherical nuclei. This
expression for A is approximate, since it relies on eq. (13.36) which, as
already noted, gives only approximately z for HON or 3D HEN of gaseous
phases. Like (13.44) and (13.45), with c3 = 36fl/(2 + cos 0W) and C„ determined
from (7.8), (7.9) or (7.11), eq. (13.46) is applicable to HEN of cap-shaped
gaseous nuclei.
In Fig. 13.4 the lines labelled 'droplet', 'crystal' and 'bubble' illustrate
the difference in the 4" dependence of the kinetic factor A for nucleation of
condensed and gaseous phases. The calculation is done for HON of water
droplets in vapours at T - 293 K, of ice crystals in water at atmospheric
pressure and of steam bubbles in water at T - 583 K. The numbers at the
symbols on the lines indicate the number «* of water molecules in the
nucleus droplet, crystal or bubble at the corresponding supersaturation (this
.—V
'co
<
104'
1040
1039
1038
1037
1036
1035
1034
1033
1032
1031
1030
bubble
1000
_ 50O~V
100N
-
-
-
400
\300~
50\
200
10O 50
\crysta1
2u\
100
droplet
10
A|WkT
Fig. 13.4 Supersaturation dependence of the kinetic factor A in HON of steam
bubbles in water at T =583 K (line 'bubble' - eq. (13.46)), of water droplets in
vapours at T = 293 K (line 'droplet'' - eqs (13-41) and (13.44)), and of ice crystals in
water at atmospheric pressure (line 'crystal' - eqs (13.41) and (13.45)). The numbers
at the symbols indicate the nucleus size at the corresponding supersaturation.
200 Nucleation: Basic Theory with Applications
number is evaluated from eqs (4.38) and (4.40) with ael = a). The A values
for the water droplets and the ice crystals are computed from cq. (13.41)
with A' from (13.44) and (13.45), respectively, and for the bubbles eq. (13.46)
is used. The corresponding supersaturation A^i is determined from (2.8),
(2.10) and (2.23). The temperature dependence of the viscosity of water is
taken into account with the help of the formula
r/(T) = 4.47x 10-7 exp [1234.6/(7- 122.3)] (13.47)
obtained by fitting the Vogel-Fulcher equation (10.56) to the r\(T) data of
Skripov and Koverda [1984] (r/ is in Pas, and T is in K). The parameter
values used are c3 = 36;r and those listed in Tables 3.1, 3.2 and 6.2, C0 in
(13.44) is calculated from (7.53), and the a(T) dependence of the ice nuclei
is ignored. Figure 13.4 demonstrates clearly the difference in the course of
A with A^i when the supersaturation is varied isothermally (lines 'droplet'
and 'bubble1) or in accordance with eq. (2.20) (line 'crystal').
Looking back at eq. (13.39), we see that it is an Arrhenius-type formula
with temperature-dependent activation energy W* and pre-exponential factor
A. According to this equation, finding /s splits up into two separate problems:
(i) determining A from (13.40) in each particular case of nucleation by model
kinetic considerations concerning/*, and (ii) obtaining the nucleation work
W* either kinetically from eq. (12.16) or (12.17) (in which n* is calculated
from (12.14)) or thermodynamically from eq. (4.5) or (4.6). For that reason,
a great deal of the theoretical studies on nucleation is and will certainly be
devoted to finding/* and W* in various concrete cases of nucleation. When
the nucleation work is obtained thermodynamically, similar to (13.30) or
(13.32), eq. (13.39) represents Js as a 'mixture' of kinetics (A) and
thermodynamics (W*). We shall now employ some of the results for W*
from Chapter 4 in order to reveal the impact of the exponential factor in eq.
(13.39) on the dependence of Js on A^i.
Let us first consider 3D nucleation of condensed phases. In this case,
regardless of whether the process occurs homo- or heterogeneously, from
(4.39) and (13.39) it follows that, classically [Hirth and Pound 1963;
Zettlemoyer 1969] (A^i > 0)
JS = A exp (-fi/Aji2). (13.48)
Here the kinetic factor A is specified by (13.40) or (13.41), and the
thermodynamic parameter B is defined as
B = 4c3i>02CJ3./27i7" (13.49)
where, according to (4.42), acS - a for HON and aeS < a for HEN.
Equation (13.48) is the known formula of Volmer and Weber [ 1926] which
was derived kinetically also by Farkas [1927], Kaischew and Stranski [1934a]
and Becker and Doring [1935], Using their own general formula represented
above by eq. (13.29), Becker and Doring [1935] were the first to arrive at eq.
(13.40) for A which remained an undetermined parameter in the previous
works. Substitution of A from (13.41) into (13.48) leads to the following
Stationary nucleation 20!
classical formula for Js in the case of 3D nucleation of condensed phases (A/y
2 0):
Js = A' exp (A/ulkT) exp (- BIAfi2). (13.50)
Here A' is given by (13.42) when Aft is not defined by (2.20) and is strictly
or nearly constant with respect to A/i when this is varied isothermally (cf. eq.
(13.44)). In this case, starting from zero at A/j = 0, Js from (13.50) is a
monotonously increasing function of A/j. When A^i is controlled by changing
7 in accordance with (2.20), however, A' is given by (13.43) and decreases
exponentially with increasing A^i. This makes 7S from (13.50) pass through
a maximum after the above-noted initial rise with A^ and then vanish with
further increase of the supersaturation. For instance, for HON of crystals in
melts when A^i is determined by (2.20), from (13.45) and (13.50) we find
that (A^i > 0)
Js = (A"lri) exp (Afi/kT) exp (- B/Afj2) (13.51)
where A" is of the form
A" = Y>[(c,okT)ll2l9t?lldl]C<ie-mT (13.52)
with C0 specified by (7.12). Without the factor exp (AfilkT), eq. (13.51)
parallels the Js(A(i) dependence of Volmer [1939]. We note also that in this
equation the product A"exp (AplkT) with A" from (13.52) is virtually constant
with respect to A/i provided y* is constant, too. Indeed, using the approximation
(2.23) for A/(, we have exp [(A^i - X)lkT\ ~ exp (- Asjk) = constant.
Let us now determine Js for 2D HEN of condensed phases. Again classically,
from (4.33), (4.36) and (13.39) we find that [Hirth and Pound 1963; Zerflemoyer
1969] (Aft >acfAa)
Js = A exp [-fi/(A/< - aefAo-)] (13.53)
for 2D nuclei on a foreign substrate (then Aa* 0) and that [Hirth and Pound
1963; Zettlemoyer 1969] (A^i > 0)
Js = A exp (- BIAfi) (13.54)
for 2D nuclei on their own substrate (then A<7 ~ 0). Here the kinetic factor A
is again given by (13.40) or (13.41), but the thermodynamic parameter B is
different from that for 3D nucleation and reads
B = b2^IAkT. (13.55)
In this formula b = 2(753ef)1/2 for disks, b = 4a'f2 for square prisms, etc., and
acf = a0 when the nuclei are of monolayer height. Using A from (13.41) in
(13.53) and (13.54), we can represent Js more instructively as
7, = A' exp (AnlkT) exp [- fi/(Aji - aefAo-)] (13.56)
Js = A' exp (A/i/W) exp (- fl/Aji) (13.57)
when the 2D nuclei are on foreign or own substrate, respectively. Here A' is
again given by (13.42) or (13.43) when, respectively, Ay is not or is controlled
202 Nucleation: Basic Theory with Applications
by T according to (2.20). Hence, as in the case of 3D nucleation, A' is
practically Afi -independent if A/i is varied isothermally, but is a strong function
of Aft when this is changed by changing T in conformity with (2.20). That is
why the character of the A/i-dependence of Js from (13.56) and (13.57) for
2D nucleation is analogous to that of Js from (13.50) for 3D nucleation. Yet,
there exists a distinction: (i) Js from (13.56) and (13.57) is a weaker function
of Afi than is Js from (13.50) because of the lower power of A/( in the last
exponential factor, and (ii) eq. (13.56) predicts occurrence of 2D HEN on
substrates which are wetted 'better' than completely even when the old
phase is undersaturated, i.e. when A/i < 0. Indeed, for such substrates
Act< 0 and from (13.56) we see that Js > 0 already when Afi > - aef| Aa|. This
kind of 2D nucleation which may be called undersaturation nucleation is
well known experimentally and is closely related to adsorption and wetting
phenomena [Kern et al. 1979; Bienfait 1980; Ebner 19861. For instance, in
electrochemical phase formation it is known as underpotential deposition
(e.g. Budevski et al. [1996]).
Finally, we consider again 3D nucleation (either homo- or heterogeneous),
but of gaseous phases. Again classically, from (4.41) and (13.39) we find
that [Volmer 1939; Hirth and Pound 1963; Skripov 1972; Baidakov 1995]
(0<p<pc)
Js = A exp [- Blip* - p)\ (13.58)
the kinetic and thermodynamic parameters A and B being given by (13.40)
and the expression
B = Ac^al,mkT (13.59)
in which cTef= a for HON and att< fffor HEN. In eq. (13.58), the pressure
p* inside the gaseous nucleus is a function of/7 as required by (4.14) andean
be approximated by the equilibrium pressure pt when peUo/kT « 1. As to A,
it is defined by eq. (13.40) and is practically /7-independent. For example,
when/* is controlled by evaporation, A is given by eq. (13.46) which shows
that, to a good approximation, this quantity is constant with respect to p.
So far, we have obtained concrete formulae for 7S in the scope of the
classical nucleation theory which is expected to run into problems when the
nucleus is not sufficiently large (see Chapter 3). It is therefore important to
find Ji in the limit of n* -» 1 when the atomistic theory of nucleation
provides a more adequate description of the process. Doing that is a simple
matter if C* from (7.47) or (7.49) is introduced in eq. (13.33). Thus, taking
into account eq. (13.40), we get
Js = A exp (-<P*lkT) exp (n*AfilkT) (13.60)
J, = A (C,/C0)°* exp [(«**, - ®*VkT\ (13.61)
where <P* is the nucleus effective excess energy, and C, is given by (7.5).
Naturally, eq. (13.60) is readily obtained also upon substituting W* from
(4.6) into (13.39).
Equations (13.60) and (13.61) are completely equivalent to eq. (13.39)
Stationary nucleation 203
and thus represent Js with the same degree of generality as does (13.39). For
that reason, like eq. (13.39), they are applicable to any kind of one-component
nucleation of condensed or gaseous phases and are valid for any n* = 1,2,
. . . . In the n* —> 1 limit eqs (13.60) and (13.61) are in fact the general
atomistic formulae for the stationary nucleation rate Js. In these formulae n*
and <J>* or n*<P\ - #* appear as free parameters, since in the atomistic theory
they are unknown quantities. As to the kinetic factor A, it is again given by
the general equation (13.40) or its particular forms (e.g. (13.41) and (13.46))
in the various concrete cases of nucleation characterized by different/* and
C0. Atomistically, z = 1 is a good approximation for the Zeldovich factor
[Walton 1962, 1969b], since the nucleus region is quite narrow (A* = 1)
when «*—>!.
Let us now exemplify the application of eqs (13.60) and (13.61) by
considering atomistic nucleation of condensed phases. Then <Pt is of the
form (cf. (3.89))
¢, = A-£|-C7sa0, (13.62)
<tf is specified by (4.49) and A is given by (13.41). Taking into account also
that by virtue of (3.86) and (7.5)
C, = C0 exp [(A/i - <P,)/kT], (13.63)
from (13.60) and (13.61) we get the equivalent expressions
Js = A' exp [(£* - hi* + o%A*)lkT] exp [(«* + l)AM/kT] (13.64)
and
Js = A' (C,/C0)"'+ ' exp ((X + £* - (n* + 1) £,
+ as[A* - (n* + 1) a0]}/kT) (13.65)
where A' is given by (13.42) or (13.43) when, respectively, Aft is not or is
controlled by 7 according to (2.20).
Equations (13.64) and (13.65) are the general atomistic formulae for the
stationary rate of nucleation of condensed phases in the cases of both HON
(then as = 0, £, = 0) and HEN (then as > 0, £, * 0) on foreign (crs * a) or
own (crs = a) substrate (we recall also that £, + (Tsa0= £des and that for 2D
HEN A* = n*a0 if the nuclei are of monolayer height). These equations
contain as particular cases the known atomistic formulae for 7S in condensation
of vapours on substrates [Walton 1962, 1969b] and in electrodeposition
[Milchev etal. 1974; Milchev 1991]. Equation (13.64) tells us that when A^i
is varied without changing T, atomistically, Js is a simple exponentially
increasing function of A^i. This is so, as then A' is A^i-independent, and n*,
E* and A* are constant, too. Also, from eq. (13.65) we see that when A;/ is
varied isothermally, Js°c C"**1. This result is merely a manifestation of the
validity of the Law of Mass Action under non-equilibrium conditions. Indeed,
in the scope of the Szilard model of nucleation Js is the frequency of appearance
of («* + l)-sized clusters which are the smallest supernuclei. As each such
supernucleus is formed by aggregation of n* + 1 monomers, the concentration
204 Nucleation: Basic Theory with Applications
of these supernuclei must be proportional to C"*+1, hence the proportionality
of7s to C" +l. The possible deviations from this proportionality, i.e. from the
Law of Mass Action, are accounted for by the Zeldovich factor in A' (see eq.
(13.42)), but they are usually negligible because of the virtual constancy of
z with respect to C,.
13.3 Particular cases
With appropriately determined supersaturation A^ and kinetic factors A, A'
orA",eqs (13.48), (13.50), (13.51), (13.53), (13.54), (13.56)-(13.58), (13.60),
(13.61), (13.64) and (13.65) give the stationary nucleation rate J, in all cases
of 3D or 2D classical or atomistic HON or HEN occurring according to the
Szilard model. Using these equations, we shall now write down concrete
formulae which are needed to describe the JS(A^) dependence in some of the
most often encountered particular cases of nucleation.
1. 3D nucleation of condensed phases
(a) Classical HON or 3D HEN in vapours or solutions [Volmer 1939; Hirth
and Pound 1963; Zettlemoyer 1969] (5 > 1):
Js = A' S exp (- fi'/ln2 S). (13.66)
This formula follows from (13.50) with the help of (2.8), (2.9), (2.13), (2.14)
and (2.16) and in it, according to (13.49), the thermodynamic parameter B'
is given by
B'= 4^0^,/21^13 (13.67)
where ocf = a for HON and 0¾ < ofor HEN (see eq. (4.42)). The supersaturation
ratio 5 is defined as
S = plpt, S = I/Ie (13.68)
for nucleation in vapours (cf. eq. (7.19)) and as
S = OCc, S = a/ac, S = n/ne (13.69)
for nucleation in solutions. The kinetic factor A' is strictly or practically S-
independent when S is varied isothermally and can be determined from
(13.42) in each concrete case of nucleation. For example, for HON in vapours
A' is given by (13.44). It must be pointed out that in the case of HON in
vapours eq. (13.66) with A' from (13.44) and C0 from (7.53) represents the
self-consistent classical formula for Js,
Js = ■f^ollZif-mtji^ipJkl? v0 exp (cuf olkT) S exp (- Win2 S),
given for spherical nuclei (c3= 36.T), e.g. by Girshick and Chiu [1990], The
correct description of the JS(S) dependence in this case thus requires the
usage of eq. (13.66), and not of the familiar expression for Js [Volmer 1939;
Hirth and Pound 1963; Zettlemoyer 1969] which contains the physically
unacceptable factor S2 in the place of S in (13.66).
Stationary nucleation 205
(b) Atomistic HON or HEN (either 3D or 2D) in vapours or solutions (S > 1):
J, = A' S""+ ' exp [(£* - Xn* + osA*)lkT\. (13.70)
This formula results from (13.64) with the help of (2.8), (2.9), (2.13), (2.14),
(2.16), (13.68) and (13.69) and in it as = 0 for HON and O", > 0 for HEN. The
kinetic factor A' is S-independent when 5 is changed isothermally and is
given by (13.42) with z ~ 1. For instance, for HON in vapours when monomer
attachment is controlled by direct impingement, according to eq. (10.3) at
p = pe and eq. (13.42) we have
A' = zf c^"[pe/(2jtm0it7)"2]n*1'3C0 (13.71)
where Q> is specified by (7.16). We note that, with e = [(1 - cos 8W)/
2v^3(ew)](36^)"3 (cf. eqs (10.3) and (10.4)) and C0 determined from (7.8),
(7.9) or (7.11), eq. (13.71) is valid also for 3D HEN of cap-shaped droplets
or crystals.
(c) Classical HON or 3D HEN in melts [Volmer 1939; Zettlemoyer 1969]
(0<AT<2TcAsJAcrJ:
J5 = [A"lrHD) exp (AstATIkT) exp (- B'lTAT2). (13.72)
This formula is obtained from eq. (13.50) with the aid of (2.23) and (13.43)
and in it, due to (13.49), the thermodynamic parameter B' is given by
B' = Ac^vlalflll kAsl (13.73)
where ae! = a for HON and acS < o" for HEN. The kinetic factor A" is defined
as
A"=zfoT]C0 (13.74)
and is 7-dependent mainly through the factor /0*r; = exp (- iP^kT) where
#1 is specified by (13.62). For instance, in the case of HON under interface-
transfer control, eq. (13.74) passes into (13.52) which shows that then A"<*
exp (- XlkT) (according to (13.62), for HON ¢, = A). It must be pointed out
that not only for HON, but also for HEN of 3D nuclei attaching monomers
which are not in contact with the substrate, we have /9t)K exp (- TJkT) (cf.
eqs (10.64) and (10.65)). For that reason, both in the case of HON and in
such cases of 3D HEN, the product A" exp (AscATIkT) in (13.72) is practically
7-independent:
A" exp (AscATIkT) ~ exp [(AicAT- X)lkT]
= exp (- Aijk) = constant. (13.75)
However, this conclusion is not valid for HEN of 3D nuclei which attach
monomers mostly to their periphery, since these monomers are in contact
with the substrate. Then we have /0*r/ « exp (- <Pt/kT) = exp [(- X + Et +
osa0)/kT] so that the product A" exp (AscATIkT) in (13.72) may depend
essentially on T:
206 Nucleation: Bask Theory with Applications
A" exp (AscAT/kT) °= exp [(AieAT - A + £, + osa0)/kT]
« exp [(£, + osa0)/kT]. (13.76)
This means that in these cases of 3D HEN the T dependence not only of the
melt viscosity 7], but also of this product may contribute to the dependence
of J, from (13.72) on the undercooling AT. The magnitude of the contribution
of this product is controlled by the energy £, + osa0 - £des-
(d) Atomistic HON or HEN (either 3D or 2D) in melts (0 < A3" < 27VW
At'p.e):
Js = [A"lt\(Tj\ exp [- (n* + \)AsJk] exp [(A + £* + a^*)lkT\. (13.77)
This expression follows from (13.64) with the help of (2.23) and (13.43) and
in it as = 0 for HON and as > 0 for HEN. The kinetic factor A" is given by
(13.74) with z, ~ 1 and, as discussed above,
A" = exp (- <PilkT). (13.78)
For example, in the case of HON (then <P\ - X) if monomer attachment is
controlled by interface transfer, according to (10.64) and (13.74) we have
A" = zr*(cvl°n*m/3 mil) (kTtv0) Cae~mT (13.79)
with C0 from (7.12). Thus, in this case the product A" exp (MkT) in (13.77)
can be treated as virtually 7-independent.
(e) Classical electrochemical 3D HEN [Volmer 1939; Vetter 1967] (A<y>> 0):
J, = A' exp (zfioA^kT) exp ( - B'/Aq?). (13.80)
This formula results from (13.50) upon using (2.27) and in it, in line with
(13.49), the thermodynamic parameter B' is given by
B' = Acclaim'i*e2a kT (13.81)
where ccf< a. The kinetic factor A'is defined by (13.42) with C0 from (7.8)
or (7.9). It depends negligibly on the overvoltage A<y> when this is changed
isothermal ly.
(f) Atomistic electrochemical 3D or 2D HEN [Milchev et al. 1974] (A<p > 0):
Js = A' exp [(£* - hi* + a^*)lkT] exp ](n* + 1)^^^71. (13.82)
This equation is obtained from (13.64) with the aid of (2.27) and in it X is the
molecular heat of dissolution. The kinetic factor A' is given by (13.42) with
z ~ 1 and is again Aip-independent when 7 is kept constant while varying A<p.
2. 2D nucleation of condensed phases of monolayer thickness
(a) Classical 2D HEN in vapours or solutions [Volmer 1939; Hirth and
Pound 1963; Zettlemoyer 1969] (In S > a„AolkT):
JS = A' S exp [- B7(ln S - aoAalkT)]. (13.83)
This expression follows from (13.56) with the help of (2.8), (2.9), (2.13),
Stalionaiy micleation 207
(2.14), (2.16), (13.68) and (13.69). In it Act* 0 for a foreign and Act = 0 for
the own substrate. According to (13.55), the thermodynamic parameter S is
of the form
B' = b2r/4k2T2 (13.84)
where b = 2(fi»0)"2 for disks, b = 4a„'2 for square prisms, etc. The kinetic
factor A' is given by (13.42) and is nearly constant with respect to S when the
supersaturation is varied isothermally. For instance, when monomer attachment
to the monolayer nucleus is controlled by surface diffusion, from (10.42) at
I = /e, (13.37) and (13.42) we find that in molecular beam condensation
A' = y*[c*kT(\n S- aoAa/kT)3'2/bK] /L2IeC„ (13.85)
where C0 is given by (7.8) or (7.9).
(b) Atomistic 2D HEN in vapours or solutions (In S > aoAo/kT):
Js = A' S"*+ ' exp ([£* + (asa0 - X)n*\lkT) (13.86)
or, equivalently,
Js = A'S (CyC0)"* exp (E*lkT). (13.87)
These formulae result from (13.64) and (13.65), respectively, upon accounting
for eqs( 13.62), (13.63), (13.68) and (13.69) and for the fact that for monolayer
nuclei A* = n*a„. Equation (13.87) is instructive, as it demonstrates the role
of the concentration C^(S) of adsorbed monomers and of the nucleus 'substrate'
binding energy E* s £s/1, defined by (7.31). Naturally, eq. (13.86) follows
also from (13.70) with A* = n*a0. The kinetic factor A' is given by (13.42)
with z ~ 1 and is independent of S when this is varied isothermally. For
instance, in molecular beam condensation, if surface diffusion controls
monomer attachment, from (10.42) at I = /„ and (13.42) we find that
A'=zr*c*X2IeC0 (13.88)
where C0 is specified by (7.8) or (7.9). We note that if we insert A' from
(13.88) into (13.87) and set z = 1 and 7* = 1, eq. (13.87) becomes the
atomistic formula for Js of Walton [ 1962,1969b]. This is so, because according
to (10.31), (10.41) and (13.68), /L2 = Dsrd, C, = /¾ and S = III,..
(c) Classical 2D HEN of crystals in melts [Volmer 1939; Zettlemoyer 1969]
(a0Ao/Aie <AT< 2reAVAcpe):
Js = \_A"lr\(T)\ exp (&scAT/kT) exp [- B'/T(AT - aoA(j/Ase)]. (13.89)
This equation is obtained from (13.56) upon using (2.23), (13.43) and (13.55)
and in it Act # 0 for a foreign and Act = 0 for the own substrate. The
thermodynamic parameter B is of the form
B' = b2^l4kAss, (13.90)
and the kinetic factor A" is given by (13.74) with C0 from (7.8) or (7.9). As
already discussed, this factor is 7"-dependent according to (13.78) with #, =
208 Nucleation: Basic Theory with Applications
X-E\- osa0, because now the monomers that join the monolayer nuclei are
in contact with the substrate. For that reason, the product A" exp (As^ATIkT)
in (13.89) depends on T in accordance with (13.76). For example, when
monomer attachment to the nucleus periphery is controlled by interface
transfer, from (4.32), (10.66), (13.37) and (13.74) we find that the A"(T)
dependence is of the form
A" = y*[b(kT)m{AseAT~ a0Acj)"2l67cd0v0]C0
exp [(-A + £, + osa0)/kT] (13.91)
where os *■ a, Aa * 0 and as = a, Aa = 0 for foreign and own substrate,
respectively, and E\ + asflo ~ ^des-
(d) Atomistic 2D HEN of crystals in melts (aoAolAs,. < AT < 2TeAsJAcpe):
Js = [A"lr](T)\ exp [- (n* + l)Asc/k] exp [(1 + E* + osaan*)lkT}. (13.92)
This formula follows from (13.64) with the help of (2.23) and (13.43), but it
results also from (13.77) upon accounting that for the monolayer nuclei A*
= n*a0. With as *■ a and as - a it is applicable to nuclei on foreign and own
substrate, respectively. The kinetic factor A" is given by (13.74) with z ~ 1
and C0 from (7.8) or (7.9). This factor depends on T according to (13.78)
with 4>| = X - E\ - O"sfl0 (the monomers which are attached to the nucleus
periphery are in contact with the substrate). For instance, from (10.66) and
(13.74) we find that A" changes with T as
A" = zt(bn*"2kT/3xd0Vo)Co exp [(- X + £, + a%aa)lkT\ (13.93)
in the case of monomer attachment controlled by interface transfer. Equations
(13.77) and (13.92) reveal the physical meaning of the parameters in the
Js(T) dependence used by Paskova and Gutzow [1993].
(e) Classical electrochemical 2D HEN [Vetter 1967] (A<y>> a(,Aa/z,eo)-
Jt = A' exp (ztoAqlkT) exp [- B'/(A<p - aoAolzfiaj\. (13.94)
This equation results from (13.56) upon using (2.27) and (13.55) and in it Aa
* 0 for a foreign and Ao = 0 for the own substrate. The thermodynamic
parameter B' has the form
B' = fcVMZifotr (13.95)
where K is the specific edge energy of the monolayer nucleus. The kinetic
factor A' is specified by (13.42) with C0 from (7.8) or (7.9) and depends
negligibly on A<p when this is varied isothermally.
(f) Atomistic electrochemical 2D HEN (Atp > a0Aalz,e0y-
Js = A' exp ([£* + (asaB - X)n*]/kT) exp [(«* + 1 )z;e0A<p/kT] (13.96)
or, equivalently,
Js = A' exp (z^AajIkT) (C,/C0)"' exp (EjlkT). (13.97)
Stationary nucleation 209
These formulae result from (13.64) and (13.65) with the help of (2.27),
(7.31) at n = n*, (13.62) and (13.63) and in them the kinetic factor A' is given
by (13.42) with z = 1 and C0 from (7.8) or (7.9). This factor is ^^independent
when the overvoltage is changed at constant T. With as *■ a and as = o", eq.
(13.96) applies to 2D HEN of monolayers on foreign and own substrate,
respectively. This equation follows also from (13.82) with A* = a0«* and
parallels the atomistic formula of Milchev et al. [1974]. As to eq. (13.97),
similar to (13.87), it reveals the role of the concentration C] of adsorbed
monomers and of the nucleus 'surface' binding energy Es* = ESJI* defined by
(7.31). It shows how knowledge about the C^Atp) dependence from adsorption
theories can help in determining /s in electrodeposition of monolayers of
condensed phases.
3. 3D nucleation of gaseous phases
(a) Classical HON or 3D HEN of bubbles in own liquid [Volmer 1939; Hirth
and Pound 1963; Skripov 1972; Blander 1979] (0 < A/> < />c):
7S = A exp (-B/A/j2). (13.98)
This equation is an approximation to eq. (13.58) and its accuracy is good
when ptV(JkT« 1. The thermodynamic parameter B is defined by eq. (13.59).
The kinetic factor A is given by eq. (13.40) with C0 from (7.12) for HON and
C0 from (7.8)-(7.11) for HEN and is practically independent of the
underpressure
Aps^-p. (13.99)
This is seen, e.g. from eq. (13.46) which gives A in the case of HON under
conditions of evaporation-controlled monomer detachment from the nucleus.
Expressions for A corresponding to other transport mechanisms are presented,
e.g. by Skripov [1972], Blander and Katz [1975], Blander [1979] and Baidakov
[1995],
(b) Atomistic HON or HEN (either 3D or 2D) of bubbles in own liquid
(0<p<pe):
J, = A exp [(- 0* + p*V* + CJsA*)ft71 exp (- V*plkT). (13.100)
This formula follows from eq. (13.60) upon using <P* from (4.50) and in it
o"s = 0 for HON and os > 0 for HEN. The practically />-independent kinetic
factor A is given by eq. (13.40) with z = 1 and C0 from (7.12) for HON and
from (7.8)-(7.11) for HEN. The quantities p*. V*. A* and tp* in eq. (13.100)
are unknown, but if they are independent of p in a given p range, the atomistic
stationary rate of bubble nucleation will increase exponentially when the
pressure p of the supersaturated liquid is lowered.
Having obtained the above most often needed formulae for Js, we can
now use some of them to represent graphically the dependence of Js on the
experimentally controllable parameters 5, A7 and A/7. The curves labelled
'HON' and 'HEN' in Figs 13.5, 13.6 and 13.7 depict the classical JJ.A/J)
dependence for HON and HEN of water droplets, crystals and bubbles,
210 Nucleation: Basic Theory with Applications
respectively. The droplets are nucleated in vapours at T - 293 K, and the
supersaturated old phase for the bubbles is water at T = 583 K. The ice
crystals are also nucleated in water, but by undercooling it under atmospheric
pressure. It is assumed that the sticking coefficient y* = I and that the nuclei
are spherical in HON and hemispherical in HEN. The foreign incompletely
wetted substrate (the wetting angle 0W is equal to Till) is considered free of
active centres and having C0= 1019 adsorption sites perm2 of its surface (see
eq. (7.8)). The numbers at the points on the Js(An) curves in Figs 13.5-13.7
indicate the number n* of water molecules in the nucleus droplet, crystal or
bubble at the corresponding supersaturation. The «* values are calculated
from the Gibbs-Thomson equations (4.38) and (4.40) with c3 = 36?r, Ay
from (2.8), (2.10) and (2.23) and with cref = a for HON and aef = (1/2)173a
for HEN (the latter equality follows from (4.42) upon accounting that according
to (3.56) y/ = 1/2 at 6W ~ kI2). As already noted many times, the resulting
rather small values of n* imply that, being classical, the J^(A^) curves in
Figs. 13.5-13.7 may have only a qualitative character.
TheJs(5) dependences in Fig. 13.5 are calculated from eqs (13.44), (13.66)
and (13,67) with the help of the parameter values listed in Table 3.1. In eq.
(13.44), c3 = 36;r and C0 = 2.6 x 1027 m"3 are used for HON, the C0 value
following from (7.53). In the case of HEN, in (13.44) c3 is replaced by 18?r
which is the value of 36rc/(2 + cos 0W) at 6W = 7tl2. Figure 13.5 shows the
sharp rise of Js with 5 (i.e. with A/j) both for HON and HEN, which is more
1020
1015
1010
105
1
10-5
io-10
io-15
:
:
i
107
5x10e
0
HEN
HEN
2
HON
3
100/
/200
/HON
50^/^
/100
40. —'
^^^ 20
Fig. 13.5 Stationary nucleation rate as a function of the supersaturation ratio in
HON of spherical and HEN of hemispherical water droplets in vapours at T =293 K
according to eq. (13.66) with A' and B' from (13.44) and (13.67). The numbers at the
circles and triangles indicate the nucleus size at the corresponding supersaturation
ratio.
Stationary nucleation 211
pronounced in the case of HON. This rise is more clearly seen in the inset in
Fig. 13.5 which shows the JS(S) dependence in linear coordinates. We see
that in the case of HON we need S > 2.85 for the nucleation of more than 1
water droplet per m3 per second. In the case of HEN this S value is smaller:
already for S > 2.35 we have Js > 1 droplet per m2 per second. In a narrow
S range (from S = 2.85 to 3.45 for HON and from S = 2.35 to 2.95 for HEN)
there is a tremendous increase of Js with 10 orders of magnitude. This means
that there exists a critical supersaturation A/(c for the nucleation of water
droplets in their vapours: below this supersaturation the process is practically
arrested (see Chapter 31). In other words, in this particular case nucleation
exhibits a threshold behaviour. What is most important, however, is that this
threshold behaviour is a general feature of the nucleation process, especially
when this process occurs homogeneously. It is that feature of the nucleation
process which is the physical reason for which a supersaturated system can
remain for a certain, in some cases practically infinitely long time in metastable
equilibrium (see Chapter 29). For instance, from Fig. 13.5 we read that in the
case of HON Js = 1CT18 m"3 s"1 at S = 2.35. This means that we have to wait
1018 s (which is about the age of our Universe) in order to witness the
homogeneous nucleation of 1 water droplet in 1 m3 of water vapour at the
chosen T = 293 K and p = Spe = 2.35/v
The calculation of the J,(AT) dependences in Fig. 13.6 is done with the
aid of eqs (13.52), (13.72) and (13.73). The parameter values used are those
Fig, 13.6 Stationary nucleation rate as a function of the undercooling in HON of
spherical and HEN of hemispherical ice crystals in water at atmospheric pressure
according to eq. (13.72) with A" and B' from (13.52) and (13.73). The numbers at the
symbols give the nucleus size at the corresponding undercooling, and the dotted line
indicates the undercooling at the glass-transition temperature.
212 Nucleation: Basic Theory with Applications
given in Table 6.2, and the 7)(7) dependence is taken into account by means
of eq. (13.47). In eq. (13.52) c3 = 36^rand C0 from (7.12) are used in the case
of HON. As for the droplets, in the case of HEN c3 in (13.52) is replaced by
1 %K, and C0 is given the value of 1019 nT2 mentioned above. The temperature
dependence of act in (13.73) is ignored and this quantity is calculated from
(4.42) with f = 1 and (1/2)"3 for HON and HEN, respectively. Figure 13.6
shows that Js increases sharply with increasing AT, just as does Js with S in
Fig. 13.5 (the threshold behaviour of the JS(AT) function is more clearly seen
in the inset in Fig. 13.6). At deeper undercoolings, however, Js first slows
down its rise, then passes through a maximum, at the glass-transition
temperature T, = 135 K [Skripov and Koverda 1984] is already less than 1
nucleus per nrper second and finally vanishes at T = 122.3 K, the temperature
at which r/ from (13.47) diverges. The descending branch of the JS(AT) curve
in Fig. 13.6 reflects the strong decrease of the kinetic factor A with AT (see
Fig. 13.4). The important point to remember is that the presence of a maximum
in the JS(AT) dependence and the virtual annulment of Js at a certain temperature
above the absolute zero is a general feature of nucleation of crystals in
condensed phases. The main reason for this feature is that increasing A/i by
lowering T leads to an impeded motion of the molecules in the old phase and
thus to a lower frequency/* of their attachment to the nuclei. The lower/*,
however, the smaller A from (13.40) and, hence, J, from (13.39).
Figure 13.7 depicts the Js(Ap) dependence predicted by eq. (13.98) for
HON and 3D HEN. The calculation is done with A and B from (13.46) and
(13.59) and the parameter values listed in Table 3.2. In eq. (13.46), c3 is
again set equal to 36?r and 18;r for the spherical and the hemispherical
bubbles in the cases of HON and HEN, respectively. Also, C0 is determined
from (7.12) for HON (as already mentioned, for HEN C„ = 10" m-2).
The asf values used in (13.59) are calculated from (4.42) with f = 1 and
(1/2)"3 for HON and HEN, respectively. As seen in Fig. 13.7, the bubble
nucleation rate is a sharply increasing function of the underpressure Ap. This
threshold behaviour of the Js(Ap) function (see also the inset in Fig. 13.7) is
analogous to that of the droplet and crystal nucleation rates with respect to
the supersaturation ratio S and the undercooling AT, respectively. Similar to
Js in Fig. 13.5 and unlike Js in Fig. 13.6, the rate of bubble nucleation is a
monotonously increasing function of the supersaturation, because in the
considered case of isothermal change of Ap the kinetic factor A is a constant.
Figures 13.5, 13.6 and 13.7 demonstrate clearly the strong stimulating
effect that the presence of a foreign substrate in contact with the old phase
can have on the nucleation process. Suppose that the supersaturated gaseous
or liquid water is in a container with volume of 1 m3 and that one of the walls
of this container is a foreign substrate free of active centres and characterized
by wetting angle 6W = Kll. If the substrate area is 1 m2, the stationary rate of
nucleation in this system is directly read from Figs 13.5-13.7. We thus see
that at lower Ap values (S < 4.4, AT< 42 K and Ap < 0.92/>e for the droplets,
the crystals and the bubbles, respectively) HEN is predominant: the HEN
curves in the figures lie above the HON curves. This means that while on the
Stationary nucleation 213
1025
Ap/pe
Fig. 13.7 Stationary nucleation rate as a function of the underpressure in HON of
spherical and HEN of hemispherical steam bubbles in water at T =583 K according
to eq. (13.98) with A and B from (13.46) and (13.59). The numbers at the circles and
triangles indicate the nucleus size at the corresponding underpressure.
substrate thousands or millions of supernuclei will be formed during a period
of 1 second, not a single supernucleus will appear in the volume of the old
phase during the same period when the supersaturation is sufficiently small.
The reason for this is the lower energy cost for the formation of a nucleus on
the substrate (W* for HEN is smaller than W* for HON, because rjef < a) and
the resulting higher equilibrium concentration C* of nuclei on the substrate
(see Fig. 7.2). However, for higher A^i values HON takes over and the
presence of the substrate is not felt any more: in Figs 13.5-13.7 the HON
curves are above the HEN ones for S > 4.4, A7' > 42 K and A/> > 0.92pc for
the droplets, the crystals and the bubbles, respectively. We thus come to a
conclusion of great importance for the experiment: if our aim is to deal with
HON, we must try to impose on the system studied the highest possible
supersaturation. Determining Js over a sufficiently wide Ay range, we can
even register the HEN-to-HON transition occurring with increasing Ay.
Experimental evidence for this transition was presented, e.g. by Butorin and
Skripov [ 1972J in the case of nucleation of ice in water.
Summarizing, we see that the kinetic treatment of nucleation provides
sufficiently general formulae, eqs (13.33), (13.39), (13.60) and (13.61), for
the stationary nucleation rate Js which are the basis for finding this quantity
in various particular cases of interest. The key parameters for Js are the
nucleation work W*, the frequency/* of monomer attachment to the nucleus
and the concentration C0 of sites on which the nuclei can form. In Chapter
4 and Sections 7.1 and 10.1 we have seen how W*, C0 and/* can be expressed
214 Nucleation: Basic Theory with Applications
in terms of quantities characterizing the nucleus and the supersaturated system.
Hence, conceptually, we do not have problems with the physical understanding
of Js beyond the problems concerning W*, C0 and/* themselves. As seen
from the formulae for Js in this section, the real difficulties come up when
we want to use these formulae for quantifying the stationary nucleation rate
in concrete systems under concrete conditions. This is due to our poor or
often no knowledge at all of the values of such important parameters entering
the formulae for J$ as the shape factors c or b, the specific energies <ref, Ao
or k, the binding energies E* or £*, the sticking coefficient y*. the number
#3 of active centres on the substrate and/or in the volume of the old phase,
etc. From a practical point of view, therefore, numerical information about
these parameters and their possible dependence on A/i is what is actually
needed for the predictive ability of the JS(A/*) formulae describing the various
concrete cases of nucleation.
13.4 Concentration of supernuclei
Experimentally, the nucleation rate is not a directly measurable quantity.
However, it can be calculated from available data for various experimental
observables which depend on the rate of the nucleation process. One such
observable is the concentration £ of all supernuclei in the system. We shall
now determine the time dependence of £ during stationary nucleation.
At the beginning of the present chapter we have seen that when nucleation
is stationary, it proceeds at a time-independent stationary rate Js, i.e. then 7(/)
= 7S. Using this result in eq. (11.10), we find easily that in this case £is a
linearly increasing function of time t:
&) = ;0 + Jst. (13.101)
Here C^ is the concentration of all supernuclei at the initial moment t = 0 (see
Section 24.3 and Appendix A3). Typically, it is negligibly small and for that
reason eq. (13.101) is usually employed in the familiar form [Volmer 1939]
£(t) = Jst. (13.102)
Equations (13.101) and (13.102) are exact formulae applicable to whatever
kind of nucleation (HON, HEN, 3D, 2D, classical, atomistic, etc.). They
show that a linear experimentally obtained £(/) dependence is an evidence
for stationary nucleation. Thus, eqs (13.101) and (13.102) are the bases for
a reliable experimental determination of Js and are widely used in studies on
nucleation.
13.5 Comparison with experiment
So far, numerous investigations have demonstrated both the ability and the
failure of the theory of nucleation to describe the experimentally obtained
Stationary nucleation 215
JS(A^/) dependences in various concrete cases of nucleation (e.g. Hirth and
Pound [1963]; Zettlemoyer [1969]; Skripov [1972, 1977]; Wagner and Strey
[1984]; Skripov and Koverda [1984]; Bedanov et al. [1988]; Katz et al.
[1988]; Hung et al. [1989]; Kelton [1991]; Katz [1992]; Viisanen et al.
[1993]; Viisanen and Strey [1994]; Strey et al. [1994]; Baidakov 11995];
Rudek et al. [1996]; Fisk et al [1998]). We shall now see how some of the
Js(Afj) formulae given in Section 13.3 can be used for analysing data for the
supersaturation dependence of the stationary rate of nucleation in vapours,
solutions and melts.
The usage of a concrete /S(A^) formula for interpretation of a given set of
experimental Js(A/i) data is justified if we have a priori knowledge about the
kind of the nucleation process (HON, 3D HEN, 2D HEN, etc.). The problem
is, however, that such a knowledge is not always available. When this is the
case, the results obtained by fitting the theoretical to the experimental Js(Afj)
dependence remain questionable. As an example, let us consider nucleation
of condensed phases. When it is known for sure that certain Js(A/i) data
correspond to HON, classically, eq. (13.48) can be employed for describing
the observed dependence of /s on A/;. Alternatively, if A and B in (13.48) are
unknown, the data are usually plotted in In Js-vs-(l/A^/2) or (1/TA^/2)
coordinates, because the classical theory suggests fitting by a straight line
with equation
ln/s = ln A-B(\lkii2) or In Js = In A - B'Asl{\iTA^i2) (13.103)
when Afi is varied isothermally or by means of T according to (2.20) (Br is
defined by (13.73)). From the best fit, the two free parameters A and B or
B'Asl are determined, and then B or B'Asl is used in (13.49) or (13.73) for
calculation of the specific surface energy a Unfortunately, eq. (13.103)
applies also to 3D HEN of condensed phases. Hence, what is actually calculated
with the help of the so-determined B or B'Asl value may be <7ef rather than
<T if the experimental conditions do not ensure HON (as it is almost always
the case, especially with nucleation in condensed old phases). However, if
the process is not HON, but HEN on a foreign substrate, we run into the
problem about the nucleus dimensionality. Indeed, while for 3D HEN eq.
(13.103) is operational, for 2D HEN it is not: on the basis of (13.53), (13.55)
and (13.90), the classical theory predicts a linearization of the Js(Ay)
dependence in In /s-vs-[l/(A^ - aefAo)] or [UT(Afj - <3efA<*)] coordinates
according to the equation
In Js = In A - B[l/{Av - aefAo)] or
In Js = In A - B'Ase[VT(Av -ae{Ao)] (13.104)
when Afi is varied isothermally or via 7 in conformity with (2.20). Here A,
B or B'Ase and <3cf Ac are free parameters and the best-fit value of B or B'Asc
can be used in (13.55) or (13.90) for calculation of the specific edge energy
k. It is worth keeping in mind that the interpretation of Js{A^t) data is
unambiguous only for HEN on the own substrate, since the process can only
216 Nucleation: Basic Theory with Application*
be 2D. Moreover, Act = 0 by definition so that, classically, the experimental
Js(Afi) dependence can be linearized in In Js-vs-(l/A/() or(\/TAfi) coordinates:
ln/s=lnA-fi(l/A^) or In J,= In A - B'Asc(llTAfj). (13.105)
This formula results from (13.54) or (13.104) and, like (13.103), contains
two free parameters: A and B or B'Asc.
It is important to note that, having found the thermodynamic parameters
B, B'Asl or B'Asc by fitting eq. (13.103), (13.104) or (13.105) to available
experimental data, we can determine easily the nucleus size n*. Indeed,
according to eqs (4.38), (4.39), (13.49) and (13.73), we have
n* = IBkTIA^ or n* = 2kB'As*/Afi3 (13.106)
for HON or 3D HEN of condensed phases when A^i is varied isothermally or
by means of T as required by (2.20). Similarly, from eqs (4.32), (4.33),
(13.55) and (13.90) it follows that
n* = BkTI(Afi - a^Aaf or n* = kB'AsJ(Afi - attAo)2 (13.107)
for 2D HEN on foreign (Act * 0) or own (Ac = 0) substrate in the respective
cases of variation of A^.
It must be pointed out, however, that even if it is known unambiguously
that the process studied is HON, 3D HEN or 2D HEN, the application of eq.
(13.103), (13.104) or (13.105) is restricted by the questionable validity of
the classical nucleation theory in the n* —» 1 limit. In this limit, the analysis
of experimental 7s(A/j) data requires the usage of eq. (13.60) of the atomistic
theory. Irrespective of the kind of the nucleation process, when «* is A/v-
independent, for nucleation of condensed phases this theory suggests a
linearization of the Js(A^i) dependence in In Js-vs-A^ coordinates when A/j
is varied isothermally. Indeed, from (13.64) there results
In Js = [In A' + (E* - In* + a,A*)/kT] + [(n* + l)/W]A/j (13.108)
where the two bracketed terms are free parameters. As seen, the nucleus size
n* can be calculated easily from the slope of the straight line. We note,
however, that eq. (13.108) predicts breaks in this straight line when, as
discussed in Section 4.4, n* changes abruptly its value in the A/( range
studied (see Fig. 4.1b).
The above method of using the classical nucleation theory for analysing
experimental 7s(A^i) data ignores the dependence of the kinetic factor A on
A/j, which is predicted by eqs (13.41 )-(13.43). Nevertheless, this method is
acceptable for nucleation experiments carried out in a sufficiently narrow A^
range (which is often the case), since then the variation of A with A^i is
usually much weaker than that of the W*-controlled exponential factor in eq.
(13.39). Clearly, a more careful analysis of a given experimental J.,(Afi)
dependence requires accounting of the fact that A may change appreciably
with A^i, especially in a wider A^ range and/or when Afj is varied by means
of T. For instance, for nucleation in vapours or solutions, it follows from
(13.66) and (13.83) that accounting for the A(A/j) dependence can be achieved
Stationary nucleation 217
with a sufficient accuracy by using In {JJS) and In A' instead of In Js and In
A in eqs (13.103)-(13.105). Similarly, in these equations In Js and In A have
to be replaced by In (TjJs) and In A" (see eqs (13.72) and (13.89)) in a more
accurate analysis of /s(A/j) data for nucleation in melts when Afj is varied
according to (2.20).
The above considerations concern nucleation of condensed phases. When
the nucleating phase is gaseous, the convenient experimental variable is the
underpressure A/? rather than the supersaturation A^i. In conformity with eq.
(13.98), classically, linearization of experimental J,(Ap) data both for HON
and for 3D HEN of bubbles in own liquid can be obtained in In 7s-vs-( l/A/>2)
coordinates provided Ap is varied isothermally:
lnJs = lnA-fi(l/A/>2). (13.109)
With the aid of the best-fit value of B it is possible to calculate ac[ and n* by
using, respectively, eq. (13.59) and the approximate relation
n* = 2Bp<./Ap3 (13.110)
which follows from (4.40) and (13.59) upon setting p* = pe. When the
condition pcviJkT « 1 is not satisfied, in (13.109) and (13.110) pc and Ap
must be replaced by p* and the difference p* - p, respectively, where p* is
a function of p according to (4.14).
Figure 13.8 exhibits the experimental Js(Afi) data of Viisanen etal. [1993]
for HON of water droplets (the circles) at T = 259 K and of Viisanen and
Strey [1994] and Strey et al. [1994] for HON of n-butanol droplets (the
triangles) at T = 240 K. The solid and the open symbols refer to droplets
nucleated in the presence of, respectively, argon and xenon as inert carrier
gas. As seen, In /s is a linear function of l/(ln S)2, i.e. of 1/A/i2. This is in
accordance with the first of eqs (13.103) and, hence, with the classical theory,
since in the narrow S range studied A = A'S~ constant (cf. eq. (13.66)). From
the best-fit slope and intercept of the straight lines drawn in Fig. 13.8 we find
BI(kT)1 =185.5 and 231.5 andA = 1.2 X 1032 and 7.5 X 102' rrrV1 for water
and n-butanol, respectively. Using these BI(kT)2 values in (13.49), c3 = 36¾
(spherical droplets), oel = a (HON), v0 = 0.03 nm3 for water and v0 = 0.145
nm3 for n-butanol results in, respectively, a= 82.5 and 28.8 mj/m2 for water
and n-butanol. These (rvalues compare with the known values of the specific
surface energy of water at T= 259 K (a= 77.6 mJ/m2 [Viisanen et al. 1993])
and of n-butanol at T = 240 K (a= 28.7 mj/m2 [Viisanen and Strey 1994]).
The theoretically predicted A values, however, do not agree with the
experimentally obtained ones. Indeed, assuming direct-impingement control
with/1 = 1, with the help of C0 from (7.53), m0 = 3 x 10~26 kg and pe = 204
Pa for water at T= 259 K [Viisanen etal. 1993] and mB= 1.2x 10~25 kg and
pc = 4.52 Pa for n-butanol at T = 240 K [Viisanen and Strey 1994], from
(13.44) it follows that A' = 1.8 x 1033 and 6.1 x 1030 m"3 s_1 for water and n-
butanol, respectively. Using the mid-range S = 8 for water and S = 11 for n-
butanol, we thus find that, theoretically, A = 1.4 x 1034 and 6.7 x 1031 trf3 s"1
for water and n-butanol, respectively. These A values are about two orders of
218 Nucleation: Basic Theory with Applications
j i : L-J> 1 j 1 i i i L_^ i i
0.12 0.16 0.20 0.24 0.28 0.32
1 / ln2S
Fig. 13.8 Dependence of the stationary nucleation rate on the supersaturution ratio
in HON of droplets in vapours: circles - data for water at T = 259 K [Viisanen el al.
1993]; triangles - data for n-butanol al T = 240 K [Viisanen and Strey 1994; Strey et
al. 1994]; solid lines - best fit according to the first eq. (13.103); dashed lines-eq.
(13.66) with A' and B' from (13.44) and (13.67). The solid and open symbols refer to
nucleation in the presence of respectively, argon and xenon as inert carrier gas.
magnitude higher than the experimental ones. The discrepancy remains (but
with respect to A') if eq. (13.66) is used for a more accurate analysis of the
Js(A^i) data in In (7s/5)-vs-l/(ln S)2 coordinates. The disagreement between
theory and experiment is visualized in Fig. 13.8 in which the dashed lines
display the 7S(S) dependence predicted by eq. (13.66) in conjunction with
(13.44) and (13.67). As established by Viisanen et al. [1993], Viisanen and
Strey [1994] and Strey et al. [1994], this disagreement persists in the whole
temperature range studied (217 to 265 K) though with different magnitude.
Possibly, theory and experiment might be reconciled by considering other
mechanisms of monomer attachment to the nucleus and/or by describing
more adequately the nucleation work W* for the rather small nucleus droplets
involved in the process (with the above B/(kT)2 values, from the first of eqs
(13.106) we find n* = 36 to 48 for the water nucleus and n* = 28 to 35 for
the n-butanol one in the S ranges studied). The possibility for having y* < 1
should not be underestimated either.
The symbols in Fig. 13.9 show experimental Js(A^i) data for nucleation of
ice in water (the circles and the triangles [Butorin and Skripov 1972]) and
for nucleation of gallium /3-phase crystals in gallium melt (the squares [Skripov
etal. 1971]). In conformity with the second of eqs (13.103) and, hence, with
the classical theory for HON or 3D HEN, In 7S is a linear function of 1/TAT2,
since in the narrow A7" range studied A = (A"lr\) exp (AscAT/kT) ~ constant
Stationary nucleation 219
-//-
,„ »7/ ' "■ ■ ''
0.55 0.60 0.65 0.70 3.0 3.2 3.4 3.6 3.8
1/T(AT)2(10"6K"3)
Fig. 13.9 Dependence of the stationary nucleation rate on the undercooling: circles
and triangles - data for ice crystals in water [Butorin and Skripov 1972 J: squares -
data for gallium ji-phase crystals in gallium melt [Skripov el al. 1971 J; lines - best fit
according to the second eq. (13.103). The triangles represent data obtained by cooling
at constant rates of 0.05 and 0.15 K/s.
(cf. eq. (13.72)) and cref is practically T-independent. The slopes and the
intercepts of the best-fit straight lines drawn in Fig. 13.9 are B' = 1.81 x 107
K3 and In (A/nrV) = 83.8 for ice and IT = 9.54 x 107 K3 and In (A/m"3 s_1)
= 87.2 for gallium. Using these B' values in eq. (13.73) with u0 = 0.0326 nm3
and Asc - 2.63k for ice and with L70 = 0.0196 nm3 and Ase = 132k for gallium
yields (% = 26.4 mJ/m2 for ice in water and rjef = 40.8 mj/m2 for gallium jS-
phase crystals in own melt (these numbers differ slightly from 28.7 of Butorin
and Skripov [1972] and 40.4 of Skripov el al. [1971]). Accordingly, A = 2.5
x 1036 nr3 s~' for ice and A = 7.4 x 1037 m"3 s"1 for gallium. The high values
of the specific surface energy and of the kinetic factor A are indicative for
HON. Indeed, theoretically, for HON of ice A = 1.7 x 1036 m-3 s_1 in the case
of interface-transfer control (this value follows from eq. (13.51) with Afj, r/
and A" calculated from (2.23), (13.47) and (13.52) at T = 240 K and the
parameter values listed in Table 6.2). With the above B" values, from the
second of eqs (13.106) we find that n* = 252 to 349 for the ice nucleus and
that n* = 122 to 156 for the gallium crystal nucleus at the experimentally
investigated undercoolings.
Figure 13.10 represents the 7,(A/j) data of Milchev and Vassileva [1980]
for electrochemical nucleation of silver crystals on a glassy carbon electrode
in aqueous solution of AgN03 at T = 333 K. Microscopic observations
[Milchev et al. 1980] reveal that the supernucleus crystals on the electrode
are cap shaped so that for the analysis of the data it seems natural to employ
30
28
\i> 26
E
---- 24
c 22
20
220 Nucleation: Basic Theory with Applications
0.10
(Ap)"2 (V2)
(A(p+0.04)"1 (V1)
Fig. 13.10 Dependence of the stationary nucleation rate on the overvoltage in HEN
on a foreign substrate: circles - data of Milchev and Vassileva 11980 J for silver
crystals on a glassy carbon electrode in aqueous solution ofAgN03 al T = 333 K;
lines - best fit according to (a) the first eq. (13.103) for 3D HEN, (b) the first eq.
(13.104) for 2D HEN, and (c) eq. (!3.10H)for atomistic nucleation.
the first of eqs (13.103) describing 3D HEN. The circles in Fig. 13.10a show
that In Js does not obey the linear dependence on 1/A(p2, i.e. on l/A/<2,
predicted by the classical theory of 3D HEN. This finding led Milchev and
Vassileva [1980] to using the first of eqs (13.104) which applies to 2D HEN
and in which A^i is again related to the overvoltage Aipvia eq. (2.27). Figure
13.10b demonstrates that the experimental Js(Ap) dependence (the circles)
can be linearized in In 7s-vs-l/(A<y>- aifAa/z;e0) coordinates (cf. eq. (13.94)).
The best-fit straight line corresponds to aef AoVz^o = -40 mV [Milchev and
Vassileva 1980], This line is drawn in Fig. 13.10b and has slope and intercept
which, according to the first of eqs (13.104), give Blz\er, = 1.04 V aud A =1.8
x 10'4 nr2 s'. With z, = 1 and at, = 0.083 nm2, from (13.55) we thus get K
= 54 pj/m for the specific edge energy of the 2D silver nucleus provided it
is a disk of monatomic thickness. The rather low value of A is an indication
for nucleation on active centres, and the value of K is reasonable though
higher than that obtained by Bostanov et al. [1983] for electrochemical 2D
HEN of silver on own substrate (see below). However, using B/zfir, - 1.04 V
and aefAo/zje0 = -40 mV in the first of eqs (13.107) results in n* = 2 to 7
silver atoms in the nucleus in the studied overvoltage range from 25 to
90 mV [Milchev and Vassileva 1980], This small nucleus size means that
the application of the classical theory to this particular case of 2D HEN is
questionable and calls for employment of the atomistic theory, i.e. of eq.
(13.108), for analysis of the experimental data. As shown by Milchev and
Stationary nucleation 221
Vassileva [1980], graphically, the dependence of In Js on A<y> is a broken
straight line (Fig. 13.10c). The slopes of this line yield n* = 4.5 ±0.2 for Aip
= 25 to 50 mV and n* - 1.2 ± 0.1 for A<y> = 50 to 90 mV (these n* values are
calculated from eq.( 13.108) with A^i from (2.27) and z\ = 1). The conclusion
of Milchev and Vassileva [1980] is that the silver nucleus changes dis-
continuously its size from 1 to 4 atoms in the Aq> range studied. Other
experiments on electrochemical HEN on foreign substrate have also produced
evidence for the applicability of the atomistic theory to this case of nucleation
(e.g. Milchev [1991]).
Nucleation on own substrate is necessarily 2D HEN and is advantageous
for theoretical description, since the kind of the process is unambiguously
known. Thus, in this case the interpretation of ^(A^) data is facilitated - for
instance, classically, this has to be done with the help of eqs (13.105). The
circles in Fig. 13.11 exhibit the experimental /S(A^) dependence obtained by
Bostanov et al. [1983] for electrochemical nucleation of silver monolayers
on the atomically smooth (100) face of a silver single crystal in aqueous
solution of AgN03 at T = 318 K. We see that, as predicted by the first of eqs
(13.105), i.e. by the classical theory. In Js is a linear function of i/A/<°= 1/A<y>.
The slope and the intercept of the best-fit straight line drawn in Fig. 13.11
are Blz^ = 335 mV and In A = 49.4, the latter giving A = 2.8 x 1021 irf2 s_1.
Since z-, = 1, using this fi/z;e0 value in eq. (13.55) results in K = 27 pj/m
[Bostanov et al. 1983] under the assumption that the 2D nucleus is square
35 I | i i i i i j
34 '- ^\_ -
^- 33 - ^v -
'in Yv
E 32 - «\. -
29 r ■ ■ ■ ' ' 1
44 46 48 50 52 54 56 58 60
1/Ap(V~1)
Fig. 13.11 Dependence of the stationary nucleation rate on the overvoltage in
2D HEN on own substrate: circles - data of Bostanov et al. [1983] for silver on the
(100) face of a silver electrode in aqueous solution ofAgN03 at T = 318 K;
lines - best fit according to the first eq. (13.105).
222 Nucleation: Basic Theory with Applications
shaped (then b = 4aJ/2 = 1.154 nm). Physically, both the k and the A values
are reasonable even though the nucleus size is rather small: setting A^/ =
Zie0A(p and Ao = 0 in the first of eqs (13.107) and using the above B/z\eG
value, we find that the 2D nucleus is constituted of n* = 19 to 32 silver atoms
in the studied overvoltage range of 17 to 22 mV.
Finally, Fig. 13.12 shows the /s(A,u) data of Skripov [1972] for HON of
bubbles of diethyl ether (the squares) and benzole (the circles) in own liquid
at 7 = 421.15 and 498.95 K, respectively. In accordance with eq. (13.109) of
the classical theory, In /R is a linear function of 1/A/?2. The slopes and the
intercepts of the best-fit straight lines drawn in Fig. 13.12 give B = 57.0
MPa2 and A = 8.7 x 1024 m~3 s"1 for diethyl ether and B = 245 MPa2 and A
= 2.9 x 1038 m~3s_1 for benzole. Using these B values in eq. (13.59) withe3
= 36?r leads to specific surface energies aei = 2.70 mj/m2 for diethyl ether
and aef = 4.65 mJ/m2 for benzole which are comparable with the independently
known a = 3.13 mj/m2 and a= 4.68 mJ/m2 for diethyl ether and benzole,
respectively [Skripov 1972]. As to the values of the kinetic factor A, they can
be compared with the theoretical value A « 1039 nT3 s"1 for HON, which
follows from eq. (13.46) under the assumption for evaporation control and
sticking coefficient y* = 1. While the agreement between the theoretical and
experimental values of A for benzole and the fact that for it aef = a are
strongly in support of HON, the smaller experimental value of A and the
obtained aef = 0.86a for diethyl ether are an indication for HEN. Since /?e =
S-*.
(A
E
in
C
21
20
19
18
17
16
15
14
0.27 0.28 0.29 0.30 0.31 0.65 0.70 0.75 0.80
1 / (Ap)2 (MPa-2)
Fig. 13.12 Dependence of the stationary nucleation rate on the underpressure:
circles - data for benzole bubbles in benzole at T = 498.95 K [Skripov 1972]; squares
- data for diethyl ether bubbles in diethyl ether at T = 421.15 K [Skripov 1972}; lines
- best fit according to eq. (13.109).
Stationary nucleation 223
1.71 MPa for diethyl ether and pe = 2.09 MPa for benzole at the respective
temperatures [Skripov 1972], using the above B values in eq. (13.110), we
find that in the A/> ranges studied the nucleus bubble contains n* = 107 to
119 diethyl ether molecules and n* = 157 to 170 benzole molecules.
Summarizing, we come to the conclusion that the classical nucleation
theory can provide at least a qualitative description of available experimental
7s(A^i) data, even when the nucleus size proves to be rather small (n* < 100).
Quantitatively, the kinetic factor A is practically always a source of uncertainty.
In many cases the analysis of the data is rendered difficult because of the
lack of unambiguous knowledge about the kind of nucleation (HON, HEN,
3D, 2D, etc.). In the next section we shall see how the nucleation theorem
can be used for avoiding this difficulty and, thereby, for making the theoretical
analysis much more reliable (see also Chapters 16, 28 and 30).
Chapter 14
First application of the
nucleation theorem
As noted in Section 13.5, when analysing experimental data for the dependence
of the stationary nucleation rate Js on Ay we are faced with the question for
the choice of a formula for Js which would provide the most plausible
interpretation. This is so because depending on the kind of nucleation (classical,
atomistic, 3D, 2D, etc.), the theory suggests different coordinates for
linearization of J& as a function of A/i. These different coordinates, however,
may lead to differences in the calculated parameters of nucleation and, in
particular, in the size n* of the nucleus. Clearly, this uncertainty will be
lessened if the Js(Ay) dependence is interpreted in coordinates which are not
specific for any particular Js(Aji) formula corresponding to a given kind of
nucleation. Such highly reliable interpretation proves to be possible thanks
to the nucleation theorem. The nucleation theorem thus finds an important
experimental application: it can be used for a universal, model-independent
determination of the excess number An* of molecules in the nucleus and of
the nucleus size n* from available JB(Afi) data [Kashchiev 1982].
Suppose we have an experimental is(A^) dependence obtained by varying
Afj isothermally. The question is how to calculate An* and n* in a most
general, model-independent way. To arrive at the needed expression for An*
and n* we can use the general 7S formula (13.39) rewritten as
-W* = kT In Js ~ kT In A. (14.1)
Differentiating this equation with respect to A/i and invoking the nucleation
theorem in the form of eq. (5.21) results in the expression
An* = kT{\ ~ p0id/pnew)[d(ln Js)/dA^ - d(ln A)/dAy] (14.2)
which, owing to (5.18), can be represented equivalently as
An* = W[d(ln 7s)%old - d(ln A)/d^0]d]. (14.3)
These formulae are applicable to all possible cases of one-component
nucleation when jUold and thus A/i is changed isothermally. They show that
for the model-independent determination of An* it is necessary to plot the /s
data in In J^-vs-Ap or In Js-v&-fi0]d coordinates: then the slope at each point
of the resulting line gives the corresponding An* value if the dependence of
In A on Afj or fioid is neglected. To a first approximation, this is possible,
because the kinetic factor A is a relatively weak function of A/i when this is
varied isothermally (see Sections 13.2 and 13.3). For a more accurate calculation
First application of the nucleation theorem 225
of An* from (14.2) or (14.3), however, we need a theoretical estimate for the
contribution of the In A derivative. Fortunately, the uncertainty in this estimate
is negligible and cannot undermine the use of eqs (14.2) and (14.3) for a
model-independent determination of An*. Indeed, for condensed phases
nucleated by changing A^i at constant T, from eq. (13.41) which expresses
the A(Afi) dependence in a very general form we find that kTd(\n A)ldA(i =
1, since d(ln A')/dA^i « 0. For nucleation of gaseous phases at isothermally
varied A^i the contribution of the In A derivative is even less important: we
can set d(ln A)/dAfi = 0, as in this case A is practically Aft -independent (see
eq. (13.46)). Hence, with an error of less than one molecule, from eq. (14.2)
it follows that
An* = (1 - pold/pnew)[H"d(ln J^/dAft - nA] (14.4)
where nA s kTd(\n A)ldAp = 1 or 0 accounts for the contribution of the kinetic
factor A when the nucleating phase is condensed or gaseous, respectively.
The An* value obtained from (14.4) is universal in the sense that it is
invariant with respect to the choice of the Gibbs dividing surface between
the nucleus and the ambient phase(s). In contrast, the determination of the
nucleus size n* is meaningful only after making such a choice. Then n* can
be calculated from (5.22) by using the An* value obtained from (14.4). As
pointed out in Section 5.1, the relation between n* and An* is simplest when
the one-component nucleus is defined by the EDS. Solely for a so-defined
nucleus, from (5.25) and (14.4) we find that
n* = [(1 - Poid/pnewVd " fWALmWddn Js)/dAu - nA). (14.5)
in the important cases of HON and HEN on weakly adsorbing substrates.
Comparing eqs (14.4) and (14.5), we see that contrary to the determination
of An*, the determination of n* requires knowledge of the molecular density
pnew of the nucleus. For nucleation of condensed phases, however, such a
knowledge is in fact unnecessary when the process occurs far enough from
the spinodal (if any), since then p„cw = pnew and eq. (14.5) simplifies to
LKashchiev 1982]
n* = lcTd(ln Js)/dAfi - 1. (14.6)
For nucleation of gaseous phases, again not too close to the spinodal, we
have pnew « p0id and pnew « pold so that eq. (14.5) can be approximated by
n* = kT( p*w/pnew)d(ln Js)/dAfi. (14.7)
Hence, the number n* of molecules in the EDS-defined gaseous nucleus
cannot be determined in a model-independent way even far from the spinodal,
because eq. (14.7) still contains the unknown density rj^ew of the gas inside
the nucleus. What can be calculated in such a way from (14.7) is not n*, but
the volume V* of the EDS-defined gaseous nucleus. Indeed, by virtue of
(5.17), eq. (14.7) transforms into
V* = (CTpnew)d(lnA)/dA/,.
(14.8)
226 Nucleation: Basic Theory with Applications
Equations (14.5)-(14.8) are valid for EDS-defined nuclei in all possible
cases of one-component HON or HEN on weakly adsorbing substrates. They
apply to nucleation at isothermally varied A/j and are generalizations of
known formulae following from the classical and the atomistic theory in
various particular cases of nucleation [Nielsen 1964; Allen and Kassner
1969;Milcheve/a/. 1974; Lewis and Anderson 1978; Anisimov et al. 1978,
1980, 1982; Baidakov et al. 1980; Kostrovskii et al. 1982; Anisimov and
Cherevko 1982; Anisimov and Vershinin 1988; Bedanov etal. 1988; Baidakov
1995], Obviously, what we need in order to use eq. (14.6) or (14.8) in a given
concrete case of nucleation is only an expression for Ap in terms of the
corresponding experimentally controllable parameter, e.g. p, /, C, etc. (see
Chapter 2). For example, with the help of (2.8), (2.9), (2.13), (2.14), (2.16),
(2.27), (10.79), (13.68) and (13.69), fromeq. (14.6) it follows that [Kashchiev
1982]
n* = d(ln 7,)/d(ln S) - 1 = d(ln 7,)/d(ln /)) - 1 = d(ln Js)/d(ln I) - 1
(14.9)
for nucleation in vapours,
n* = d(ln 7s)/d(ln 5) - 1
= d(ln Js)/d(ln C) - 1 = d(ln J,)/d(ln a) - 1 = d(ln Js)/d(ln IT) - 1
(14.10)
for nucleation of condensed phases in solutions,
n* = (kT/z,e0)<i(in Js)/dA<p- 1 = -ikT/z,eo)d(ln Js)ld<p -1 (14.11)
lor electrochemical nucleation of condensed phases and
n* = (*77Aue)d(ln J,)IA(p-pc) - 1 = (A;T/Aue)d(ln Js)/dp - 1 (14.12)
for nucleation of crystals in melts (or during polymorphic transformation)
when Ap is varied by p at constant T. Similarly, under the condition p^VglkT
« 1, treating the new gaseous phase as ideal gas (then p„ew = plkT) and
using (2.10) and (13.99), from eq. (14.8) we find that
V* = kTd(\n 7s)/dA/> = -fcTdQn 7s)/d/> (14.13)
for nucleation of gaseous phases at isothermally varied supersaturation. It is
instructive to verify that the corresponding classical or atomistic formulae
for Js in Section 13.3 are consistent with eqs (14.9)-(14.13).
The model-independent determination of An* and n* with the help of the
nucleation theorem does not seem feasible when the experimental Js(Ap)
data are obtained by changing Ap. non-isothermally. This is so, since then
(i) eq. (14.1) has to be used in conjunction with the complicated form (5.36)
of the nucleation theorem and (ii) the contribution of the In A derivative may
be significant because of the relatively strong dependence of the kinetic-
factor A on Ap through T. Nevertheless, in the experimentally important case
of changing Ap by varying T at constant pressure, the formula for n* is also
First application of the nucleation theorem 227
simple if the nucleus is defined by the EDS. Indeed, again for HON or HEN
on weakly adsorbing substrates, differentiating (14.1) with respect to Afi and
applying the nucleation theorem in the form of eq. (5.39), we find
n* = i[(l - J„ew/j„id)/( 1 - sL,lsMW(.T In Js)ldAy - d(7" In A)/dAy].
(14.14)
This equation tells us that in addition to the knowledge of the In A derivative,
for the calculation of n* we need also to know the entropy j„w of a molecule
in the EDS-defined nucleus. Unfortunately, since snew is generally unknown,
eq. (14.14) is practically usable only in its approximate form [Kashchiev
1982]
n*=kd(Tlt\Js)/dAii-nA (14.15)
corresponding to the assumption s„m. = jnew and, hence, to the nucleation
theorem in its simplest representation (5.40). The problem with the usage of
eq. (14.15) is that, unlike in eqs (14.6) and (14.7), now the contribution of nA
= kd(T In A)ldAfi to n* can be significant. For instance, for nucleation of
crystals in melts when Ay is changed according to (2.20), the analysis shows
[Kashchiev 1982] that nA may depend considerably on Ay and be a negative
number with absolute value as high as 100 which is not negligible with
respect to n*. This is so because as follows from eqs (13.41), (13.43) and
(13.74), in this case A is given by the expression
A = (A"lr)) e*"kT (14.16)
in which despite that the product A" exp (AylkT) may be virtually constant,
the melt viscosity r) is a strong function of T (see eq. (10.56)), i.e. of A^i. The
conclusion is, therefore, that due to the theoretical uncertainty in nA, i.e. in
the kinetic factor A, care must be exercised when using eq. (14.15) for
determination of the size n* of the EDS-defined nucleus from Js(Afi) data
obtained at non-isothermally varied Ay.
The nucleation theorem has a valuable application also to multicomponent
nucleation. With the aid of 7s(/j0]d,) data, in this case it is possible both the
excess number An, of molecules of component i in the nucleus and the
composition of the nucleus to be determined again in a model-independent
way. When the chemical potential yoldl of component i in the old phase is
varied isothermally, the formula for An; reads [Strey and Viisanen 1993;
Viisanen et al. 1994; Oxtoby and Kashchiev 1994]
An* = kTd(ln Js)ldyoW - nAi (14.17)
where nA, = &T<?(ln A)/dy0\di is a number from about 0 to 1 irrespective of
the theory for the kinetic factor A. Equation (14.17) is a generalization of
(14.3) and is obtained by differentiating (14.1) with respect to yMi and
employing the nucleation theorem in the form (5.44). The usage of eq. (14.1)
is again possible, since in multicomponent nucleation Js can also be presented
by the general formula (13.39) (e.g. Reiss [1950]; Sigsbee [1969]; Wilemski
[1975]; Stauffer [1976]). In a number of studies on binary and ternary nucleation
228 Nucleation: Basic Theory with Applications
of liquids in vapours, eq. (14.17) has recently been employed for determination
of the size and the composition of the nucleus droplets [Anisimov et al.
1987; Anisimov and Vershinin 1988; Strey and Viisanen 1993; Viisanen et
al. 1994; Strey et al. 1995; Viisanen and Strey 1996; Laaksonen 1997; Luijten
1998], In this way, Viisanen and Strey [1996] were able to register for the
first time phase separation in the ternary nucleus droplet of water-nonane-
butanol despite that this droplet contained a total of only some 20 to 40
molecules.
Summarizing, we see that eqs (14.2)-(14.13) and (14.17) give auniversal
method for determination of An*, n* and V* from experimental 7s(A|i) data
without using any concrete theory for the nucleation process. To that end it
is only necessary the data to be plotted in the universal In /s-vs-A^ or ^0)d
coordinates. The slope of the resulting line then yields An* or n* with an
accuracy of about 1 molecule. Thus, the A^ dependence of An*, n* and V*
can be obtained experimentally in a model-independent way and then used
for verification of any concrete theory for this dependence.
Figure 14.1 illustrates the applicability of the above method to the Js(A^i)
dependence obtained by Weeks and Gilmer [ 1979] by a Monte Carlo simulation
of one-component nucleation of crystalline monolayers on a perfect (100)
crystal face at 7= constant. The circles in Fig. 14.1a show the simulation
data for In (JJIC) as a function of A^i/kT (Ie is the impingement rate at
equilibrium), and the curve represents the best-fit function In (7s//e) =
- 13.43262/(A^/H) - 3.38439 + 1.93868(A/j/W). The circles in Fig. 14.1b
depict the n*(A^i) dependence calculated with the help of this function and
of eq. (14.6) which applies to the simulation experiment. Now, as the simulation
involves 2D HEN on own substrate with square symmetry, it is of interest to
compare the so-obtained 'experimental' n*(A/<) dependence for the EDS-
defined nucleus with the classical Gibbs-Thomson dependence of n* on A/;
for 2D square nucleus with monolayer height. Since in this case b = 4do and,
to a first approximation, K = e,/2d0 [Kaischew and Stranski 1934a], from
(4.35) we have
n* = (£lIAfif (14.18)
where £, is the energy of molecular interaction between nearest neighbours
in the monolayer. The curve in Fig. 14.1 b represents this equation with £tlkT
= 4, the value used in the simulation experiment. As seen from this figure,
the classical Gibbs-Thomson formula (14.18) is in agreement with the
'experimental' finding for the size of the EDS-defined 2D nucleus even
though the nucleus is constituted of a few molecules only. This agreement is
particularly notable because of the absence of free parameters in eq. (14.18).
In real experiments, eq. (14.6) finds a wide application in atomistic nucleation
[Robinson and Robins 1974; Paunov and Harsdorff 1974; Lewis and Anderson
1978; Stenzel et al. 1980; Meyer and Stein 1980; Milchev and Vassileva
1980; Milchev 1991]. Equation (14.6) was used also for analysis of/s(A^i)
data for electrochemical nucleation [Scharifker and Wehrmann 1985] and
for droplet nucleation in vapours [Allen and Kassner 1969; Anisimov et al.
First application of the nucleation theorem 229
Fig. 14.1 (a) Dependence of the stationary nucleation rate on the supersaturation in
2D HEN on own substrate: circles - Monte Carlo simulation data [Weeks and
Gilmer 1979}; line - best-fit function used for calculating the derivative in eq. (14.6).
(b) Dependence of the corresponding nucleus size on the supersaturation: circles -
data obtained according to eq. {14.6); line-Gibbs-Thomson eq. (14.18).
1978, 1980, 1982; Bedanov et al. 1988; Viisanen et al. 1993; Viisanen and
Strey 1994; Strey et al. 1994], The solid circles in Fig. 14.2 represent the
experimental n*(Afi) dependence obtained by Viisanen et al. [1993], Viisanen
and Strey [1994] and Strey et al. [1994] with the help of eq. (14.9) from
J^Afj) data for HON of water (Fig. 14.2a) and n-butanol (Fig. 14.2b) droplets
in vapours at different temperatures. It is seen that these circles follow closely
the curves in the figure, drawn through the open circles which represent the
respective n* values calculated by the authors from the classical Gibbs-
Thomson equation (4.10) without using adjustable parameters, since in this
230 Nucleation: Basic Theory with Applications
uu
50
40
, 30
c
20
10
1
60
50
40
c 30
20
10
1
; ' ' '
;\
:
:N#
r
r
■ i • ■
i^0»
i
^-8^»
. i . .
-r-i
. . I
•
I 1 1
•
•
1 1
•
. 1
•
. 1
1—1—I—I—
(a) :
;
• "
~°~^
:
(b) ':
:
-;
:
:
:
10
15
S
20
25
Fig. 14.2 Dependence of the nucleus size on the super saturation ratio in HON of
(a) water and (b) n-butanol droplets in vapours at different temperatures (adapted
from Viisanen et al. [1993 J; Viisanen and Strey [1994J; Strey ex at [1994]): solid
circles - data obtained according to eq- (14.9); open circles - n* values predicted by
the Gibbs-Thomson eq. (4.10). The lines are drawn through the open circles to
guide the eye.
case the macroscopic values of vq and o are known. We thus have the first
reliable experimental finding that this equation can be applicable to EDS-
defined liquid nuclei even of some 30 molecules.
Chapter 15
Non-stationary nucleation
Mathematically, nucleation is non-stationary when the cluster concentration
Z„ depends on time t. This means that the fluxj,, from (9.6) is also a function
of time. The master equation (9.1) shows that Z„ will vary with t and, hence,
nucleation will be non-stationary when the transition frequencies/,,,, (i.e. the
supersaturation, the temperature, etc.) and/or the non-aggregative fluxes K„
and L„ are time-dependent. The point is, however, that even under conditions
ensuring stationarity, the nucleation process cannot become stationary right
after the imposition of these conditions; due to the limited speed of the
molecular motion, a certain time will elapse before the establishment of the
stationary population of clusters in the system. The easiest to study is perhaps
the case of isothermal non-stationary nucleation in a closed system at constant
supersaturation A^i, as then/K/n and n* are time-independent. In this case,
after the initial moment t = 0 at which the system becomes supersaturated,
the concentration of the initially existing clusters begins to evolve towards
the corresponding stationary cluster concentration X„. During the period
necessary for the establishment of Xn nucleation proceeds in non-stationary
regime characterized by time-dependent both cluster concentration Z„(t)
and nucleation rate 7(/). This non-stationary cluster size distribution Z„(?)
(nT3 or nT2) is the solution of the master equation (9.4) or (9.10) (with
r-independent fnm because of the constancy of A/j), which satisfies the initial
condition (9.2). The respective non-stationary nucleation rate J{t) (nT3 s_l
or m~2 s_1) can be determined from the expression
J(t)=j*(t) (15.1)
where j*(t) is given by (11.6) or (11.7) with time-independent/„„ and n*.
This equation follows from (11.5) upon taking into account that dn*/6t = 0
when A^i does not change with time.
In this section we shall consider only non-stationary nucleation at constant
supersaturation. In addition, we shall confine the analysis to a process which
occurs according to the Szilard model, i.e. solely by monomer attachment to
and detachment from the one-component clusters. After the seminal work of
Zeldovich [1942], many theoretical papers were devoted to this problem
[Turnbull 1948; Probstein 1951; Kantrowitz 1951; Wakeshima 1954; Collins
1955; Lyubov and Roitburd 1958; Courtney 1962; Nielsen 1964; Andres and
Boudart 1965; Chakraverty 1966; Hile 1969; Abraham 1969; Kashchiev
1969a; Andres 1969; Walton 1969a; Lyubov 1969, 1975; Frisch and Carlier
1971], More recently, new findings have also been reported [Kanne-
Dannetschek and Stauffer 1981; Kelton et al. 1983; Volterra and Cooper
232 Nucleatian; Basic Theory with Applications
1985; Trinkaus andYoo 1987; Kozisek 1988, 1989, 1990, 1991; Shizgal and
Barrett 1989; Shi et al. 1990; Shi and Seinfeld 1991a, 1992; Shneidman and
Weinberg 1991,1992a, 1992b; Wu 1992a, 1992c; Miloshev 1992; Demo and
Kozisek 1993, 1996; Kozisek and Demo 1995].
15.1 Non-stationary cluster size distribution
Our task now is to find the size distribution Zn(t) of the clusters during
isothermal non-stationary nucleation occurring at constant Ap according to
the Szilard model. In doing that we shall again treat the cluster size n first as
a discrete and then as a continuous variable.
When n is allowed to assume only integer values, Zn{t) is the solution of
the Tunitskii equation (9.18) along with the initial condition (9.2). Hereafter,
we shall consider the particular case when at the initial moment / - 0 of
supersaturating the system, in it there are only monomers whose concentration
Z{ equals their equilibrium concentration Ct corresponding to the imposed
supersaturation. The initial condition (9.2) then simplifies to
Z,(0) = C,; Z„(0) = 0, (n = 2, 3, .. . , M). (15.2)
Further, to ensure stationarity of the kind analysed in Sections 13.1 and
13.2, we shall require Z^t) and ZM(t) to remain constant with time and
satisfy eqs (13.4) and (13.5). In other words, in solving (9.18) we shall have
Z,(r) = C, (15.3)
ZM(t) = 0. (15.4)
Thus, eq. (9.18) becomes
dZj/d/ =/,C, - <f2 + g2)Zj(0 + g3Z3(/)
dZ„/dr = /„ _ ,Z„ _ , (/) -(/■„ + g„)Z„(t)
+ &,4iZ„ + 1(f), (« = 3,4, ..., M- 2) (15.5)
dZ/u- ,/d/ =fM-2ZM_2(t) -(/■«_! + g«_ i)ZM_ ,(/).
This is a set of M - 2 ordinary linear differential equations of first order
in the M - 2 unknowns Z2, Z3, . . ., ZM _ 1, which has to be solved under the
initial condition (15.2). This can be done in various ways, e.g. by using
matrix formalism [Ree et al. 1962; Kelton et al 1983] or integral transforms
such as the Laplace one [Korn and Korn 1961]. However, the exact solution
of eq. (15.5) can be obtained also in a straightforward manner.
The first step is to homogenize eq. (15.5) by presenting Zn(l) in the form
(n = 2. 3, W-l)
Z„(t) = X„ + y„(t) (15.6)
where X„ is the stationary cluster size distribution (13.16), and y„(t) is the
Non-stationary nucleation 233
unknown deviation of Z„(r) from Xn. Substituting Z„ from (15.6) in (15.5)
and allowing for (13.6) results in
dy2/dr = -(/2 + g2)y2(t) + g &>,(t)
dy^dr =/,.,¾ _,(t)-(/"„ + f„)>-„W
+ ^, + ,^+,(0, (« = 3,4, ..., M- 2) (15.7)
dy»_ ,/dr =SM_1yM_1{f) ~ (fM-\ + gin- i)>'»- i(')
where, according to (15.2), y„ satisfies the initial condition (n = 2, 3, . . . ,
M- 1)
y„(0) =
(15.8)
Equation (15.7) is a set of M - 2 already homogeneous ordinary linear
differential equations of first order with time-independent coefficients. This
means that for any fixed n = 2, 3, . . . , M - 1 we shall have M - 2 linearly
independent particular solutions ynt{t) of the form (i = 2, 3, . . . , M - 1)
y„i(') = «,..■ <MP (-V)-
(15.9)
Here a„, is a constant, and X; > 0 is the ith eigenvalue, i.e. the (th root of the
characteristic equation
f h
0 -/3
0 0
0 0
0 0
X
0
-gi
/a
+gt-
0
0
0
X
0
0
0
/m-3
+ S»-i -
~/m - 3
0
X
0
0
0
-gM-2
/m-2
+ gM-2 -A.
~fu-2 .
0
0
0
0
-Sm
JM-
= 0
(15.10)
of the set (15.7). This equation, in which X is any of the A;'s, is obtained by
substituting yni from (15.9) in (15.7) and requiring that not all ani equal zero
simultaneously (this requirement ensures the linear independence of yni [Korn
andKorn 1961]).
The above determinant represents a polynomial of degree M - 2. It can be
shown [Ree et al. 1962] that this polynomial has M-2 simple roots X^, X^,
234 Nucleatitm: Basic Theory with Applications
..., XM _ [. The next step is, therefore, to find these roots. Having found them
from (15.10), for each i = 2, 3, . . . , M - 1, we are able to determine the
constants a„, with the help of the recursion formulae
(/2 + #2 - -¾¾ - S3a3i = °
~fn - 1«» - l.i + (/» + £»" ^i)a„i -g„+ ,a„ + M = 0,
((1 = 3,4,.. .,M-2) (15.11)
~/m -2aM -2,1 + (/a/-! + gM - 1 - ^i)aM-l,i = 0-
These formulae result from substituting yni from (15.9) in (15.7) and in them,
without loss of generality, it is convenient to set (i = 2, 3, . . . , M - 1)
1.
(15.12)
We can now use the M - 2 linearly independent solutions ym from (15.9)
in order to represent the general solution yn(t) of (15.7) as a linear combination
of them (n = 2, 3, . . . ,M- 1):
y,W = £ c,-fl„;exp(-V)-
(15.13)
Thus, the last step is to find the M - 2 unknown constants c,. They are the
solution of the linear algebraic set of M- 2 equations (n = 2, 3, . . ., M- I)
Z Cja„, = -X„ (15.14)
resulting from using the initial condition (15.8) in (15.13). In accordance
with the Cramer rule [Korn and Korn 1961], c,- is given by (i = 2, 3, . . .,
M- 1)
Cj = d,/d' (15.15)
where d' and d; are the following determinants of order M - 2:
'" a2,M-l
d'--
a}2
a 23
«33
3M-1,2 aM - 1,3
a2i a2l
032 a33
'■< 2,,-1
33.i-l
"2./-I
a 3.i-I
yM-l,M-l
a2,M-l
a3,M-l
(15.16)
(15.17)
It remains now only to insert >„(() from (15.13) into (15.6) in order to
terminate the solving of the problem. Using also (15.14) and (15.15), we
thus find that (n = 2, 3,. . ., M - 1)
Non-stationary nucleation 235
Z„(r) = X„ + I (d,W) a„,-exp (-V) (15AS)
i=2
or, equivalently,
l-[Z4a„,]-' I d,a„, exp(-A,r)[. (15.19)
Equation (15.18) or (15.19) represents the sought time-dependent cluster
size distribution during non-stationary nucleation which begins at no
preexisting clusters in the system and proceeds at constant supersaturation. We
note that, as it should be, Z„(r) —> X„ for t —> °° and that eq. (15.18) is similar
to the Z„(t) formula of Shizgal and Barrett [1989], Mathematically, Z„(r)
from (15.18) or (15.19) is the exact and complete solution of eq. (15.5) along
with the initial condition (15.2). Physically, however, eq. (15.18) or (15.19)
is not informative, for it gives only implicitly the dependence of the non-
stationary cluster concentration on the transition frequencies fn and gn through
the constants A,, a„,andrf,(as seen from (15.19), knowing d'from (15.16) is
in fact unnecessary for the determination of Z„). The real difficulty with the
usage of eq. (15.18) or (15.19) is of purely mathematical character: regretfully,
as mathematics does not give general formulae for the roots of polynomials
of degree higher than 4, we can find analytically the exact expressions for
the roots A, of eq. (15.10) only for A/-2 <4, i.e. for M < 6. This means that
for M > 6 we must resort to numerical methods for the determination of the
A,'s as functions of/„ and g„. Fortunately, for the reasons mentioned in
Sections 13.1 and 13.2, with practically no loss of accuracy, in the sums in
(15.18) and (15.19) it is possible to replace M by Mc, = n2. Hence, eqs
(15.18) and (15.19) are particularly useful for the analytical description of
atomistic non-stationary nucleation characterized by n* S 5 and A* ^ 2 so
that for the right end of the nucleus region we have n2 = "* + A*72 < 6.
Indeed, the sums in (15.18) and (15.19) do not contain more than 4 summands
(since M = Mef = n2) and after some algebra \, ani and ds are obtainable from
(15.10)-(15.12) and (15.17) as exact and explicit functions of/„ and g„.
Let us now exemplify the above results in the concrete case of n2 - 4,
which corresponds to nucleation characterized by n* = 2 or 3 and A* S 2.
Setting M = MCf = n2 = 4, from the characteristic equation (15.10) (which is
now a quadratic one) we find that its two roots A2 and A3 are given by
h= (1/2)(½ +/3 + «2 + ft) - [(/2 +/3 + g2 + ft)2
- 4(M+ ?/3 + ftft)]1/2} (15.20)
A3 = (1/2)(½ +/3 + ft + g3) + [(/2 +/3 + ft + ft)2
-4(//3 + s/3 + ftft)]1/2}. (15.21)
Accordingly, from (15.12) we have
"32 = "33 = 1,
(15.22)
236 Nucleation: Basic Theory with Applications
and from the last equation in the set (15.11) (this equation is now the only
independent one in the set) it follows that
«22 = (/3 + S3 ~ V)lh, «23 = % + f3 - A3)//2. (15.23)
Using these expressions for a22. «2.3* «32 and «33 in eq. (15.17) yields
d2 = -X2 + (f, + g,- JyXj//2. d3 = -¾ - (/3 + S3 - ^)X3//2 (15.24)
where, in line with (13.16), (15.20) and (15.21),
*2 =/1(/3+ S3)CW3, X3 =/1/-^,/^3 (15.25)
are the stationary concentrations of the dimers and the trimers, respectively.
Substituting the a„,'s and the d;'s from (15.22)-(15.24) in eq. (15.19) and
accounting for (15.25), after some algebra we thus find that the non-stationary
concentrations of the dimers and the trimers are given by
7 (,) = X l"i _ (/3 + ^3-¾)¾ xv . (/3 + g3 - A3U2 ,„ 1
1 ' 2[ (/3 + 83)(^-^) (/3 + ^3)(^-^) J
(15.26)
Zl(t) = Xi\\-j^j-e-^ + T^re-^\ (15.27)
-A2
These equations, along with Z,(l) = C{ and Z4(I) = 0 which follow from
(15.3) and (15.4), are the exact solution of the non-stationary problem in the
exemplified case of nucleation with M = Mef = 4. They represent Z2 and Z3
as explicit functions of t, C, and the monomer attachment and detachment
frequencies/,,/2,/3, gi and g3. It is interesting to note that the relative time
variations Z2(/)/X2 and Z3(/)/X3 of Z2 and Z3 do not depend on the monomer-
to-monomer attachment frequency /,. Also, we see that Z2(I) °= t and Z3(I)
0^ r2 in the limit of t —» 0. Hence, by analogy, we can expect that at the very
beginning of non-stationary nucleation, regardless of the value of M, Z„ will
scale according to (n = 2, 3, . . ., M - 1)
Z„(I) = ("-'. (15.28)
Curves 2 and 3 in Fig. 15.1 illustrate, respectively, the time dependence
of Z, and Z3 from eqs (15.26) and (15.27) for HON of water droplets in
vapours at T = 293 K and a rather high supersaturation ratio S = plpe = 50.
The used transition frequencies are/2 = 3.08 nsJ, g2 - 5.33 ns~' and/3 = g3
= 4.02 ns"1. These values are calculated from eqs (10.3), (10.72) and (10.73)
with c = (36^r)"3 and /„, va, m0, pc and a from Table 3.1. We stress, however,
that since the validity of these equations for the smallest clusters is questionable,
physically, the dependences illustrated in Fig. 15.1 are more or less qualitative.
According to eq. (12.14), the fact that/3 = g3 means that at the chosen S value
the nucleus droplet is constituted of n* = 3 water molecules. Hence, curve 3
in Fig. 15.1 visualizes the evolution of the actual concentration Z*(r) = Z3(?)
of nuclei in the water vapours. As seen, Z* first increases parabolically with
t (in conformity with (15.28) Z* <* f~' - t2) and then slows down its rise
Nan-stationary nucieation 237
m
X
-^
NT
><"
CN
N
I.U
0.8
0.6
0.4
0.2
n
2^' ^-^
/ s^
/ 3/
/ /
0.2
0.4 0.6
t(ns)
1.0
Fig. 15.1 Time dependence of the concentration ofdimers and trimers in non-
stationary HON of water droplets in vapours at T = 293 K and p/pe = 50.
Curves 2 and 3 represent eqs (15.26) and (15.27), respectively.
until reaching 95% of its stationary value X* = X3 at time t = 0.75 ns which
is comparable with the average time l//3 = 1//* = 0.25 ns between two
successive events of monomer attachment to the nucleus.
We now turn to the problem of finding the non-stationary cluster size
distribution Z(n, f) when n is treated as a continuous variable. Then Z(n, t) is
the solution of the Zeldovich equation (9.27) along with the initial and
boundary conditions
Z(n, 0) = 0, (1 <n<M)
Z(l,r) = C!
Z(M, r)=0
(15.29)
(15.30)
(15.31)
which correspond to those specified by eqs (15.2)-(15.4) and are particular
forms of eqs (9.2), (9.32) and (9.33). In finding the unknown Z(n, t) function
it is possible to employ various mathematical methods [Chakraverty 1966;
Kashchiev 1969a; Trinkaus and Yoo 1987; Shi el al. 1990; Shneidman and
Weinberg 1992b; Demo and Kozisek 1993]. Following Kashchiev [1969a],
we shall now see how Z(n, t) can be obtained exactly by the method of
separation of the variables n and t in eq. (9.27).
Analogously to (15.6), we first represent Tin, I) in the form
Z(n, t) = X(n) + C(n)y(n, t) (15.32)
in order to homogenize the boundary condition (15.30). Here C(n) and X(n)
are the equilibrium and the stationary cluster size distributions (7.4) (or
238 Nucleation: Basic Theory with Applications
(12.5)) and (13.18), and y(n, /) is the unknown deviation of the Z(n, t)IC(n)
ratio from the X(n)IC(n) one. Substituting Z(n, t) from (15.32) in the Zeldovich
equation (9.27) and the initial and boundary conditions (15.29)-(15.31) results
in (1 <n<M)
dt - C(n) dn[f(n)C(n) dn \
y(n, 0) = -X(n)IC(n), (1 < n < M)
y(l,t) = 0
y(M, t) = 0,
(15.33)
(15.34)
(15.35)
(15.36)
because by virtue of (13.18) we have d[/(n)C(n)d[X(n)/C(n)]/dn}/dn = 0,
X(1) = C, andX(Afi = 0.
As seen, eq. (15.33) admits separation of the variables. Hence, analogously
to (15.9), we shall have linearly independent particular solutions y,(n, t) of
the form
y,(n, 0 = a^n) exp (- A,r) (15.37)
where, however, 1 = 1,2 °°, since now the number of these solutions
is infinite. In (15.37) A, > 0 and a,(n) are, respectively, the eigenvalues and
the eigenfunctions of the equation of Sturm-Liouville [Kom and Korn 1961]
(i= 1,2, . . . ,oo)
[fl,-(«)]}■
£ \f(n)C(n) ^ [a, («)] \ + l,C(«)fl,(«) = 0 (15.38)
which follows from (15.33) upon substituting y,(n, r) from (15.37) in it.
According to (15.35)-(15.37), the boundary conditions which must be satisfied
by the unknown at(n) function in eq. (15.38) are (i = 1, 2, . . . , °°)
0,(1) = 0, a,(Afl = 0. (15.39)
Having found A, and a^n) by solving eqs (15.38) and (15.39), we can use
yi(n,t) from (15.37) in order to represent the general solution y(«, t) as (1 S
n<M)
y(n,/)= I c,-a,(n) exp (-AjO (15.40)
which parallels eq. (15.13). Here only the constants e, are unknown so that
the last step is to determine them from the equation (1 S n < M)
C(n) Z Ciakn)=-X(n) (15.41)
which is the analogue of (15.14) and results from setting equal y(n, 0) from
(15.40) to y(n, 0) from the initial condition (15.34). The determination of c,
can be done with the help of the relation ((,4=1,2 =»; i * t)
Non-stationary nucleation 239
C(«)a,(«K(«) dn = 0 (15.42)
which expresses the fact that the eigenfunctions of the Sturm-Liouville problem
of the kind specified by eqs (15.38) and (15.39) are orthogonal [Korn and
Korn 1961] (C(n) is the weight function corresponding to eq. (15.38)).
Indeed, multiplying both sides of (15.41) by ak(n), integrating over n from
n = 1 to M and using (15.42) reduces the sum in (15.41) to the single
summand ck C(n) aj(n) dn so that (i = 1, 2, . . . , «>)
Ci=-\ C(n)af(n)dn\ X(n)ai(n)dn (15.43)
where the subscript ;' stands instead of k.
As we now know all of the quantities Xi,at{n) and q in (15.40), combining
(15.32), (15.40) and (15.41), we finally obtain (1 < n < M)
Z{n, t) = X(n) + C(n) £ c,- flj-(n) exp ( - Afr) (15.44)
or, equivalently,
Z(n,t) = X(n) \\-\ Zc;Q;(b) S cia,{n) exp {-X,t) L (15.45)
Equation (15,44) or (15.45) represents the sought non-stationary cluster
size distribution. The analogy of these equations with eqs (15.18) and (15.19)
is obvious and again Z(n, t) —» X(n) when /—»«>. This demonstrates that
considering n as a continuous variable leads to adequate results also in the
case of non-stationary nucleation. We note also that Z(n, t) from (15.44)
parallels the Z(n, t) function of Shizgal and Barrett [1989]. It must be
emphasized that since in deriving eqs (15.44) and (15.45) we made no
approximations, mathematically, either of them is the exact and complete
solution of the Zeldovich equation (9.27) under the initial and boundary
conditions (15.29)-(15.31). However, as in the case of eqs (15.18) and (15.19),
the practical usage of (15.44) and (15.45) requires knowledge of A,-, a,(n) and
c, as functions of the parameters controlling the monomer attachment frequency
j{n) and the equilibrium cluster concentration C{n). Unfortunately, in general
none of A„ a,<«) and c,- can be found exactly by solving the Sturm-Liouville
eq. (15.38) and performing the integrations in (15.43). What we can therefore
do is to seek only approximate formulae for A;, «,(«) and c, and use them in
(15.44) or (15.45) to obtain corresponding, approximate expressions for
Z{n,t).
Let us now exemplify the application of eq. (15.44) to the problem of
finding the Z(n, t) function only for n = «*, i.e. the non-stationary concentration
only of the nuclei and of those clusters which are nearly of their size. This
problem is relatively simple, because rigorous mathematical considerations
240 Nucleation: Basic Theory with Applications
[Kashchiev 1969a] show that the boundary conditions (15.39) can be 'shifted'
to the left and right ends «,' and n2 of a size region around the nucleus size
n* and that for the n values inside this region the Sturm-Liouville eq. (15.38)
can be given the approximate form (('=1,2,...,°°)
/*d2a,(n)/dn2 + A,a;(tt) = 0. (15.46)
The boundary conditions to this equation, which replace (15.39), thus
read
fli(n,) = 0, o,-(nl) = 0. (15.47)
Here
«,' = «* - A72, n2 = n* + A72, (15.48)
and A' is specified by [Kashchiev 1969a]
A' = 4//3 = (4/7t"2)A* (15.49)
where ji is given by (7.38).
As seen from (15.49), the width A'of the size region in which eq. (15.46)
is valid is about twice greater than the width A* of the nucleus region
defined by (7.43). Thus, physically, A' and A* are analogous. This analogy
makes physically understandable eqs (15.46) and (15.47) which approximate
(15.38) and (15.39). Indeed, from Sections 7.1 and 10.1 we know that C(n)
~ C* and^n) «/* inside the nucleus region (this transforms (15.38) into
(15.46)). Also, in view of eqs (13.21) and (13.25) we may expect (i) that
Z(«, t) will evolve practically instantaneously from Z(«, 0) = 0 to Z(«, t) =
X(n) for all subnuclei far enough from the nucleus region (i.e. for 1 £ n
< n't), and (ii) that Z(n, () will be always vanishingly small for all supernuclei
of size n > n2. Since Z(n, t) and «,(«) are related by (15.44), this expectation
is quantified by the replacement of (15.39) by (15.47).
The solution of eqs (15.46) and (15.47) is known [Korn and Korn 1961]
(i= 1,2 ,-):
A, = i V/*/A'2 (15.50)
a,(n) = sin [in(n - nf)/A']. (15.51)
Introducing these eigenfuncuons in (15.43), 'shifting' the limits of integration
from 1 to n[ and from M to n2 and using the fact that within these limits C(«)
« C* and X(n) = C*l(l/2) - ()3/;r"2)(fl -n*)] (see eqs (7.37) and (13.25)), we
find that c, is given approximately by the formula (1 = 1,2,...,»)
c, = -(2/i;r)[sin2 (M2) + (Alnm) cos2 (in/2)]. (15.52)
Finally, substituting the above A,, at(n) and c, in (15.44) and accounting for
(13.25), (15.48) and (15.49), we arrive at the following approximate expression
for the non-stationary concentration of clusters of size n ~ n* [Kashchiev
1969a]:
Non-stationary nucleation 241
Z(n, t) = C(n)\\l2 - (!5lKm)(n - «*)]
- (2ln)C(n) Z (l/i)[sin2 (Htl2)
+ (4/jt"2) cos2 (inl2)\ sin [(i!tfilA)(n - n* + 2//3)]
x exp (- iVp2f*t/l6). (15.53)
It must be emphasized that this expression is not applicable to all clusters
inside the nucleus region, but only to those for which n is so close to n* (e.g.
\n - n*\ < A*/4 = /r"2/4j3) that the linear dependence (13.25) of the X(n)IC(n)
ratio on n is a sufficiently accurate approximation to the exact dependence,
eq. (13.18), of this ratio on the cluster size. This is so, because the constant
c, is very sensitive to the approximate C(n) and X(n) dependences used for
performing the integrations in eq. (15.43). In obtaining c, from (15.52) we
have assumed constancy of C(n) and linearity of X(n) with respect to n and
it is this assumption that limits the applicability of eq. (15.53) only to clusters
whose size n is close enough to «*. In fact, eq. (15.53) is most accurate at n
= n*, which allows employing it for the determination of the time dependence
of the non-stationary concentration Z*(t) = Z(n*, t) of nuclei in the system.
Setting in this equation n = rc* and recalling (13.23), mutatis mutandis, we
get
Z*(t) = X* jl + (ilK) Z [(- l)'7(2i - 1)] exp [-(2i - l)2t/4T][ (15.54)
where T(s), called often the time lag of nucleation (see Section 15.3), is
given by [Kashchiev 1969a]
z = 4lx2p2f*. (15.55)
In some cases it may be more convenient to useeq. (15.54) in its equivalent
presentation [Carslaw and Jaeger 1959]
Z*(t) = 2X* Z (- 1)' - ' {1 - erf [(2( - 1 )mml2tm]) (15.56)
;=i
where erf(jt) is the error function defined by eq. (10.15). This formula is
particularly suitable for describing the initial evolution of the nucleus
concentration. Indeed, for t —» 0 only the first 1 - erf summand in (15.56) is
significant and since 1 - erf(x) = (1/71¾) exp (- x2) for x —> «> [Carslaw and
Jaeger 1959], Z*(t) takes the simple form
Z*(t) = X*(16r/ff3T)"2 exp (- K2xlAt). (15.57)
This equation approximates Z*(/) from (15.56) with an error of less than
30% for I < 2x.
Equation (15.54) shows that, in the scope of the approximations made in
obtaining Xb at(n) and c„ the Z*(t)IX* ratio is a universal function of the
dimensionless time tlz. This function is depicted in Fig. 15,2 by the solid
242 Nuclealion: Basic Theory with Applications
curve. As seen, eq. (15.54) predicts correctly the sigmoidal shape of the
Z*(r) dependence, expected for n* > 2 on the basis of eq. (15.2S). Also, Z*
= 0 at t = 0 and Z* —■> X* as t —» «. For t > lOr the nucleus concentration is
already practically equal to its stationary value X*, which means that, physically,
T is the time constant for reaching the steady state. Since eq. (15.54) is
derived by considering n as a continuous variable, according to (9.13), it is
mathematically more justified to use it for sufficiently large nuclei, e.g. of
size n* > 10. It is therefore of interest to check if Z*(r) from (15.54) is a
reasonable approximation to the exact Z*(I) dependence (15.27) corresponding
to the case of n* = 3. To do that we can replot curve 3 from Fig. 15.1 by
changing the time axis to the tit one where x - 0.158 ns is calculated from
(15.55) with/* = 4.02 ns"1 and /3 = nmz = 0.799 (the z value used is obtained
from the formula z = gigi/ififi + gz/3 + ftft) which follows from (13.33)
with C* and Js given by (12.3) at n = n* = 3 and by (13.29)). The result is
shown in Fig. 15.2 by the dashed curve 3. As seen, qualitatively, eq. (15.54)
is in agreement with the exact Z*(r) dependence. Quantitatively, however,
eq. (15.54) is a rather crude approximation to eq. (15.27). The dashed curves
27 and 71 in Fig. 15.2 display the Z*(t) dependences obtained numerically
by Demo and Kozisek [1993] and by Abraham [1969], respectively. Curve
27 pertains to HON of Li20-2Si02 crystals in their vitrified melt under
conditions of n* = 27, and curve 71 is for HON of water droplets in vapours
t/x
Fig. 15.2 Time dependence of the non-stationary concentration of nuclei: solid curve
-eq. (15.54); dashed curve 3 - eq. (15.27) for HON of water droplets in vapours
under conditions ofn* = 3; dashed curve 71 - numerical dam of Abraham [1969]
also for HON of water droplets in vapours, but under conditions ofn* = 71; dashed
curve 27 -numerical data of Demo and Kozisek [1993] for HON ofLi20-2Si02
crystals in vitrified melt under conditions ofn* = 27.
Non-stationary nucleation 243
when n* - 71. We see that eq. (15.54) describes qualitatively the numerical
data, but the quantitative agreement is not good enough, especially at shorter
times. Demo and Kozisek [ 1993] showed that their numerical Z*(?) dependence
(curve 27) is described well by the expression
Z*(r) = X*[\ ~ erf (3/3n*(l - n*-m)
x [1 - exp (- 80//7rc2?)]-"2 exp (- 40r/7;r2T))] (15.58)
which follows from their general formula for Z(n, t).
15.2 Non-stationary rate of nucleation
When we know the cluster size distribution, it is a simple matter to determine
the non-stationary nucleation rate J(t). Indeed, according to (11.8) and (15.1),
with the help of eq. (13.3) (used at n = n*) and eq. (15.18) we find easily that
for n* = 2, 3, . . . , W - 1
M-\
./(0 = ./,+ I «/rf')(/*aD-,-f„.+ia„.+,,,)exp(-A,r) (15.59)
where the rate Js of stationary nucleation is given by (13.29). From (11.8)
and the initial condition (15.2) we have 7(0) = 0 for n* > 2. This allows
elimination of d! from (15.59) and representation of this equation in the
equivalent form (n* = 2, 3, . . . , M - 1)
J(t) = J, 1
2 <*;(/*«„.,-
M-l -I
Z dL(f*ann - s„.+lfl„.+,if) exp (-A,0 | (15.60)
which can be obtained also by using eq. (15.19) in (11.8) and (15.1). We note
that when n* = 1, J(t) is given by the formula J(t) = /, Cx - gi^t) with Z2(/)
from (15.18) or (15.19).
Equation (15.59) or (15.60) represents the exact time dependence of the
non-stationary rate of nucleation occurring according to the Szilard model.
These equations show that J —» Js for t —> <*>. As already noted in Section
15.1, the stumbling block when using them is the determination of A,, a„„ d'
and dj which are quantities obtainable only approximately from (15.10),
(15.11), (15.16) and (15.17) whenM= Wef>6.
Let us consider again the case of M = Mef = n2 = 4 and n* = 3, for which
we have explicit expressions for A,, ani and djt eqs (15.20)-(15.24). Using
these equations in (15.60) and taking into account that now, due to (13.5),
(15.4), (15.6), (15.8) and (15.9), we have a„,+ ,, = am = 0, we can find the
corresponding J(t) dependence in a straightforward way. However, it is easier
to get the same result merely by substituting Z3(r) from (15.27) into the
244 Nuclealion: Basic Theory with Applications
relation J{t) =/3Z3(/) which follows from (11.8), (15.1) and (15.4). Recalling
also that now Js -f^, we thus obtain
where Js, ^ and A3 are given by (13.29), (15.20) and (15.21) as functions of
the attachment and detachment frequencies.
Equation (15.61) is the exact formula for the time dependence of the non-
stationary nucleation rate in the exemplified case of M - Mei ~ 4 when the
trimers are nuclei (n* = 3). The dashed curve 3 in Fig. 15.3 illustrates this
dependence in 7/7s-vs-r/Tcoordinates. Since x, A^ and A3 are given the values
used for Figs 15.1 and 15.2, curve 3 in Fig. 15.3 corresponds to HON of
water droplets in vapours at T= 293 K and S = 50 when, classically, «* = 3.
As noted in Section 15.1, the calculation of the values of T, Xi and A3 by
means of classical formulae for the monomer attachment and detachment
frequencies implies that these values are questionable and that, thereby, the
J{i) dependence represented by curve 3 in Fig. 15.3 is more or less qualitative.
Nevertheless, this curve shows that J is a sigmoidally shaped function of t
initially, J = 0 and after a lag in its rise it tends to its stationary value Js as
t —» °°. It is worth noting that the J{t)/Js ratio and, hence, the position of
Fig, 15.3 Time dependence of the non-stationary rate of nucleation: solid curve ~ eq.
(15.64); dashed curve 3 - eq. (15.6/) for HON of water droplets in vapours under
conditions of n* = 3; dashed curve 71 ~ numerical data of Abraham [1969] also for
HON of water droplets in vapours, but under conditions of n* ~ 71; dashed curves 23
and 27 ~ numerical data of respectively, Kelton et al. [1983] and Demo and Kozisek
[1993} for HON ofLi20-2Si02 cr\'stals in vitrified melt under conditions of n* = 23
and 27.
Non-stationary nucleation 245
curve 3 in Fig. 15.3 does not depend on the monomer-to-monomer attachment
frequency/|.
The above considerations are valid when n is allowed to assume only
integer values. Let us now see what is the formula for J(t) when n is treated
as a continuous variable. This formula is obtained easily by employing eq.
(15.44) for calculating the derivative in (11.9): in conformity with eqs (13.31)
and (15.1) we get
J(t) = Js -f*C* I c,a;* exp (-kit) (15.62)
where the stationary nucleation rate Js is given by (13.32), and a'* s [da,(n)/
dn]„ = „*. Since J = 0 at t - 0, this equation can be cast into the equivalent
form
J(f) = 75|l- Zcj0/* Ic/ expH-iOl. (15.63)
Equation (15.62) or (15.63) is the sought exact expression for the non-
stationary rate of nucleation when n is considered as a continuous variable
and shows explicitly that J —> Js for t —> «. The analogy between these
equations and eqs (15.59) and (15.60) becomes obvious when we recall that
/„ and g„ + i are related by (12.2) and that the finite difference in (15.59) and
(15.60) corresponds to first derivative with respect to n. We note that eq.
(15.62) is analogous to the J(t) formula of Shizgal and Barrett [19891. As in
the case of eqs (15.59) and (15.60), the real difficulty with the application of
(15.62) and (15.63) is in the determination of Ar, a,'* and ct. When we have
approximate expressions for these quantities, we can use them in (15.62) or
(15.63) in order to obtain the corresponding approximate J(t) dependence.
To exemplify this procedure let us make use of X;, a;(n) and c{ from eqs
(15.50)-(15.52). From (15.48), (15.49) and (15.51) it follows that a'* =
(ilcf}/4) cos (in/2) so that, due to (15.50) and (15.52), eq. (15.62) yields
7(r) =JS + (2Pf*C*/n"2) Z cos (in/2) exp (- ;Vj32/**/16)-
Hence, invoking (13.33), (13.35) and (15.55) and rearranging the series, we
get
J(t) = Js [1 + 2 Z (- 1)' exp (- hit)]. (15.64)
This approximate formula for the non-stationary nucleation rate was derived
by Kashchiev [1969a] with the help of Z(n,/) from (15.53). It shows that in
the scope of the approximations used for the determination of A,-, a^ri) and c,-,
the J(t)IJ^ ratio is a universal function of the dimensionless time tlx, the
nucleation time-lag T being given by (15.55). As seen from Fig. 15.3 in
which the solid curve depicts the J(t) dependence (15.64), X sets up the time
scale of the attainment of steady state: J = 0.95JS at t = 3.7t. At short times
246 Nuclealion: Basic Theory with Applications
(t < T) the series in (15.64) converges rather slowly. Then it is convenient to
use eq. (15.64) in the equivalent form (see, e.g. Carslaw and Jaeger [1959])
Jit) = Js(4jct/i)"2 X exp [- (2/ - 1)Vt/4/] (15.65)
i=i
which shows that for / < 4-r, with an error of less than 1 %,
J(t) = Js(Anxltf2 exp (- T^xlAt). (15.66)
As noted above, according to Kashchiev [1969a], the time lag Tin (15.64)-
(15.66) is defined by eq. (15.55). This makes J(t) from (15.64) numerically
different from the J(f) dependences of Kantrowitz [1951], Collins [1955]
and Walton [1969a] which formally coincide with it, but have time lags
which are not given by (15.55). For instance, comparison of eq. (15.64) with
eq. (18) of Collins [1955] shows that according to him T= 1/47T/32/* (this
expression corrects for the missing factor 4 in front of t in Collins's eq. (18)).
Thus, Collins's T is about 5 times less than Tfrom (15.55). A comparative
analysis of the above-mentioned and other approximate J(t) dependences
was made by Kelton et al. [1983] who concluded that eq. (15.64) with Tfrom
(15.55) provides a good description of their numerical results for the J(t)
function. If, however, the same eq. (15.64) is employed with Collins's Tas
given above, and not as used by Kelton et al. [1983], it cannot describe
satisfactorily their numerical J(t) data.
In obtaining eq. (15.64) we have treated n as a continuous variable. This
means that the application of this equation, like that of the Z*(r) dependence
(15.54), is mathematically more justified when the nucleus size is large
enough, e.g. when «* > 10. It is therefore interesting to see what is the
correspondence between eq. (15.64) and eq. (15.61) which describes exactly
the J(t) dependence when n* = 3. Figure 15.3 shows that eq. (15.64) (the
solid curve) is in qualitative agreement with eq. (15.61) (the dashed curve
3): it predicts correctly the sigmoidal shape of the exact J(t) curve.
Quantitatively, however, there exists a discrepancy mostly in the range of the
shorter times. The reliability of eq. (15.64) was tested with the aid of numerically
obtained data for J(t) when n* > 10 [Kelton et al. 1983; Kozisek 1990;
Miloshev 1992; Demo and Kozisek 1993], Depending on the nucleation
conditions and the system studied, it was found that eq. (15.64) describes the
corresponding numerical j(t) data with a different degree of accuracy. This
is seen in Fig. 15.3 in which the dashed curves 23, 27 and 71 represent J(t)
dependences obtained numerically by Kelton et al. [1983], Demo and Kozisek
[1993] and Abraham [1969], respectively (curves 27 and 71 correspond to
the Z*(t) curves 27 and 71 in Fig. 15.2). Demo and Kozisek [1993] showed
that their J(t) data are in good agreement with the following approximate J(t)
expression derived by them:
J(t) = Js [1 - exp (- 80//77T2T)r,/2
x exp (- [3/3n*(l -n*~ 1/3)]2[1 - exp ( - 80//7Jr2T)]-'
exp(-80;/77r2T)}. (15.67)
Non-stationary nucleation 247
At present, eq. (15.64) is the most widely used formula for J(t) in both
theoretical and experimental studies on non-stationary nucleation (for reviews
see, e.g. Toschev [1973a]; Gutzow [1980]; James [1982]; Kelton etal. [1983];
Kashchiev [1984a]; Gutzow etal. [1985]; Kelton [1991]). Figure 15.4 exhibits
the experimental J(t) data of Koster [ 1978] for crystal nucleation in amorphous
Si layers at T = 824 K (the circles), 831 K (the squares) and 920 K (the
triangles). We see that the data symbols are grouped around the master curve
drawn according to eq. (15.64). The observed good agreement between theory
and experiment was obtained by treating Js and Tin (15.64) as free parameters.
In this way Koster [1978] found that Js = 2.9 x 1012, 3.7 X 1013 and 3 x 1015
nr3 s'1 and x= 2.5 x 105, 6 x 104 and 2.5 x 103 s at T = 824, 831 and 920
K, respectively.
0.6 | y
0.5 - / -
04: X :
to . /
~Z 03 : r
—> /
0.2 - /
0.1 - ^i \
o r i i -» i ,^-fTT , i •
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
t/T
Fig. 15.4 Time dependence of the non-stationary rate of nucleation: circles, squares
and triangles - data for crystal nucleation in amorphous Si layers at T = 824, 831
and 920 K, respectively [Koster 197SJ; curve-eq. (15.64).
In obtaining eqs (15.59)-(15.64) for J(t) we have used the results for
Z(n,t) in Section 15.1. However, just like the stationary nucleation rate Js
(see Section 13.2), J(t) can be found without any knowledge of the cluster
size distribution. To do that it is only necessary to solve the set
dj,ldt = -g2[Ji(.t)-J2(t)]
djjdt = f„ [jn M (/) - ;„(/)] - g„ + , [j„(t)
-y'n + i(')l, (n = 2, 3, ...,A/-2) (15.68)
dJM - l Idt = /„ _ , [jM _ 2(0 - Jm - l (')]
which results after (i) differentiating (9.19) with respect to t, (ii) allowing for
248 Nucleation: Basic Theory with Applications
eqs (15.3) and (15.4) and the time independence of/„ and g„, and (iii)
eliminating the appearing dZ„/d/ derivatives with the help of (9.7). According
to (9.19) and (15.2), the initial condition to (15.68) is
./,(0)=/,(:,, j„(0) = 0. (« = 2, 3, ..., W-l) (15.69)
Equation (15.68) is a set of linear ordinary differential equations of first
order in the unknown flux jn(t) and is convenient for numerical calculation
of the non-stationary nucleation rate J't) =j„*(t) when one is not interested
in the cluster size distribution Z„(r). Mathematically, eq. (15.68) is completely
equivalent to the set of equations (15.7) so that its exact solution can be
found, e.g. in the way we used to solve (15.7). Naturally, the integral transform
technique is also suitable for solving (15.68). In Appendix Al, the exact
formula (15.59) or (15.60) for J(t) is obtained by solving (15.68) with the aid
of the Laplace-Carson integral transform.
It remains now only to see what is the analogue of eq. (15.68) when n is
treated as a continuous variable. Employing (12.2) in order to exclude g„
from (15.68) and replacing the finite differences by first derivatives by means
of the truncated Taylor expansions
j(n - 1, 0 =j(n, t) + [dj(n, t)ldn][(n - 1) - „]
[C(n + i)T'dj(n + 1, t)ldn = [C'n)T]dj(n, t)/dn
+ (d{[C(n)]-'dj(n, t)ldn}ldn)\(n + 1) - n]
about point n, we get (1 ^ n ^ M)
^=/^^¾¾ (1"0)
This basic equation for j(n, I) can be obtained by other methods as well
[Nowakowski and Ruckenstein 1991; Slezov and Schmelzer 1994]. It is a
single partial differential equation of second order and replaces the set (15.68)
of M — 1 ordinary differential equations of first order. Its solution must
therefore satisfy one initial and two boundary conditions. The initial condition
reads (cf. eq. (15.69))
;(n,0)=/,C|(5D(n- 1) (15.71)
where (¾ is the Dirac delta-function [Kom and Kom 1961 ], and the boundary
conditions ensuring stationarity for t —> » are
[dj(n, t)/dn]„ = , = 0, [dj(n, t)/dn]„ = M = 0. (15.72)
The determination of the non-stationary nucleation rate J(t) s /(n*, t)
is straightforward when the solution j(n, 0 of eqs (15.70)-(15-72)is known.
In Appendix A2 it is shown how the approximate J(t) dependence
(15.64) can be obtained by solving these equations rather than the Zeldovich
equation (9.27) subjected to the initial and boundary conditions (15.29)-
(15.31).
Non-stationary nucleation 249
15.3 Time lag of nucleation
As noted in Sections 15.1 and 15.2, the time scale of the approach to the
steady state of a system nucleating isothermally at constant supersaturation
A^i is set up by the time lag T of nucleation. In Chapter 17 we shall see that
T plays a major role also in determining the condition for quasi-stationary
nucleation. Thus, along with the stationary nucleation rate Js, the nucleation
time lag T appears as another basic quantity characterizing the kinetics of
nucleation. According to Kashchiev [1969a], Tis defined by eq. (15.55), but
other approximate solutions of the Zeldovich equation (9.27) under the initial
and boundary conditions (15.29)-(15.31) would lead to other definitions of
T (see, e.g. Zeldovich [1942]; Wakeshima [1954]; Collins [1955]; Walton
[1969a]). This point was discussed, for instance, by Abraham [1974a], Kelton
et al. [1983] and Kelton [ 1991 ] and will not be addressed here. The reliability
of T from (15.55) was checked numerically, e.g. by Kelton et al. [1983],
Volterra and Cooper [1985], Kozisek [1989], Shi and Seinfeld [1991a] and
Miloshev [1992] (see also Section 15.4).
Using eqs (7.43) and (13.35), we can represent f from (15.55) in the
following equivalent forms [Kashchiev 1969a]
T = 4/7r¥/* = 4A*Wf*. (15.73)
The last of these equalities allows a simple physical interpretation of T
[Feder et al. 1966], because it is an Einstein-type formula (cf. eq. (10.31))
relating Tto the width A* of the nucleus region and the attachment frequency
f* of monomers to the nucleus. Namely, we can view T as the mean time
needed for a cluster to 'cross' the nucleus region by 'walking' randomly
along the size axis from n = ri[ to n = n2 with a size-independent 'diffusion
coefficient'/* (see also Section 9.2). According to eqs (15.55) and (15.73),
the nucleation time lag Tis largely determined by/*, since the effect of the
other transition frequencies (fn and gn for n *■ n*) is amalgamated into the
single parameter /3 (or z or A*) which is virtually independent of the peculiarities
of the nucleation process. This implies that for an express, albeit rough,
estimate of T we may use the approximation
T= 10//* (15.74)
which results from (15.73) with typical z - 0.1. The conclusion is, therefore,
that T is more or less a direct measure of the reciprocal of/* and for that
reason is very sensitive to the concrete mechanism of monomer attachment
to the nucleus. This distinguishes Tfrom the other basic nucleation parameter
Js which is much more strongly dependent on, and thereby more informative
about, n* or W* than about/* (see, e.g. eqs (14.9)-(14.12)).
Let us now see what is the Ay dependence of T both in general and in
some particular cases of nucleation. Using (10.109) and (10.110) with t-
independent A/v and T, from (15.73) we find that for nucleation of condensed
phases, in general, T is given by
x=(4lj:iz2f*)e-'L'"kT (15.75)
250 Nucleation: Basic Theory with Applications
when A^i is not controlled by T according to (2.20) and by
T=(4/ff3zVo)e"A^'T (15.76)
when 7" is the parameter used to change Ari in conformity with (2.20). These
expressions for T are valid regardless of the kind of the nucleation process
(HON, HEN, 3D, 2D, etc.) and in them f*=fejl* and /0* =/0,„, are the values
of/* at Aij = 0. Being derived by treating n as a continuous variable, eqs
(15.55), (15.73)-(15.76) are mathematically more acceptable for n* > 10.
Nonetheless, with z = 1, these equations may be expected to describe
satisfactorily the t(Aju) dependence even in the atomistic limit of n* —> 1.
This expectation is supported by the reasonable correspondence between the
exact (n* = 3) and the approximate Z*(t) and J(t) dependences in Figs 15.2
and 15.3. That is why we shall regard eqs (15.55), (15.73)-(15.76) as applicable
to both classical and atomistic nucleation of any kind.
Equation (15.75) tells us that by virtue of the relatively weak dependence
of z and/e* on Aii the nucleation time lag T decreases monotonously with the
increase of the supersaturation A^i provided this is varied isothermally. This,
however, is not the case with rfrom eq. (15.76) which shows that when A^i
is changed by means of 7as required by (2.20), T first decreases with increasing
A^i (i.e. lowering /), but then passes through a minimum and rises sharply as
A^i goes on increasing by further lowering of T. This rise of T at lower
temperatures is due to the decrease of /0* with T. For instance, in crystal
nucleation in melts /0* °= 1 /7) and this rise is largely governed by the exponential
T dependence of the viscosity 7] (see eq. (10.56)) which thus appears as a
quantity of major importance in the pre-exponential factor in (15.76). In
what follows we shall exemplify the general T(A^i) behaviour in some particular
cases of nucleation and for various mechanisms of monomer attachment to
the nucleus by using concrete expressions for A/j, z and/* (or /e* and /0*)
in eqs (15.73), (15.75) and (15.76).
1. Nucleation of liquids or solids in vapours
(a) HON or 3D HEN under direct-impingement control
From (15.73), with the help of (2.8), (2.9), (10.3)-(10.5) and (13.68) we find
that
T= A(2miikT)"2l^llz2Yfcmv^ptn*mS (15.77)
where cDi is a numerical shape factor. For HON cDI = c with c = (36;r)1/3
for spheres, c = 6 for cubes, etc. For 3D HEN cDI * c. For example, cra =
(367t)"3(l - cos ew)/2i^2/3(6w) for caps on a substrate contacting the vapours
(then S = plPc) and cDI = (36;r)"3 sin2 8„/4i^\e„) for caps with 6W < nil on
a substrate during molecular beam condensation (then S - ///e). Equation
(15.77) gives directly the r(S) dependence in the case of atomistic nucleation,
as then z ~ 1 and n* - 1,2,... is ^-independent in a given S range. Classically,
however, with n* and z from (4.38) and (13.36), eq. (15.77) leads to (S > 1)
T= 16c(2m0)"2o-ef/ff3/27*cDI(A:r)"2/>(;S(lnS)2 (15.78)
where ae! = ffforHON and acf< dforHEN (seeeq. (4.42)). Apart from the
Non-stationary nucleution 251
numerical factor, with cDI = c and aef = a (i.e. in the case of HON) this TiS)
dependence becomes the one given by Toschev [1973a].
Equation (15.78) shows that the time lag is infinitely long at S = 1 (i.e. at
saturation) and that it diminishes monotonously with increasing supersaturation
ratio 5. From eq. (15.77) we see that, atomistically, the T(S) dependence is
a broken straight line when plotted in T-vs-(lAS) coordinates. The breaks in
the slope of this line can manifest themselves only if the S range is large
enough to ensure the stepwise diminishing of «* with increasing S (see
Section 4.4). From eqs (15.77) and (15.78) we see also that in HON, at the
same T and 5 values, T is greater for substances with lower vapour pressure
pe and/or higher specific surface energy o. Another point to emerge from eq.
(15.78) is that the time lag THEN in HEN is related to the time lag tHon m
HON (then cDI = c and acf = a) by the formula
Then=(c/cd1)'PtHoN (15.79)
where Y, a number between 0 and 1, is the activity factor appearing in eq.
(4.42). In particular, V= y"3(8w) for cap-shaped nuclei, y/(9„) being given
by (3.56). Hence, for such nuclei we have
%en = [2v(0„)/(l - <=os ew)]THON (15.80)
when the substrate is in contact with the vapours and (8W < 7tl2)
%en = [4v(6w)/sin2 6J TH0N (15.81)
during molecular beam condensation. Equation (15.80) was obtained by
Toschev and Gutzow [1967a] who concluded that tHBn < ?hon when the
nucleus/substrate wetting angle 8W is small enough. The relation between
THEN and THON in other cases of nucleation was also considered by these
authors.
Curve DI in Fig. 15.5 illustrates the T(S) dependence (15.78) for HON of
water droplets in vapours at 7= 293 K. The calculation is done with cD1 = c,
CJef = a and the parameter values given in Table 3.1. The numbers at the
circles on the curve indicate the number «* of water molecules in the nucleus
droplet at the corresponding supersaturation. As seen, in this case T < 64 ns
in the typical experimental range of S > 3.5. According to eq. (15.74), this
rather small value of Tis due to the relatively high frequency/" of monomer
attachment to the nucleus droplet (see Fig. 10.2).
(b) 3D or 2D HEN under surface-diffusion control
In this case, when/* is given by (10.42) or (10.43), from (2.8), (2.9), (13.68)
and (15.73) it follows that
?=4ln3z2r*c*^IcS (15.82)
where /e = pJ(l7tm^kTfa. With z~\, this equation represents the atomistic
T(S) dependence in the cases of both 3D and 2D HEN. Classically, however,
Tis a different function of S in these two cases. Indeed, accounting for z from
(13.36) and (13.37) transforms (15.82) into (S > 1)
252 Nucleation: Basic Theory with Applications
SD Dl
100
\70
\v^50
250 ^
100
~~*^-^^_ 50 35 25
u i l i " ' ■ <l
12 3 4 5
s
Fig. 15.5 Time lag of nucleation as a function of the supersaturation ratio in HON
of spherical and HEN of hemispherical water droplets in vapours at T = 293 K
according to eq. (15.78) for direct-impingement control {curve Dl) and eq. {15.83)
for surface-diffusion control {curve SD). The numbers at the symbols indicate
the nucleus size at the corresponding supersaturation ratio.
T= (Acivlal,l9it:ir*c*(kTf^ilcS(\nS)i (15.83)
for 3D HEN and into (In S > atfAo/kT)
x = i,blKLl7v,y*c*(klf)illeSiya S - a^alkTf (15.84)
for 2D HEN on a foreign (Act * 0) or own (Act = 0) substrate.
Equations (15.82)-(15.84) reveal that the time lag X decreases again
monotonously with increasing 5 and that it is shorter for longer mean distance
A, of monomer surface diffusion. Curve SD in Fig. 15.5 depicts the t(S)
dependence (15.83) for HEN of hemispherical water droplets on a foreign
substrate held at 7= 293 K in vapours and characterized by a relatively small
ls = 10 nm. The calculation is done withe = (36¾) "3,c* = 1.9, a]t =(1/2)03
and the parameter values listed in Table 3.1. The numbers at the triangles on
curve SD indicate the n* values at the corresponding supersaturations. We
observe that T is less than several nanoseconds for S > 2.5, which reflects the
fact that /* is rather high also when monomer attachment is controlled by
surface diffusion (see Fig. 10.6). The quite general conclusion is, therefore,
that under both direct-impingement and surface-diffusion control, typically,
the time lag T in the cases of both HON and HEN is negligibly small. This
means that nucleation in vapours at time-independent supersaturation can
usually be regarded as occurring practically in stationary regime. However,
it is worth keeping in mind that if 7* « 1, i.e. if monomer sticking to the
80
60
40
20
Non stationary nucleation 253
nucleus is strongly impeded, Tcould be significantly long even under typical
experimental conditions.
2. Nucleation of condensed phases in solutions
(a) HON or 3D HEN under volume-diffusion control
We can now employ eqs (10.20) and (10.24) at time-independent C to find
with the aid of (2.14), (13.69) and (15.73) that
T= VKll2z2y*c%v'(1" DC,n*mS (15.85)
where cVD is a numerical shape factor. For HON cVD = c with c = (36;r)l/3 for
spheres, c = 6 for cubes, etc., and for 3D HEN of caps with 0W > nlA,
approximately, cVD = (36;r)"3(l - cos 6w)2/4l/^'3(ew). Equation (15.85) gives
the t(S) dependence for atomistic nucleation, since then z = 1 and n* is S-
independent. We see that due to abrupt changing of the nucleus size «*, as in
the case of nucleation in vapours, this dependence may appear as a broken
straight line in T-vs-(l/S) coordinates. Classically, however, n* and z depend
on the supersaturation ratio Saccording to (4.38) and (13.36) when nucleation
is 3D so that from (15.85) we obtain (S > 1)
T= 16c2D0cey37rV2y*cVD"2(/t7O2DCcS(lnS)3 (15.86)
where o"ef = a for HON and aet = y/"3(9„)o for 3D HEN of caps (see eq.
(4.42)).
Qualitatively, the r(S) dependences (15.85) and (15.86) are similar to
those for 3D nucleation in vapours. We note also that the time lag is longer
for nucleation in solutions in which the monomers have lower solubility and/
or diffusivity. Curve VD in Fig. 15.6 represents the f(5) dependence (15.86)
for HON of spherically shaped crystals of sparingly soluble salts in aqueous
solutions at T = 293 K. The parameter values used are those given in
Table 6.1 and it is taken into account that for HON o"ef = <J and cVD = c =
(36.T)"3. As seen from Fig. 15.6, X is of the order of microseconds in the
typical experimental range of S = 5 to 20.
(b) HON or 3D HEN under interface-transfer control
In this case we have to use eqs (10.60) and (10.61) with time-independent C
for the determination of/* in (15.73). Recalling again (2.14) and (13.69), we
thus obtain
x = 44/ff3z27*c1Tu2/3DCen*2/3S. (15.87)
In HON, for the numerical shape factor err we have cn = c with c = (36;r)"3
for spheres, c = 6 for cubes, etc. In HEN, if the nuclei are cap shaped, qT =
(36;r)"3(1 - cos ew)/2v^'3(ew). With z = 1 and S-independent n*. eq. (15.87)
is the atomistic T(5) dependence which, graphically, is again a straight (possibly
broken) line in T-vs-(l/S) coordinates. Classically, using n* and z from (4.38)
and (13.36) transforms (15.87) into (S > 1)
T= lecdaajT^y^cnkTDQSQn S)2 (15.88)
where o"ef = o" for HON and cvf < O for HEN.
254 Nucleation: Basic Theory with Applications
10
a
IT \
\
-
_
vd\
\l50
VlOO
\100
■ .
\80
^80
i i 1
^\50
- 50
10
S
15
20
Fig. 15.6 Time lag of nucleation as a function of lite supersaturalion ratio in HON
of crystals of sparingly soluble salts in aqueous solutions at T = 293 K according to
eq. (15.86) for volume-diffusion control (curve VD) and eq. (15.88) for interface-
transfer control (curve IT). The numbers at the circles indicate the nucleus size at
the corresponding supersaluration ratio.
Comparison of eqs (15.86) and (15.88) shows that the <S) dependence is
nearly the same under volume-diffusion and interface-transfer control. In the
latter case, however, Tis less sensitive to the e0ective specific surface energy
a„f of the nucleus/solution interface. Also, from eq. (15.88) it follows that
when monomer attachment is controlled by interface transfer, the connection
between Them for 3D HEN of caps and TH0N for HON of spheres is again
given by eq. (15.80). The t(S) dependence (15.88) is illustrated in Fig. 15.6
by curve IT for HON of spherically shaped crystals of sparingly soluble salts
in aqueous solutions. The calculation is done with the help of the parameter
values from Table 6.1. As seen, the time lag is again in the microsecond
range. The numbers at the circles on curves IT and VD represent the
corresponding n* values.
(c) 2D HEN of monolayer nuclei under interface-transfer control
Since now/* is given by eq. (10.63) at n = n* and constant C, owing to
(2.14) and (13.69), from (15.73) we find that
T = 4vol7??f>bDKcCji*lnS
(15.89)
where b = 2(jrfl0)1/2 for disks, b = Aa^2 for square prisms, etc. With z •» 1
and S-independent n*, this equation is in fact the T(S) dependence of
atomistically small 2D nuclei of monolayer height (T vs IIS is again a straight
line with possible breaks). To obtain the classical dependence of x on S we
Non-Stationary nucleation 255
setaef = a0in eqs (4.32) and (13.37) forn* and z and combine them with eq.
(15.89), the result being (In S > aoAalkT)
X - iv0Kl7r1'y*kTDKcCcS(\}\ S - a0Ao/kT)2 (15.90)
where Aa * 0 for foreign and Ao" = 0 for own substrate. As seen, this t(S)
dependence is analogous to those given by eqs (15.86) and (15.88). In fact,
for 2D nuclei on own substrate Tfrom (15.90) shortens with increasing S in
precisely the same way as does Tfrom (15.88) although eq. (15.88) applies
to 3D nuclei. With K= d^o, KCCC = 0.001 (according to (10.47), this corresponds
to 0.1% adsorption coverage of the substrate at C = Ce), Ao - 0 and the
parameter values in Table 6.1, eq. (15.90) predicts T = 18 ns at S = 5. The
rather general conclusion is, therefore, that for both HON and 3D or 2D
HEN of condensed phases in liquid solutions the time lag is usually negligible
when monomer attachment is controlled by volume diffusion or interface
transfer. However, if y* « 1 (i.e. if monomer sticking to the nucleus is
impeded), rcould have considerably high values. Also, in the case of nucleation
in solid solutions or in more viscous liquid ones the time lag may be of the
order of hours or days, since in such solutions the monomer diffusion coefficient
D can be quite small. We note as well that when it is more convenient, eqs
(15.85)-(15.90) can always be used with the solution viscosity T\ introduced
in the place of D by means of the Stokes-Einstein formula (10.54).
3. Nucleation of crystals in melts
(a) HON or 3D HEN under interface-transfer control
In this case, from eqs (10.64), (10.65) and (15.73) it follows that
t=\L\2dlv^r\(T)lK1z1y*cukTn*m} e(1-"» (15.91)
where clT is the shape factor appearing in (15.87), X = 7eAse, and A^i is given
by (2.20). We recall that when applied to 3D HEN, eq. (15.91) implies that
none of the molecules joining the nucleus is in contact with the substrate.
This equation is a concrete form of eq. (15.76). Since the melt viscosity rj
depends on T(see eq. (10.56)) and thereby on A/j, eq. (15.91) shows that the
7/(7) function has a strong effect on the dependence of T on Ay. Again, with
z~ 1 and A/u -independents* eq. (15.91) is the atomistic formula for the time
lag t when attachment of monomers from the melt to the nucleus is controlled
by interface transfer. The corresponding classical formula is readily obtained
from (15.91) with the aid of n* and z from (4.38) and (13.36) (A(i > 0);
T= [48crf02u0CJefr/(r)/n27*cITA^2] ea-^m. (15.92)
If the undercooling ATs Tc - 7 is related to Afi by (2.23), this equation leads
to the following explicit dependence of T on T (0 < AT < IT^AsJAc^):
x = Aicdlv0 exp (AsJ^aanCDI^c-aAs^AT1. (15.93)
This expression applies to both HON and 3D HEN provided in the latter
case the nucleus attaches only molecules that are not in contact with the
substrate. In the case of HON (then c% = a, clT = c) eq. (15.93) was used by
Toschev [1973a], but with a different numerical factor and without the
256 Nucleation: Basic Theory with Applications
exponential one. As seen from (15.93), the connection between THEN and
tHon is again given by eq. (15.80), i.e. for small enough nucleus/substrate
wetting angle the time lag in 3D HEN of crystals in melts is also shorter than
that in HON. Equation (15.93) reveals as well that at T= T„ (then AT = 0) X
is infinitely long and that with lowering T it first shortens due to the AT* term.
However, as r\ is higher at lower 7", after passing through a minimum x
begins to increase with further undercooling. Thus, in contrast to the time lag
for nucleation in vapours and solutions at isothermally varied A/;, X is not a
monotonous function of A^i when this is controlled by T in accordance with
(2.20).
Equation (15.93) tells us that T increases linearly with the melt viscosity.
As evidenced experimentally (e.g. Gutzow et al. [1968]; Koverda et al.
[1974]; James [1974]; Skripov [1977]; Kalinina et al. [1977, 1980, 1997];
Fokin etal. [1977,1981, 1997]; Koster [1978]; Gutzow [1980]; Skripov and
Koverda [1984]; Schiffner and Pannhorst [1987]; Kelton [1991]; Deubener
et al. [1993]), this means that the time lag may extend over hours and days
during nucleation at such low temperatures (e.g. close to the glass-transition
temperature rg) at which the melt viscosity is sufficiently high. The curve in
Fig. 15.7 depicts the t(A7*) dependence (15.93) for HON of spherical ice
crystals in water at atmospheric pressure. The calculation is done with c[T =
c, (% = a (assumed ^-independent), t}(T) from (13.47) and the parameter
values listed in Table 6.2. As seen, for AT between 5 and 60 K the time lag
140
Fig. 15.7 Time lag of nucleation as a function of the undercooling in HON of ice
crystals in water at atmospheric pressure according to eq. (15.93) for interface-transfer
control. The numbers at the circles give the nucleus size at the corresponding undercooling,
and tire doited line indicates the undercooling at the glass-transition temperature.
Non stationary nuclealion 257
Tis shorter than a microsecond, but increases sharply with T approaching the
glass-transition temperature Tg ~ 135 K [Skripov and Koverda 1984] of
water. The numbers at the circles in Fig. 15.7 represent the n* values following
from eq. (4.38) at the corresponding undercoolings.
(b) 2D HEN of monolayer nuclei under interface-transfer control
Combining eqs (10.66) and (15.73), we now obtain
T= [\2dtsv(sr\(T)lKiz2Y'bkTn*m\ exp [(A-E, - asa0 - Ap)lkT\. (15.94)
We note again that with z = 1 and ^-independent n* this is in fact the
atomistic formula for ras a function of Ay. Classically, n* and z change with
Aft according to eqs (4.32) and (13.37) so that eq. (15.94) becomes (A^i >
a0Ao)
x= [24(i0yo")(7)/^7*(A/J-a0AcJ)2]exp [(A- E, - asaB - Ap)lkT\
(15.95)
where Ao * 0 for foreign and Ao = 0 for own substrate. Hence, in terms of
the undercooling AT, using (2.23) we find {a^AolAs, < AT < 2TeAs„/Acp e)
T = [24d0v0 exp (Asjk) KTi(T)l7C2y*A s2 (AT - aoAo/Ase)2]
exp [- (E, + osa0)lkT]. (15.96)
Equations (15.94)-( 15.96) are analogous to (15.91)-(15.93), which means
that all of the above conclusions concerning the T(A/() dependence for HON
or 3D HEN are valid for 2D HEN, too. The new knowledge about f from
(15.94)-(15.96) is the presence of the exponential factor which contains the
energy term E| + asflo and thus takes into account that the molecules joining
the nucleus periphery are adsorbed on the substrate. Physically, E^ + osa0
corresponds to the desorption energy £des of a molecule (see Section 10.1).
4. 3D nucleation of gaseous phases under evaporation control
In this case the time lag Tis also given by eq. (15.73) so that using (10.9) at
n - n* and recalling that the nucleus vapour pressure pefl* is equal to the
pressure p* inside the nucleus yields
z= 4(2m0)"2/;r5'2rycEV(A:D"V"3n*2'3. (15.97)
Here cEV is a numerical shape factor: for HON cEV = c with c = (36;r) "3 for
spheres, c - 6 for cubes, etc., and for 3D HEN of caps cEV = (36;r)i/3(l - cos
ew)/2i^'3(ew). As noted in Section 13.2, the Zeldovich factor z for HON or
3D HEN of gaseous phases is given approximately by eq. (13.36). Thus,
with the aid of the first equality in (13.36) and of eqs (4.40) and (4.41) we
find that (0 < p < pe)
T= \bc[2m(s)mp*aj7ci'2y*cEy(kT)m(p* -pf (15.98)
where oef = a for HON and oel < a for 3D HEN. Since p* depends on p
according to (4.14) in which Ap 5/>e -p, whenpeu0 « kT, in (15.97) and
(15.98) we can use the approximations /J* - pe and p* - p = Ap.
25S Nucleation: Basic Theory with Applications
Equation (15.98) shows that the time lag Tis infinitely long at A/> = 0 (i.e.
at saturation) and that it shortens monotonously if the underpressure A/> is
increased isothermally. This behaviour of T is analogous to that for nucleation
of condensed phases in vapours and solutions when the supersaturation ratio
S is varied at constant T. Equation (15.98) leads to the conclusion that THEN
and thon for HON of spherical and 3D HEN of cap-shaped gaseous nuclei
under evaporation control are connected also by eq. (15.80). The curve in
Fig. 15.8 illustrates the T(A/>) dependence (15.98) for HON of steam bubbles
in water at T = 583 K. The calculation is done with cEV = c, o-ef = a, p* - pc
and the parameter values listed in Table 3.2. The numbers at the circles on
the curve indicate the number n* of water molecules in the nucleus bubble
at the corresponding underpressure. Since evaporation is a relatively fast
process, according to (15.74), the corresponding rather high value of/* is
the reason for the negligibly short time lag T in bubble nucleation controlled
by evaporation. It must be pointed out, however, that for other mechanisms
of mass transport Tmay have much higher values. For instance, under viscous-
flow control, analogously to (15.91 )-(15.96), Tis proportional to the viscosity
t] of the liquid [Carlson and Levine 1975] and can be as long as, e.g. an hour
or a day.
Ap/pe
Fig. 15.8 Time lag of nucleation as a function of the underpressure in HON of steam
bubbles in water atT = 583 K according to eq. (15.98) for evaporation control. The
numbers at the circles indicate the nucleus size at the corresponding underpressure.
15.4 Delay time of nucleation
In Section 15.2 we have seen that, graphically, at no pre-existing supernuclei
the J(t) dependence at constant supersaturation is a sigmoidally shaped curve
Non-stationary nucleation 259
starting at the coordinate origin and saturating as time goes on (Fig. 153).
The practically complete saturation of J at its stationary value Js occurs after
a period of the order of t. The time lag Tis thus the parameter characterizing
the duration of the non-stationary regime in isothermal nucleation at constant
Aji. It is clear, however, that this duration can be equivalently characterized
by other time parameters such as, e.g., the time r1/2 at which J = (\/2)Js. This
is so because these parameters are unambiguously related to X. For instance,
Fig. 15.9 Time dependence of (a) the non-stationary rate of nucleation and (b) the
concentration of supernuclei. The arrows indicate the nucleation time lag T, the time
tia at which J = (1/2)JS and the delay time 6 of nucleation, the latter being defined by
the equality of the hatched areas A and B or by the intercept of the t —> °° asymptote
{the dashed line) of the tft) function.
260 Nucleation: Basic Theory with Applications
with 7(r) from (15.64) we calculate that tm = 1.38T. The time tm is indicated
in Fig. 15.9a where it is seen also that there exists a characteristic time ft
called often delay time ofnucleation, which makes equal the area A between
the J(t) curve and the time axis in the 0 < t < 9 range and the area B between
the J(t) curve and the J, line from (= 9 to t = . A mathematical expression
of the equality of these areas is the following definition of the delay time 9
[Andres and Boudart 1965]
6= f [l-V(0/7s]dt. (15.99)
Jo
Physically, at no pre-existing supemuclei in the system 8 determines a
thought process of non-stationary nucleation occurring at zero rate from t -
0 to t = 9 and at stationary rate Js for t > 9 (see Fig. 15.9a). In other words,
this thought non-stationary nucleation is a delayed stationary nucleation, the
time of the delay being ft We note that because of (11.10), when ^, = 0 (no
pre-existing supemuclei), 9 from (15.99) can be defined equivalently by
((t) = Js(t-8) atr-»», (15.100)
i.e. 9 is merely the time intercept of the long-time asymptote of the C,(t)
function (Fig. 15.9b). Equation (15.100) tells us that in the t —> °° limit the
concentration Js(t-9) of supemuclei formed in the delayed stationary process
is equal to their actual concentration £(t) resulting from the real non-stationary
process.
From eq. (15.99) we see that the delay time 9 can be determined if we
know explicitly J as a function of t. For example, using J(t) from eq. (15.59)
or (15.60) in eq. (15.99) yields readily the following exact formula for the
delay time («* = 2, 3, . . . , M- 1):
M-\ M-\
9 = [ Z d,</*a„.,- - g„, + ,a„. +,,,-)]-' I (d,/A,)(/*a„,, - g„, + ,a„. + ,.,)
i=2 i=2
M-\
= -(lW'7s) Z WA)(/*fl„.,■-£„• + ,a„.+l ;). (15.101)
( = 2
The same result is obtained also with the help of J(t) from (A 1.12), but then
9 is expressed in terms of the polynomial PM _„«_,.
The problem with the application of eq. (15.101) is that A,, ani and d,- are
obtainable only approximately from (15.10), (15.11) and (15.17) when M =
Mef > 6. Thus, in general, the dependence of A,-, a„, and d; and, thereby, of 9
on the attachment and detachment frequencies /„ and g„ cannot be revealed
easily. To get a feeling about the relation between 9 and these frequencies we
can consider the particular case of M = Wet = 4 and n* = 3 analysed in
Sections 15.1 and 15.2. Either by using in (15.101) the results obtained
in these sections for A„ a„, and d, or by introducing J(t) from (15.61) in
(15.99) and performing the integration, owing to (15.20) and (15.21) we find
that
Non-stationary nucleaiion 261
6 = (/2 + /3 + Si + mWifi + gifi + gigi) (15.102)
where/3 = /*. We observe that ft does not depend on the monomer-to-
monomer attachment frequency/. Also, with the/2,/3, g2, #3 and rvalues
given in Section 15.1 we calculate 9 = 0.298 ns and ft/T = 1.89 in the case
considered there of HON of water droplets in vapours.
Equations (15.101) and (15.102) represent the delay time when the cluster
size is considered as a discrete variable. If n is treated as varying continuously,
the analogue of (15.101) is the expression
0= [I c,a'*Yl X (c,/X;)a;* = (f*C*IJ,) X (e,/A,)a,'* (15.103)
; = l . = 1 i = l
which follows after using J(t) from (15.62) or (15.63) in eq. (15.99) and
which parallels the formula of Shizgal and Barrett [1989].
Clearly, if we want to employ eq. (15.101) or (15.103) for practical purposes,
we should transform it into a sufficiently simple approximate formula for ft
Instead of doing that, however, we can obtain such a formula directly from
eq. (15.99) with the aid of available approximate J(i) dependences. We shall
now make use of J(t) from (15.64) in order to find an approximation to ft
from eq. (15.103). Performing the integration in (15.99) with J(t) from (15.64),
taking into account that [Gradshtein and Ryzhik 1962]
X (-l)7i2 = -;r2/12 (15.104)
1=1
and recalling eqs (15.55) and (15.73) leads to the formula [Kashchiev 1969a]
ft= (^/6)7= 2/3/32/* = 2BlK2f*. (15.105)
Equation (15.105) shows that in the scope of the approximations involved
in the derivation of J(t) from (15.64) the distinction between the delay time
ft and the time lag T is only in a numerical factor: ft is merely ;r2/6 »1.6 times
greater than T. This implies that all results and conclusions in Section 15.3
concerning T are directly applicable to ft from (15.105) after allowing for the
presence of the factor 7r2/6. In particular, eq. (15.105) says that ft is in fact
determined only by the frequency/* of monomer attachment to the nucleus
- the effect of/, and gn for n * n* is minor and is taken into account by the
virtually constant factor f) or z. From practical point of view eq. (15.105) is
quite useful, because it allows an easy, albeit approximate, determination of
the time lag t from experimental data for the delay time 9 [James 1974,
1982; Kalinina et al. 1977; Gutzow 1980; Fokin et al. 1981; Penkov and
Gutzow 1984]. Nowadays, eq. (15.105) is used extensively for analysing
non-stationary effects in various nucleation experiments (for a review see,
e.g. Kelton [1991]).
The above considerations show that although leading to the exact formulae
(15.101) and (15.103), in general, the method of direct usage of J(t) for
determination of ft from (15.99) does not allow the exact dependence of ft on
the transition frequencies/, and gn to be revealed explicitly. Impossible as it
262 Nucleation: Basic Theory with Applications
may seem, this dependence can be obtained from (15.99) explicitly without
having any idea about the concrete form of the 7(/) function. This 'indirect'
method of finding B was used by Andres and Boudart [1965], Hile [1969],
Frisch and Carlier [1971], Shizgal and Barrett [1989], Wu [1992a, 1992c]
and Shneidman and Weinberg [1992a, 1992b]. Following closely the elegant
derivation of Wu [1992a], let us now determine the exact and explicit
dependence of 9 on/„ and gn by considering n as a continuous variable.
Our goal is to find out a representation of the integrand in eq. (15.99) in
the form of a time derivative, for then performing the integration is trivial.
We start by integrating the Zeldovich equation (9.27) from n = m to n = n*:
j" Z(n,t)dn = J" ^\f{n)C(n)^-[Z(n,t)/C(n)]\dn. (15.106)
),(13.18),(13.31),(
) we can write
(£{f(n)C(n)
d_
dt
Using eqs (11.9), (13.18), (13.31), (13.32) and (15.1), for the integral on the
right of (15.106) we can write
n)^[Z(n,t)IC(n)]\in
= -7(0 - f(m)C(m)d[Z(m, t)/C(m)]/3m
= -J(t) + JsldiX/Q/dmT'idiZ/QIdm]
so that (15.106) becomes
-3(ZJC)l3m + [J(t)lJs][d(XIC)ldm]
'-""■'5
f
Zdn
[d(X/C)/dmH
We now integrate this equality from m = 1 to m = M and get
- f -^- (ZIC) dm + [ J(t)IJs] [ -^(XlC)dm
- -(ws)i (r [ch [t(x/c>] "4 (,5-io7a>
Employing the boundary conditions (13.4), (13.5), (15.30) and (15.31), we
find
dm
(Z/Q dm = [Z(m, t)/C(m)}m = M - [Z(m, t)/C(m)]m = , =
■^ (X/C) dm = [X(m)IC(m)]m = M - [X(m)/C(m)]m , , = -1,
Non-stationary nucleation 263
nfH[^(x/c)]dm
[f
[X(m)IC(m)]
Z(n,t)dn
[X(m)/C(m)J
[X(m)/C(m)][-Z(m, /)] dm
Jl
= - Z(n, /)dn+ [Z(n,/)X(n)/C(n)]dn.
Hence, eq. (15.107a) transforms into
1 - 7(()/7, = (1/7,) -^- | [ Z(n, t)dn- [ [Z(n,/) X(n)/C(n)] dn L
(15.107b)
This form of 1 -J{t)IJs is ready for integration in accordance with (15.99).
Thus, finally, owing to the initial condition (15.29) and the fact that Z(n, «>)
= X(n), we arrive at the sought exact formula for the delay time 9 [Wu 1992a;
Shneidman and Weinberg 1992a]:
9 = (1/7,)
\( X(n)dn- f [X2(n)IC(n)]dn\
(15.108)
Here C(n) is given kinetically or thermodynamically by eq. (12.5) or (7.4),
respectively, andX(n) and 7, are specified by eqs (13.18) and (13.32). Hence,
depending on the chosen description of C(n), eq. (15.108) represents 9 either
purely kinetically or as a 'mixture' of kinetics and thermodynamics.
In analogy with similar results in Chapters 9, 12 and 13 we may expect
that if the integrals in eq. (15.108) are replaced by sums, this equation will
become the exact formula for 9 in the case when n is treated as a discrete
variable. This is indeed so - in this case there holds [Hile 1969; Wu 1992a,
1992c; Shneidman and Weinberg 1992b]
6= (1/7,) [Z Xn- Z (X„2/C„)].
(15.109)
When here C„,X„ and 7, are given by (12.3), (13.16) and (13.29), eq. (15.109)
represents 9 as an explicit function of the transition frequencies /„ and g„.
264 Nuclealion: Basic Theory with Applications
For instance, in the particular case of M = Met = 4 and n* = 3, from the
above-mentioned equations we have C2 = C{f1/g2, C3 = C^gi X2 = ^1/1(/3
+ #3)/(/2/3 + gifi + gigs), X3=XJ2/(f3 + g3) and J, = /3X3 so that eq. (15.109)
passes into eq. (15.102) obtained from (15.99) by direct integration of the
corresponding exact 7(0 function (15.61). It is worth noting again that,
alternatively, if C,„ Xn and Js are expressed by means of eqs (7.4), (13.17)
and (13.30), eq. (15.109) gives 8 as a 'mixture' of kinetics and thermodynamics.
Inspection of eqs (15.108) and (15.109) shows also that, as already mentioned
in respect to 6 from (15.102), the delay time is independent of the monomer-
to-monomer attachment frequency/,.
Equations (15.108) and (15.109) are complementary in the sense that
while the latter is more convenient for numerical calculations, the former is
more easily handled analytically. In particular, eq. (15.108) can be used for
finding approximate, but physically more informative expressions for 9 [Wu
1992c; Shneidman and Weinberg 1992a, 1992b], Following Wu [1992c], let
us employ the approximations (7.37) and (13.21) for C(n) andX(n) in order
to perform the integrations in (15.108). In lieu of (15.108) we get
r°
2PJS&C* = exp(*2)[l -erf(;c)]dx
rf(»-»<)
-(1/2) exp(jc2)[l -erf(jc)]2djr
exp (x2) erf(jc) [1 - erf (jr)] dx
Jo
-(1/2) exp(x2)n -erf(x)]2djr. (15.110)
J0(n*-ll
Since this result is valid under the condition (13.22), i.e. for large enough
nucleus size «*, using the approximations erf (x) = (2/k]/2)x for x < 1 and 1-
erf {x) = (l/Jr"2*) exp J-*2) for;c> 1 [Carslaw and Jaeger 1959] and recalling
that exp (x2) = 1 + x2 for x < 1, we find that
exp (jc2) erf (jc) |1 - erf (x)] dt
Jo
fl (•/!>'*
= (2/?r"2) (\+x1)x(\-lK-mx)Ax+(\lnm) \ (lft)dx
Jo Jl
= ](45;r"2 - 64)/30^] + (\lnm) In (0n*),
expU2)[l -erf(x)]2dbt = (\ln) \ (1/x2) exp (-;c2)djr
= Olrfn*) exp (- 0V2) - (I/O [1 - erf (#1*)] = 0.
Non-stationary nucleation 265
With these results for the last two integrals in (15.110) and with the help
of eqs (13.33), (13.35), (15.55) and (15.73) we finally obtain [Wu 1992c;
Shneidman and Weinberg 1992a]
6= [In (fin*) + 0.3]/2jS2/* = [In (7t"2zn*) + 0.3]/2;rz2/*
= (n2[ln(j3n*) + 0.3]/8!T. (15.111)
This approximate formula for the delay time 6 is valid for fin* > 1 and
shows again that 0 is proportional to the time lag r (cf. eq. (15.105)). The
distinction between (15.105) and (15.111) is only in the proportionality factor:
according to (15.111) this factor is a function of the nucleus size «*, rather
than just a number as predicted by (15.105). It is worth noting that eq.
(15.111) parallels the formula of Lyubov and Roitburd [1958] for the time
necessary for Z*(t) to reach 99% of its stationary value X* (see also Lyubov
[1969]). From eqs (7.39) and (7.40) we find that, classically, fin* = (W*l
3kT)"2 and fin* = (W*/4kT)m for 3D and 2D nucleation, respectively. In
view of eqs (4.39) and (4.33), this means that for the dependence of fin* on
the supersaturation Ay we have fin* ^ 1/A/y for 3D and fin* x 1/(A/; -
aefAcr)l/2 for 2D nucleation. Since under typical experimental conditions 10
< W*/kT< 60, fin* usually has values from 1.8 to 4.5. Hence, the value of the
&x ratio from (15.111) is typically between 1.1 and 2.2 and is comparable
with the value rf/6 = 1.64 predicted by eq. (15.105). This result is important,
because it is an evidence for the reliability of the time lag x defined by
(15.55) or (15.73).
Figure 15.10 displays 6Cras a function of fin* according to eq. (15.111)
(the solid line) and eq. (15.105) (the dashed line). As seen, the agreement
between (15.105) and (15.1 II) is reasonable, especially in view of the many
uncertainties in the theoretical determination of the monomer attachment
frequency/*. The dotted line in Fig. 15.10 visualizes the prediction &x- t?I
96 ~ 0.32 following from the first equality in (15.105) with the time lag r
= \Hnfif* found by Collins [1955] (see Section 15.2). We see that this
prediction does not agree well with that from eq. (15.111). The cross in Fig.
15.10 illustrates the exact Six value calculated from eq. (15.102) in then* =
3 case considered in Sections 15.1 and 15.2. The down triangle and the
squares, circles, up triangles and stars also represent exact Bit values: they
were obtained numerically by Shizgal and Barrett [1989], Abraham [1969],
Kelton etal. [1983], Miloshev [1992] and Shneidman and Weinberg [1992b],
respectively. Most of these values are less than 50% different from those
predicted by eqs (15.105) (the dashed line) and (15.111) (the solid line). All
in all, therefore, it can be concluded that, trading accuracy for simplicity, we
may employ eq. (15.105) with a sufficient degree of confidence both for
theoretical analyses and for interpretation of experimental data. Equation
(15.111) is important for more elaborate considerations and in this respect it
must be noted that higher order approximate formulae for the 9/x ratio are
also available [Shneidman and Weinberg 1991, 1992a, 1992b],
Figure 15.11 displays the experimental 8(AT) data of Gutzow el al. [1968]
(the circles) and of Koverda et al. [1974] (the squares), obtained in the case
266 Nucleation: Basic Theory with Applications
fin*
Fig, 15.10 Dependence of the 6/x ratio on the thermodynamic parameter /3«*- solid
line - eq. {15.111); dashed line - eq. (15.105); dotted line-approximation of Collins
[1955]; cross-exact value according to eq. (15.102); down triangle-numerical
finding ofShizgal and Barrett [1989]; squares -numerical data of Abraham [1969];
circles-numerical data of Kelton et al. [1983]; up triangles-numerical data of
Miloshev [1992]; stars-numerical data of Shneidman and Weinberg [1992b].
of nucleation of, respectively, NaP03 crystals in NaP03 glass and ice in
vitrified water. We note, however, that since the data of Koverda et al.
[1974] (see also Skripov [1977]) are not obtained directly with the help of
£(r) dependences by using eq. (15.100), they have to be regarded as representing
with some uncertainty the actual delay time 9 of the nucleation process. The
dashed and the solid curves in the figure depict the 6{AT) dependences
calculated, respectively, from eqs (15.105) and (15.111) with the help of r
from eq. (15.93) describing 3D nucleation under interface-transfer control.
In eq. (15.93) itself the r\{T) function is introduced by means of eqs (10.56)
and (13.47), and the parameter values used in the calculation are listed in
Tables 6.2 and 15.1.
As seen from Fig. 15.11, the agreement between theory and experiment is
reasonable and the difference between the simple and the improved 91%
formulae (15.105) and (15.111) is experimentally insignificant. What must
be noted is that this agreement is a result of treating y* in eq. (15.93) as a
free parameter and of assuming that y* = 2 x 10~3 and 2x10"* for the ice and
the NaP03 crystal nuclei, respectively (the latter y* value is comparable with
the value of 10"6 reported by Gutzow and Toschev [1968]). If it is assumed
that y* = 1, curves 2 x 10~3 and 2 x 10"6 in Fig. 15.11 'fall down' orders of
magnitude (they become curves 1) and, quantitatively, theory and experiment
disagree strongly. This finding emphasizes the important role of the monomer
Non-stationary nucleation 267
10°
105
10"
2 103
102
10
80
100
Fig. 15.11 Delay time of nucleation as a function of undercooling: squares-data for
ice crystals in vitrified water [Koverda et at. 1974; Skripov 1977]; circles - data for
NaPO$ crystals in NaPO^ glass [Gutzow et at 1968/; solid curves-eq. (15.111);
dashed curves - eq. (15.105). The numbers at the curves give the y* values used in the
calculation, and the dotted lines indicate the undercoolings at the glass-transition
temperatures of water and NaPO^.
Table 15.1 Values of various quantities used for calculation of the delay time 6 in
3D nucleation of crystals in NaPOi glass.
(-1)
0.066
d„
(nm)'
0.50
(K)
898
Asjk
3.0
(mJ/m2)
100
(mPa.s)
12
E,/k
(K)
17S2
(K)
493
acalculaled from d0 = (bVo/Jl)m
sticking coefficient y* in non-stationary nucleation. In this respect it is worth
recalling that, via the kinetic factor A or A' in Js, 7* plays a significant role
also in stationary nucleation (see Sections 13.2 and 13.3). We note that
values of 7* much less than unity were found by Penkov and Gutzow [ 1984]
and Stoyanova et al. [1994] in fitting theoretical 0(A7) and7s(AD dependences
to experimental data for crystal nucleation in, respectively, Li20 • 2Si02 glass
and water.
15.5 Concentration of supernuclei
In Section 13.4 we have seen that the time dependence of the concentration
f of all supernuclei formed during stationary nucleation gives direct information
268 Nucleation: Basic Theory with Applications
about the stationary nucleation rate Js. In the case of non-stationary nucleation
at constant supersaturation the £(r) dependence contains one more piece of
information - it allows determination also of the delay time 6 (see Fig.
15.9b) and, thereby, of the time lag t.
Let us consider again nucleation at no pre-existing clusters in the system.
Then, according to (11.2) or (11.3) and (15.2) or (15.29), for the initial
concentration £0 of all supernuclei in the system we have £0 = 0 and the exact
£(f) dependence can be found easily by using J{t) from (15.59), (15.60),
(15.62), (15.63) or (Al. 12) in conjunction with eq. (11.10). For instance, with
the help of eq. (15.59) the integration in (11.10) is performed without difficulty
and the resulting £(r) function is of the form (n* = 2, 3, . . . , M - 1)
M-\
ZV) = Jst+ I (rf,/A,<i')(/*a„.,-f„.+ ,a„.+ i.,)[l -exp(-A,t)].
;=2
(15.112)
In conformity with eq. (15.100), this exact formula shows that in the limit
of t —» °o the concentration of supernuclei increases linearly with time (see
Fig. 15.9b). The slope of this asymptotic linear £(?) dependence is merely the
stationary nucleation rate (just as in the case of stationary nucleation), and
the time intercept is nothing else but the delay time 6 defined by eq. (15.101).
We can therefore say that no delay in the £(r) linearity (i.e. 0=0) is characteristic
for stationary nucleation. Indeed, when 0 = 0, eq. (15.100) passes into eq.
(13.102) describing nucleation in a stationary regime. Also, in the particular
case of M = Mei = 4 and n* = 3 eq. (15.112) takes the simple form
^'^■['-^OT^'^'-ro^)^'] (15-113)
which follows from employing J(r) from (15.61) in (11.10). Here hi, A3 and
©are given by eqs (15.20), (15.21) and (15.102) as explicit functions of the
monomer attachment and detachment frequencies. We note as well that in
eq. (15.112) the cluster size can assume only integer values. When n is
treated as a continuous variable, the exact f(r) dependence has a similar form
(see eq. (24.45)), since it results from integration of J(t) from (15.62) according
to (11.10) at 5, = 0.
Equation (15.112) is exact, but its usage is hampered by the mathematical
difficulties associated with the determination of A,, a„, and dt when M = Wef
> 6 (see Section 15.1). Hence, in addition, we need a simpler, albeit approximate
expression for the time variation of £. Such an expression is easily found
with the aid of J(t) from (15.64). Integration in accordance with (11.10),
accounting for (15.104) and setting £j = 0 yields TKashchiev 1969a]
CM = Js{t- (^/6)t- 2t Z [(- l)7i2] exp (- i2tlr)). (15.114)
Non-stationary nucleation 269
This approximate formula is valid for whatever kind of nucleation (HON,
HEN, 3D, 2D, classical, atomistic, etc.). It shows that for t —> °o (i.e. when
/» X), as required, £(/) tends asymptotically to £(/) from (15.100) with 9
given by (15.105). Equation (15.114) can be represented in the equivalent
form
£(/) = 2Jsx Z ((4OT/T)"2 exp [ - (2( - 1)Vt/4/]
-(2/- l);r2(l -erf [(2/- D^t"2^'2]}} (15.115)
which follows from using 7(/) from (15.65) in (11.10). Equation (15.115) is
convenient for analysing £(/) in the t —> 0 limit. For instance, retaining only
the first summand the infinite sum in (15.115), we find that for t < 0.25T,
with an error of less than 15%,
£(/) = %Jsx(tlJtxf12 exp (- if-ilit). (15.116)
Fromeq. (15.114) we see that the QJ^X ratio is a universal function of the
dimensionless time tlx. Since the £(/) dependence is accessible to a direct
experimental determination, this means that various £(/) data plotted in (£/
Jst)-vs-(//t) coordinates should produce a master curve. Figure 15.12 illustrates
the validity of this prediction: we observe a good agreement between the
theoretical £(/) dependence (15.114) (the curve) and available experimental
£(/) data for nucleation of crystals in Li20 • 2Si02 glass at T = 727 K (the
tlx
Fig. 15.12 Time dependence of the concentration of supernuclei in non-stationary
nucleation: circles - data of Kalinina et at. [1980] for nucleation of crystals in
2Na2OCaO-3Si02 glass at T = 753 K; squares — data of James [1974] for nucleation
of crystals in Li20-2Si02 glass at T = 727 K; curve - eq. {15.114).
270 Nucleation: Basic Theory with Applications
squares [James 1974]) and in 2Na2OCa0-3SiO2 glass at T = 753 K (the
circles [Kalinina et al. 1980]). Non-stationary ((1) dependences are often
observed in experiments on nucleation of droplets or crystallites in solids
[Hammel 1967; Gutzow et al. 1968; James 1974; Larikov and Brick 1977;
Kalinina etal. 1977, 1980, 1997; Fokin etal. 1977, 1981, 1997; Penkov and
Gutzow 1984; Schiffner and Pannhorst 1987; Schlesinger and Lynch 1989;
Filipovich et al. 1996; Sycheva 1997, 1998a, b; Potapov et al 1998] (for
reviews see, e.g. Gutzow and Toschev [1971]; Gutzow [1980]; James [1982];
Gutzow et al. [1985]; Kelton [1991]; Gutzow and Schmelzer [1995]). At
present, eq. (15.114) is used extensively for analysing not only such
dependences, but also other non-stationary effects in nucleation kinetics.
15.6 Suggestion
The considerations in the preceding sections and the results of existing
numerical and experimental investigations lead to the conclusion that the
approximate eqs (15.54), (15.55), (15.64), (15.73), (15.105) and (15.114)
allow both simple and reasonably accurate description of the time dependences
of Z*, J and f and of the Aft dependences of x and ft The fact that eq.
(15.111) provides a higher-order approximation to the ft(A^) dependence
suggests replacing the time lag Tin eqs (15.54), (15.64) and (15.114) by the
delay time ft and using these equations in the form
Z*(t) = X* 11 + (ilK) S [(- 1 )7(2/ - 1)] exp [- (2/ - 1)V//240]}
(15.117)
J(t) = Js[l+2 X (-l)'exp(-(2r7/6ft)] (15.118)
r=i
C,(t) = Js{t-9-(\2lit2)6 I [(- l)7/2] exp (-/^//66)).(15.119)
When ft is considered as related to T by the simplest approximation (15.105),
these equations are merely an equivalent representation of (15.54), (15.64)
and (15.114). If, however, Sin (15.117)-(15.119) is regarded as given by the
higher-order approximation (15.111), these equations have the potential to
describe the time variation of Z*, J and £more accurately than eqs (15.54),
(15.64) and (15.114), but not at the expense of becoming mathematically
more complicated than them.
15.7 Finding the equilibrium concentration of nuclei
Equation (13.40) shows that the main problem with the theoretical determination
of the kinetic factor A in the general formula (13.39) for the stationary
nucleation rate Js is the lack of detailed knowledge about f* and C0 under
concrete experimental conditions. Moreover, the attachment frequency/* of
Non-stutionary nucleation 271
monomers to the nucleus and the concentration C0 of nucleation sites are
coupled into a single parameter, the product/*C0, and it is not easy to reveal
separately their role in case of discrepancy between a theoretically expected
and experimentally obtained value of A. Clearly, if additional information
about either /* or C0 is available experimentally, we can split the product
f*C0 and, thereby, reduce the theoretical uncertainty in A. Such an information
about/* is contained in the nucleation time lag Tfromeq. (15.55) or (15.73)
and, accordingly, in the delay time 9 from eqs (15.105) and (15.111). As
pointed out by Kashchiev [1972a, 1972b], this means that experimental data
for the A/u dependence of the product Jsx or Js6 are very convenient for
theoretical interpretation, because this product does not contain f* and is
thus insensitive to the kinetic peculiarities of the nucleation process. Indeed,
multiplying Js from (13.33) and Xfrom (15.55) and accounting for (7.44) and
(13.35), we find that
JST = (Al7fap)C* = (4l7?z)C* = (4/7c\)C„ exp (- W*lkT). (15.120)
This relation between the product /ST and the equilibrium concentration
C* of nuclei (or the nucleation work W*) was derived by Kashchiev [ 1972a,
1972b], and in another context a similar /srformula was used by Zeldovich
[1942]. Since the factor Mtv'z is practically A^i-independent and is a number
usually between 0.5 and 5, eq. (15.120) tells us that the experimental
determination of the product isT is equivalent to a direct and model-independent
determination of the equilibrium nucleus concentration C* with an accuracy
of about one order of magnitude. For example, with the Js and T values of
Koster [1978], given in Section 15.2, from eq. (15.120) with assumed 4/t^z
= 1 we find that C* = 7 x 1017, 2 x 1018 and 8 x 1018 crystal nuclei per m3
of amorphous Si at T = 824, 831 and 920 K, respectively. Recently, eq.
(15.120) was used by Penkov and Gutzow [1984], Weinberg and Zanotto
[1989b] and Paskova and Gutzow [1993] for analysis of experimental data
on non-stationary nucleation.
Experimentally, it is more convenient to deal with the JSS product than
with the /ST one. This is so because Js and 9 are obtainable directly from the
slope and the time intercept of a given experimental £(t) curve without using
any concrete theory (see Sections 15.4 and 15.5 and eq. (15.119)). From eqs
(7.44), (13.33), (13.35), (15.105) and (15.111) it follows that, analogously to
(15.120),
Js6 = E.C* = K exp (- W*/kT). (15.121)
Here the pre-exponential factor K (nr3 or m-2) is defined by
K = $C0, (15.122)
and the numerical factor £ is given by
{ = 2/3^^ = 2/3¾ (15.123)
to a first approximation and by
t, = [In (fin*) + 0.3]/2jt1/2j8 = [In (nmzn*) + 03]/2kz (15.124)
272 Nucleation: Basic Theory with Applications
to a higher-order approximation. Similar to the bracketed factor in (15.120),
£ is practically A/j-independent and is a number typically between 0.5 and 5.
Hence, as in the /sTcase, the calculation of the product Js6 from available
£(?) data is in fact a direct way for a model-independent experimental
determination of the equilibrium nucleus concentration C* with a theoretical
uncertainty of less than one order of magnitude. This is so because like
(15.120), eq. (15.121) is applicable to any kind of nucleation (HON, HEN,
3D, 2D, classical, atomistic, etc.).
Thanks to the analogy between eqs (13.39) and (15.121) (or (15.120)),
when analysing JB6 (or 7ST) data we can use directly all results concerning
the stationary nucleation rate Js. Namely, in all these results (see Sections
13.2, 13.3 and 13.5) it suffices merely to replace the kinetic factor A by the
factor K which, due to the absence of/*, is virtually A^-independent and
roughly equal to the concentration C0 of nucleation sites. For instance, for
classical HON or 3D HEN of condensed phases we have (cf. eq. (13.48))
(A^>0)
Js6 = K exp (-B/Afi2) (15.125)
where B is defined by (13.49). When this kind of nucleation occurs in vapours
or solutions, the above formula reads (cf. eq. (13.66)) (S > 1)
Ji8=Ktxp(-B'lln2S), (15.126)
B' being given by (13.67). Accordingly, when the process takes place in
melts, similar to (13.72), there holds (0 < AT < 2TCAse/Acp c)
Js6 = K exp (- B'lTAT1) (15.127)
where B' is specified by (13.73).
Another example is classical 2D nucleation of condensed phases. Then,
analogously to (13.53) and (13.54) we have (Aft > acfAo)
Js6 = K exp [- BI(Afi - acfAo)] (15.128)
where Aa* 0 or Aa= 0 for 2D nuclei on foreign or own substrate, respectively,
and B is given by (13.55).
Again for condensed-phase nuclei, but atomistically, we can write (cf. eq.
(13.60))
Js6 = K exp (- &*lkT) exp (n*A^//tr) (15.129)
where the nucleus effective excess energy <P* is specified by (4.49). Obviously,
with properly defined supersaturation Aft and K (i.e. C0, see Section 7.1),
eqs (15.125), (15.128) and (15.129) are applicable to nucleation in vapours,
solutions, melts, etc. We note also that the atomistic formula (15.129) predicts
breaks in the linear dependence of ln(/s8) on Ap when the supersaturation is
varied isothermally and the stepwise changes in the value of «* occur in the
Ap range studied (see Sections 4.4 and 13.5).
Finally, we note the classical Js 9 formula for HON or 3D HEN of gaseous
phases, which is the analogue of (13.58) (0 <p ipe):
Non-stationary nucleation 273
Js6 = K exp [- Blip* - pf\
(15.130)
Here/)*, B and K are given by (4.14), (13.59) and (15.122), and/)* -p = Ap
is a good approximation when pev^kT « 1.
The above expressions show that when analysing an experimentally obtained
A/; dependence of Js9 we can apply directly the methods used with respect
to 7s(A/j) data (see Section 13.5). The great advantage now is that the pre-
exponential factor K is completely insensitive to the particular kinetics of
monomer attachment to the nucleus. Figure 15.13 depicts the dependence of
Js9 on A7", obtained by Koverda et al. [1974] (see also Skripov [1977]) for
ice nucleation in amorphous water. The 9{AT) data are those already used for
Fig. 15.11. As seen from Fig. 15.13, the experimental data (the circles) are
fitted well by the straight line representing the classical formula (15.127) in
lnt^-vs-d/TAT2) coordinates. The best-fit values of B' and K are B' =
19.23X 106 K3 and K = 9.2 x 1029 m-3. Since K~ C„, and for HON in water
Co = 1/1¾ = 3.1 X 1028 m"3, vis-a-vis the uncertainty in the experimental
determination of 7S and 6, this value of K is reasonable and indicative for
HON. Also, with c3 = 36;r, u0 = 0.0326 nm and Asjk = 2.63, in accordance
with (13.73), the above B' value leads to <7ef = 27.0 mj/m2 which compares
with CTet- = 26.4 mj/m2 found in Section 13.5 with the help of classical
analysis of the Js(Aji) data for ice in Fig. 13.9. Using the value of B' and eq.
(2.23), from the second of eqs (13.106) we find as well that n* = 10 to 13
water molecules in the studied range of temperatures from 160 to 170 K.
Thus, even though in this range the ice nucleus is quite small, the classical
nucleation theory remains in agreement with experiment.
60.0
59.5
59.0
58.5
58.0
0.48
0.50
0.52
0.54
0.56
1/T(AT)2(10"
'rC3)
Fig. 15.13 Dependence of the Js6 product on the undercooling: circles-data for
nucleation of ice crystals in vitrified water (Koverda et al. 1974; Skripov 1977],
line-best fit according to eq. (15,127),
Chapter 16
Second application of the
nucleation theorem
In Chapter 14 we have seen that the nucleation theorem can be used for a
universal, model-independent determination of the nucleus size from
experimental /s(A/() data. The uncertainty in this determination is associated
with the presence of the kinetic factor A ineq. (13.39) and is negligible when
the supersaturation A/i is varied isothermally, but may be considerable when
the temperature is the parameter used to change A/i. We shall now see that
this uncertainty can be reduced significantly if in addition to /s(A/i) data we
dispose with corresponding independently obtained 0(A/i) data so that we
know also the product Js9 as a function of A/i. The application of the nucleation
theorem to the Js6 product is highly advantageous, for it eliminates the
necessity to seek a concrete formula for Js9 (see Section 15.7) which would
provide the most plausible interpretation of the available dependence of Js6
on A/i. Most importantly, the An* and n* values resulting from this application
are free of any theoretical assumptions about the kind of nucleation (classical,
atomistic, 3D, 2D, HON, HEN, etc.).
Our task now is to find formulae allowing the determination of An* and/
or n* in a general, model-independent way from a known A/i dependence of
Js9. To that end we rewrite eq. (15.121) in the form
- W* = kT la (J,9)-kT In K (16.1)
and consider first the case when the Js9 data are obtained by varying A/i
isothermally. Differentiating both sides of this expression with respect to A/i,
taking into account that d(ln X)/dA/i = 0 (see eqs (15.122)-(15.124)) and
employing the nucleation theorem in the form of eq. (5.21), we get
An* = kT(\ - p„,d/p„5„)d[ln (Jse)]/dA/i (16.2)
or, in an equivalent presentation resulting from using (5.18),
An* = £7"d[ln (7s0)]/d/iold. (16.3)
These formulae are applicable to all possible cases of one-component
nucleation when /ioid and thus A/i are changed isothermally. We see from
them that for the model-independent determination of the excess number
An* of molecules in the nucleus the Jfi data must be plotted in In (Js8)-vs-
A/i or /i0i(i coordinates. Then the slope at each point of the resulting line
gives An* at the corresponding value of A/; or /i0,d. The obvious analogy
between eqs (16.2) and (16.3) and eqs (14.2) and (14.3) implies that eqs
(14.4)-(14.13) are directly applicable to the calculation of An* or n* also
Second application of tile nucleation theorem 275
from 7S6 data if in them Js is replaced by Js0 and nA or the unity is omitted.
For instance, for the determination of the number n* of molecules in the one-
component EDS-defined nucleus we have the universal formula
n* = kT[0 - p„id/p„e„)/(l - p„id/p„ew*)]d[ln (J,e)]/dAu (16.4)
which parallels (14.5) and is valid for HON or HEN on weakly adsorbing
substrates. When nucleation occurs far enough from the spinodal (if any),
this formula is well approximated by (cf. (14.6))
n* = kTd[ln (Vse)]/dAji (16.5)
for condensed-phase nuclei and by (cf. (14.8))
V* = (i77pnew)d[ln (JM'dAp (16.6)
for gaseous nuclei. Analogously to (14.9)-(14.13), from (16.5) and (16.6)
we find in particular that
n* = d[ln (7se)]/d(ln S) (16.7)
for nucleation of condensed phases in vapours and solutions,
n* = (tr/jie0)d[ln (Js6)]/dA<p (16.8)
for electrochemical nucleation,
n* = (A:7-/Ai>e)d[ln QMWp - P„) 06.9)
for nucleation of crystals in melts (or during polymorphic transformations)
when A^i is varied by means of p at constant T, and
V* = kTd[ln (JsS)]/dA/> (16.10)
for nucleation of gaseous phases behaving as ideal gas. It is worth reiterating
that all of the above equations are universal in the sense that they are valid
for whatever kind of one-component nucleation (classical, atomistic, 3D,
2D, etc.) as long as the J56 data are obtained by isothermal variation of the
supersaturation. From eqs (16.2), (16.3), (16.5)-(16.10) we see that there is
no uncertainty in the determination of An*, n* or V*, because the right-hand
sides of these equations contain only the experimentally obtainable derivative
of In (Js6) with respect to A^ or ^0,d.
Let us now consider the case when the /s0 data are obtained at different
temperatures. When A^i is changed by varying T at constant pressure, the
formula for n* is of practical significance again only for EDS-defined nuclei
and reads:
n* = k[(\ - Ws„id)/(1 " s*ew/i'„M)]{d[rin (Jsff)]ldAp
-diTlnfOldAfi}. (16.11)
This formula results from differentiating eq. (16.1) with respect to A^i and
applying the nucleation theorem in the form of eq. (5.39). Naturally, due to
the analogy between (13.39) and (15.121), it can be obtained also when Js
and A in (14.14) are replaced by ./,0 and K, respectively.
276 Nucleation: Basic Theory with Applications
Equation (16.11) is valid for any kind of one-component HON or HEN on
weakly adsorbing substrates, but its usage is hampered by the lack of knowledge
of the entropy $* w of a molecule in the EDS-defined nucleus. To a first
approximation, however, it can be assumed that j*w = jncw. Then eq. (16.11)
simplifies to (cf. eq. (14.15))
n* = kd[T In (Js0)]/dAjU + nK (16.12)
where nK = - kd(T In K)/dAfj, and K is expressed in the units of Js9.
Compared with (14.15), eq. (16.12) is much more convenient for a model-
independent determination of n*, since nK can be estimated with a much
greater confidence than nA. Indeed, recalling that Kis, virtually A/i-independent
and close to C0 (see eq. (15.122)), we find that nK is well approximated by
nK = -k{dT/dAv) In C0 (16.13)
where C0 cannot exceed 1029 m "3 for HON or HEN in the bulk of the old
phase and 1019 irf2 for HEN on a substrate (see Section 7.1). Thus, eq.
(16.13) sets an upper limit of the value of nK and, hence, of the theoretical
uncertainty in the usage of eq. (16.12). For example, for nucleation of crystals
in melts, according to (2.20) we have dApldT = -As(T) ~ - Ase so that,
approximately,
nK=(kf&se)\t\C0. (16.14)
If Js0is measured in m~3 orm-2, this formula gives n& = 25 &ndnK = 17,
e.g. for nucleation of ice in liquid water (in this case Ase/k = 2.63) when the
ice nuclei appear, respectively, during HON (then C0 = 3.1 x 1028 nT3) and
during HEN on a substrate free of nucleation-active sites (then C0= 1019 m"2).
We note also that when the undercooling AT satisfies the condition AT <
27eAje/Acpe, Ap is given by (2.23) andeq. (16.12) can be used in the simpler
form
n* = (k/Ase)d[T In (Js0)]/dAT+ nK
= ~(k/Ase)d[T\n {JsB)]/dT + nK (16.15)
where nK is given by (16.14), and ./s0and C0 must be expressed in the same
units.
The above considerations show that even when A/i is varied by means of
T as required by eq. (2.20), the size n* of the one-component EDS-defined
nucleus can be determined in a model-independent way from the slope of the
curve representing the An dependence of the product Js0 in [T In (7s0)]-vs-
Afj (or AT or T) coordinates. When /s0 is measured in m"3 or m~2, the
respective uncertainty in this determination is less than 70 or 45 molecules -
these numbers are the values of nK from (16.14) with the rather low Asjk = 1
(typically, Asjk = 1 to 5) and with the maximum possible C0 = 1029 nT3 or
C0=10,9nr2.
The circles in Fig. 16.1a represent the experimental T In (JSG) data of
Koverda et al. [1974] (see also Skripov [1977]) for nucleation of ice crystals
in vitrified water, which we have already used in Fig. 15.13. The dependence
Second application of the. nucleation theorem 277
1.00
o
'E
CD
0.98
0.96
15
10
5 -
' '■ ' 1 1 l-I—I—1 1 '1' I-l ■ 1 ■
H (a) '
1
.
(b) :
1
102 104
106 108 110
AT(K)
112 114
Fig. 16.1 (a) Dependence of the T In (J Jl) product on the undercooling: circles - data
for nucleation of ice crystals in vitrified water [Koverda et al. 1974; Skripov 1977];
line - best-fit function used for calculating the derivative in eq. (16.15).
(b) Dependence of the corresponding nucleus size on the undercooling: circles - data
obtained according to eq. (16.15) with nK= 25. line-Gibbs - Thomson eq. (4.38).
of T In (J,8) on AT is fitted by the function T In (Jsff) = 104(- 49.72278/AT
+ 2.3042 - 0.00805AT) (the curve in Fig. 16.1a) allowing the evaluation of
the d[Tln (Js0)]/dATderivative. Considering nK (i.e. C0) as a free parameter
in eq. (16.15), with the aid of this derivative and Asc/k = 2.63, we can use the
limiting value of C0 = 3.1 x 1028 m"3 which corresponds to HON in order to
calculate the highest possible n* values in the AT range studied. These n*
values are represented by the circles in Fig. 16.1b. We can now check if the
so-obtained model-independent n*(AT) dependence is in agreement with the
classical dependence for 3D nucleation. This dependence is depicted by the
curve in Fig. 16.1b and is calculated from the Gibbs-Thomson equation
278 Nucleation: Basic Theory with Applications
(4.38) with Ay = As<.AT, c = (36;r)1/3 (spherical or cap-shaped nuclei) and oef
= 26.4 mj/m2 (this is the cef value obtained from the analysis of the JS(AT)
data in Fig. 13.9). As seen from Fig. 16.1b, the classical theory provides a
good description of the experimental n*(AT) dependence even though the
nucleus is constituted at most of only 9 to 12 water molecules.
Summarizing, we see that when Afj is varied by T in accordance with
(2.20), due to the presence of the quantity nK in eqs (16.12) and (16.15), the
absolute value of n* cannot be determined experimentally in a fully theory-
independent way. What can be done in such a way is in fact the determination
of the dependence of n* on A/a, AT or T, since nK either has no effect on this
dependence when it is constant with respect to T (see eq. (16.14)) or can
affect this dependence in a known way through the factor dA^/dr = - As(T)
(see eq. (16.13)). The so-determined rc*(A/i) dependence can then be confronted
with various theoretical dependences such as those given by the classical
Gibbs-Thomson equations (4.32) and (4.38). Resulting good quantitative
agreement between theory and experiment allows a reliable order-of-magnitude
determination of the concentration Co of nucleation sites in the system.
Worth noting in this respect is that at present there exists no other method for
such a determination of Q>.
Chapter 17
Nucleation at variable supersaturation
When the supersaturation Ay in the system varies with time t, the transition
frequencies fnm in the master equation (9.1) are time-dependent. For that
reason, the cluster concentration Z„ and, hence, the flux./,, from (9.6) are also
functions of ?. Moreover, such a function is the nucleus size «*, too. According
to eq. (11.5), this means that the nucleation rate J is also a time-dependent
quantity. In other words, under conditions of variable supersaturation nucleation
is always non-stationary in the sense that J is a function of t. Physically,
however, this non-stationarity is different from that at constant Ay. While
the latter is only temporary and terminates after a period of the order of the
time lag Tnecessary for the transformation of the initial cluster size distribution
into the stationary one (see Section 15.1), the former is permanent and due
to the continuous reorganization of the actual cluster size distribution Z„ into
the stationary one Xn(t) = X„[Ay(t), 7(/)] which corresponds to the momentary
value of Ay. It is clear, therefore, that when Ay varies sufficiently slowly, Z„
will have time to adjust itself to Xn(t), i.e. then Z„{t) = Xn(t). Accordingly,
nucleation will then occur at quasi-stationary rate Jqs{t) given approximately
by
JqM = JM (17.1)
where Js(t) = Js\Afi(t), T(t)] is the stationary nucleation rate corresponding to
the value of A^i at time t.
It seems that Tunitskii [1941] was the first to use eq. (17.1) in a theoretical
study of nucleation at variable supersaturation. The main question to be
answered when using (17.1), however, is what is the condition warranting
the validity of this equation. Following Kashchiev [1970], in this chapter we
shall find the answer to this question under the presumption that nucleation
takes place according to the Szilard model (see Section 9.2). More recently,
the kinetics of nucleation at variable supersaturation were considered
theoretically by Trinkaus and Yoo [1987], Kozisek [1990] and Kozisek and
Demo [ 1993a]. It should be pointed out that such considerations are important,
as they widen significantly the possibilities for application of the nucleation
theory. This is so because in a real nucleating system the very process of
formation of nuclei and growth of supernuclei brings about changes in the
supersaturation imposed on the system. Hence, without a special control on
the constancy of A/<, nucleation occurs practically always at time-dependent
supersaturation.
280 Nucleation: Basic Theory with Applications
17.1 Quasi-stationary cluster size distribution
As already stated above, we consider nucleation which takes place in a
closed system according to the Szilard model. In addition, we presume that
the absolute temperature T and the supersaturation A^i are known functions
of time t and that the cluster size n can be treated as a continuous variable.
The unknown size distribution Z(n, /) of the clusters is then the solution of
the master equation (9.26) in which the monomer attachment frequency
/(«, t) and the quasi-equilibrium cluster concentration C(n, t) are also known
functions of/ (see Sections 10.1, 10.2, 10.5 and Chapter 12). As in Section
15.1, when the nucleation process begins at no pre-existing clusters in the
system, Tin, /) must obey the initial condition (15.29) and the boundary
conditions (15.31) and
Z(l,/) = C,(r), (17.2)
the latter being the analogue of (15.30). Thus, introducing the new unknown
function
Y(n, t) = Z(n, t)IC(n, /), (17.3)
we can represent eqs (9.26), (15.29), (15.31) and (17.2) in the following
form [Kashchiev 1970] (1 < n < M):
3Y(n, /) dln[C(n,t)]
dt dt n"'"
Y(n, 0) = 0, 7(1,/)=1, Y(M, /) = 0. (17.5)
The exact solution of eq. (17.4) under the conditions (17.5) is unknown
and we must look for an approximate solution. As noted in Section 15.1, for
n values inside the critical region, i.e. for nx(t) < n < n2(t), the r.h.s. of (17.4)
is approximately equal tof*(t)d2Y(n, t)/dn2, because for these n values f(n, /)
=/[n*(r), t\ =f*(t) and C(n, /) = C[n*(t), /] = C*(t). The latter relation allows
also to approximate In [C(n, t)} on the left of (17.4) by In [C*(t)]. In addition,
if Afi varies with t sufficiently slowly, the boundary conditions can be 'shifted'
to the ends «i(r) and n2(t) of the nucleus region (we note, however, that
now n, and n2 which are defined by (7.41) and (7.42) depend on time through
A^i and, possibly, T). Thus, with the help of (7.44), considering only cases
when the concentration C0 of nucleation sites is /-independent, we can replace
(17.4) and (17.5) by the equations («,(/) < n < n2(t))
Y(n, 0) = 0, Y[nl(t),t] = \, Y[n2(t), t] = 0. (17.7)
Nucleation at variable supersaluration 281
Here W*(t) = W[n*(t), t] is the nucleation work corresponding to the values
of Afi and T at time t and can be calculated from (10.86) at n = «*(/). The
nucleus size «* is also a known function of t because of the presumption that
we know the temporal variation of A^i and T.
Equation (17.6) is much simpler than eq. (17.4), but its exact solution
under conditions (17.7) is unknown either (note that it has 'moving' boundary
conditions). Nevertheless, we may try to find its quasi-stationary solution yqs
which is characterized by dY^Jdt = 0, since it depends only implicitly on t
through Afi, Tand their time derivatives. It is clear that the function yqs will
be physically acceptable after elapsing of sufficiently long time /qs from the
beginning of the process so that the initial condition will have no effect on
it. Also, yqs will describe the dependence of the ratio Z(n, t)IC(n, t) on n and
t, which is due only to the change of both the length and the position of the
nucleus region. Thus, setting Y(n, t) = Yqs(n) and dismissing the initial condition,
from (17.6) and (17.7) we get (n,(t) <n< n2(0)
d%s(«)/dn2 + %(«) = 0 (17.8)
V»i)=l, y«2) = 0- (17.9)
Here the dimensionless quantity 2 is defined by
\t) = [Hf(tW[W*(t)lkT(t)]/dt (17.10)
and may be positive or negative depending on whether A/; diminishes or
increases with time. It depends on t implicitly through Afi(t) and 7(0 and
must have a sufficiently small absolute value in order for the quasi-stationary
solution yqs to be physically realistic (the = 0 limit corresponds to stationary
nucleation at constant supersaturation). It is important to note also that in all
time-dependent quantities characterizing yqs, the initial moment t — 0 must
be regarded as chosen at time t > Jqs. Evidently, /qs cannot be defined exactly
in the scope of the considerations in this section, but generally speaking, it
is of the order of the nucleation time lag rcalculated from (15.55) or (15.73)
at suitably chosen A/( and T values.
Before proceeding further let us obtain an equivalent, but simpler expressiou
for applicable to all cases of nucleation of condensed phases when A^t
varies with time (e.g. through the pressure/>(/) in the system), but Tremains
constant. Invoking the nucleation theorem in the form of eq. (5.29) and using
the relation dW*/dt = (dH™/dA^)(dA,u/dr), from (17.10) we find that for
EDS-defined nuclei [Kashchiev 1970]
2(/) = -n*(t)A^i'(t)lkTf(t) (17.11)
where A//(r) s dA^i/dr. This formula tells us that | 21 is small enough and,
hence, quasi-stationary nucleation is possible only when Ap varies not too
rapidly (then | A/j' | == 0) and/or the monomers attach themselves to the nucleus
with a rather high frequency f*.
Now, being a linear ordinary differential equation of second order with n-
independent coefficient 2, eq. (17.8) has an exact solution [Korn and Korn
1961], In view of the boundary conditions (17.9), this solution is of the form
282 Nucleation: Basic Theory with Applications
Yqs(n) = cos [ ( n - n,)] - [cos ( A*)/sin ( A*)] sin [ ( n - n,)] (17.12)
where A*(J) = n2(t) - n | (t) is the variable width of the nucleus region. According
to (7.38) and (7.43), A* is given by
A*(t) = xu2iP(t) = {2nkT(t)ll-d2W(n, /)/dn2]„ = „.<„)l/2 (17.13)
and is an implicit function of t through A^i and T. It must be pointed out that
y„s from (17.12) is physically meaningful not only for 2 > 0, but also when
< 0. Indeed, from mathematics [Korn and Korn 1961], in the latter case
we have = i'| |, cos (i | | x) = cosh ( | | x) and sin (;' | | x) = i sinh
(| | x) so that then the r.h.s. of (17.12) is also real (i is the imaginary unity).
In conformity with eq. (17.3), the quasi-stationary cluster size distribution
Zqs(«) in the nucleus region is given by Zqs(«) = C{n,t)Y^{n). This relation
can be used for determination of the quasi-stationary concentration Z*s(t) =
Zqs[n*(J)] = C*(t)Yqs[n*(t)] of the nuclei in the system. Setting n = n*(t) and
accounting that n,(t) = n*(.t) - A*(()/2, from (17.12) we get
Z*(0 = C*(i)/2 cos [ ( /)A*(t)/2]. (17.14)
As already noted above, the quasi-stationary approximation is physically
realistic only for sufficiently small | | values. This means that in eq. (17.14)
we can regard the product | | A*/2 as smaller than unity. Hence, employing
the approximations cos (x) = 1 - *2/2 and (1 - x2!!)'1 = 1 + *2/2 which are
valid for x < 1 [Kom and Korn 1961], from (17.14) we find that
Z*(r) = (1/2)C*(0P + 2«A*2(r)/8] (17.15)
provided
| d[W*(t)/kT(t)]/dt | < 4f(t)/&*\t) = 16/ttV). (17.16)
Here 2 is specified by eq. (17.10), and x(t) s r[A^(/), T(t)\ is the time lag
corresponding to the momentary values of Aft and T. It is defined by (15.55)
or (15.73) and is an implicit function of t through A^i and, possibly, T.
The above inequality represents the condition guaranteeing quasi-stationarity
when one-component nucleation occurs at variable supersaturation. For
nucleation of condensed phases, in the case of isothermal variation of A^i
with time t, (17.16) simplifies to
| VC) I < 4i7J*(/)/n*(r)A*2(/) = 16H7;rV(/)T(r), (17.17)
since in this case 2 in eq. (17.15) is given by the equivalent (17.11) of eq.
(17.10). With a slightly different numerical factor, this condition for the rate
A/j' of change of the supersaturation at constant T was obtained by Kashchiev
[1970],
Both eqs (17.16) and (17.17) show that quasi-stationary nucleation can
take place only in systems possessing the ability for a sufficiently quick
reorganization of the size distribution of the clusters in them. As the time lag
T is a measure of this ability (see Section 15.3), it appears as the most
important parameter in the condition for quasi-stationarity. In the limiting
case of x = 0 (then = 0), according to (17.15), the quasi-stationary
Nucleation at variable aupersaturation 283
concentration Z* (/) of nuclei is exactly equal to their stationary concentration
X*(t) = (l/2)C*(r) corresponding to the momentary values of Aft and T (cf.
eq. (13.23)).
17.2 Quasi-stationary rate of nucleation
Having obtained the quasi-stationary size distribution Zqs(n) of clusters in
the nucleus region, we can now determine the quasi-stationary nucleation
rate Jqs(t). Using Yqs(n) from (17.12) in order to calculate the derivative of
the ratio Z(n, t)IC(n, t) = Zqs{n)IC(n, t) = Yqs(n) in eq. (11.9), we find that the
quasi-stationary flux j*s(t) through the 'moving' point «*(/) on the size axis
is given by
;,* (') =f*(t)C*(t) ( r)/2 sin [ ( t)A*(t)ll\. (17.18)
Since the value of | | is restricted by the condition (17.16) (i.e. by | | A*
< 2), we can simplify (17.18) with the aid of the approximations sin (x) = x
- x3/6 and (1 - x2lb)" ' = 1 + jc2/6 valid for x < 1 [Korn and Korn 1961].
Hence, approximately,
7*s = [f MC*(r)/A*(r)][l + 2(,)A*2(/)/24]. (17.19)
Setting Z* and;* in eq. (11.5) equal to Z* and ;'* from (17.15) and (17.19),
we finally obtain the following expression for the quasi-stationary rate of
nucleation:
Jv(t) = JjLt) (1 - A*(t)n*'(t)/2j*(t)
+ [ 2(0A*2(/)/24][l - 3A*(/)n*'(r)/2/*(r)]}. (17.20)
Here n*' = dn*/d/, 2 is specified by eq. (17.10), and 7S(0, defined in analogy
with (13.33) and (13.39), by
UO = z(tV*V)C*(t) = A(t) exp [- W*(t)lkT(t)\ (17.21)
is the stationary nucleation rate corresponding to the momentary values of
tyi and T (the Zeldovich factor z(t) = 1/A*(r) and the kinetic factor A(t) =
z(0/*(')C0, cf. eqs (13.35) and (13.40), depend implicitly on t through A^i
and, possibly, T).
Equation (17.20) describes the time variation of 7qs when | 2(J) | A*2(t)
< 4, i.e. when A^i is changed at a rate sufficiently low for the fulfilment of
the inequality (17.16) (or (17.17) in the case of constant T). It thus gives the
sought answer to the important question about the conditions warranting the
validity of the approximation (17.1). Since in*' = (dn*ldAfi)Afj', from eq.
(17.20) we see that eq. (17.1) is valid when the rate of change of the
supersaturation has absolute value | Aft' \ satisfying not only (17.16)
(respectively (17.17)), but also the inequality
| An'(t) I < 2/*(')/A*(r) | dn*(Wty(t) I
= 8A*(r)/^T(t) | dn*(t)/dAAi(0 |. (17.22)
284 Nucleation: Basic Theory with Applications
This inequality ensures that A*(()|n*'(J)|/2/*(r) < 1 and in it x(i) is again the
time lag from (15.55) or (15.73) corresponding to the momentary values of
Ay and T. It turns out that in the scope of the classical theory of nucleation
this inequality can be given a more informative presentation. Namely, using
the Gibbs-Thomson equations (4.32) and (4.38) to calculate the dn*/dAy
derivative, with the help of (7.43) and the last equalities in (7.39) and (7.40)
we find that (17.22) is of the form
| Ay'(t) | < AitkT(t)f*(t)IA*\t) = 16£T(/)/;r2A*(/)T(/) (17.23)
both for 3D and 2D nucleation of condensed phases.
Let us now consider the case when Ay varies with time t, but T remains
constant. Inspection of the inequalities (17.17) and (17.23) shows then the
r.h.s. of the former is always smaller than the r.h.s. of the latter (this is so
because always A* < «*). This leads to the important conclusion that in the
case of nucleation of condensed phases under isothermal variation of Ay
with time t the inequality (17.17) is the condition for both quasi-stationarity
and validity of eq. (17.1): when this inequality is satisfied, (17.23) is necessarily
satisfied, too. In other words, in this case Jqi(t) is always given by eq. (17.1)
to a first approximation and by the equation
Jqs(t) = JMV - „*(,)A*2(,)Ay'(t)/24kTf*(t)]
= Js(t)[\ - 7rV(0i-WA/(/)/96A:r] (17.24)
to a higher approximation. This formula follows from (17.20) upon neglecting
the n*' terms and using (15.73) and (17.11). It is valid under the condition
(17.17) and parallels the known /qs(/) formula [Kashchiev 1970] (the difference
in the numerical factor in this formula and in eq. (17.24) is due to the present
usage of A* from (7.43) instead of A' from (15.49) as done originally). As
seen from (17.24), the smaller the nucleus size n*, the time lag T and the
absolute value | A// | of the rate at which Ay is changed, the easier for the
system to nucleate in quasi-stationary regime. If it were possible for the
system to react instantaneously to the changes in Ay (this is the T = 0 limit),
/qs(J) would be strictly equal to J5(t) even at the highest rate at which these
changes could occur.
We now turn to the case of quasi-stationary nucleation at time-dependent
temperature. As we shall see in Section 17.3, in this case the n*' terms in eq.
(17.20) are not necessarily smaller than unity when the inequality (17.16) is
satisfied. This means that the 7qs(/) function will be different depending on
whether | Ay' | obeys or violates the condition (17.22) (or (17.23)). In the
former case, i.e. when both (17.16) and (17.22) (or (17.23)) are satisfied, we
shall have the formula
V) = ■/*(<)(! + [A*2(t)/24f(t)]d[W*(t)/kT(t)y6t}
= Js(t){l + [^T(0/96]d[»*(/)/A:r(/)]/d/} (17.25)
which is the analogue of (17.24) and follows from (17.20) after neglecting
the «*' terms and accounting for (17.10). Hence, in this case eq. (17.1) is
Nuclealion at variable supersaluralion 285
again a good approximation for Jqs(t) as long as T and A^i change with time
at such a low rate that not only (17.16), but also the condition (17.22) (or
(17.23)) is fulfilled.
In the opposite case, however, i.e. when the inequality sign in (17.22) is
reversed, the 2 term in eq. (17.20) is negligible with respect to the n*' term
in front of it and the Jqs(r) dependence becomes
V> = •WN " A*Wn*'(W(0] = UOU - n3T(t)n*'(t)/8A*(0]. (17.26)
Using the classical theory of nucleation, we can again represent this expression
in a form which is more convenient for practical work. Since n*' - (dn*/
dAjuJA/, with the aid of eqs (4.32), (4.38), (7.43) and the last equalities in
(7.39) and (7.40), from (17.26) we obtain
V) = •«')[! + A*3(/)VW/4*A:n«y*(0]
= ys(/)[l + ^A*(t)T(OA^'(0/16t7(/)]. (17.27)
This formula applies to both 3D and 2D nucleation of condensed phases
at time-dependent temperature and is valid when (17.16) is still satisfied, but
| A^i' | is already so high that it obeys the inequality
| V(f) I > 4*T(/)/*(/)/A*3(I) = \6kT(t)ln2b.*(t)x(t) (17.28)
which is merely the opposite of (17.23). Clearly, Jqs(t) from (17.27) can
deviate considerably from Js(t), the deviation being proportional to the rate
of change of A/i. Hence, despite the fulfilment of the condition (17.16) for
quasi-stationarity, eq. (17.1) cannot be used as a good approximation to the
yqs(I) dependence when (17.28) is in force.
17.3 Condition for quasi-stationarity
The general condition necessary for quasi-stationarity and, hence, for the
applicability of the first- and higher-order approximations (17.1), (17.15),
(17.20), (17.24)-(17.27) for Zqs and 7qs is the inequality (17.16) which takes
the form (17.17) for isothermal nucleation of condensed phases. In addition,
when the process occurs at time-dependent 7, the inequality (17.22) (or
(17.23)) must also be satisfied if we want eq. (17.1) to be valid. With
appropriately defined A^i and W* these inequalities can be used to assess if
quasi-stationarity is possible during one-component nucleation in vapours,
solutions, melts, etc. We shall now apply (17.16), (17.17) and (17.23) to
some most often encountered particular cases of nucleation.
1. Isothermal nucleation of condensed phases in vapours or solutions
In this case the condition for both quasi-stationarity and validity of eq. (17.1)
is the fulfilment of the inequality (17.17), since it is stronger than (17.23).
With the help of (2.8), (2.9), (2.13), (2.14), (2.16) (13.68) and (13.69), from
(17.17) we find that when
| dS(r)/dr | < 16S(/)/7rV(r)T(r), (17.29)
286 Nucleation: Basic Theory with Applications
eq. (17.1) with Js(t) from (17.21) is a reliable approximation for the Jqs(f)
dependence. We thus see that the greater the supersaturation ratio S and/or
the smaller the nucleus size n* and the time lag X, the easier for the nucleation
process to proceed in quasi-stationary regime. To get an idea about the
numbers in (17.29), with the lowest value of S = 1 and the high enough value
of n* = 50, in view of (13.68) and (13.69) we obtain
\dp(t)/dt\<Pe/i00x(t) (17.30)
| dC(t)/dt | < Ce/100T(/) (17.31)
for nucleation of condensed phases in vapours or solutions, respectively.
Considering HON as an example, with the high enough values of t= 100 ns
and t= 10,us for these two cases (see Figs 15.5 and 15.6) and with, e.g. /?e
= 1 kPa and Cc = 1023 rrf3 (see Tables 3.1 and 6.1), for the r.h.s. of (17.30)
and (17.31) we find 100 MPa/s and 1026 m"3 s"\ respectively, for water
vapours and aqueous solutions of sparingly soluble salts at room temperature.
Such rapid changes of the pressure of the vapours or the concentration of the
solution do not occur in typical nucleation experiments. The conclusion is,
therefore, that nucleation at isothermally varied supersaturation in both vapours
and not too viscous solutions is practically always quasi-stationary. We thus
have a posteriori justification of the pioneering usage of eq. (17.1) by Tuni tskii
[1941] in his study of vapour condensation.
2. Isothermal electrochemical nucleation of condensed phases
In this case the inequality (17.17) is again stronger than (17.23) and is the
condition for both quasi-stationarity and validity of eq. (17.1). Substituting
Ay from (2.27) in (17.17), we find that this condition is of the form
[ dA<p(t)/dt | < \6kT/7pZie0n*(t)x(t). (17.32)
To estimate the r.h.s. of (17.32) for nucleation in electrolytic solutions, we
can use again the high enough n* = 50andr= 10 ps so that, e.g. for bivalent
ions {z\ = 2) in solutions at room temperature we get
ldA<p(*)/d?|< lOV/s. (17.33)
Experiments on electrochemical nucleation in galvanostatic regime (then
the overvoltage A<p varies with time) [Schottky 1962; Klapka 1971; Bostanov
et al. 1972] evidence that the rate of change of A<p can have absolute values
comparable or surpassing that in (17.33). Hence, verifying the fulfilment of
(17.32) before the usage of the /qs(f) formula (17.1) is necessary in each
concrete case of electrochemical nucleation at variable overvoltage.
3. Isothermal nucleation of bubbles in own liquid
As in the above cases, now the inequality (17.16) is also the condition for
both quasi-stationarity and validity of eq. (17.1), because it is stronger than
the inequality
| dp(t)/dt | < 2/*(/)/A*(0 | dn*(t)fdp(t) |
= 8A*(')/7rV) I dn*(t)/dp(t) | (17.34)
Nucleation at variable supersaluration 287
which is the appropriately modified (17.22). To express (17.16) in terms of
the dp/dt derivative it is convenient to use the nucleation theorem for EDS-
defined nuclei in the form of eq. (5.28). With ^oli =^c + v0(p - pt) for the
chemical potential of the liquid (cf. eq. (2.6)), p^w = p*lkT (ideal-gas
approximation) and p0(d = 1/l70, after neglecting the unity with respect to the
/WAitw ratio we find from (5.28) that dW*/dp = n*kTlp*. Thus, with the
help of the relation AW*lit = (dW*ldp)(dp/dt), (17.16) becomes
| dp(t)/dt | < 4p*(t)f*(t)/A*1(t)"*(r) = 16/>*(/)/;rV(/)T(r) (17.35)
where the pressurep* =p*\p(t)] inside the nucleus is an implicit function of
t according to (4.14). For practical purposes this condition can be used in the
form of the inequality
| dp(t)ldt | < pJlOOOr(t) (17.36)
which parallels (17.30) and results from approximating/)* by pt and setting
n* = 500. For evaporation-controlled detachment of molecules from the
nucleus bubble the value of X is similar to that of the time lag for nucleation
of droplets in vapours (see Fig. 15.8). Hence, with exemplary pt = 1 MPa
and the high enough T = 100 ns for this type of control, it follows that the
r.h.s. of (17.36) is equal to 10 GPa/s. As such a high absolute value of the
rate of pressure change is not typical for nucleation experiments, we cart
conclude that isothermal bubble nucleation at variable pressure under
evaporation control usually proceeds in quasi-stationary regime. It must
be emphasized, however, that quasi-stationarity may require much lower
] dpldt | values for other mechanisms of mass transport, because then the
time lag rcan be quite long (e.g. under viscous-flow control Tis proportional
to the viscosity r/ [Carlson and Levine 1975] and can be seconds or hours if
the liquid is sufficiently viscous).
It remains now to check if (17.35) is stronger than (17.34). This is easy to
do in the scope of the classical nucleation theory under the approximation p*
= /7C. Using (4.40) to calculate the dn*fdp derivative, with the help of A* =
1/z, (4.41) and the first equality in (13.36) we find that the r.h.s. of (17.34)
is equal to (l6p*/n3n*x)(7Cpen*lp*A*). This quantity is greater than that on
the right of (17.35), because always /?e £ p* and n* ^ A*. This proves that
(17,35) is stronger that (17.34).
4. Isobaric nucleation of crystals in melts
In this case both T and A^i vary with time and either of the inequalities
(17.16) and (17.22) can be the stronger. We can employ the classical nucleation
theory in order to transform (17.16) and compare it with (17.23). Using eqs
(4.38) and (4.39) and the relations d(W*/kT)/dt = [d(W*lkT)ldAfi](dA(i/dt)
and dA/i = - AsdT, from (17.16) we get
| dr(/)/d/1 < 8/tr2(/)/*(/)/n*(r)A*2(I) | Aji(() - 2T(t)As(t) \
= 32kT2(t)'^"*(t)-c(t) | Aji(r) - 2T(t)As(t) | (17.37)
where As(t) s As[T(t)], and Afi is given by (2.20). This inequality changes
288 Nucleation: Basic Theory with Applications
only a little if instead of (4.38) and (4.39) which apply to HON or 3D HEN
we use eqs (4.32) and (4.33) valid for 2D HEN: its r.h.s. is divided by 2, and
Afj in the denominator is replaced by A^i - aelAo" (the latter change disappears
for 2D HEN on own substrate, as then Ac = 0). Hence, we can consider
(17.37) as applicable to both 3D and 2D nucleation of crystals in melts at
constant pressure. With the help of A/y from (2.23) and the approximation As
= Ase we can give (17.37) the simpler and sufficiently accurate form
| dT(t)ldt | < 8/t7"2(r)/*(/)/Asen*(/)A*2(() | 3T(t) - Tt \
= 12kT\i)l%iAs<,n*(i)x(i) | 37-(0 - Tc |. (17.38)
Now, to be able to compare this inequality with (17.23) we must express
the latter in terms of the cooling (or heating) rate dT/dt. Again by means of
A/< from (2.23), we rewrite (17.23) as
| AT(t)ldt | < 47tknt)f*(t)/AssA*\t) = l6kT(t)/n?AseA*(t)T(t). (17.39)
Inspection of (17.38) and (17.39) reveals that, as always A* < «*, the
former is stronger than the latter if 2T < n\3T- Te\. This means that when 0
<T< 0.3Te or 0.47V < T< r„, the condition for the validity of the approximation
(17.1) for the Jq,(t) dependence during nucleation of crystals in melts is the
inequality (17.38). Only when T is in the relatively narrow range between
0.37V and 0.47V will (17.39) be stronger than (17.38) so that then it is the
fulfilment of (17.39) that guarantees the accuracy of eq. (17.1). Usually,
however, crystals nucleate in melts at temperatures in the range of applicability
of (17.38) and that makes this inequality the condition of actual importance.
It is, therefore, desirable to simplify (17.38) to a form more convenient for
practical purposes. With T - Tc and the high enough n* = 50, from (17.38)
we find that when T is varied between 0.47V and Te, crystal nucleation in
melts will be quasi-stationary and, in addition, eq. (17.1) will be valid provided
| dT(r)/d/1 < We/100Ase*(r). (17.40)
It is worth noting also that the conditions (17.37)-( 17.40) are directly applicable
to nucleation of crystals during isobaric polymorphic transformations at time-
dependent temperature below Te.
As seen from (17.40), the higher the melting temperature 7e and/or the
smaller the melting entropy Ase and the time lag T corresponding to the
momentary value of T, the higher the cooling (or heating) rate at which the
crystals in the melt can be nucleated in quasi-stationary regime. When (17.40)
is fulfilled, eq. (17.1) is a reliable approximation tothe7qs(r) dependence for
T in the range of 0.47V <T<Te. If, however, in this range the | d77dr | value
is close to or somewhat higher than the value of the r.h.s. of (17.40), iqs(r)
may deviate noticeably from Js(r) and may need the more accurate description
provided by eq. (17.20) or its particular form (17.25). Such a deviation was
found by Kozisek and Demo [1993a] who studied numerically crystal
nucleation in Li20-2Si02 melt at oscillating temperature. In their study Tt =
1313 K, As,Jk = 5, T=25 s (at the chosen average temperature of 850 K), and
the maximum value of the rate of temperature variation is | d77df | = gw^osc
Nucleation at variable tupersaluralion 289
= 0.125 to 3 K/s (coosc and 7"osc are, respectively, the oscillation frequency and
amplitude). Comparing these | d77dr1 values with the corresponding value of
0.1 K/s of the r.h.s. of (17.40), we see that this inequality is more strongly
disobeyed for higher co0!x and/or 70sc values. This is in line with the finding
of Kozisek and Demo [ 1993a] that the deviation of Jqs(t) from 7s(r) increases
linearly with coOBC and 7osc and is considerable only at high enough oscillation
frequency and/or amplitude.
As an example of the application of (17.40) to a real system we may
consider HON of ice in water at atmospheric pressure. If nucleation occurs
in the range of temperatures between, e.g. 210 and 270 K when T < 1 /is
(see Fig. 15.7), with Tt = 273 K and Aselk = 2.6 we find from (17.40)
that the process will be quasi-stationary and eq. (17.1) will be valid when
| dT(t)/dt | < 106 K/s. The experimental data of Butorin and Skripov [1972]
(the triangles in Fig. 13.9) evidence that, indeed, constant cooling rates | d77
dr1 of 0.05 and 0.15 K/s have no effect on the JS(AT) dependence. If, however,
nucleation takes place at lower temperatures, e.g. for T < 165 K when T > 1
s (see Fig. 15.7), according to (17.40), | d77d/1 should not exceed the much
lower value of 1 K/s in order for quasi-stationarity to be preserved. This
result exemplifies the crucial role of the time lag T in limiting the regime of
quasi-stationary nucleation of crystals in melts.
This Page Intentionally Left Blank
Part 3
Factors affecting nucieation
This Page Intentionally Left Blank
Chapter 18
Seed size
In Sections 3.2 and 4.3 we have seen that in HEN on a substrate the main
effect of the substrate is to reduce the work IV for cluster formation and,
thereby, the nucleation work W*. In the case of 3D HEN this effect is quantified
by the activity factor f in the effective specific surface energy 0"ef defined by
eq. (4.42). As f is sensitive to the geometry of both the cluster and the
substrate, it may be expected that the substrate size and curvature could
affect the nucleation activity of the substrate. We shall now consider 3D
HEN on a finite-size substrate in the form of a spherical particle called a
seed. Such seeds originating, e.g. from dust or smoke, are always present in
the atmosphere and play an important role in raining and snowing [Boucher
1969]. Foreign microparticles in the bulk of insufficiently purified solutions
and melts are also examples of such seeds whose presence in most cases
eliminates the occurrence of HON in the old phase. The question about the
effect of the seed size was addressed first by Krastanov [1941, 1947/48], but
the complete answer was given by Fletcher [1958] for spherically shaped
seeds and clusters. Later, more complex seed/cluster geometries were also
studied [Kaischew and Mutaftschiev 1963; Miloshev 1963; Krastanov el al.
1965; Krastanov 1970].
Following Fletcher [1958], we consider a moon-shaped condensed-phase
cluster of radius R on a spherical seed of radius iJs (Fig. 18.1). After determining
the areas of the cluster/old phase, cluster/seed and seed/old phase surfaces
with the help of elementary geometry, classically, for the respective total
surface energies we can write (cf. eqs (3.51), (3.53) and (3.54))
(*>
Fig. 18.1 Moon-shaped cluster of radius R on a spherical seed of radius Rs.
294 Nucleation: Basic Theory with Applications
<I>s0 = 4ko,R2 (18.1)
0s(fi) = AitotRl -iKO^lW -h(x)(j;-cos 6W)] (18.2)
0(K) = 2koR2[1 + u(x)(\ - x cos 6W)]
+ Ino, R2U - u(x)(x - cos 6W)]. (18.3)
Here a, as and c* are the specific surface energies defining the wetting angle
0W in the Young equation (3.55), and the dimensionless reciprocal cluster
radius x and the factor u{x) are specified by
x = RJR (18.4)
u(x) = (1 - 2x cos 6W + x2)-"2. (18.5)
Since we are considering a condensed-phase cluster, according to (3.15),
the excess energy Gcx of the cluster is also given by eq. (18.3). Hence,
employing eqs (18.1)-(18.3) in (3.87), we find that the cluster effective
excess energy reads
0(R) = 27toR2{[l + u(x)(\ -xcos 8-J]
+ *2cos B*[u(x)(x - cos 9W)- 1]} (18.6)
provided that the cluster is defined by the EDS.
Using this result in eq. (3.86) leads to the following formula for the work
for cluster formation:
W(R) = - A^(7tf3i>o)fi3(2 + 3«U)(1 - x cos «,) - «3(jc)(1 - x cos 6W)3
- *3[2 - 3u(x)(x - cos 6„) + u\x)(x - cos 6W)3]}
+ 2koR2{[\ + u(x)(\ -*cos ej]
+ /cos ew[«(jc)(jc - cos 6W)- 1]). (18.7)
In writing this formula we have utilized eq. (3.13) to replace the number n
of molecules in the cluster by the cluster volume which is given by the
product of (TtlTjR? and the expression in the curly brackets behind B*. We
note that eq. (18.7) takes the simple form
W(R) = - A^(4^/3u0)(«3 - Rl) + Ana(R2 - Rl) (18.8)
in the limiting case of 8„, = 0 (complete wetting of the seed) when the cluster
is a spherical shell around the seed (Fig. 18.2d).
Now, the standard procedure to determine the nucleus radius R* and size
n* and the nucleation work W* requires calculation of the position and the
value of the maximum of the W(R) function (18.7) (see eqs (4.1) and (4.2)).
However, most of this tedious work can be saved if, following Fletcher
[1958], we note that the nucleus radius R* is necessarily given by the Gibbs-
Thomson equation (4.9) (otherwise the nucleus cannot stay in equilibrium
with the old phase). Thus, substituting R* from (4.9) into eq. (18.7) yields
the sought nucleation work in the form [Fletcher 1958]
Seed size 295
(b) (c)
(e) (f)
Fig. 18.2 Spherical nucleus of radius R* on (a), (b), (c) seeds with different radii,
and (d), (e), (J) seeds with different wetting angle.
W* = 16xv%ol,BAlll- (18.9)
Here the effective specific surface energy c% (J/m2) is defined by (4.42), and
the activity factor •P is given by
f3= (1/2)(1 + u*\\ -x* cos S„)3 + 3jc*2cos S„[«*(x* - cos 8„) - 1]
+ jc*3[2 - iu*(x* - cos S„) + k*V - cos 6W)3]] (18.10)
where
x*=RJR* (18.11)
u* = (1 - 2x* cos S„ + jc*2)-"2. (18.12)
Comparing eqs (18.9) and (4.39), we see that the former is a particular
case of the latter. There is an important difference, however: while for caps
OVf is entirely determined by the wetting angle 6U, now 0"ef is a function not
only of 6W, but also of the supersaturation Aft. This is so, because the activity
factor f from (18.10) depends on the nucleus radius R* which varies with
A/j as required by eq. (4.9). The implication is that, classically, the familiar
dependence of W* on 1/A^i2 for HON of spheres, cubes, etc. or 3D HEN of
caps, lenses, etc. (see eqs (4.8), (4.25) and (4.31)) is not in force for 3D HEN
of moon-shaped nuclei. In fact, HON of spheres and 3D HEN of caps appear
as limiting cases of 3D HEN of moons. Indeed, when 8W is fixed, from
(18.10) we find that V= 1 at Rs = 0 and that f = ^(¾) at /?s = ■», y<0„,)
being given by (3.56). Hence, these two limiting cases are HON of spheres
(Fig. 18.2a) and 3D HEN of caps (Figs. 18.2c), respectively. Alternatively,
when it, is fixed, but the wetting angle 6W has the limiting values 6W = 0
296 Nucleation: Basic Theory with Applications
(complete wetting) or 6W = it (complete non-wetting), from eq. (18.10) it
follows that
f = (1 - 3x*2 + 2**3)"3 (18.13)
in the former case [Krastanov 1947/48] (this is 3D HEN of shells, Fig.
18.2d) or that ¥ = 1 in the latter case (this is HON of spheres, Fig. 18.2f).
Equation (18.10) reveals also that, geometrically, unlike in HEN of a cap or
a lens (see Section 4.3), the activity factor fin HEN of a moon-shaped
nucleus is not equal to the ratio between the volume of the nucleus and the
volume (4/3)rcK*3 of the corresponding homogeneously formed spherical
nucleus. As noted by Fletcher [1958], this distinction between f for caps or
lenses and f for moons is a result of the lack of proportionality of the moon
volume to /J3 and of the moon surface area to R2. Nonetheless, as in the case
of caps and lenses, the greater the volume which has to be subtracted from
the volume of the sphere to make it a moon (see Figs 18.2a, b, c), the smaller
the value of fand, hence, the greater the nucleation activity of the considered
seed. This means that diminishing the seed radius Rs from infinity (flat
substrate) to zero (no substrate) makes the seed less and less active and can
even eliminate it from the nucleation process in the Rs —> 0 limit.
Using the fact that R* is given by (4.9), we can determine also the nucleus
size n*. Since the number n of molecules in a moon-shaped cluster of radius
R is given by the expression multiplying A^i in eq. (18.7), setting in this
expression R = R*, x = x* and u = u* and allowing for (4.9), we get
n* = (8otJcj3/3A^3)(2 + 3«*(1 - x* cos 6W)
- «*3(1 - x* cos ew)3 - **3[2 - iu*(x* - cos 6W)
+ «*3(x*-cos 9W)3]}. (18.14)
This is the Gibbs-Thomson equation for the moon-shaped nucleus. It
shows that, due to the A^i dependence of x* and »*, the number n* of
molecules in such nucleus is a rather complicated function of the supersaturation
&/j. As it should be, in the limits of fis = 0 or 8W = 7teq. (18.14) passes into
eq. (4.10) for HON of spheres, and in the Rs —> °° limit it turns into eq. (4.24)
for 3D HEN of caps. Also, it can be verified that W* and n* from (18.9) and
(18.14) satisfy the nucleation theorem in the form of eq. (5.29).
Now that we know the nucleation work for the considered case of 3D
HEN on seeds we can analyse the effect of the seed radius on the stationary
nucleation rate 7s(m"2 s"1). Thus, classically, using eqs (4.42), (13.40) and
(18.9) in (13.39), we find that [Fletcher 1958],
Js = zf*CB exp [- ¥3(An, Rs, 6„)B/Aji2] (18.15)
where the thermodynamic parameter B is given by eq. (13.49) with c =
(36tt)1/3 and (Tef = a. The Zeldovich factor z changes relatively slightly with
the seed radius Rs (through n* and W*) and may be treated as virtually /?s-
independent. This is valid also for the frequency/* of monomer attachment
to the nucleus if the mechanism of this attachment remains the same for
Seed size 297
different Rs values. According to eq. (15.73), this implies that the nucleation
time lag X cannot be affected considerably by the seed radius under otherwise
equal conditions.
Figure 18.3 illustrates the JS(RS) dependence (18.15) at a given fixed
wetting angle (as indicated). The calculation is done for nucleation of water
droplets in vapours at 7= 293 K and supersaturation ratio Ssp/pe - 2.5, and
the other parameter values used are listed in Table 3.1. According to (2.8),
(4.9) and (13.49), under these conditions for the supersaturation we have Ay
= 0.92kT, the nucleus droplet is of radius R* = 1.2 nm, and B = S9(kTf. The
kinetic factor A = z/*C0 is expressed as A'S (cf. eqs (13.40)-(13.42)), and A'
is assumed to have the exemplary value of 1026 nT2 s_1 corresponding to C0
1030
1025
1020
1015
^ 1010
™ 105
E
^ 1
10"5
10-10
io-15
io-20
0.1 1 10 100
Rs/R*
Fig. 18.3 Dependence of the stationary nucleation rate on the seed radius according
to eq. (18.15) for HEN at T = 293 K and p/pe = 2.5 of water droplets in vapours on
seeds with different wetting angle {as indicated).
298 Nucleation: Basic Theory with Applications
= 1019 m-2 (seed surface free of active centres) and to typicalz = 0.1 and /e*
= 108 s"1. As seen from Fig. 18.3, better seed wetting leads to higher nucleation
rate as long as the seed radius Rs is sufficiently greater than the nucleus
radius R*. In fact, given the 0W value, such a large seed already acts as a flat
substrate with the same wetting angle (Fig. 18.2c). Diminishing Rs lowers Js,
but the effect is strong only for fls = fl*. When the seed is sufficiently smaller
than the nucleus, 7S is virtually independent of both fls and 0W and equal to
the nucleation rate in the case of complete non-wetting. This is so because
such a small seed is practically inactive for nucleation: regardless of its
wetting, energetically, its presence is not 'felt* by the almost spherical nucleus
(Fig. 18.2a). We can say, therefore, that when the old phase contains an
ensemble of seeds of various size and the same wetting angle, it is the
nucleus that 'chooses' whether a given seed is or is not important for the
nucleation process. Indeed, approximately, all seeds with Rs < R* are effectively
out of the process, and all seeds with Rs > R* behave as flat substrates. Since
R* depends on Ay, this means that the number of the seeds which can affect
the process will vary with variation of the supersaturation of the system.
The Js(Ay) dependence predicted by eq. (18.15) is visualized in Fig. 18.4
in the case of nucleation of water droplets in vapours at T = 293 K. The
calculation is done with the parameter values used for Fig. 18.3 and with Ay
= kT In 5 as required by eq. (2.8). The seeds are assumed to have the same
radius R& = 2 nm, but different wetting angle (as indicated). Figure 18.4
shows that, due to the dependence of the activity factor ffrom (18.10) on
the supersaturation, the stationary nucleation rate is a more complicated
function of Ay for 3D HEN on seeds (curves 30°, 60° and 90°) than for 3D
HEN on a flat substrate (curve 180°). This has the important implication that
an experimental Ay dependence of the stationary nucleation rate in the presence
of seeds cannot be linearized in the classical In Js-vs-( XIAy)2 coordinates for
3D nucleation if the Ay range studied corresponds to nuclei of size comparable
with that of the seeds. In this situation a reliable interpretation of the Js(Ay)
data is possible only in the scope of the first application of the nucleation
theorem (Chapter 14). Indeed, employing eq. (14.6) allows a theory-
independent determination of the Ay dependence of the nucleus size n*. The
so-obtained experimental dependence can then be confronted with that predicted
by the Gibbs-Thomson equation (18.14). A good description of the experimental
n*(Ay) data by eq. (18.14) would be an evidence for 3D HEN of moon-
shaped nuclei on seeds.
As already noted above, the smaller seeds are less active with respect to
nucleation, because the nucleus corresponding to HON loses less volume
when formed on a smaller seed. This geometrical rule can be invoked for
understanding the nucleation activity of substrates with complex surface
relief. The basic elements of such a relief are convex, concave and flat parts
of the substrate surface. At a given Ay, whereas the 3D nucleus corresponding
to HON loses maximum of its volume when it is situated on the concave
surface, the loss is minimal when it is on the convex surface. The conclusion
of practical significance is, therefore, that the nuclei in 3D HEN are most
Seed size 299
Fig. 18.4 Dependence of the stationary nucleation rate on the supersaturation ratio
according to eq. (18.15) for HEN at T = 293 K of water droplets in vapours on
equally sized seeds with different wetting angle (as indicated).
likely to appear on the concave parts of the substrate surface. The convex
parts are least active for 3D HEN, and the flat parts are with intermediate
activity. It should be kept in mind, however, that the substrate curvature can
be 'felt' by the nucleation process only when, locally, the radius of this
curvature is comparable to the nucleus radius R*. We note as well that,
qualitatively, the above results and conclusions about the effect of the seed
size and the substrate curvature on 3D HEN are valid also for gaseous nuclei.
Chapter 19
Line energy
Already Gibbs [1928] pointed out that the energy balance for a system in
which three or more phases contact along a line requires accounting for the
energy arising from the presence of the contact line. The classical theory of
3D HEN of cap- or lens-shaped nuclei, however, ignores the contribution of
the line energy (see Sections 3.2 and 4.3) although the periphery of the cap
or the lens represents a line of three-phase contact. Gretz [1966a, 1966b] was
the first to consider the effect of the line energy on nucleation of caps. Later,
the problem was approached by Navascues and Tarazona [1981], Navascues
and Mederos [ 1982,1983], Toshev [ 1993] and Talanquer and Oxtoby [ 1996].
In the case of HEN of lenses, the effect of line energy was investigated both
theoretically and experimentally by Scheludko el al. [1981], Toshev et al.
[1988], Toshev and Scheludko [ 1991], Chakarov et al. [ 1991 ], Alexandrov et
al. [1991, 1993] and Alexandrov and Avramov [1993]. We shall now see
how the line energy can influence the kinetics of 3D HEN of cap-shaped
nuclei of condensed phases.
As in Section 3.2, let us consider a cap-shaped cluster of radius R and
wetting angle 6W in the range from 0W = 0 (complete wetting) to 6W = n
(complete non-wetting) (Fig. 19.1). The total surface energies 0s?oand 0S of
the substrate before and after the cluster formation are again given by eqs
(3.51) and (3.54), respectively. Compared with 0 from (3.53), however, now
the total surface energy tp of the cluster contains an extra term equal to the
total line energy k'ItsR sin 6W of the cluster/substrate contact line whose
length is 2nR sin 6W:
<MiJ) = o-2^rfi2(l - cos 6W) + cyr/?2sin2 0W + k'ItcR sin 0W. (19.1)
Here k' (J/m) is the specific line energy of the cluster periphery and, in the
Fig. 19.1 Cap-shaped cluster on a substrate.
Line energy 301
spirit of the classical theory, it is treated as independent of the cluster size.
Although the k" term in (19.1) is analogous to the /derm in 0 for 2D HEN
of disks (see eq. (3.68)), there exists a basic distinction between k* and K.
Namely, whereas the disk specific edge energy K is always positive, the cap
specific line energy k" can be either positive, zero or negative. This distinction
between K"' and K is a manifestation of their different physical meaning.
Since ids the energy per unit length of a 2D phase boundary (the disk edge)
separating two phases in two dimensions, it must be positive: otherwise the
condition for thermodynamic stability is not satisfied and the two 2D phases
cannot coexist [Toshev 1991]. In contrast, k" is the energy per unit length of
the line at which three phases (the substrate, the cluster and the old phase)
meet and it can be of either sign or zero, since this cannot cause the
disappearance of the three-phase contact line representing the cap periphery
[Toshev 1991]. The possibility for the line energy to be negative was noted
by Gibbs [1928], and Platikanov el al. [1980] demonstrated experimentally
that t? can change sign in dependence of the molecular interactions in the
contact-line region (concerning the sign of t? see, e.g. Exerowa et al. [1994]).
Since we are considering condensed-phase clusters, according to (3.15),
the cluster excess energy Gcx is given by 0 from (19.1). Therefore, using
(3.51), (3.54) and (19.1) in eq. (3.87) leads to the following expression for
the effective excess energy # of the EDS-defined cap-shaped cluster [Gretz
1966a]:
*(/?) = 2koR2(\ - cos 6W) + K(a, - as)R2 sin2 6W + InnfR sin fl^. (19.2)
In principle, R and #w can be treated as independent variables, but for a
given cluster volume their values are most likely to be those corresponding
to the equilibrium shape of the cluster. This shape is defined by the condition
for minimum of the cluster effective excess energy <P from (19.2) at a fixed
cluster volume V„ from (3.52). Hence, assuming hereafter that R and Sw are
the radius and the wetting angle characterizing the equilibrium shape of the
cap, we can find them as the solution of the simultaneous equations d4> = 0
and dV„ = 0. Doing that yields the formula [Shcherbakov and Ryazantsev
1964; Gretz 1966a]
cos 6W = (os-o,)lo- k'IRosin 6» (19.3)
which can be represented equivalently as
cos ew = (o-s - 0,)/0- (4;r/3u0)"Vv"3(ew)/n"3osin 0W (19.4)
because of the relations (3.13) and (3.52) between R and the number n of
molecules in the cap-shaped cluster (y(8w) 's defined by (3.56)).
Equation (19.3) or (19.4) is a generalization of the Young equation (3.55)
in the sense that it reduces to (3.55) at k" = 0. Understandably, it passes
into (3.55) also in the limit of R or n -» «°. For clusters of smaller size,
however, the effect of the line energy is considerable if Ik^I is high enough:
for such small clusters 8W is already a function of R or n and can depart
substantially from the Young wetting angle defined by (3.55). To avoid
302 Nucleation: Basic Theory with Applications
confusion, we shall denote the Young wetting angle by 6y and rewrite (3.55)
in the form
cos 8Y = (as - a, )1 a.
(19.5)
It is clear that, physically, 6Y is the angle of wetting of the substrate by a
macroscopically large new phase.
The 6w(n) dependence (19.4) for water droplets in vapours at 7= 293 K
on a substrate with macroscopic wetting angle 0Y = 80° is depicted in Fig.
19.2. The calculation is done with u0 and a from Table 3.1 and with assumed
CD
■o
n
Fig. 19.2 Size dependence of the cluster wetting angle: lines - 20, 0 and 20 - eq.
(19.4) for cap-shaped water droplets with specific line energy - 20, 0 and 20 pJ/m,
respectively. The droplets are on a substrate in vapours at T = 293 K, and the
macroscopic wetting angle is 80°.
Line energy 303
K-" = - 20, 0 and 20 pj/m (as indicated in the figure). We see that in the k' <
0 case 0W is an unambiguous function of n and that the smaller the cap-
shaped cluster, the smaller its wetting angle. For that reason, the equilibrium
shape of the caps of less than, say, 10 molecules may require a height which
is below the minimal possible height of one molecular diameter. This means
that too small clusters with the corresponding equilibrium shape of caps
have no physical reality. When k" > 0, the smallest clusters cannot be with
the corresponding equilibrium shape either. Indeed, Fig. 19.2 shows that the
6W has no value for n < 11. Besides, in the k' > 0 case a given cluster of large
enough size can have two wetting angles. As we shall see below, the smaller-
angle clusters require less work for their formation so that they are more
likely to be involved in the nucleation process than the greater-angle ones.
Substituting <1>(R) from (19.2) in eq. (3.86), we can now find the work for
formation of a cap-shaped cluster whose size and wetting angle are related
by eq. (19.3) or (19.4). Using eqs (3.13) and (19.3) to replace the number n
of molecules in the cluster by the cluster volume and the energy difference
(Tj - <7S by the line energy and recalling that the cluster volume is given by
(3.52) leads to
W(R) = y[0w(«)][ - (4jrA,u/3u0)fi3 + 4tzoR2] + kk?R sin [6W(R)] (19.6)
where y/(9w) is defined by (3.56). In terms of n, this expression can be given
the equivalent form
W(n) =-nA/i + (36tfu02)"3 (J\i/',[en(n)]nm
+ (3jtW4)"V sin [6U«)lV~ "3[e„(n)]n"3. (19.7)
We note that these two equations turn into (3.59) and (3.60) in the particular
case of k' = 0 when 9W is ^-independent and equal to 0Y.
Figure 19.3 exhibits the W(n) dependence (19.7) for condensation of water
vapours at 7= 293 K on a substrate again with 8V = 80°. The calculation is
done with the parameter values used for Fig. 19.2 and with Apt determined
from (2.8) at supersaturation ratio S = plps = 2.2. Curves - 20, 0 and 20
correspond to k' = - 20, 0 and 20 pj/m, respectively. As seen, in the k" < 0
case W(n) displays a minimum and a maximum. While the position of the
maximum determines the size n* of the nucleus, the minimum corresponds
to a cluster in stable thermodynamic equilibrium with the old phase. However,
as already noted above, when it is too small, this cluster and thus a part of the
initial portion of the W(ri) curve have no physical reality. For instance, the
minimum of curve - 20 in Fig. 19.3 is at n = 0.3, and the whole initial part
of this curve up to n = 10 is physically irrelevant, because such small clusters
cannot have the shape of a cap with wetting angle of less than 55° (this is the
Sw value which we read in Fig. 19.2 at n = 10). In the opposite case of k" >
0 (curve 20 in Fig. 19.3), W(n) has two branches, each of them with a
maximum. The lower and the upper branches give W for a cluster with the
same n, but with the smaller and the greater of the two possible wetting
angles 0W, respectively (see curve 20 in Fig. 19.2). The effective nucleus is
the cluster with size given by the position of the maximum of the lower
304 Nucleation: Basic Theory with Applications
branch. This is so since its formation is more likely than that of the greater-
angle nucleus with size determined by the maximum of the upper W(n)
branch. The common point (at n = 11) of the two branches of curve 20 in Fig.
19.3 corresponds to the 'nose' of the 8w(n) curve 20 in Fig. 19.2.
160 p 1
-20 r ' '. i ■ . .' i .>* I
1 50 100 150 200 250 300 350 400
n
Fig. 19.3 Dependence of the work for cluster formation on the cluster size: curves -
20, 0 and 20 - eq- (19.7) for cap-shaped water droplets with specific line energy - 20,
0 and 20 pj/m, respectively. The droplets are on a substrate in vapours at T = 293 K
and pipE = 2.2, and the macroscopic wetting angle is 80°.
Equations (19.6) and (19.7) and Fig. 19.3 show that, depending on its
sign, the specific line energy t? increases or diminishes the work for cluster
formation and in this way changes also the nucleation work W*. To find W*,
as prescribed by eqs (4.1) and (4.2), we must determine the position and the
value of the maximum of W from (19.6) or (19.7) at the same time taking
into account that Sw depends on R or n according to eq. (19.3) or (19.4). This
involves a great amount of labour which we need not do if we recall that the
nucleus radius R* is necessarily given by the Gibbs-Thomson equation (4.9)
(if not so, the nucleus cannot be in equilibrium with the old phase). Thus,
setting R = R*,n = n* and 6* s 0w(fi*) in eqs (3.13), (3.52) and (19.3) and
using (4.9) and (19.5), we find that the number n* of molecules in the cap-
shaped nucleus and the nucleus wetting angle 9* are given by
n* = y/(9*) 32;ti>0zcj3/3 A,u3 (19.8)
cos 6* = cos 6V - K'&fi/2v0a2 sin 6*. (19.9)
Regretfully, these two equations express only implicitly the dependence of
n* and 9* on the supersaturation A^i. Yet, as sin 9* > 0 in the physically
relevant range of 9* = 0 to 9* = 7t, eq. (19.9) tells us that 9* > 9W or 9* < b\
Line energy 305
for k' > 0 or k' < 0, respectively. Accordingly, the Gibbs-Thomson equation
(19.8) shows that when k" > 0 or k' < 0, the nucleus size n* is, respectively,
larger or smaller than that at k' = 0. This is visualized in Fig. 19.3 and follows
from the monotonous rise of y/v/ith the wetting angle (see Fig. 3.6). Also, eq.
(19.9) reveals that the effect of the line energy on the nucleus wetting angle 8*
(and, thereby, on n*) is increasingly important at higher supersaturations and
for nuclei with higher 11? | and/or lower <J. Moreover, sufficiently high values
of the parameter | k" \Afil2vQ01 make the r.h.s. of eq. (19.9) either less than-
1 (when k' > 0) or greater than 1 (when k" < 0). This equation then has no
solution with respect to #*, which is a mathematical expression of the fact that
for such high values of this parameter none of the clusters is a nucleus, indeed,
then Wfrom (19.6) or (19.7) has no maximum-it only decreases with increasing
R or n so that all clusters on the substrate are supernuclei.
Figures 19.4a and 19.4b depict, respectively, the n*(A/<) and 8*(Afi)
dependences (19.8) and (19.9) again for nucleation of water droplets in
vapours at T = 293 K on a substrate with 0y = 80°. The calculation is done
with the parameter values already used for Figs 19.2 and 19.3. As seen, when
k' * 0 (curves -20 and 20), some 20° departure of 9* from fly already
changes considerably the nucleus size n* with respect to that in the k' = 0
case (curve 0). We observe also that each of the curves - 20 and 20 in Fig.
19.4 has upper and lower branches which visualize the two values of n* and
8* satisfying eqs (19.8) and (19.9) at k' * 0 when A^i is sufficiently low. In
the k" < 0 case, whereas the greater n* and 6* values characterize the nucleus,
the smaller ones correspond to the cluster in stable thermodynamic equilibrium
with the old phase (we recall, however, that such a cluster does not exist in
reality if its size is too small). In the opposite case of K1 > 0 the nucleus
which is effective at the given supersaturation is characterized by the smaller
n* and 9* values. Figure 19.4 shows also that when k' * 0, it is not possible
to speak about nucleus (eqs (19.8) and (19.9) have no real solutions with
respect to n* and 0*) when A^i is higher than the supersaturation corresponding
to the 'noses' of curves -20 and 20. This means that in the range of these
high Aft values nucleation of caps is not involved in the condensation process
on the substrate.
Now, knowing that R* and 6* are given by (4.9) and (19.9), it is a simple
matter to obtain the sought nucleation work W* by substituting them in eq.
(19.6). The result can be represented in the form of eq. (4.39), namely,
W* = 16lcv$olt/3An2 (19.10)
where cref = ¥a in accordance with (4.42), and the activity factor f is
defined by
V3 = v(9*) + (3k-78d0cj2)A,u sin 8* = y(8*)
+ (3/4)(cos Sy-cos 9*)sin2e*. (19.11)
In a different appearance, eq. (19.10) was obtained by Gretz [1966a], At
k" = 0 (then 8* = fly) it passes into eq. (4.25) for W* for cap-shaped nuclei
306 Nucleation: Basic Theory with Applications
Fig. 19.4 Dependence of (a) the nucleus size, and (b) the nucleus wetting angle on
the supersaturalian ratio for cap-shaped nucleus droplets of water on a substrate in
vapours at T = 293 K: curves - 20, 0 and 20 - eqs (19.8) and (19.9) for droplets with
specific line energy -20, 0 and 20 pj/m, respectively. The macroscopic wetting angle
with zero line energy, but when t? > 0 or k1 < 0, it predicts nucleation work
higher or lower, respectively, than that in the k" = 0 case. When k" > 0, W*
from (19.10) has two values, the smaller one being the effective nucleation
work (see Fig. 19.3). Equation (19.10) shows that the line energy affects the
nucleation work of caps by making it decrease with A^i in a rather complicated
way, and not just proportionally to Ap ~2 as required by (4.25). This is due
to the fact that now c% is a function of A^i through the activity factor V
which is controlled not only by the Young wetting angle 8Y as 'n the K-1 = 0
case (then y(0*) = V(9y) and f = i//'3(0Y)), but a's0 by the A^-dependent
Line energy 307
wetting angle 6* of the nucleus. Thus, with regard to their A/j dependence,
atf in (19.10) and V from (19.11) are congenial with c% and f for moon-
shaped nuclei (cf. cqs (18.9) and (18.10)). It should be noted that a more
accurate calculation of W* in the t? < 0 case may require accounting that in
this case, if large enough, the stable clusters which correspond to the minimum
of W(n) (curve -20 in Fig. 19.3) could act as seeds for the nuclei [Chakarov
etal. 1991], Also, it can be verified that n* and W* from (19.8) and (19.10)
satisfy the nucleation theorem in the form of eq. (5.29).
Having obtained W* for caps with K'V 0, we are now in a position to
determine the corresponding stationary nucleation rate Js (irf2 s~'). Thus,
classically, substituting W* from (19.10) in the general formula (13.39) and
accounting for (4.42) and (13.40) yields [Gretz 1966b]
Js = zf*C0 exp [- «P3(Aij, ey)BIAM2] (19.12)
where the thermodynamic parameter B is given by (13.49) with c =
(36s:)"3 and acf= a, and the activity factor f is specified by (19.11). Owing
to the relatively weak dependence of the Zeldovich factor z, and the frequency
/* of monomer attachment to the nucleus on W* and/or «*, both z and/* can
be treated as practically unaffected by the specific line energy k'. We note
that, in view of eq. (15.73), this implies also that /r'has virtually no effect on
the nucleation time lag r. Depending on the mechanism of monomer attachment,
however, some effect of k" on/* and, thereby, on X may be expected when
the value of K* is such that the nucleus wetting angle #* is close to its limits
0* = 0 or #* = K corresponding to complete wetting or non-wetting,
respectively.
Shown in Fig. 19.5 is the Js(A[i) dependence for condensation of water
vapours at T = 293 K on a substrate which, macroscopically, is characterized
by fly = 80°. The calculation is done according to eq. (19.12) with kinetic
factor A = zf*C0 expressed as A'S (see eqs (13.40)-( 13.42)), A' is assumed to
have the exemplary value of 1026 rrr2 s~' corresponding to C0 = 1019 m2 (no
active centres on the substrate) and to typical z = 0.1 and /e* = 108 s"1. The
supersaturation A/y is determined from (2.8) in dependence of the super-
saturation ratio S = plpe. The variation of the activity factor f with A/j is
taken into account with the help of eq. (19.11) in which 6* is considered as
a function of A/j as required by eq. (19.9). The parameter values used are
given in Table 3.1 (according to (13.49) they result in B = %9(kT)\ and the
specific line energy k' of the cap-shaped nuclei is assumed again to be -20,
0 and 20 pj/m (as indicated in the figure). We see from Fig. 19.5 that while
t? < 0 leads to higher nucleation rate in comparison with that in the *? = 0
case, fc" > 0 acts in the opposite direction, the effect being considerable under
the chosen conditions. The line energy affects also the character of the 7s(A/j)
function: due to the A/j dependence of the activity factor f from (19.11) at
k" *■ 0, 7s(A/j) from (19.12) is not practically linear in the classical In 7s-vs-
(1/A/j)2 coordinates when | kt' | is sufficiently high. This means that a non-
linearity in such coordinates of the plot of experimental 7s(A/j) data for 3D
HEN on a substrate might be caused by non-vanishing line energy. Whether
308 Nucleation: Basic Theory with Applications
1020
1015
'co
V 1010
105
1
12 3 4 5
s
Fig, 19.5 Dependence of the stationary nucleation rate on the supersaturation ratio
for HEN of cap-shaped water droplets on a substrate in vapours at T = 293 K: curves
- 20, 0 and 20 - eq. (19.12) for droplets with specific line energy - 20, 0 and
20 pJ/m, respectively.
this is the case or not can be answered reliably only in the scope of the first
application of the nucleation theorem (Chapter 14). Namely, according to eq.
(14.6), we can calculate in a theory-independent way the nucleus size «*
from the slope of the experimental /s(Aju) curve in In Js-vs-Afj coordinates.
The so-obtained experimental n*(Afj) dependence can then be juxtaposed
with the Gibbs-Thomson one, eq. (19.8), in order to check if the nucleus line
energy is indeed the reason for the observed non-linearity of In /s as a
function of A,u"2.
Finally, we note that line energy effects of the kind considered above may
be expected to be of significance also for 3D HEN of cap-shaped bubbles
provided the radius R* of the nucleus bubble is sufficiently small.
-
;
-
-
-
/-20
|/
. i 1. .
/°
. /i .
/20
Chapter 20
Strain energy
Most generally, nucleation is a process accompanied with volume changes.
Such changes are of practically no significance for the cluster formation
when it occurs in a fluid phase, but give rise to strain fields both inside and
outside the cluster when it is formed in a solid matrix. This is the reason for
which the strain energy can be an important factor in nucleation, for instance,
in crystals, vitrified melts and solid solutions. We shall now consider the
effect of strain energy on the nucleation work W* and, thereby, on the stationary
rate Js of nucleation of condensed phases in solid matrices. The analysis is
restricted to elastically deformed isotropic clusters and matrices. More
complicated cases involving clusters either coherent or incoherent with the
matrix are discussed elsewhere (e.g. Hornbogen [1969]; Lyubov [1969];
Christian [1975]; Russell [1980]; Gutzow and Schmelzer [1995]).
As known [Landau and Lifshitz 1965], when deformation takes place
isothermally and reversibly, the strain energy adds to the free energy of the
undeformed phase. Accordingly, the expression for the work Win) to form a
strained /z-sized cluster in an initially unstrained matrix must contain as a
summand the total energy ¢5,,(/1) of the strain fields generated in the cluster
and the matrix as a result of the cluster formation [Fisher etal. 1948]. Hence,
in lieu of eq. (3.86) we shall have
W(n) = -n&fi + #(n) + ¢,,,(/1). (20.1)
This formula is of most general validity and with appropriate expressions
for the supersaturation A^i, the cluster effective excess energy $ and the total
strain energy ¢,,, it can be used to describe classically or atomistically the
thermodynamics of one-component nucleation involving strain in any particular
case of interest. We shall confine our analysis to classical HON of EDS-
defined condensed-phase clusters in a solid matrix such as vitrified melt,
one-component crystal or solid solution. In this case A^i is determined
approximately by eqs (2.14), (2.23) and (2.25), and ¢(/1) is obtainable from
(3.15) and (3.87) at <ps = 0sO = 0 so that the actual problem is the calculation
of ¢,,,(/1). Let us see how ¢,,,(/1) can be calculated in the particular case of
a spherical cluster in a solid matrix, both being elastically isotropic and
under dilatational strain (Fig. 20.1).
The dilatational strain inside and outside the cluster is due to the volume
misfit mv between the cluster and the matrix, which is defined as
">v = (vm - v0)lv0 (20.2)
where v0 and vm are the volumes per molecule in the cluster and the matrix,
310 Nucleation: Basic Theory with Applications
Fig. 20.1 Cross-section of a dilatationally strained spherical cluster in solid matrix
(after Hornbogen [1969]).
respectively. The density of the energy of elastic strain in the cluster, £sltc,
and in the matrix, £^tr>II1, depends only on the distance r from the cluster
centre because of the spherical symmetry of the strain fields in and around
the cluster. In the framework of the continuum approximation of the theory
of elasticity E,Kf and £M7]T1 are given by (e.g. Christian [1975]; Ulbricht et al.
[1988])
£fl7,c = 2(1 - 2vJmiE$MEVfFJW - 2^¾.
+ (1 + vJEyJ2, (0<r<R) (20.3)
ESB.m= {(1 + vJmiE^EVM/3Pii -2vc>EYjn
+ (1 + vrn)£'Y,c]2)(Rfr)6, («<r<~). (20.4)
Here £Y,c and vc are, respectively, Young's modulus and Poisson's ratio for
the cluster, and £Y m and vm are the same quantities, but for the matrix. Both
£Y,c and vc are treated as independent of the cluster radius R. Equations
(20.3) and (20.4) show that the (Mutational strain is uniformly distributed
over the volume of the cluster and that it decays quickly outside the cluster.
Consequently, interactions between the strain fields around the clusters are
negligible provided the clusters are a few cluster radii away from each other
[Ulbricht et al 1988].
Now, #st7 is readily obtained by integration of E^t[c and £str,m in accordance
with
<2>at = 4k \ £SMr2 dr + Ak \ £wr2 Ar (20.5)
Jo Jr
and by using eqs (3.13) and (3.22) which relate R and n. The result is
Strain energy 311
[Nabarro 1940a, b; Christian 1975; Ulbricht et al 1988]
«««) = <rW (20.6)
where 0str (/), given by
0str = m^o£Y.c£y.m/3L2(l - 2vc)EYm + (1 + vm)EYJ, (20.7)
is the strain energy per molecule of the cluster. This energy increases
quadratically with the volume misfit mv and is thus always non-negative.
From eqs (20.6) and (20.7) we see also that in the considered case of
dilatationally strained isotropic spherical cluster in isotropic matrix 0^ is
independent of the cluster size n (since the possible dependence of EYc and
vc on n is ignored) so that <Ps(r is proportional to n, i.e. to the volume of the
cluster. It turns out that the size independence of 0str is preserved also for
non-spherical clusters as long as their shape is fixed - then 0str is given again
by eq. (20.7), but with a corresponding numerical shape factor in the r.h.s. of
this equation [Nabarro 1940a, b; Robinson 1951; Eshelby 1957; Christian
1975; Russell 1980]. Only in the particular case of isotropic elasticity and
equal elastic constants of the cluster and the matrix (£Y,c = E\.m and vc = vnl)
is 0str independent of both shape and size [Nabarro 1940a, b; Christian
1975]. The point to remember is that the independence of 0str of the cluster
shape and, even more importantly, of the cluster size n is not a general
property. This means that the total strain energy <Pstr(«) is by far not always
linear with n, i.e. it is not always a volume-type contribution to the work for
cluster formation (for a discussion of various <Psu(n) dependences see, e.g.
Christian [1975]; Russell [1980]).
According to eqs (3.15) and (3.87), in HON *(n) is equal to the cluster
total surface energy <j>(n) when the cluster is defined by the EDS. Hence,
using in eq. (20.1) the classical <p(n) dependence (3.20) and <Pstr(«) from
(20.6) yields [Fisher et al. 1948; Christian 1975; Russell 1980]
W(n) = -(Aji - 0stt)n + (36;r)"3 v2f Onm. (20.8)
This is the classical formula for the work to form a spherical condensed-
phase cluster of n molecules in a solid matrix under dilatational strain. It is
valid for EDS-defined elastically isotropic clusters in a matrix with isotropic
elasticity and its applicability is restricted by the usual requirement for large
enough cluster size. When there is no volume misfit between the molecules
of the cluster and the matrix (mv = 0), according to (20.7), 0str = 0 and eq.
(20.8) passes into the classical formula (3.39) for HON in the absence of
strain. Since 0slr > 0, eq. (20.8) reveals that the dilatational strain energy
always acts against the supersaturation imposed on the system. In other
words, under dilatational strain, HON of condensed phases in solid matrices
occurs always at an effective supersaturation A/v - 0srr which is lower than
the actual supersaturation &/j. As a consequence, even if the old phase is
supersaturated, the occurrence of the process is possible only when A,u > <j>str.
This is seen also from the expressions for the nucleus size n* and the nucleation
work W*, which follow upon using W(n) from (20.8) in eqs (4.1) and (4.2)
(e.g. Russell [1980]):
312 Nucleation: Basic Theory with Applications
n* = 32s^o3/3(A^-0str)3, (20.9)
W* = 16^(^/3(^1 - &„)2. (20.10)
Equation (20.9) is the Gibbs-Thomson equation in the considered case of
HON accompanied with dilatational strain. As it should be, in the case of no
volume misfit (then 0slr = 0) eqs (20.9) and (20.10) pass, respectively, into
eqs (4.10) and (4.11) for HON of condensed phases in the absence of strain.
It is worth noting as well that n* and W* from (20.9) and (20.10) satisfy the
nucleation theorem in the form of eq. (5.29). This is understandable in view
of the fact that <Pm(n) from (20.6) is treated as A/i-independent.
In order to see what is the effect of the dilatational strain energy on the
stationary rate Js of HON of condensed phases in solids we can employ the
formula (e.g. Russell [1980]) (A/i > 0nr)
Js=?fC0exp [-fi/(Aj, - 0slr)2] (20.11)
which is obtained from eq. (13.39) with the help of (13.40) and (20.10). This
formula gives Js in the scope of the classical nucleation theory and in it the
thermodynamic parameter B is defined by eq. (13,49) with c = (36^)"3
(spherical nuclei) and oe! = a (HON). As seen, the strain energy retards
nucleation by effectively lowering the supersaturation and can even arrest
the process when the condition A/i ^ 0str is fulfilled. The effect is largely due
to the exponential factor in (20.11), because the Zeldovich factor z and the
frequency /* of monomer attachment to the nucleus are relatively weak
functions of 0str through n* and W* (see eqs (13.36) and (13.40)-(13.43)). It
must be noted, however, that f may be directly affected by the strain field
generated by the nucleus if the transport of molecules across the strained
zone around the nucleus and/or the nucleus/matrix interface is altered strongly
in the presence of strain. This can even lead to a change in the transport
mechanism controlling monomer attachment.
According to eq. (15.73), in all cases when/* is influenced significantly
by the presence of strain, the nucleation time lag x will also be very different
from that corresponding to no strain. For example, for HON of condensed
phases in solid solutions under volume-diffusion control the general formula
(15.73) transforms into eq. (15.85) with cm = c = (36^)"3. Using the first
equality in (13.36) and eqs (20.9) and (20.10) to express z and n*"3 in
(15.85) through 0str and accounting for eq. (2.14), we obtain (kT In S> 0str)
? = 32u0o2l7i2y*(kT)2DCeS(lt\ S - tpstTlkT)3 (20.12)
where S = C/Cc is the supersaturation ratio. This equation corresponds to eq.
(15.86) for HON of spheres: it turns into (15.86) at 0stI = 0. Clearly, when the
sticking coefficient y* and/or the diffusion coefficient D of the solute molecules
are affected strongly by the strain field around the nucleus, the above
dependence of T on 0slr will change considerably.
Figure 20.2 illustrates the effect of the strain energy on the A/* dependence
of the stationary rate Js of isothermal HON of condensed phases in solid
solutions. The curves are drawn according to eq. (20.11) with 0slr = 0, 1.2 x
Strain energy 313
10~21 and4.8x 10JI J. These values are calculated from (20.7) with assumed
volume misfit ± mv = 0, 0.05 and 0.1 (as indicated in Fig. 20.2) and with
typical EXc = EYm = 100 GPa, vc = vm = 0.3 and v0 = 0.03 nm3. The
supersaturation Ap is determined from eq. (2.14) as a function of the
supersaturation ratio S, and B is given the exemplary value of 1.36 x 10~39
J2 corresponding to cr= 100 mj/m2 and T= 800 K in eq. (13.49). The kinetic
factor A = zf*C0 is represented as A =A'S (see eq. (13.41)), and A' is assumed
to be A^i-independent and equal to 3 X 1031 m"3 s"1. According to eq. (13.42),
this A' value corresponds to z = 0.1, C0 = 3 x 1028 m~3 and fc =104 s_l (this
rather small value of /* reflects the low diffusivity of the solute in solid
solutions).
As seen from Fig. 20.2, the strain energy has a strongly inhibiting effect
on the stationary nucleation rate. It leads also to a change in the course of the
1020
1015
'co
<? 1010
C/3
105
1
1 2 3
s
Fig. 20.2 Dependence of the stationary nucleation rate on the supersaturation ratio:
curves 0, 0.05 and 0.1 - eq. (20.1!) for HON of condensed phase under dilatational
strain in solid matrix at volume misfit mv = 0, ± 0.05 and ± 0.1, especfively.
314 Nucleation: Basic Theory with Applications
Js(Ay) dependence: this dependence cannot be linearized in the classical In
Js-vs-(l/A/i)2 coordinates discussed in Section 13.5. This means that non-
linearity of the plot of experimental 7s(A/i) data in these coordinates can be
a manifestation of the effect of strain energy. Since n* and W* from (20.9)
and (20.10) satisfy the nucleation theorem, it is advantageous to interpret the
data in the scope of the first application of this theorem (see Chapter 14).
Namely, if the data are obtained by varying A/i isothermally, plotting them
in In Js-vs-A^/ coordinates allows a reliable experimental determination of
then*(A^/) dependence with the help of eq. (14.6). This dependence can then
be confronted with the Gibbs-Thomson dependence (20.9) in order to check
the correspondence between theory and experiment. It must be emphasized
that this procedure is applicable to any kind of strain-influenced nucleation
(HON, HEN, 3D, 2D, etc.) and regardless of the physical nature and the size
dependence of the total strain energy #str(«) in eq. (20.1) as long as this
energy is A^/-independent. Indeed, the general form (5.4) of the nucleation
theorem remains unchanged, because there is no contribution from the partial
A^/ derivative of <Pstt(n*). This implies validity of eq. (5.29) and, hence,
legitimacy of the usage of both the first (Chapter 14) and the second (Chapter
16) applications of the nucleation theorem for a model-independent
experimental determination of the nucleus size n* also in cases in which the
strain energy is expected to be a factor affecting the nucleation process.
Chapter 21
Electric field
The electric field is a carrier of energy and can, therefore, be a factor affecting
nucleation by changing the nucleation work W*. Best known is, perhaps, the
role played by the electric field in nucleation on ions and other electrically
charged nanoparticles [Thomson 1906; Tohmfor and Volmer 1938; Volmer
1939; Hirth and Pound 1963; Russell 1969; Boucher 1969; Chernov and
Trusov 1969; Stoyanov etal. 1970; Kortzeborn and Abraham 1973; Castleman,
Jr. 1979; Rusanov 1979; Kuni et al. 1983; Kuni 1984a; Rusanov and Kuni
1984; Shchekin et al. 1984; Shchekin and Warshavsky 1996; Warshavsky
and Shchekin 1999a, b]. The influence that an externally applied electric
field may exert on the nucleation process has also been studied both
experimentally (e.g. Kozlovskii [1962]; Jalaluddin and Sinha [1962]; Parmar
and Jalaluddin [1973]; Basu [1973]; MacKenzie and Brown [1975]; Kozlovskii
et al. [1976]; Isard et al. [1978]; Gattef and Dimitriev [1979, 1981];
Chianadzhev et al. [1982]; Kanter [1983]; Kanter and Neizvestnyi [1983];
Shablakh et al. [1983]; Shichiri and Araki [1986]) and theoretically (e.g.
Sirota [1969]; Kashchiev [1972a, 1972b, 1987]; Lychev et al. [1977]; Isard
[1977]; Isard et al. [1978]; Pastushenko et al. [1979]; Brainin and Smolyak
1980; Smolyak [1980]; Chizmadzhev et al. [1982]; Cheng [1984]; Exerowa
and Kashchiev [1986]; Warshavsky and Shchekin [1999a, b]). Clearly, the
electric field may be expected to affect not only the energetics (i.e. IV*) of
the nucleation process, but also the kinetics of monomer attachment to the
nucleus (i.e./*). In this chapter we shall consider the impact of the electric
field on the stationary nucleation rate Js and on the nucleation time lag T by
accounting for the effect of the field only on the nucleation work W*. The
analysis is restricted to isotropic phases for which the vectors E (V/m) and
D (C/m2) of, respectively, the electric field and the electric displacement are
related by D = e0eE, where £ is the E-independent relative permittivity or
dielectric constant, and £0 - 8.85 pF/m is the permittivity of empty space.
21.1 General formulae
When a cluster of n molecules is formed in a pre-existing electric field, the
work Win) for its formation is again defined by eq. (3.4), but the Gibbs free
energy of the system both in the absence and the presence of the cluster is
affected by the electric field. Thermodynamics tells us that [Guggenheim
1957; Landau and Lifshitz 1982]
G = G0±(l/2) f E(r)-D(r)dr (21.1)
316 Nuclealion: Basic Theory with Applications
where G0 is the value of the Gibbs free energy at zero field, r is the position
vector, dr = dx dy dz is elementary volume, the dot denotes dot product, and
the integration extends over all space (or that part of space where E ?t 0). A
most important point concerning eq. (21.1) is that while the plus sign must
be used when the system is of fixed charges, the minus sign is in force for a
system held at fixed potentials. Indicating by subscripts 1 and 2 the values of
G, G0, E and D in the absence and presence of the cluster, respectively, and
recalling that the work to form a cluster at zero field is given most generally
by eq. (3.86), from (3.4) and (21.1) we get
W(n) =-nAfi + <P(n) ± <Pd(n). (21.2)
This is the general formula for the work to form an n-sized cluster in a
pre-existing electrostatic field and in it the plus or minus sign refers,
respectively, to a system of fixed charges or potentials. In this formula
«>el(n) = (1/2) j [E2(r) • D2(r) - E,(r) ■ D,(r)] dr (21.3)
is the change in electrostatic energy due to the replacement of the old electric
field E] by the new field E2 which appears in the system as a result of the
cluster formation. Unfortunately, usually it is hard to calculate 4>e| from
(21.3), because the integration is over all space and it is necessary to know
E2 both inside and outside the cluster. When the system is of fixed charges,
however, eq. (21.3) can be given the equivalent form [Landau and Lifshitz
1982]
<Z>d(«) = -(1/2) f E^rJ-ID^rJ-eo^WE^ldr (21.4)
where £m is the dielectric constant of the medium (one-component old phase,
liquid or solid solution, etc.) in which the cluster is formed. The usage of eq.
(21.4) for determination of 0e|(n) is facilitated, since the integration is only
over the volume Vn of the cluster. What we need, therefore, is knowledge of
the electric field only in the space occupied by the cluster. Recalling that
within the cluster, by assumption, D2 = %£CE2 and treating £m and the
dielectric constant £c of the cluster as r-independent, we can rewrite (21.4)
as
"%(«) = (V2X£m " *t) f E,(r) • E2(r) dr. (21.5)
As seen, when the dielectric constant £c of the cluster (i.e. of the new
phase) is greater than that of the medium, 4>C| is negative. Since this quantity
enters eq. (21.2) with plus sign (eq. (21.5) is valid for a system of fixed
charges), this means that the electrostatic field stimulates the cluster formation.
There is no effect if £c - £„,, and the process is inhibited by the field when
£ < £
Electricfleld 317
21.2 Nucleation on ions
We shall now apply the above results to the important case of HEN of
condensed phases on ions [Thomson 1906; Tohmfor and Volmer 1938; Volmer
1939; Hirth and Pound 1963; Russell 1969; Kortzeborn and Abraham 1973;
Castleman, Jr. 1979; Rusanov 1979; Kuni etal. 1983; Kuni 1984a; Rusanov
and Kuni 1984; Shchekin etal. 1984; Warshavsky and Shchekin 1999a, b].
The ion is regarded as a sphere of radius R± and fixed charge Q (C), and the
cluster of n molecules is considered as a greater sphere, radius R, having the
ion in its centre (Fig. 21.1a). This geometry implies that the ion acts as a
spherical seed which is wetted either completely (see Chapter 18) or 'better'
than completely. Limiting the considerations only to the former case, for <P
in eq. (21.2) we can use the classical expression
01 K\ - 4jkt< R -R,*> (216.1
(a) <b)
Fig. 21.1 Spherical cluster ofn molecules (a) on a positively charged ion, and (b) in an
externally applied uniform electric field (in both cases £c > Em).
which follows from (18.6) at 0W = 0 and Rs = Rt and which is valid for EDS-
defined clusters. We note also that n and R are related by
n = (47tftv0XR3-R;). (21.7)
The electric field about the ion before and after the cluster formation is
spherically symmetrical so that for the dot product Ej • E2 we have Ej • E2
= ErlEr2, Er , and Er2 being the radial components of Ej and E2 in the
space occupied by the cluster. According to electrostatics,
Erl = QI4K£,yEmrl, (Ri<r<R) (21.8)
Er2 = Q/4Ke^,.r2, (Ri < r < R) (21.9)
where r is the distance from the cluster centre. Since in this case the system
is of fixed charge, we can use eq. (21.5) to find 4>el. Substituting Er t and Er 2
from (21.8) and (21.9) in this equation, accounting that dr = Anfiir and
integrating over r from Rt to R yields
318 Nucleation: Basic Theory with Applications
0JR) = (l/£c - l/em)(e2/8OT0)(l«i - MR). (21.10)
Thus, from eqs (21.2) (taken with +d>el), (21.6), (21.7) and (21.10) we find
that the work to form a spherical BDS-defined condensed-phase cluster on
an ion is given by (R > /¾)
W(R) = -A/i(47r/3u0)(«3 - R?) + i7to(R2 - R})
+ (l/ec - 1/^)(62/8^(,)(1/^ - \IR). (21.11)
Owing to (21.7), in terms of n this formula takes the form
W(n) = -nAfi + AnaRf([1 + Ovq/411 R3)n]2'3 - 1)
+ (1/¾ - l/£m)(e2/8»£ofij)(l - 1/[1 + ^411 R?)n]"3). (21.12)
At £m = 1 eq. (21.11) is the formula of Russell [1969], This equation and
eq. (21.12) show that if the cluster specific surface energy a is not affected
by the ion charge, Wdepends quadratically on Q and is, therefore, insensitive
to the sign of Q provided the positive and negative ions have the same radius
Rt (for the effect of the sign of Q see, e.g. Russell [1969]; Castleman, Jr.
[1979]; Rusanov [1979]; Rusanov and Kuni [1984]; Shchekin el al. [1984]).
The ion stimulates the cluster formation when £c > Em, as then the Q term in
(21.11) or (21.12) is negative. In the opposite case of £c < £m this term is
positive and the cluster formation is inhibited. Naturally, the electric field
about the ion has no effect on W when £c = £m. We note also that eq. (21.11)
or (21.12) is fully applicable to electrically charged completely wetted spherical
nanoparticles acting as seeds, since for them Er 1 and Er 2 are also given by
eqs (21.8) and (21.9). At Q = 0 the electric field'vanishes and (21.11) passes
into (18.8) taken at fis = Rv
Figure 21.2 displays the W(n) dependence (21.12) in the practically more
interesting case of ec > £m. The calculation is done for water droplets (£c =
80) in vapours (£c = 1) at T = 293 K and supersaturation ratio S = plpt = 2.5.
The supersaturation A/< is evaluated according to eq. (2.8), the pe, u0 and a
values used are listed in Table 3.1, and it is assumed that the ion is with
charge Q = ± e0 and radius R{ = 0.15 nm (e0 = 1.6 x 10'" C is the electronic
charge). As seen in Fig. 21.2, W(n) has a minimum and maximum which
determine the size of a smaller and larger cluster in stable and unstable
thermodynamic equilibrium, respectively. Whereas the smaller cluster is
formed barrierlessly, i.e. spontaneously, the larger cluster is the nucleus.
Hence, in this respect nucleation on ions at £c > £m is congenial with HEN
of cap-shaped clusters with negative line energy (see Chapter 19 and curve
-20 in Fig. 19.3). Figure 21.2 thus shows that before the formation of the
nucleus on the ion this is not 'naked', but with a 'jacket' of molecules of the
old phase. In this figure we read that at the chosen S = 2.5 while the nucleus
is of «* = 203 water molecules, the ion 'jacket' is constituted of nmin - 14
such molecules. In fact, it is the 'jacketed' rather than the 'naked' ion which
acts as a charged completely wetted seed in the £c > £m case. For that reason,
as illustrated in Fig. 21.2, in this case the nucleation work W* is given by the
Electric field 319
1 50 100 150 200 250 300 350 400
n
Fig. 21.2 Dependence of the work for cluster formation on the cluster size at £c > £m
according to eq. (21.12) for water droplets on unit-charge ions in vapours at T = 293 K
and pipe = 2.5. The double arrow indicates the height of the nucleation barrier.
difference between the values Wmilti and WmirL of the maximum and minimum
of W(R) or W(n), i.e. we have
»™ = ^raax - WW (21-13)
In the case of £c < em, however, just like in HON, W from (21.11) or (21.12)
displays only a maximum at the nucleus size n* so that then W* is given by
W* = IVraai in accordance with eq. (4.2).
Having obtained W(R), we can now determine the nucleus radius R* and
the nucleation work W*. Differentiation of W from (21.11) with respect to R
under the classical assumption for constancy of the cluster dielectric constant
£c with respect to the cluster size n (or radius R) and using the result in (4.1)
leads to (e.g. Voltner [1939]; Hirth and Pound [1963])
R* = 2u0o/Av + (l/ec - l/£m)e2L;0/32A0A^fi*3. (21.14)
In view of (21.7), for the number n* of molecules in the shell-shaped nucleus
on the ion we get the expression
n* = (ilKvla^B A//3)[l + (l/ec - l/em)
x Q2mx£0v0cj(n* + AnRll'iva)f-A7CRp'iva (21.15)
which follows also from (4.1) and (21.12).
Equations (21.14) and (21.15) give only implicitly the dependence of R*
and n* on Ay and are the Gibbs-Thomson equations for spherical condensed-
phase nuclei on completely wetted ions or electrically charged seeds. At Q =
0 or £c = Em they turn into eqs (4.9) and (18.14), respectively, the latter taken
-60
-70
-80
-90
-100
-110
.ion
320 Nucleation: Basic Theory with Applications
at 8W = 0 and Rs = /?;. The curves in Fig. 21.3 represent the n*(A/j) dependence
(21.15) for water droplets in vapours at T= 293 K and supersaturation Aft =
kT In S. The parameter values used are those for Fig. 21.2. Curve 'HON'
refers to the case of Q = 0 and R{ = 0 corresponding to HON, and curve 'ion'
is for nuclei on ions. This curve thus illustrates eq. (21.15) in the £c > £m
case. As seen from Fig. 21.3, at a given sufficiently low value of A^i > 0 there
are two n* values satisfying the Gibbs-Thomson equation (21.15) at Q * 0
and £c> em (the same is true for/?* fromeq. (21.14)). While the greater n*
value gives the number of molecules constituting the nucleus, the smaller
one represents the number «min of molecules of the stable cluster, i.e. of the
'jacketed' ion. As already noted, this means that in the £c > £m case nucleation
on ions is analogous to HEN of cap-shaped clusters with negative line energy
(cf. curves '-20' and 'ion' in Figs. 19.4a and 21.3). In this case, when the
supersaturation is higher than that corresponding to the 'nose' of curve 'ion'
in Fig. 21.3, nucleation is not involved in the process of new-phase formation
on the ions - this process occurs barrierlessly. Clearly, the radius Rmin and
the size nmiTL of the stable cluster which exists in the case of ec > £m can be
determined from eqs (21.14) and (21.15) upon replacing R* and n* by Rmjn
and «min:
Kn,in = 2v0°/Af + (1/¾ " l/S^eV32*2^ RL (21.16)
nmin = (32^t.02fj3/3Au3)[l + (l/£c - 1/¾)
x e2/48OT0u0otnmi|1 + 4jtR?/3v0)Y - 4jcR?/3v0.
400
100
Fig. 21.3 Dependence of the nucleus size on the supersaturation ratio: curve 'ion'
eq. (21.15) at £c > £mfor nucleus droplets of water on unit-charge ions in vapours
at T = 293 K; curve 'HON'- the corresponding Gibbs-Thomson eq. (4.10) for
homogeneously formed nucleus droplets.
Eectricfield 321
In the £c < £m case, however, the course of the n*(Afi) or R*(Afi) dependence
is quite different. Then there exists only one n* or R* value which satisfies
eq. (21.15) or (21.14) at a given value of A/i, because n* or R* diminishes
monotonously with increasing A/j just as it does in HON (see curve 'HON'
in Fig. 21.3). This n* or R* value determines the size or the radius of the
nucleus on the ion and is greater than that for the respective homogeneously
formed spherical nucleus.
Our next step is the determination of the nucleation work W*. To do that
in the £c > £m case, as required by eq. (21.13), we have to find WmwL and
Wmjn by setting R = R* and R = Rmin in (21.11) and using R* and R^ from
(21.14) and (21.16). After some algebra we obtain (Q * 0, £c > £m)
W* = (4/t/3)otfi*2 - O + (1/¾ - l/£mXe2/6^)(l/fimin - 1«*).
(21.17)
In a similar way, in the £c < £m case, from eqs (21.11) and (21.14) we get
W* = 47cRlAfiBvQ + (4^3)ot«*2 - 3/¾2)
+ (1/¾ - l/£,m)(G2/6aE())(3/4Ri - i/R*), (21.18)
since now W* = W(R*).
Equations (21.17) and (21.18) represent the Aft dependence of W* only
implicitly through R* and Rmin from (21.14) and (21.16). They show again
that the effect of the electrostatic field about the ion (or the charged seed) is
controlled by the relation between £c and £^ When £c > £^ the field stimulates
the nucleation process - then the Q term in (21.17) is negative. The opposite
is true when £c < £m, since then this term in (21.18) is positive. Also, at Q =
0, i.e. in the absence of the field, eq. (21.18) passes into eq. (18.9) with f
from (18.13). This has to be so because the ions are treated as completely
wetted seeds with radius Rs = Rr Naturally, at Q = 0 and R{ = 0, W* from
(21.18) becomes identical with W* from (4.11) for HON of spheres. In the
particular case of £,. » £m = 1 eq. (21.17) is the formula used by Volmer
[1939]. We note also that, as can be verified, W* from (21.17) and (21.18)
satisfies the nucleation theorem. This is compulsory, because the sum ^(n)
± 4>e](«) in (21.2) plays the role of a A^-independent <5(n) in (3.86). When
£c < Em, n* and W* from (21.15) and (21.18) are related through eq. (5.29),
since we consider EDS-defined nuclei of condensed phases. In the £c > £m
case, however, W* from (21.17) obeys the nucleation theorem in the form of
eq. (5.21) with An*/(1 - p0|d/pnl:w) = n* - nmin. This is so because in this
case W* is given by (21.13), and for WmliJi and Wmill eq. (5.29) applies in the
form dWmax/dAfi = - n* and dW^JdAfi = - «m[n.
Using W* from (21.17) and (21.18), we can now determine the stationary
rate 7S (m~3 s-1) of HEN of condensed phases on ions with concentration Ci
(m-3). Treating the ions as nucleation-active sites or, equivalently, as seeds
having a single active site each (cf. eqs (7.10) and (7.11)) allows setting C0
= Cj so that with the help of (13.39) and (13.40) we find that
322 Nucleation: Basic Theory with Applications
Js = z/*Cj exp {- (47to/3kT)[R*2(ty) ~ *LCVO]
- (Uec~ V£j{Qy6x£0knVRmm(^) ~ VR*W]) (21.19)
in the case of ec > em and that
Js = zf*Cx exp (-(47^3/3v0kT)Av - (47to/3kT)[R*2(Av) - 3K,2]
- (l/ec - l/£m)((>2/6?r^r}r3/4/?i - 1/**(4m)]}
when £c < em. At £m = 1 eq. (21.19) is the formula of Russell [1969]. The
Zeldovich factor z and the frequency/* of monomer attachment to the nucleus
change only weakly with n* and IV* and can be regarded as practically
independent of both Q and /^. According to eq. (15.73), this means that the
nucleation time lag t is influenced relatively little by the ion charge and
radius provided that the mechanism of monomer attachment is not affected
by the presence of the electric field about the ion.
The above equations represent implicitly the Js(Afi) dependence for HEN
of condensed phases on ions (or charged seeds) and show that the stationary
rate of the process is a complicated function of the supersaturation. Curve
'ion' in Fig. 21.4 depicts the is(A^/) dependence for HEN of water droplets
on unit-charge ions in vapours at T= 293 K. The calculation is done according
to eq. (21.19) with R* and flmin from (21.14) and (21.16), and the kinetic
factor A = zf*C-t is expressed in the form of A'S (cf. eqs (13.40)-(13.42)). The
factor A' = zf* C{ is assumed to have the exemplary value of 1016 m~3 s-1
corresponding to Cy - 10y m~3 and typical z = 0.1 and /e* = 108 s~]. The
used values of Q, Rv £c, £m, v0, aand/?e are those given above or in Table
3.1. The dotted portion of curve 'ion' represents the rate A'S of barrierless (at
W* = 0) formation of water droplets on the ions (this occurs in the S range
on the right of the 'nose' of curve 'ion' in Fig. 21.3). For comparison, in Fig.
21.4 the Js(Af*) dependence for HON under the same conditions is displayed
by curve uHON' already shown in Fig. 13.5. As seen from Fig. 21.4, at lower
supersaturations Js for HEN on ions is by far greater than 7S for HON. This
means that ion-containing vapours can condense when they are relatively
little supersaturated. At higher supersaturations, however, as already noted
in Section 13.3, HON takes over HEN, because even though the droplets
require no work for their formation on the ions, the ion concentration Cj is
much lower than the concentration C0 of nucleation sites for HON.
Experimentally, the presence of ions in supersaturated vapours is known to
stimulate the nucleation process, but the quantitative agreement between
theory and experiment does not seem firmly established [Volmer 1939; Hirth
and Pound 1963; Boucher 1969; Russell 1969; Castleman, Jr. 1979]. In this
respect it must be pointed out that available experimental JJAf*) data for
HEN of condensed phases on ions can be analysed reliably in the scope of
the first application of the nucleation theorem (Chapter 14). Namely, when
£c > £m, using eq. (14.4) with A«*/(l - po)d/pnew) = n* - nmin and nA = 1
allows a model-independent determination of the Afj dependence of the
difference n* - «min between the number of molecules in the EDS-defined
nucleus and in the EDS-defined stable cluster on the ion and verification of
Electric field 323
1015
1010
105
1
-
-
-
ion 1
i . il i i/ i .
/HON
. . . i....
12 3 4 5
s
Fig. 21.4 Dependence of the stationary nucleation rate on the supersaturation ratio:
curve "ion'-eq. (21.19) at ec > £m for HEN of water droplets on unit-charge ions
in vapours at T = 293 K; curve 'HON' — the corresponding eq. (13.66) for
homogeneously formed water droplets.
the Gibbs-Thomson equation (21.15). In the opposite case of £c < £m such a
determination is possible with the help of eq. (14.6). Also, we note again
that, though obtained for ions, upon setting R^ equal to the seed radius, the
above results are directly applicable to completely wetted spherical seeds
with a fixed charge Q of their surface.
21.3 Nucleetion in external electric field
We shall now consider the effect of an externally applied electric field on the
324 Nucleation: Basic Theory with Applications
nucleation process, confining the analysis to HON in uniform electrostatic
field E (Fig. 21.1b). The clusters are assumed to be spherical and their
elongation in the direction of the field is ignored, although accounting
for this effect is also possible [Cheng 1984; Warshavsky and Shchekin
1999a, b]. We first consider a system of fixed charges (e.g. an old phase
filling a capacitor with parallel plates whose charge remains the same during
nucleation) and then a system at fixed potentials (e.g. the same old phase in
the same capacitor, but with plates held at constant voltage).
For a system with fixed charges we can employ eq. (21.5) to find the
electrostatic energy change 4>el which enters (21.2) with a plus sign. The
electric field E2 inside a spherical body in uniform external field E is also
uniform and given by (e.g. Landau and Lifshitz [1982])
E2=[3£m/(£c + 2£m)]E- (21-20)
Substituting E2 from (21.20) in (21.5), setting E, = E and accounting that E2
= E2 readily yields
*el<"> = [3%<=m<£m " ¢=)/¾¾ + 2£m)]E2 V„ (21.21)
where E (Vim) is the strength of the externally applied electrostatic field.
This expression for ¢,,, was obtained by Isard [1977] (see also Cheng [1984];
Warshavsky andShchekin [1999a, b]). It corrects the similar formula for <Z>e|
[Kashchiev 1972a, 1972b], in which the factor 3 is absent and £m - £c enters
as £c - £m because of inaccurate choice of the limits of integration in the
general eq. (21.3). As seen from (21.21), ¢^ < 0 when £c > £m, i.e. the
cluster formation decreases the electrostatic energy of the system.
For a system at fixed potentials the usage of eq. (21.5) is illegitimate and
we must resort to eq, (21.3). The calculation is easy if we view the old phase
as occupying the volume V, of a parallel-plate capacitor held at constant
potentials <p, and <p2 of its plates. Electrostatics tells us that for such a
capacitor, if C (F) is its capacitance, E • D = 0(¾ _ <P\)2/VC inside and E =
0 outside, provided edge effects are negligible. Hence, since E, = E2 = 0
outside the capacitor, the integration in eq. (21.3) is in fact only over the
volume of the capacitor. Taking into account that <p, and tp2 are the same
before and after the cluster formation, neglecting the possible change of Vc
and setting E, ■ D, = 0,(¾ - <p])2IVc and E2 ■ D2 = C2(<p2 - <p{)2IVc leads
to
¢,, = (1/2)(02-0,)(^-^)2. (21.22)
Now, as the capacitance C2 in the presence of a small enough spherical
cluster in the capacitor is related to the capacitance C, of the cluster-free
capacitor by [Landau and Lifshitz 1982]
C2 = CX- 3£0£m(£m - ec)Vn/(ec + 2£m) d\, (21.23)
for ¢,, from (21.22) we get
*el(n) = -|3£0£m(£m - £c)/2(£c + 2em))E2V„ (21.24)
Electric field 325
where E = ((¾ - q>\)ldc is the strength of the uniform electric field between
the plates of the capacitor, and dc is the distance between them. This equation
shows that <Pel > 0 when ec > £m, which means that the electrostatic energy
is increased by the cluster formation. This is in contrast with 0gl for a system
with fixed charges (cf. eq. (21.21)). The absolute value of <Pel from (21.24)
is, however, precisely equal to that of <2>el from (21.21).
We can now substitute in eq. (21.2) either &el from (21.21) or &ei from
(21.24) in order to find the work for formation of spherical clusters during
HON in uniform externally applied electrostatic field. Recalling that the plus
and minus sign in (21.2) refers to <£e] from (21.21) and (21.24), respectively,
we obtain
W(n) = -nty + 0(n) + [3¾¾¾ - £c)/2(£c + 2em)]E2Vn. (21.25)
This formula reveals that the energy contribution of the field to W(n) is
the same regardless of whether the system is of fixed charges or at fixed
potentials (this is so, however, only for clusters of radius R « dQi since the
usage of (21.23) is justified only under this condition [Landau and Lifshitz
1982]). We see also that, like the strain energy &slT (see Chapter 20), the
change #el in electrostatic energy is a volume-type contribution to W. The
proportionality of <Pei to Vn, however, is a consequence of the assumptions
for fixed cluster shape and rc-independent cluster dielectric constant £c. We
note that eq. (21.25) is a particular case of the more general formula of
Cheng [1984] for spheroidally shaped clusters whose eccentricity varies
with the field strength E. This equation is valid for both condensed-phase
and gaseous clusters and tells us that the electrostatic field stimulates HON
when £c > £m (then the E term in (21.25) is negative). There is no effect at
£c = £„,, and the process is inhibited for £c < em. Qualitatively, the externally
applied field thus plays the same role as the natural electrostatic field about
ions or charged seeds.
In conformity with eq. (3.87), &(n) in (21.25) is given by the r.h.s. of
(3.15) for EDS-defined condensed-phase clusters and of (3.16) for so-defined
gaseous ones. Hereafter, we shall confine the analysis only to clusters of
condensed phases. In this case <P(n) is merely equal to the cluster total
surface energy 0(h) so that accounting for eqs (3.13) and (3.20) transforms
(21.25) into [Isard et al. 1978]
W(n) = -(Afi + ceE2)n + (36^3^ an273. (21.26)
Here the parameter c£ (F ■ m2) is defined by
c£ = 3EQem(ec - em)vG/2(ec + 2em) (21.27)
and corrects the analogous parameter c in the identical formula for W(n)
obtained by a similar analysis [Kashchiev 1972a, 1972b]. Thus, all c-containing
expressions resulting from the analysis of Kashchiev [1972a, 1972b] remain
usable if everywhere in them the parameter c is replaced by c£ from (21.27).
Equation (21.26) shows that the change c^E1 in the electrostatic energy
per molecule is additive to A^. Since c£is n-independent, this means that the
326 Nucleation: Basic Theory with Applications
formation of spherical condensed-phase clusters in uniform electric field
occurs at an effective supersaturation Afj + ceE2 which is a function of the
field strength E. Depending on the sign of cr i.e. of the difference £c - em,
this supersaturation can be smaller (when cE < 0, i.e. £c < em) or greater (for
c£ > 0, i.e. £c > em) than the actual supersaturation A/;. The magnitude of
CgE2 even at strong fields of E - 1 to 100 MV/m is relatively small. For
example, for HON of water droplets (£c = 80, v0 = 0.03 nm3) in vapours (£m
= l)atE= 1 MV/m, with the help of (21.27) we find that c^2 = \0~7kTat
T - 293 K. Hence, in this case the field does not contribute to Afj. If,
however, both £m and E are higher, the contribution may be of significance:
for metal clusters (£c = «>) formed, e.g. in a medium of £m = 1000, with u0
= 0.03 nm3 it follows that c^E2 = 0.01 W at the above temperature and E =
10 MV/m.
The simplicity of the dependence (21.26) of W on n allows an easy
determination of the nucleus size n* and the nucleation work W* as functions
of A/i and E. Using (21.26) in conjunction with eqs (4.1) and (4.2) yields
[Isaidetai 1978]
n* = n7tvl<r>I3(Afi + c^E2)3 (21.28)
W* = 16^^(T3/3(A^ + ceE2)2. (21.29)
These expressions apply to HON of spherical EDS-defined nuclei of
condensed phases, and (21.28) is the respective Gibbs-Thomson equation.
They have the form of those of Kashchiev [1972a, 1972b], but contain the
correctly determined parameter c£ from (21.27). At E - 0 they pass into
(4.10) and (4.11) and show that for c£ > 0, i.e. when ec > em, HON is
stimulated by the applied electric field, since then both n* and W* are smaller
than at zero field provided Ap is kept the same. Also, it can be verified that
n* and W* from (21.28) and (21.29) obey the nucleation theorem in the form
of eq. (5.29). This has to be so because of the postulated A/i-independence
of the sum <P(n) ± &e[(n) in (21.2) (this sum is the analogue of <&(«) in eq.
(3.86)).
We are now in position to quantify the effect that an externally applied
electrostatic field exerts on the stationary rate Js (m~3 s_1) of HON of condensed
phases. With the help of eqs (13.39), (13.40) and (21.29) we get (Aji >
-cE£2)
Js = zf*C0 exp [-B/(Afj + cEE2)2] (21.30)
where the thermodynamic parameter B is specified by (13.49) with c ~
(36;r)1/3 (spherical nuclei) and oc( = cr(HON). Equation (21.30) represents
Js in the scope of the classical theory and shows that at a given supersaturation
the stationary nucleation rate is a strong function of E when the contribution
of the electric field to A/; is appreciable. The Zeldovich factor z and the
frequency /* of monomer attachment to the nucleus can be treated as E~
independent, since they vary relatively little with E through n* and W* (see
eqs (13.36) and (13.40)-(13.43)). It must be pointed out, however, that the
external field could have a direct influence on /*, e.g. by affecting the
Electric field 327
monomer diffusion coefficient in volume-diffusion control or by changing
even the very mechanism of monomer attachment. Then in eq. (21.30) the
factor/* may take over the thermodynamic exponential factor in governing
the /s(A^i) dependence. Also, the possible dependence of the specific surface
energy a (i.e. of B) on E [Smolyak 1980] can alter considerably the JS(E)
dependence (21.30) and make Js sensitive to the presence of comparatively
weak electric fields.
According to eq. (15.73), when/* is influenced only indirectly by E
through n*, the nucleation time lag X will be a relatively simple function of
the field strength. For instance, for HON of condensed phases in solutions
under volume-diffusion control the general formula (15.73) transforms into
eq. (15.85) with cVD = c - (36;r)"3. Expressing z and n*1'3 in (15.85) as
functions of £ with the help of the first equality in (13.36) and eqs (21.28)
and (21.29) and accounting for eq. (2.14), we obtain (kT In S > - c^2)
T= 32vaa2/jfiy*(.kTi1DCeS(\n S + cfilkT? (21.31)
where S = C/Cs is the supersaturation ratio. This equation corresponds to eq.
(15.86) for HON of spheres and passes into it at E = 0. Obviously, if the
monomer sticking coefficient /*, the nucleus specific surface energy <J and/
or the solute diffusion coefficient D are affected considerably by the applied
electric field, T will depend differently and more strongly on E than as
predicted by eq. (21.31).
Figure 21.5 displays the 7s(A^i) dependence (21.30) for HON of condensed
metal phase (£c = <») in a dielectric solid solution with £^, = 1000. The
system is at T= 293 K and in externally applied electrostatic field of strength
E = 0 or 15 MV/m (as indicated). The supersaturation is expressed as A/i =
M" In S (cf. eq. (2.14)), the thermodynamic parameter B = 3.73 x 10"42 J2 is
calculated from (13.49) with c = (367t)"3 (spheres), v0 = 0.03 nm3 and a =
10 mj/m2, cE is evaluated from (21.27), and the kinetic factor A = zfC0 is
represented as A = A'S (see eq. (13.41)) with A/j-independent A' = 3 x 1031
m-3 s_1. According to (13.42), this A' value corresponds to z = 0.1, C0 = 3 x
1028 m~3 and /* = 104 s_1 (this rather small value of /* accounts for the
low diffusivity of the solute molecules in solid solutions at room temperature).
As seen from Fig. 21.5, the effect of the electric field is more strongly
manifested at lower supersaturations. Since we consider the case of £c > £^,,
i.e. ce > 0, the field stimulates the nucleation process. In this case the field
can even make a saturated old phase (then A^i = 0) nucleate at a certain,
though quite low rate. We note that a reliable experimental determination of
the nucleus size n* from Js(A[i) data at different fixed values of E is again
possible in the scope of the first application of the nucleation theorem (Chapter
14). This determination can be done with the help of eq. (14.6). The«*(A^i)
dependence obtained in this model-independent way can then be confronted
with that predicted by eq. (21.28) or any other Gibbs-Thomson equation.
Figure 21.6 illustrates the ./.(£) dependence (21.30) at 5 = 1.06 and 1.07
(as indicated). The calculation is done with the parameter values used for
Fig. 21.5. The threshold character of the JS(E) dependence is clearly seen:
328 Nucleation: Basic Theory with Applications
1.02
1.06 1.08 1.10
Fig. 21.5 Dependence of the stationary nucleation rate on the super saturation ratio:
curves 0 and 15 - eq. (21.30) at Ec > em for HON of condensed metal phase in
dielectric solid solution at T = 293 K in externally applied uniform electric field of
strength E = 0 and 15 MV/m, respectively.
below a certain critical field strength, Js is practically unaffected by the
presence of the electric field and keeps its value corresponding to zero field
[Kashchiev 1972a, 1972b]. Only high enough values of £ can stimulate the
nucleation process (or inhibit it if £c < £m).
The rather general conclusion from the above considerations is that, typically,
electrostatic fields of strength E < 1 MV/m can hardly influence the stationary
rate of HON. Indeed, no significant effects were observed in the experiments
of Isard et at [1978] on nucleation of glass ceramics and Shichiri and Araki
[1986] on nucleation of ice at fields weaker than about 1 MV/m. On the
other hand, in other cases fields in the range of 0.01 to 1 MV/m were found
Electric field 329
100
E (MV/m)
Fig. 21.6 Dependence of the stationary nucleation rate on the strength of
externally applied uniform electric field in the case of £c > em: curves 1.06 and 1.07 -
eq. (21.30) for HON of condensed metal phase in dielectric solid solution at
T - 293 K and supersaturation ratio S = 1.06 and 1.07, respectively.
to strongly change 7S or other characteristics of the nucleation process
[Kozlovskii 1962; Kozlovskii et al. 1976; Gattef and Dimitriev 1979, 1981;
Kanter 1983; Kanter and Neizvestnyi 1983; Shablakh et al. 1983]. Moreover,
there exists experimental evidence [Jalaluddin and Sinha 1962; Parmar and
Jalaluddin 1973; Basu 1973; Shablakh et al. 1983] for field-stimulated
nucleation in systems with £c < £m, which is even in qualitative disagreement
with theory. This suggests that the influence of the electric field on other
parameters (e.g. o and/*), and not only on W*, has also to be accounted for
by the theoretical analysis. We note, too, that extension of the analysis to
HEN on seeds [Kashchiev 1972a] gives another possibility to widen
substantially the applicability of the theory.
Chapter 22
Carrier-gas pressure
In experiments on one-component HON of condensed phases in vapours,
usually inert carrier gases such as hydrogen, helium, argon or other noble
gases are utilized to keep the process proceeding at constant temperature
(hereafter we shall call inert a carrier gas whose molecules are completely
absent from both the surface and the volume of the clusters of the new
phase). Although the presence of inert carrier gas makes the nucleating
phase a two-component gaseous mixture, the common practice is to interpret
the so-obtained experimental data with the aid of the corresponding formulae
for one-component nucleation. We, too, have done that with regard to the
Js(A/i) and n*(Ay) dependences in Figs 13.8 and 14.2. As it is by far not
obvious that the role of the inert carrier gas is only to maintain constancy of
temperature, experiments at different pressures of the carrier gas and/or with
various carrier gases were carried out to reveal whether nucleation is affected
by the presence of the carrier gas. While Katz et aL [1992], Heist et al.
[1994] and Kane and El-Shall [1996] observed a considerable effect of the
carrier-gas pressure on the stationary nucleation rate Js, Wilemski et al.
[1992], Wagner et al. [1992], Viisanen et al. [1993], Muitjens [1996] and
Luijten [1998] found that 7S is not significantly influenced by the pressure
and/or the nature of the inert carrier gas.
In a recent theoretical study Oxtoby and Laaksonen [1995] analysed the
effect of carrier-gas pressure on nucleation by treating the process as binary
nucleation and by using the nucleation theorem in the form of eq. (5.44).
Their model-independent conclusion about the smallness of this effect
confirmed that of Ford [1992a, b] drawn with the help of the classical nucleation
theory. Other theoretical studies on the subject were carried out by Wilcox
and Bauer [1991], Bauer and Wilcox [1993], Fisk and Katz [1996] and
Kashchiev [1996]. Following the latter, in this section we shall analyse the
influence that the pressure P of an inert carrier gas can exert on the isothermal
nucleation (either HON or HEN) of a condensed phase whose vapours are
mixed with such a gas. The analysis is confined to the effect of P only on the
supersaturation Afj, since Afj is the major parameter controlling the various
characteristics of the nucleation process - nucleus size «*, nucleation work
W*, stationary nucleation rate Js, etc.
Let us have a gas mixture of condensing vapours of species 1 and carrier
gas of species 2 with partial pressures p and P, respectively. We assume that
the system is held at constant absolute temperature 7 and consider solely the
case of one-component condensed phase nucleating in the gas mixture. This
means that the clusters of the new liquid or solid phase are built up of species
Carrier-gas pressure 331
1 only and that there are no adsorbed species 2 on the cluster surface. The
supersaturation Ap is again defined by eq. (2.1) in which now ,u0,d and ^new
are the species 1 chemical potentials in the gas mixture and in the condensed
phase, respectively. Hence, in order to determine A^i as a function of p and
P we must know separately the pQ\d(p, P) and ^new(p, P) dependences.
According to the thermodynamics of binary gas mixtures [Guggenheim 1957],
^0,d can be written down as
Ho\dp, P)=Vc + kTIn (p/pe) + bn(P + p-pe)
-(bn-2bn + b21)P2l(P + p). (22.1)
Here ^e and pe are the equilibrium chemical potential and pressure of the
condensing vapours in the absence of carrier gas (i.e. at P = 0), 6,, (m3) and
£>22 (m3) are tne second virial coefficients of the condensing vapours and the
carrier gas, respectively, and bl2 (m3) is the second mixed virial coefficient
of the gas mixture of species 1 and 2. These coefficients depend on T and
account for the interactions between pairs of molecules of type 1 and 1, type
2 and 2 and type 1 and 2, respectively. At P - 0 eq. (22.1) passes into eq.
(2.5) provided the condensing vapours behave as ideal gas (then bn = 0).
The p, P dependence of ^new is also simple when the new condensed
phase of species 1 can be treated as incompressible. This is so, since eq. (2.6)
holds again, but with p replaced by the total pressure P + p of the gas
mixture. Hence,
ri«tJp, P) = MC + va(P + p-pj. (22.2)
Combining eqs (2.1), (22.1) and (22.2) thus yields the sought formula for
the supersaturation in the presence of carrier gas [Kashchiev 1996]
Aji(p, P) = kT In (plpe) + (6,,- v0)(P + p - pt)
- (b,, - 2fc,2 + b^KP + p). (22.3)
This formula shows that when both the condensing vapours and the carrier
gas behave as ideal gases (then ft,, = bn - 0) and when there is no interaction
between the molecules of type 1 and 2 in the gas mixture (then bn = 0), the
presence of the carrier gas (P > 0) results in a decrease of the supersaturation
with respect to its value Apip, 0) at P = 0. Indeed, then eq. (22.3) reduces to
Ap(p, P) = ty(p, 0) - v0P (22.4)
where Afi(p,0) is given by the r.h.s. of eq. (2.7). The same decrease of A^i(p,
P) with P is in force when only the condensing vapours behave as ideal gas
(then 6,, = 0), but the gas mixture as a whole is ideal (by definition, for such
a mixture bn - 26,2 + 622 = 0 [Guggenheim 1957]).
Equation (22.3) says that for high enough carrier-gas pressures, i.e. for
P » p, Ap depends linearly on P according to [Kashchiev 1996]
Ap(p, P) = kT In (j>/pe) + vefP. (22.5)
Here vef (m3) is an effective molecular volume, defined by
332 Nucleation: Basic Theory with Applications
Vtr=2bn-b22-v0, (22.6)
which depends on T and takes account of the nature of the carrier gas through
612 and 622. It must be emphasized that eq. (22.5) is a very good approximate
formula for practical use, since the experiments on nucleation in the presence
of carrier gas are carried out typically under the condition P»p. Moreover,
eq. (22.5) describes A^i adequately even in the limiting case of P = 0: indeed,
it then passes into eq. (2.8) which is a highly accurate approximation to
(2.7). For that reason, instead of (22.3), in what follows we shall use eq.
(22.5) when expressing A^i as a function of p and P.
As seen from eq. (22.5), the sign of vcC controls the character of the
change of A/u in the presence of carrier gas: uef>0oruef<01eads,respectively,
to higher or lower A^i with respect to A(i(p, 0) = kT In (p/pe). However, since
usually | uef | < 1 nm3, only relatively high carrier-gas pressures (e.g. tens or
hundreds of the atmospheric pressure) can cause such a departure of Aft from
A^i (p, 0) which is experimentally relevant (e.g. of the order of kT).
Inspection of eq. (22.3) or (22.5) reveals also that condensing vapours of
pressure p = /?e which are saturated in the absence of carrier gas (then ^ = 0
and A^i - 0) become either super- or undersaturated in the presence of such
a gas, for then P > 0 and Ap * 0. Therefore, the condensing vapours should
have equilibrium pressure peP * pe if they have to be in the state of saturation
after mixing with carrier gas. The dependence of />e P on P is readily obtained
from the condition for saturation, APi{pcP, P) = 0, with the aid of Aft from
(22.3) [Kashchiev 1996]:
kT In (Pz.plp,) + 'bu - v0)(P + pcP - pc)
- >bn - 2b12 + b22)P1/(P + p^p) = 0. (22.7)
This equation expresses only implicitly peP as a function of P for any P > 0.
In the practically important case of P » pc it can be solved with respect to
pcP to yield explicitly the dependence of this quantity on P in the approximate
form
/>e,P=/7eexp(-DefP/W) (22.8)
which follows also upon using (22.5) in the saturation condition Aft (pc P, P)
= 0.
Equations (22.7) and (22.8) are in fact generalizations of the thermodynamic
formula (e.g. Glasstone [1956]) for the vapour pressure ptP of one-component
liquid in the presence of carrier gas. Equation (22.8) was obtained by Beattie
[1949] and with some loss of accuracy it can be used even in the P < pe
range, since it satisfies the requirement peP = /?e at P - 0. When there are no
interactions between the gas molecules (bu = bl2 - b22 = 0) or when only the
condensing gas behaves as ideal (bu = 0), but the gas mixture itself is ideal
(6, | - 2bl2 + b22 = 0), owing to eq. (22.6) va = - v0 and both eqs (22.7) and
(22.8) take the form of the known dependence of pcP on P [Glasstone 1956],
It must be noted that ptP is a relatively weak function of P because of the
rather small absolute value of ve!. For example, with ve[ = - v0 = - 0.03 nm3
Carrier-gas pressure 333
for water vapours at T - 293 Kin ideal mixture with a carrier gas having the
atmospheric pressure P = 0.1 MPa eq. (22.8) predicts an increase of pc P over
pe by a factor of only 1.0007. If, however, the carrier gas is with pressure P
= 10 MPa, this increase is already appreciable:pcplpc = 1.08. It is instructive
to note as well that combining (22.5) and (22.8) allows representing the
supersaturation in the form of Ap from (2.8):
Ap(j>, P) = kT In (plp^P). (22.9)
This expression is an acceptable approximation even when P <p, for it gives
correctly Ap in the P = 0 limit (provided, of course, the v0 term in eq. (2.7)
is negligible). Physically, it implies that the carrier-gas pressure affects Ap
mainly by changing the equilibrium pressure of the condensing vapours.
Once the supersaturation of the condensing vapours in the presence of
carrier gas is known, we can employ the general results of the nucleation
theory in order to quantify the effect of the carrier gas on the nucleation
process. We shall restrict the analysis to HON or 3D HEN on a substrate, but
it is a simple matter to repeat it for 2D HEN. From eqs (4.38), (4.39), (13.39)
and (13.40), with the help of Ap from (22.5) we find that, classically, the
numbers* of molecules in the one-component EDS-defined nucleus of species
1, the nucleation work W* and the stationary nucleation rate JB are given by
[Kashchiev 1996] (kT In S > - vt!P)
n* = Sc3vlcrl(m(kTinS + uefP)3 (22.10)
W* =Acivlalsl21(kT\nS + vtfPf (22.11)
Js = tf*C0 exp [- B7(ln S + vcfP/kT)2]. (22.12)
Here S = p/pe is the supersaturation ratio of the condensing vapours in the
absence of carrier gas (i.e. at P = 0), and the effective specific surface energy
(Tef and the thermodynamic parameter B' are specified by eqs (4.42) and
(13.67) ((% = a for HON). As known [Slowinski, Jr et al. 1957; Rusanov
1967, 1978; Luijten 1998], for one-component condensed phases in contact
with binary gas mixtures the possible change of o with P is small and in what
follows we shall treat cref and Br as constants with respect to P. According to
eq. (13.36), the Zeldovich factor z is a weak function of n* and W* and can
be considered as practically /^-independent. As to the frequency f* of
monomer attachment to the nucleus, it can also be regarded as independent
of P in the case considered here of formation of one-component nuclei
constituted of species 1 only. That is why, as in the absence of carrier gas
(see eqs (13.41) and (13.42)), the kinetic factor A = zf*C0 can be represented
in the usual form A = A'S where A' = zf* Co is virtually constant with regard
to bothp and P. For instance, A' is given by (13.44) if monomer attachment
is controlled by direct impingement. It must be kept in mind, however, that
if the presence of the carrier gas affects the mechanism of monomer attachment,
f* can be quite sensitive to the carrier-gas pressure. Then it may not be
adequate to treat A or A' as ^-independent parameters. Another point to note
is that n* and W* from (22.10) and (22.11) satisfy the nucleation theorem in
334 Nucleation: Basic Theory with Applications
the form of eq. (5.29), since now Aii is given by eq. (22.5). This has to be so,
because eqs (22.10) and (22.11) are valid for EDS-defined nuclei of condensed
phases.
When the mechanism of monomer attachment is unchanged by the presence
of the carrier gas,/* depends on P only through n* and, according to eq.
(15.73), the nucleation time lag T is a relatively weak function of the carrier-
gas pressure. For instance, under conditions of direct-impingement or surface-
diffusion control the classical dependence of t on P for HON or 3D HEN can
be obtained from eqs (15.77) and (15.82) with the help of z, n* and W* from
(13.36), (22.10) and (22.11). Obviously, the same result follows from replacing
In S in (15.78) and (15.83) by In S + v^P/kT and in this way we find easily
that (kT In 5 > - vc,P)
t= 16c(2m0)"2ae/ff3'VcDi(47)"2peS(ln S + vcfPlkT)2 (22.13)
for HON (then cD1 = c and o"ef = a) or 3D HEN controlled by direct impingement
and that (kT In S > - vt!P)
-r = (Acivf)olf/97t2y*c*(kT)3tt[cS(lnS+ v^P/kT)" (22.14)
for 3D HEN under surface-diffusion control. These formulae show that the
sign of the effective molecular volume vcC governs the effect of the carrier-
gas pressure Pont: while vel > 0 leads to shortening of Twith increasing P,
when vc( < 0, r is longer at higher pressures.
Going back to eq. (22.12), we see that it passes into eq. (13.66) at P = 0.
It is seen as well that the sign of vt! is decisive for the effect of P also on the
stationary nucleation rate. Indeed, depending on it, Js can either increase
(when vti > 0) or decrease (if uef < 0) with increasing carrier-gas pressure.
According to (22.12), the effect is stronger for smaller S values and is
temperature-dependent. This means inter alia that, as uef is a function of T,
the same carrier gas can stimulate the nucleation process at a given temperature
and inhibit it at another temperature if at this temperature v^ has the opposite
sign. The more general conclusion is that in a T range in which the v^T)PI
kT ratio is considerable with respect to In S, an experimentally obtained J^(T)
dependence for nucleation in the presence of carrier gas may be found to
differ substantially from that predicted by the commonly used JS(T) dependence
(13.66) in which the effect of the carrier gas is not accounted for. Equation
(22.12) shows also that the classical linear dependence of In Js on (1/ln S)2
can be violated in the presence of carrier gas. Nonetheless, a reliable
interpretation of experimental JS(S, P) data is again possible in the scope of
the first application of the nucleation theorem (Chapter 14). Namely, using
eq. (14.9) or the formula [Kashchiev 1996]
n* = (A:r/uef)d(ln Js)ldP (22.15)
when such data are obtained at fixed P or S, respectively, we can determine
in a model-independent way the size n* of the EDS-defined nucleus as a
function of S and P and verify the Gibbs-Thomson equation (22.10) or any
other theoretical dependence of n* on S and P. Equation (22.15) is a general
Carrier-gas pressure 335
result valid for whatever kind of one-component condensed-phase nucleation
(HON, HEN, 3D, 2D, classical, atomistic, etc.) in the presence of carrier gas,
since it follows from (14.5) upon setting dA/; = v^P (see eq. (22.5)) and
using the approximations pn*w = pnew and nA = 0. As can be verified, eq.
(22.15) is satisfied by n* and Js from eqs (22.10) and (22.12) which describe
classically the concrete case of HON or 3D HEN. In the particular case of b12
= 0 (ideal-gas behaviour of the carrier gas) eq. (22,15) reduces to the expression
derived by Ford [ 1992a, b] and Oxtoby and Laaksonen [ 1995]. This equation
shows that in a given P range the effect of P on Js is more pronounced at
smaller values of the super saturation ratio S, for then n* is greater.
Figure 22.1 displays the dependence of Js on S for HON of water droplets
at 7'= 293 K in a carrier gas for which vef = 0.1 or-0.1 nm3 (curves 0.1 and
- 0.1, respectively). The calculation is done according to eq. (22.12) with P
= 10 MPa and B' = 89 (this B' value follows from (13.67) with tref = tr, c =
(36?r)1/3 and u0 and o from Table 3.1). The kinetic factor A = z/*C0 is
represented as A = A'S, and A' is given the value of 5.5 x 1034 m~3 s"1 used
for curve 'HON' in Fig. 13.5. For comparison, the 7S(5) dependence at P =
0 is depicted by curve 0. However, it must be noted that, being valid for
isothermal nucleation, eq. (22.12) is likely to give with some error Js in the
P = 0 limit, especially when used for HON. Indeed, as shown by Barrett et
al. [ 1993], at low carrier-gas pressures non-isothermal effects may be important,
because the latent heat of condensation cannot be easily removed from the
appearing subnuclei. This can result in a drop of the actual nucleation rate at
too small P values. For that reason, curve 0 in Fig. 22.1 should be regarded
as a convenient reference curve corresponding to the theoretical P = 0 limit
of 7S from (22.12) under isothermal conditions. As evidenced in Fig. 22.1,
the effect of the carrier-gas pressure is controlled by the sign of uef: positive
or negative uet- makes Js higher or lower, respectively, the effect being stronger
at lower supersaturation ratios.
The dependence of Js on P for HON of water droplets at T = 293 K is
illustrated in Fig. 22.2. Curves 0.1 and-0.1 refer to vef = 0.1 and-0.1 nm3,
respectively, and correspond to gas mixtures at S = 3 and 3.5 (as indicated).
The calculation is done according to eq. (22.12) with the parameter values
used for Fig. 22.1. It is seen that only carrier-gas pressures higher than about
ten times the atmospheric pressure can increase (when uef > 0) or decrease
(when uef < 0) the nucleation rate by more than one order of magnitude with
respect to the Js value at P = 0 (as mentioned above, for HON this value is
only a theoretically convenient reference value). The JS{P) dependence has
a threshold character and the stronger effect that the carrier-gas pressure
exerts on nucleation in gas mixtures containing less supersaturated condensing
vapours is again clearly demonstrated.
The main conclusion from the above considerations is that increasing the
carrier-gas pressure stimulates or inhibits the nucleation process when the
effective molecular volume vcf is, respectively, positive or negative depending
on the nature of the carrier and the condensing gases. The effect of P on the
stationary nucleation rate Js is relatively small and in the case of HON Js is
336 Nucleation: Basic Theory with Applications
Fig. 22.1 Dependence of the stationary nucleation rate on the supersaturation ratio:
curves -0.1, 0 and 0.1 - eq. (22.12) for HON of water droplets at T= 293 K in a carrier
gas which is under pressure P = 10 MPa and for which ve{ = ~ 0.1, 0 and 0.1 nm3,
respectively.
practically independent of both P and the nature of the carrier gas when P is
not much greater than the atmospheric pressure and the gas is inert (i.e.
insoluble in the nucleating condensed phase). This conclusion is supported
by me experimental findings of Wilemski era/. [1992], Wagner era/. [1992],
Muitjens [1996] and Luijten [1998] (see also Fig. 13.8), but is not by those
of Katz et al. [1992], Heist et al [1994] and Kane and El-Shall [1996].
Recalling that HEN occurs at smaller S values than HON, due to the stronger
change of Js with P at lower super saturations, we can expect the carrier-gas
pressure to influence more strongly HEN than HON. Also, it is worth keeping
in mind that a relatively strong effect of P on Js cannot be conceived as
Carrier-gas pressure 33 7
P (MPa)
Fig. 22.2 Dependence of the stationary nucleation rate on the carrier-gas pressure:
carves -0.1 and 0.1 - eq. (22.12) for HON of water droplets at S - 3 and 3.5
(as indicated) and T ~ 293 K in a carrier gas for which ve/ = -0.] and 0.1 nmJ,
respectively.
theoretically ruled out. Indeed, if o and/or/*, i.e. the thermodynamic and/of
the kinetic parameters B' and A = ^/*C0 in eq. (22.12), depend sufficiently
strongly on P, the change of Js with P can be much greater than in the
considered case of P-independent B' and A.
Chapter 23
Solution pressure
According to thermodynamics, the chemical potential of the solute molecules
in a liquid or solid solution and in the new phase nucleated in the solution
depends on the pressure P of the solution. For that reason, the solution
pressure can be a factor affecting the nucleation process. The role played by
P in nucleation of condensed phases in solutions is analogous to that of the
carrier-gas pressure in nucleation in vapours and can be quantified by repeating
the analysis in the preceding chapter. Following a recent study [Kashchiev
and van Rosmalen 1995], we shall now see what is the effect of P on nucleation
in bulk solutions and in solutions which are in pores or are dispersed into
droplets. The analysis is restricted to isothermal one-component HON or
HEN of condensed phases.
We consider a liquid or solid solution of fixed composition at constant
absolute temperature T. When the solution is at a given pressure P0, in
accordance with eqs (2.1) and (2.13) the supersaturation is expressed as
Aji(a, P0) = iU0id(a. ^o) ~ A'newta, Po) = kT In (a/ae) (23.1)
where a is the solute activity, and ae is the equilibrium activity at pressure
P0. Now, the question is: what is the supersaturation of the solution when the
solution is put under another pressure P? Regarding as P-independent the
partial volume us of a solute molecule in the solution and the volume u0 of
such a molecule in the nucleating one-component condensed phase allows
using the known thermodynamic formulae [Lewis and Randall 1923;
Guggenheim 19571
^ow(a. p) = *,idfo Po) + vs(P - Po) (23.2)
^new(a, P) = ^/new(a, P0) + v0(P - P0). (23.3)
With the help of eqs (2.1), (23.1)-(23.3) we thus find that [Kashchiev and
van Rosmalen 1995]
*v(a, P) = kT In (fl/fle) + (vs - u0)(P - P0). (23.4)
This is the needed formula for the dependence of the supersaturation in
liquid or solid solutions on the solute activity a and the solution pressure P.
It says that the character of the effect of P on Ap is controlled by the sign of
the difference between vs and u0. In the u0 < us case A^/ increases with
increasing P, but when v0 > us, higher pressures lead to lower supersaturations.
Examples of these two cases are benzene solutions of naphthalene for which
vs - vo - 0.022 nm3 [Lewis and Randall 1923] and aqueous solutions of
CaC03 for which us-vQ -- 0.075 nm3 [Stumm and Morgan 1981]. At room
Solution pressure 339
temperature, if P0 is equal to the atmospheric pressure, the rather small
absolute value of vs - v0 (usually \v, - v0\ < 0.1 run3) makes the effect of the
solution pressure P on A/j of practical significance only when P is more than
ten or one hundred times the atmospheric pressure. Indeed, only then can the
pressure term in (23.4) be comparable with or greater than the activity term
which is often, especially in HON, close to kT. We note also that A/<(a, P)
from (23.4) can be represented in the form of Aii(a, P0) from (23.1), i.e. as
[Kashchiev and van Rosmalen 1995]
&fi(a, P) = kT la (a/acP), (23.5)
if the equilibrium activity aeP of the solute at the new pressure P of the
solution is defined by
ac P = ac exp [- (vs - v0)(P - P0)lkT\. (23.6)
Physically, eq. (23.6) expresses nothing else but the thermodynamic
dependence of the equilibrium activity of the solute on the solution pressure
P. For sufficiently dilute solutions the activities a, ae and ae P can be replaced
by the respective solute concentrations C, Ce and Ce P and eq. (23.6) turns
into the known thermodynamic formula [Lewis and Randall 1923] for the
change of the equilibrium solute concentration (i.e. solubility) with the pressure
of the solution. The obvious analogy between eqs (23.4), (23.5) and (23.6)
and eqs (22.5), (22.9) and (22.8) makes congenial the effects of the solution
pressure and of the carrier-gas pressure in the respective cases of nucleation.
Having obtained the a, P dependence of the supersaturation of the solution,
as in the preceding chapter, we can use the general formulae for the size n*
of the EDS-defined nucleus, the nucleation work W*, the stationary nucleation
rate Js and the nucleation time lag t in order to analyse the effect of P on
these basic quantities. We shall again restrict the considerations to classical
HON or 3D HEN on a substrate, but they can be easily repeated for 2D HEN.
Thus, combining eqs (4.38), (4.39), (13.39), (13.40) and (23.4) results in the
expressions [Kashchiev and van Rosmalen 1995] (In 5 > - (us - v0)(P - P0)l
kT)
n* = &c3v$ol,mikT In S + (vs - v0)(P - P0)f (23.7)
W* =4c3vlal,m[kT In S+(vs-v0)(P-P0)f (23.8)
Js = zf*C0 exp (- B7[ln S + (vs- v0)(P - P0)lkT]2} (23.9)
where the supersaturation ratio S at the pressure P$ is defined by (13.69).
The effective specific surface energy <7cf and the thermodynamic parameter
B' are given by (4.42) and (13.67) (aet = a for HON) and can be treated as
/^independent when the possible o(P) dependence [Rusanov 1967, 1978] is
neglected. As in eq. (13.66), the kinetic factor A = zf*C0can be represented
in the form A'S where A' = zf* Ca is practically independent of both S and
P provided the mechanism of monomer attachment to the nucleus does not
change with the solution pressure. This is so since/* and /e* vary relatively
little with P only through n* from (23.7). The similar weak dependence of
340 Nucleation: Basic Theory with Applications
the Zeldovich factor z on P through n* and W* (see eq. (13.36)) can also be
ignored. Taking into account that now Aft is given by (23.4), it is easy to
verify that n* and W* from (23.7) and (23.8) satisfy the nucleation theorem
in the form of eq. (5.29). This is understandable upon recalling that (23.7)
and (23.8) are valid for one-component EDS-defined nuclei of condensed
phases.
As seen from eq. (23.9), depending on the sign of the volume difference
vs - v0, the nucleation rate can either increase (when v0 < vs) or decrease (if
l>o > vs) with increasing pressure P of the solution, the effect being stronger
for smaller S values. Hence, having in mind that HEN occurs at lower
supersaturations than HON, we can expect the influence of the solution
pressure to be more pronounced in HEN than in HON. We note also that at
P = P0 eq. (23.9) takes the form of eq. (13.66).
Curves -0.1 and 0.1 in Fig. 23.1 illustrate the JS(S) dependence (23.9) for
HON of condensed phases in liquid solutions for which us - va = -0.1 and
0.1 nm3, respectively. The calculation is done with T - 293 K and with P/P0
= 1 (curve 1) and P/P0 = 100 (curves 100). The pressure P0 is chosen to be
equal to the atmospheric pressure 0.1 MPa. In the calculation B' = 635 is
used (this value is obtained from (13.67) with the help of c3 = 367t, r% = a
and the u0 and (rvalues in Table 6.1), zf*CG is expressed as A'S, and A' is
given the exemplary value of 3 x 1035 m~3 s_1 which corresponds to z = 0.1,
f* = 10s s_l and C0 = 3 x 1028 m-3. As seen from Fig. 23.1, when the
solution is under high enough pressure, the nucleation rate can be orders of
magnitude higher or lower (for us - va > 0 or us - v0 < 0, respectively) than
that at the atmospheric pressure. This is evident also in Fig. 23.2 which
exhibits the JS(P) dependence (23.9) at two values of S (as indicated). The
calculation is done with vs - v0 = 0.1 and -0.1 nm3 (these values are also
indicated), and the other parameter values are those used for Fig. 23.1. The
JS(P) dependence has a threshold character and it is seen that pressures about
twenty times higher than the atmospheric pressure decrease or increase Js by
more than one order of magnitude with respect to Js at P = 0.1 MPa which
is the value of P0. The stronger effect of P on 7S in less supersaturated
solutions is also clearly demonstrated.
Concerning the JS(S) and Ja(P) dependences, it must be noted that if they
are experimentally available, in the scope of the first application of the
nucleation theorem (see Chapter 14) they can be used for a model-independent
determination of the size n* of the EDS-defined nucleus. This is so, because
while for JS(S) data at fixed P eq. (14.10) is in force, when we have JS(P) data
at fixed 5, the formula
n* = [kTI(vs - u0)]d(ln Js)ldP (23.10)
is applicable. This formula is general, i.e. valid for whatever kind of nucleation
(HON, HEN, 3D, 2D, atomistic, etc.). It follows from (14.5), since dAji = (vs
- Do) dP (see eq. (23.4)) and, approximately, p*e„ = pnew and nA = 0. In the
particular case of classical HON or 3D HEN eq. (23.10) is easily obtained
from (23.7) and (23.9). An experimental n*(S, P) dependence determined
Solution pressure 341
1020
1015
'co
?E 1010
105
1
1 10 20 30 40 50
s
Fig. 23.1 Dependence of the stationary nucleation rate on the supersaturation ratio:
curves 1 and 100 - eq. (23.9) for HON at T = 293 K of condensed phase in a liquid
solution for which v<- v0 = -0.1 and 0.1 nm3 (as indicated} and which is under
pressure P = P0 and 100P(I. respectively, P0 being the atmospheric pressure.
with the aid of eqs (14.10) and (23.10) thus allows a reliable check of the
Gibbs-Thomson equation (23.7) or of any other theoretical formula for n*
as a function of 5 and P.
It remains now to see what is the effect of P on the nucleation time lag r.
The conclusion based on the general formula (15.73) is that t is a comparatively
weak function of P when, as noted above, the mechanism of monomer
attachment to the nucleus is not affected by changes in the solution pressure.
For example, when volume diffusion or interface transfer controls/*, we can
determine the S, P dependence of X by employing eqs (13.36), (23.4), (23.7)
and (23.8) to express z and «* in (15.85) and (15.87) as functions of S and
342 Nucleaiion: Basic Theory with Applications
1020
1015
'en
\ 101°
105
1
0.01 0.1 1 10 100
P (MPa)
Fig. 23.2 Dependence of the stationary nucleation rate on the solution pressure:
curves -0.1 and 0.1 - eq. (23.9) for HON of condensed phase at S = 20 and 30 (as
indicated) and T — 293 K in a solution for which us - un = -0.1 and 0.1 nm3.
respectively.
P. We can, however, avoid the tedious algebra, if we replace In S by In S +
(us - v0)(P- P0)/W directly in eqs (15.86) and (15.88). We thus find that (In
S>-(us-i>„)(P-Po)/W)
T = \6c2viicjl!l3ir''1tcs,Din(kT)2 DCe5[ln S + (i>s - v0)(P - P0)/k7f
(23.11)
for HON (then cVD = e and c% = a) or 3D HEN under volume-diffusion
control and that (In S > - (v, - v0)(P - P0)/kT)
x = IbcdoajT^yc^kTDC^ln S + (us - v0)(P - Pa)lkT]2 (23.12)
-
-
-
-
—
-
-
-
30
20
0.1/ j
^-0.1 /
0.1 / \
x0-1 I
Solution pressure 343
for HON (then c1T = c and c*.f = (f) or 3D HEN controlled by interface
transfer. These formulae show that increasing the solution pressure P results
in either shorter (for v0 < vs) or longer (for 1¾ > us) nucleation time lag.
The above general results for the effect of P on Aft, «*, W*, Js and Tfind
a straightforward application to HON or 3D HEN of condensed phases in
liquid solutions confined in pores or dispersed into droplets [Kashchiev and
van Rosmalen 1995]. Due to capillary effects, in both pores and droplets the
pressure of the solution can differ considerably from that of the corresponding
bulk solution. Suppose we have a single cylindrical pore or spherical droplet
of radius r. In addition, let the pore, like the droplet, have no contact with the
corresponding bulk solution and let gravity effects on the pressure along the
height of the solution in the pore be negligible. Owing to the Laplace effect
of curvature, the pressure P of the solution in the pore or the droplet is given
by (e.g. Guggenheim [1957])
P = P0-2o„l cos 9„M/r (23.13)
where P0 is the pressure of the fluid (e.g. a gas) with which the solution in
the pore or the droplet is in contact, o"so, (J/m2) is the surface tension of the
solution/fluid interface, and 8wsoi is the wetting angle. It characterizes the
wetting of the pore walls by the solution: 8W soi = 0 and Sw sol = K correspond
to complete wetting and non-wetting, respectively, 0WSOi between 0 and nil
results in a concave meniscus, and when 0wsol is between nil and n, the
meniscus is convex. Since the pressure of the solution in the droplet is
formally described by eq. (23.13) in the limiting case of complete non-
wetting (then cos 0ws„i = -1), this equation is applicable to both pores and
droplets. We see from (23.13) that the pressure P of the solution in a pore
characterized by a concave meniscus (then cos 6W s0l > 0) decreases below P0
with diminishing pore radius r. The opposite is true for a pore with solution
with a convex meniscus or for a droplet: as then cos 0W,SO| < 0, P increases
above P0 with diminishing r. The case of bulk solution, the pressure of which
equals the pressure PG of the fluid contacting the pore or the droplet, is
described in the limit of r —> «. We can, therefore, refer to Pq as the pressure
of the bulk solution and use the r —> °° limit for comparing a given result for
nucleation in a solution in pores or droplets with the respective result for
nucleation in the corresponding bulk solution.
We can now substitute P from (23.13) in eqs (23.4), (23.6)-(23.9), (23.11)
and (23.12) in order to see how Aft, aeP,n*, IV*, Js and T are affected by the
pore or droplet size. For instance, by virtue of (23.9) and (23.13), the S,r
dependence of the stationary nucleation rate for HON or 3D HEN in solutions
in pores or droplets can be written down as [Kashchiev and van Rosmalen
1995] (lnS>-r<i/r)
}, = zf*C0 exp [- B7(ln S + ro/r)2]. (23.14)
The kinetic factor A = zf*C0 can again be represented in the form A'S with
practically S, r-independent A' = zf*C0, and the characteristic 'radius' r0 is
either positive, zero or negative, since it is given by
344 Nucleation: Basic Theory with Applications
r0 = 2(u0 - ljs)cjS0| cos ewsol/kT. (23.15)
Equation (23.14) reveals the effect of the pore or droplet size on Js provided
the nucleation process occurs in pores or droplets of radius sufficiently greater
than the nucleus radius which, typically, is below 1 nm. For large enough
pores or droplets (i.e. in the r —> «> limit) it turns into eq. (13.66) for the rate
of stationary nucleation in the corresponding bulk solution. Equation (23.14)
shows that, depending on the sign of r0, Js increases or decreases with
decreasing r when, respectively, r0 > 0 or r0 < 0. The r0 > 0 case is exemplified
,23
10'
10'
10'
10^ -
£ 10,a -
10
101' -
10
10
)
22
21
20
It*
18
17
16
15
bulk
-
\ pore
/ droplet
0.01
0.1
r(jim)
Fig. 23.3 Dependence of the stationary nucleation rate on the pore or droplet
radius: curves 'pore'and 'droplet'- eq. (23.14) for HON at T = 293 K and S = 50 of
CaC03 crystallites in an aqueous solution which, respectively, is in pores and has a
concave meniscus or is dispersed into droplets; line 'bulk'— eq. (23.14) for HON
under the same conditions in the corresponding bulk solution.
Solution pressure 345
by cos 9WSQ] > 0 (pore with solution with concave meniscus) and u0 > vs, and
the r0 < 0 one by cos 0WiSOi = - 1 (droplet) and also v0 > vs. Obviously, the
effect is stronger when the solution in the pore or the droplet is less
supersaturated. We see as well that even if the solution is saturated (then S
= 1), nucleation in the pore or the droplet is possible (Js > 0) provided r0 is
positive.
Displayed by curves 'pore' and 'droplet' in Fig. 23.3 is the Js(r) dependence
(23.14) for HON in solutions which are characterized by v0 > vs and which,
respectively, are in pores and have a concave meniscus or are dispersed into
droplets. The calculation is done with 5 = 50 and r0 = 3.6 nm for the pores
and r0 = - 3.6 nm for the droplets. These r0 values are illustrative for aqueous
CaC03 solutions in contact with air, since they are calculated from eq. (23.15)
with T = 293 K,vs-v0 = -Q.\ nm3, asol = 73 mJ/m2 and cos 0wsol - 1 for
the pores and cos 0wsoi = - 1 for the droplets. In the calculation, the kinetic
factor A = ^/*C0 is again expressed as A = A'S, and A' and B' are treated as
S, r-independent parameters with values equal to those used for Figs 23.1
and 23.2. As seen from Fig. 23.3, while nucleation in the pores is stimulated,
in the droplets it is inhibited with respect to the process in the corresponding
bulk solution (the Js value for the bulk solution is indicated by line 'bulk').
It must be emphasized, however, that although in the illustrated case of v0 >
vs the pores and droplets act as promoters or inhibitors, respectively, their
role will be the opposite for nucleation of condensed phases for which, under
the same conditions, the inequality v0 < us holds.
Chapter 24
Pre-existing clusters
In Chapter 15 we have considered non-stationary nucleation under the condition
that only monomers are present in the system at the initial moment t = 0 at
which the old phase is put at a constant supersaturation Aii > 0 and cluster
formation begins. However, as the process can also begin in the presence of
clusters formed previously in the system, we may expect these pre-existing
clusters to be a factor affecting the kinetics of nucleation. For instance, it is
clear that if the pre-existing clusters are supernuclei, they will be able to
grow spontaneously right after the initial moment and will thus cause an
increase in the non-stationary nucleation rate at the earliest stage of the
process. Also, the pre-existing clusters can exert influence on the concentration
of supernuclei in the system and the nucleation delay-time. Pre-existing
clusters can be created, e.g. if prior to the imposition of the working
supersaturation A/j the old phase is held long enough at a fixed under- or
supersaturation Afi0< A/j. If A/j0 <0, the old phase is previously undersaturated
or saturated and the concentration of pre-existing clusters of size n is represented
merely by the equilibrium cluster size distribution C(n, Aii0) corresponding
to A/j0 (see Section 7.1). When A/j0 > 0, however, the old phase is
presupersaturated and this concentration is described by the respective
stationary cluster size distribution X(n, A/j0) considered in Section 13.1. In
this case Aii0 must be sufficiently low, as only then can virtually no nucleation
take place before putting the old phase at the working supersaturation A/j.
The possibility for the existence of previously formed crystal clusters in
melts was pointed out first by Kaischew [1937] and then by Frenkel [1955]
and Fisher et al. [1948]. Later, other authors considered the effect of
preexisting clusters on the rate of non-stationary nucleation [Kashchiev 1969c],
on the concentration of supernuclei [Ziabicki 1968; Kashchiev and Kaischew
1969; Kashchiev 1969c] and on the delay time of nucleation [Andres and
Boudart 1965; Andres 1969; Hile 1969; Kashchiev 1969c; Frisch and Carlier
1971; Kelton et al. 1983; Shizgal and Barrett 1989; Wu 1992a, 1992c],
Extending the analysis in Chapter 15, we shall now see how a pre-existing population
of clusters in the old phase can affect the process of non-stationary nucleation
which occurs at constant supersaturation according to the Szilard model, i.e.
solely by monomer attachment and detachment to and from the clusters.
24.1 Non-stationary cluster size distribution
We first consider the number n of molecules in the one-component cluster as
Pre-existing clusters 347
a discrete variable. The problem is to find the solution of the Tunitskii
equation (9.18) again under the boundary conditions (15.3) and (15.4), but
under the non-zero initial condition (9.2) which now replaces (15.2). This
means that we can repeat the entire mathematics in Section 15.1 from eq.
(15.5) down to eq. (15.18) by taking into account that solely (15.8) and
(15.14) change, respectively, into (n = 2, 3, . . . , M - 1)
v„(0) = Z„,0-X„, (24.1)
M-\
Z c;ani = Zn0-X„. (24.2)
i'=2
Hence, the sought non-stationary cluster size distribution Zn(t) coincides
with Zn{t) from (15.18), i.e. we can write again (n = 2, 3, . . . , M - 1)
M-\
Zn(t) = Xn + Z (di/if) aBi exp (- X-t) (24.3)
where X»aH\ and d' are specified by eqs (15.10)-(15.12) and (15.16), andX„
is the stationary cluster size distribution (13.16). The only parameter affected
by the non-zero initial cluster size distribution Z,l0 is df. now it is given by
the equation
dt =
«22 «23 ■"■ «2,,-l Zifi ~X2 ■•■ A2.M-I
fl32 «33 ■" «3.f-l Z30 - X3 •■■ fl3_M_]
«/W-L.2 <*M-l,3 '"«M-1.(-1 ZM_hQ-XM_i '■■aM.1M.
(24.4)
which is a generalization of (15.17) and, as required, passes into it in the
limit of ZH o = 0 (no pre-existing clusters in the old phase).
Mathematically, Z„(t) from (24.3) is the exact and complete solution of
the problem. However, as noted in Section 15.1, for mathematical reasons
we can use it to find analytically the exact explicit dependence of Z„ on the
transition frequencies /„ and gn and on the concentration Z,l0 of pre-existing
clusters only when M = Xfet- < 6.
Now, let n be allowed to vary continuously. Then the problem is to solve
the Zeldovich equation (9.27) again under the boundary conditions (15.30)
and (15.31), but under the non-zero initial condition
Z{n, 0) = Zo(«) (24.5)
which replaces (15.29) and corresponds to (9.2) [Kashchiev 1969c]. Again,
the analysis in Section 15.1 from eq. (15.32) to eq. (15.44) remains fully
valid except that, due to (24.5), eqs (15.34) and (15.41) become (1 <n<M)
>in, 0) = [Z0(«) - X(n)]/C(n) (24.6)
348 Nucleation: Basic Theory with Applications
C(n) X ciai{n) = Z0(n)-X(n). (24.7)
Consequently, the sought non-stationary cluster size distribution Z{n, t) is of
the form of Z(n, t) from (15.44) (1 < n < Af):
Z(n, t) = X(n) + C(n) S c; a;(n) exp (- ^0- (24.8)
Here A,- and a,(n) are again the eigenvalues and the eigenfunctions of the
Sturm-Liouville equation (15.38) with boundary conditions (15.39), and
C(ri) and X{n) are, respectively, the equilibrium and stationary cluster size
distributions (7.4) (or (12.5)) and (13.18). The only difference between Z(n,
t) from (24.8) and (15.44) is that c, in (24.8) is defined by the equation
■rr
C(n)af(n)dn\ [Z0(n) - X(n)] a,(n) An (24.9)
which turns into eq. (15.43) at no pre-existing clusters (then Z0(n) = 0). We
note also that eq. (24.8) is analogous to the Z(n, t) formula of Shizgal and
Barrett [1989].
Although Z(«, t) from (24.8) is the exact and complete solution of the
Zeldovich equation under any non-zero initial condition (24.5), in general,
mathematical difficulties do not allow the exact determination of Xj, a,(«)
and c,. To express Z(n, t) from (24.8) in a physically instructive form we
must, therefore, resort to approximate results for these quantities. For example,
we can substitute A, and at(n) from (15.50) and (15.51) into eq. (24.8) in
order to approximate Z(«, t) for n in the vicinity of the nucleus size n*
[Kashchiev 1969c], Accordingly, 'shifting' the limits of integration in
(24.9) from 1 to n," and from M to nj, with the help of a^n) from
(15.51), the approximation C(n) = C* (see eq. (7.37)) and the truncated
Taylor expansion
Zo(n) - X(n) = (Z0* - X*) + (Z{* - X**)(« - „*) (24.10)
we find from (24.9) that c/ is given approximately by
q = (2lin)[(ZtlX* - 1) sin2 (inn.)
+ (4/ff"2)(Zo*IX'* - 1 ) cos2 (inll)] (24.11)
where Z* = Zo(n*) and Zq* = (dZo/dn)n = „*. To a good accuracy, from
(13.25) for X'* = (d»dn)„ = „. we have
X'* = -(p/Km)C*, (24.12)
and the stationary concentration X* of nuclei can be approximated by C* or,
more accurately, by (1/2)C* (see eqs (13.23) and (13.24)). Thus, using (13.25),
(15.48)-(15.51), (15.55) and (24.11) in (24.8) yields
Pre-existing clusters 349
Z(n, t) = C(n)[V2 - (Plitm)(n - n*)]
+ (Vn)C(n) Z (l/i)[(Z*/X* - 1) sin2 (Mr/2)
+ {AI%m)(ZC IX'* - 1) cos2 (wr/2)]
x sin \(iKpiA)(n - n* + 2//¾] exp (- i2t/4r). (24.13)
Like eq. (15.53) into which it passes when Z0(«) = 0 (no pre-existing
clusters), this expression is applicable only for | n - n* \ < 7Tl/2/4/3, i.e. only
for clusters of near-nucleus size. As it is most accurate just for the nuclei,
upon setting in itn = n*, accounting for (13.23) and rearranging, we find that
the time dependence of the non-stationary concentration Z*(r) = Z(n*, t) of
nuclei is given by
Z*(t) = X*{\ +(4/^)(1 -Z0*/X*) I [(- 1)7(2(- 1)]
i=\
exp [-(2/- 1)2//4t]). (24.14)
Equivalently, this equation can be represented as (cf. eq. (15.56))
Z*(r) = Zo*+ 2(X*-Z„*)Z (-1)M(1-erf [(2i-l)OT1/2/2r"2]) (24.15)
r=l
so that, analogously to (15.57), for t < X
Z*(r) = Z„* + (X* - Zj)(16//^T)"2 exp (- ifilAt). (24.16)
Equations (24.14)-(24.16) show that while initially (at t = 0) Z* = Zt, in
the t —> <*■ limit Z* —> X*. This is expected, since the pre-existing clusters
cannot affect the stationary cluster concentration. They can only shorten the
time needed for the establishment of this concentration: from eq. (24.14) we
see that in the special case of Z0 = X*, the actual concentration Z* of nuclei
is equal to their stationary concentration X* at any time t > 0. From eqs
(24.13)-(24.16) it is seen as well that the smaller the Z0*/X* ratio, the
weaker the effect of the pre-existing clusters on the time evolution of the
concentration of clusters with size n~n*.
Equations (24.3), (24.8), (24.13)-(24.16) are valid for arbitrary pre-existing
cluster size distribution Z„0 = Zo(n). We shall now consider the particular
case of pre-existing clusters formed in the system when prior to the imposition
of the working supersaturation A^i at t = 0 the system is held long enough at
a fixed A/i0 - A/i. As already noted above, in this case we shall have [Kashchiev
1969c; Kashchiev and Kaischew 1969]
Zo(n) = C(n, Afj0, T0) = C0 exp [- W(n, A^0, T0)lkTa] (24.17)
when Aji0 < 0 (then the system is previously undersaturated or saturated) and
Zo(n) = X(n, Afj0, T0) (24.18)
when Afia > 0 (presupersaturated system). Here T0 is the fixed absolute
350 Nucleation: Basic Theory with Applications
temperature at which the system is kept before the initial moment t - 0, and
C is expressed in terms of the work W for cluster formation in conformity
with (7.4). The important point is that even if A/i0 > 0* e9- (24.17) is a good
approximation for Z0(n) provided n< nj [Kashchiev 1969c] (n* = n*(Afi0,T0)
is the nucleus size at A/io and 7q). This is so, because according to (13.24)
(see also Fig. 13.1), in this size range the pre-existing stationary cluster size
distribution X(n, A^0,T0) differs less than 50% from the corresponding
equilibrium cluster size distribution C(«, A^0, To)- For that reason, hereafter
we shall use eq. (24.17) regardless of the sign of Afj0. Thus, with the help of
(4.6), (7.44), (13.24) (used at n = n*) and (24.17), for the Z*/Z* ratio we
obtain
Z*/X* = exp [-n*(Afi/kT - Avo/kTr,)
+ &*(Afi, T)lkT- <t>*(&na, TB)/kT0]- (24.19)
This expression takes a very simple form for EDS-defined clusters of
condensed phases in all cases when 70 = T so that the change of A^0 to A/v
is isothermal. Indeed, according to (5.20), for such clusters in either HON or
HEN on weakly adsorbing substrates the nucleus effective excess energy $*
does not depend explicitly on A^i and (24.19) reduces to
Z0*/X* =exp (-„**) (24.20)
where % > 0, defined by
Z=(Afl-AfJ<i)lkT, (24.21)
is the experimentally controllable dimensionless difference between the working
and the pre-existing supersaturation (when negative, A^0 is in fact the
preexisting undersaturation). For instance, the usage of eqs (2.8), (2.9), (2.13),
(2.14) and (2.16) in eq. (24.21) leads to
* = ln(p/po), * = In (///„) (24.22)
for nucleation in vapours and to
* = In (C/Co), X= In (a/a0), * = In (n/n0) (24.23)
for nucleation in solutions, p0, /0, C0, a0 and n0 being the pre-existing
pressure, impingement rate, solute concentration, solute activity and solute
activity product, respectively.
Equation (24.20) is free of assumptions concerning <P* and is therefore
applicable to any kind of nucleation (classical, atomistic, 3D, 2D, etc.) when
T0 = T so that % is varied isothermally. However, with redefined % it can be
used also when 70 * 7, but only in those concrete cases of nucleation in
which, on the basis of model representation, **(A^0,70) can be treated as
practically equal to <P*(A[i, T). For example, using $>(n) from Table 3.3 with
account of eqs (4.38), (4.39) and (4.42), for classical HON or 3D HEN of
condensed phases we have <P*(A/j, T) - co«(T) i>„/3 {T)n*m = (3/2)n*Afi and
<P*(Ati0Jo) = cacf(T0) v^ (T0)n*m so that under the assumption <P*(A/j0,T0)
= 0*(A^i, T) eq. (24.19) leads again to (24.20), but with % given by
Pre-existing clusters 351
X = (3777b - 1 )Afi/2kT - AnolkTf,. (24.24)
Another example is classical 2D HEN of condensed phases. Then, under the
same assumption concerning 0*, for % we have
X = (2777(, - 1 )Afi/kT - A/<(//tT0 + (a*Aolk)( UT - \IT„), (24.25)
since in conformity with <P(n) from Table 3.3 and eqs (4.32) and (4.33)
<P*(Afj, T) = n*acfA(j + bKn*m = n*(2Afi - aefArj) in the cases of both
foreign (Ac * 0) and own (A<7 = 0) substrate. In particular, when the
supersaturation is controlled by the temperature according to eq. (2.20), in
the scope of the approximation (2.23) x from (24.24) becomes (T0 > T)
X = (AsJ2k)(TJT0 -TJT+3- 3777b). (24.26)
This formula was used by Kelton etal. [ 1983] in a study of the effect of
preexisting clusters on HON of crystals in vitrified melts, but it is applicable
also to classical 3D HEN. Similarly, from (2.23) and (24.25) we find that (7b
>T)
% = (AsJkXTJTo -TJT+2- 277¾ + (acfAa/k)(UT ~ 1/7¾) (24.27)
for classical 2D HEN of condensed phases on foreign (Aa *■ 0) or own (Aa
= 0) substrates.
Equation (24.20) shows that the magnitude of the Z*,IX* ratio, i.e. the
effect of the pre-existing clusters, is controlled by one single quantity, X*
which can be varied experimentally by means of A^i0 and/or T0 in accordance
with eqs (24.21 )-(24.27) in the respective cases of nucleation at fixed working
supersaturation Afj and temperature T. As seen, Z0 /X* = 1 at ^ = 0 so that
then the impact of the pre-existing clusters on the nucleation process is
strongest. Theoretically, the opposite limit of x —> °° corresponds to no such
impact, since then Zj/X* -> 0. Actually, however, already x > 5/n* suffices
to practically prevent the pre-existing clusters from playing a perceptible
role in the process. For example, when the 7 and Ay values are such that n*
= 50, if x> 0.1, the Z*/X* ratio (24.20) is much less than unity: Z*/X* <
0.007. This means that, e.g. for nucleation in vapours, in view of (24.22), the
pre-existing pressure p0 must be at least e5ln* times lower than the working
pressure p when the pre-existing clusters are required to exert practically no
influence on the process (in the exemplified case of n* ~ 50 the result is /?0
< pie51"' = 0.90/)).
Figure 24.1 illustrates the effect of pre-existing clusters on the time course
of the non-stationary concentration Z* of nuclei. The dashed curves are
drawn according to the formula (x - 0)
Z*(t) = X*{1 + (4/7t)(l-e-"'x) I [(-1 )7(2i - 1)] exp [- (2« - 1 ft/4-c]}
(24.28)
with n*x~ 0, 1 and 3 (as indicated). This formula follows from (24.14) and
(24.20), and the n*x values used correspond to Z0/Z* = 1, 0.37 and 0.05,
respectively. The solid curve represents the Z*(r) dependence (15.54) (or
352 Nucleation: Basic Theory with Applications
x
N
1.0
0.8
0.6
0.4
0.2
:
:
:__
-
0
\y
3 //
y
---"'""i^**"*"^
-"" ''/^
/y
10
t/t
Fig. 24.1 Time dependence of the non-stationary concentration of nuclei: solid line -
eq. (15.54) for no pre-existing clusters; dashed lines 0, 1 and 3 - eq. (24.28) for
preexisting clusters at n*% ~ 0. 1 and 3. respectively.
(24.28) at n*x = =») describing non-stationary nucleation at no pre-existing
clusters. We see that, as already pointed out, only sufficiently small n*%
values (n*x < 5) result in concentration Z0 of pre-existing nuclei, which is
high enough (Zo > 0.01X*) to be of practical significance for the process.
It is important to note that the accuracy of eqs (24.13)-(24.16) and (24.28)
could be improved if in them the time lag T is replaced by the delay time 6
in the way discussed in Section 15.6. For instance, eq. (24.14) becomes (cf.
eq. (15.117))
Z*(t) = X*[ 1 + (4/;r)(l - Z0*IX*) I [(- l)'/(2i - 1)]
exp [-(2!-l)Vr/24fJ]}
(24.29)
with Zo IX* specified by (24.19) and (24.20) in the cases considered above.
When 9 is regarded as expressed by (15.105), eq. (24.29) is an equivalent
representation of (24.14) or (24.28). If, however, 6 in (24.29) is viewed as
given by the more accurate formula (15.111), eq. (24.29) is likely to describe
the Z*(t) dependence with an accuracy higher than that of eq. (24.14) or
(24.28). It may be noted as well that when A/j0 is sufficiently less than A/(,
eqs (24.19) and (24.20) are better approximations for the Z0/X* ratio if they
are used with pre-exponential factor 2 instead of 1. The factor 2 appears
upon employing eq. (13.23) rather than (13.24) for the determination of X*
in this ratio. For instance, when /tyo£ 0 (previously undersaturated or saturated
system), the expression zJ/X* = 2e'"'x is more accurate than eq. (24.20).
Pre-existing clusters 353
Accordingly, eq. (24.28) will provide a better description of the Z*(t)
dependence, if it is used with 2e~~""x instead of e~"*x when A/(0 is not too
close to A/i.
24.2 Non-stationary rate of nucleation
Following the procedure in Section 15.2, we can now determine the non-
stationary rate J{t) of nucleation at pre-existing clusters in the system. From
eqs (11.8) and (15.1), with the aid of (13.3) (used at n = «*) and (24.3) we
find that J(t) is again given by eq. (15.59), i.e. by (n* = 2, 3 M - 1)
J(t) = Js+ I (^')(/*a„.,-f„. + ,a„-ti,)exp(-A,r), (24.30)
but with dj specified by (24.4). Similarly, when n is treated as a continuous
variable, using (11.9), (13.31), (15.1) and (24.8) yields (cf. eq. (15.62))
J(t) = Js-f*C* Z ciB;* exp (-A,-0 (24.31)
where now c, is determined by (24.9).
Equations (24.30) and (24.31) in which J, is given by (13.29) and (13.32),
respectively, are the exact solutions of the problem for the non-stationary
rate of nucleation at pre-existing clusters, and the latter parallels the J(t)
formula of Shizgal and Barrett [1989]. For the reasons noted in Section 15.2,
to reveal the effect of these clusters on J(t) we must simplify (24.30) and
(24.31) by means of appropriate approximations for the i-dependent quantities
in them. For example, using A,, a,(n) and qfrom (15.50), (15.51) and (24.11)
and taking into account (13.33), (13.35), (15.48), (15.49) and (15.55), from
(24.31) we find that J(t) can be represented approximately as
J(t) = JSU + 2(1 - Z'0*IX*) I (- 1)' exp (- i2tlx)} (24.32)
i=i
or, equivalently, as (cf. eq. (15.65))
J(t) = J, {ZglT* + (1- Z'*IX'*)(A7CXlt)m
£ exp [-(2/- l)2;r2T/4l]}. (24.33)
Of course, eq. (24.32) is obtainable also by using Z(n, t) from (24.13) in
(11.9) and (15.1). We note as well that for t< T the sum in (24.33) converges
rapidly so that, initially,
J(t) = JS[Z!,*/X'* + (1- Zi*IX'*)(Antlt),n exp (- n2xlAt)\ (24.34)
Equations (24.32)-(24.34) tell us that the effect of the pre-existing clusters
on the non-stationary nucleation rate is controlled by the ratio Z'^IX'* of
354 Nucleation: Basic Theory with Applications
the derivatives (at n = «*) of the pre-existing and the stationary cluster size
distributions. This effect is strongest at the beginning of the process: the
presence of clusters of nucleus size n* already at t = 0 makes possible their
overgrowth right after the imposition of the working supersaturation A^i so
that even initially the nucleation rate is not zero. According to (24.34), the
initial nucleation rate ./(0) is given by
7(0) = (ZJ*/X'») Js. (24.35)
Naturally, the stationary nucleation rate /s is not affected by the pre-existing
clusters: from (24.32) we see that J -» Js for t —> <*■. As it has to be, in the
case of no such clusters (then Z'a* = 0) eqs (24.32)-(24,34) turn into eqs
(15.64)-(15.66), and ./(0) = 0.
Equations (24.30)-(24.35) are valid for whatever pre-existing cluster size
distribution Ztfji) and for any kind of nucleation (classical, atomistic, HON,
HEN, etc.). We shall now consider the case of Z0(n) given by (24.17) or
(24.18), assuming again that eq. (24.17) can be used regardless of the sign of
A/;0 (this assumption implies that A^i0 is sufficiently less than A/v). Thus,
with the help of (4.6), (7.44), (24.12) and (24.17) we find that (Aj,,, < A/i)
Zi*IX'* = (KmipkTa)[W(n, A/<0, T0)/dnJ„ = „«
x exp [- n*(Ap/kT - A/j0lkT0)
+ <P*(A/i, T)/kT- 0*(Auo, T0)lkT0]. (24.36)
Similar to (24.19), this expression becomes very simple for EDS-defined
clusters of condensed phases in all cases of nucleation when To = T. Owing
to (3.86) and (5.6) we then have [dW(rc, A/j0, 70)/d«]n = n. = - A/y0 + A^, and
the exponential factor is given by the r.h.s. of eq. (24.20). Hence, eq. (24.36)
takes the form (x > 0)
Zi*IX'* = (itvlXm exp (-n*X) (24.37)
where x >s specified by (24.21) in general and by (24.22) and (24.23) in
particular. Substitution of this expression for Zq*/X'* into eq. (24.32) yields
(X > 0) [Kashchiev 1969c]
J(t) = Js[\ +2[l-(7r"V/$e~"*rl £ (-l)'exp(-i2//T)j. (24.38)
Accordingly, from (24.35), for the initial nucleation rate ./(0) we get (% > 0)
J(0) = (7C],2x'P)Jse-"% (24.39)
The important point with eqs (24.37)-(24.39) is that in them x cannot
assume its limiting value of zero [Kashchiev 1969c]. Actually, in them x is
restricted by the condition x > 1/«*, since only then does Zq*/X'* from
(24.37) increase with decreasing x- This restrictive condition for x, like its
stronger version [Kashchiev 1969c], reflects the fact that eq. (24.36) is an
acceptable approximation for the Zq*/X'* ratio only when A/;0 is sufficiently
less than A^i, e.g. when the system is previously undersaturated or saturated
Pre-existing clusters 355
(then A^0 < 0). Experimentally, however, of greatest interest are the smallest
X values: from Section 24.1 we know that the smaller x> the greater the role
of the pre-existing clusters. To avoid the inconvenience with respect to the
above restriction for x we can trade accuracy for generality by neglecting the
pre-exponential factor in (24.37) and representing this equation as (x £ 0)
Zi*/X'* = exp(-n*X).
(24.40)
Hence, eqs (24.38) and (24.39) can be used in the less accurate, but more
general form (% > 0)
J(t) = Js [1 + 2(1 - e ~ "**) 2 (- 1)' exp (- A/T)] (24.41)
J(0)=Jse-"'*. (24.42)
Equations (24.40)-(24.42) have another important advantage over eqs
(24.37)-(24.39). Like (24.20) and (24.28), with £ defined by (24.24)-(24.27),
eqs (24.40)-(24.42) are applicable also when 7¾ *■ T provided the temperature
dependence of the cluster effective excess energy <P is negligible so that
0*(A|iO, T0) ~ <t>*(Afj, T). These equations show again that the effect of the
pre-existing clusters is strongest at % = 0 (then ZB(n) = X(n, An, T) and,
expectedly, ](t) = ./(0) = Js) and that this effect is of practically no significance
when x > 5ln* (then ./(0) < 0.017,, and J(t) from (24.41) is virtually identical
with J(t) from (15.64) which is valid for nucleation at no pre-existing clusters).
The effect of pre-existing clusters on the rate of nucleation is visualized in
Fig. 24.2 in which the dashed curves display the J(t) dependence (24.41) at
Fig. 24.2 Time dependence of the non-stationary rate of nucleation: solid line -
eq. {15.64} for no pre-existing clusters; dashed lines 0, 1 and 3 - eq. (24.41) for
pre-existing clusters at n*y = 0, 1 and 3. respectively.
356 Nucleatian: Basic Theory with Applications
n*X = 0. ' ant* 3 (as indicated). The solid curve represents the limiting J(t)
dependence (15.64), i.e. (24.41) at % - «>.
As suggested in Section 15.6, the accuracy of eqs (24.32), (24.38) and
(24.41) may be improved by replacing in them the time lag x with the delay
time 0. Analogously to (15.118), we can therefore write
J(t)=Js [I + 2(1 -Z^/X'*) X (-1)' exp(-/27r2z/60)] (24.43)
where Zq*/X'* is given by (24.36), (24.37) or (24.40) in the cases noted
above. This formula is equivalent to (24.32) when 9 is expressed through x
by means of eq, (15.105), but can be used also with 9 from (15.111) when a
more accurate description of the J(t) dependence is necessary.
24.3 Concentration of supernuclei
The pre-existing clusters can have an effect also on the concentration £(?) of
supernuclei in the system. Since we already know the non-stationary nucleation
rate J(t), for the determination of the £(?) dependence it is suitable to use eq.
(11.10). Integration of J(t) from (24.30) and (24.31) in accordance with
(11.10) yields (n* = 2, 3 , M- 1)
«0=&+^+ X (rff/V)(/*fli.*i-^ + i«- + u)[l-exp(-^)]
(24.44)
when n assumes only integer values and
f(/) = Co + J,t -f*C* i {c^r/X^l - exp (- V)] (24.45)
when n is treated as a continuous variable.
These equations represent the exact Qj) dependence for nucleation at
non-zero concentration of pre-existing clusters whose effect is taken into
account by the quantities dh c,- and £0 from (24.4), (24.9) and (24.46) or
(24.47). When no such clusters are present in the system (then Zy(n) = 0 and
£o = 0) eq. (24.44) coincides with (15.112). We note also that eq. (24.45) is
similar to the £(r) formula of Shizgal and Barrett [1989]. As seen from
(24.44) and (24.45), while at t —> «> the concentration of supernuclei increases
again linearly with time (cf. eq. (15.112)), initially, it does not vanish: £(0)
= £o * 0. The latter is so, because the sufficiently large pre-existing clusters
turn out to be supernuclei at the very beginning of the nucleation process.
According to eqs (9.2), (11.2), (11.3) and (24.5), the initial concentration £0
of supernuclei is given by
M
?o= £ Z„,0 (24.46)
Pre-existing clusters 357
or by
f„= J tZ0(n) dn (24.47)
when n is considered as a discrete or continuous variable, respectively.
As already noted in Section 15.1, the mathematical difficulties associated
with the determination of the various quantities in the sums in eqs (24.44)
and (24.45) call for finding approximate £(/) formulae. One such formula
can be obtained either from (24.45) with the help of A,, a^n) and c, from
(15.50), (15.51) and (24.11) or by integrating /(() from (24.32) as required
by (11.10). The result is
?(») = fo + J, {' ~ (*2/6)(l - Zf/X'*) X
- 2 (1 - Z'a*IX'*) X £ [(- 1)' li2] exp (- ihlx)} (24.48)
where f0 is given by (24.47). Equivalently, we can write (cf. eq. (15.115))
?(f) = ?0 + (A*IX'*) Jst + 2(1 - Zi*IX'*) jsx
x Z [(4m/T)"2expl-(2;-l)Vi/4t]
- (2i - 1)712) 1 - erf [(2i - \)jcil2l2tia]}) (24.49)
so that for t < < x we have
?(r) = ?0+(Z6*«'*)V
+ 8(1 -Zo'lX'*)Jsx(t/jcx)312 exp(-jf2t/4t). (24.50)
As it has to be, in the case of no pre-existing clusters (then Zq* = 0), eqs
(24.48)-(24.50) turn into eqs (15.114)-(15.116).
All of the above equations are valid for any kind of nucleation occurring
at time-independent supersaturation A^i in the presence of non-zero initial
concentration of clusters in the system. Again in the particular case of
preexisting cluster size distribution specified by eq. (24.17) or (24.18) the Zq* IX'*
ratio in them is given by eq. (24.36) in general and by eq. (24.37) or (24.40)
under the conditions discussed in Sections 24.1 and 24.2. For example, with
the aid of Z(,*/X'* from (24.40), eq. (24.48) becomes (/>0)
«/) = f0 + 7s(/-(^/6)(l-e-"*^)T
- 2(1 - e-""*)x I [(- 1)7;2] exp (- ft/x)} (24.51)
where ^is defined by eqs (24.21 )-(24.27) in the respective cases of nucleation.
Similarly, employing Z'0*IX'* from (24.37) in (24.48) leads also to f(t)
from (24.51), but with the quantity 7t"1xlfi as a multiplier of e~"'x [Kashchiev
1969c]. Though more accurate than (24.51), this (^^/^-containing formula
for f(J) has the disadvantage of being applicable only in the range of x > '/
n* and when T0 = T.
35S Nucleation: Basic Theory with Applications
To reveal fully the effect of the pre-existing clusters on the £(/) dependence
in the case when ZQ(n) is specified by eq. (24.17) or (24.18), it remains now
only to determine the initial concentration of supernuclei, £0, which appears
in (24.51). As this has already been done by treating the cluster size n as a
discrete variable [Kashchiev and Kaischew 1969], we shall do it below
under the condition that n varies continuously. In view of (24.47) we can
therefore write
6, = I C(n, A^0, T0) dn (24.52)
when A^/0 < 0 and
Co = J X(n, A^0, T„)dn = J C(n. Afi0. T0) dn (24.53)
when Afi > 0. The approximation of the first integral in (24.53) by the second
one is based on eq. (13.24) and is reasonable when the presupersaturation
Afi0 is not too close to the higher working supersaturation Aft, since only
then is the size n^ of the nucleus at A^0 sufficiently larger than the nucleus
size n* at Aft. In both (24.52) and (24.53) C diminishes exponentially with
increasing n within the limits of integration (see eq. (24.17)). Hence, for the
calculation of Co from (24.52) and (24.53) we can invoke the formula [Zeldovich
and Myshkis 1972]
j '/(x) dx = /2(*,)/| (d/7d*)„„ | (24.54)
where/> 0 is an exponentially decaying function of x between xt and xi, and
*2 is sufficiently greater than xx or, possibly, x2 = °°. Applying (24.54) to both
(24.52) and (24.53) leads to the same approximate expression, namely,
Co = CV, A^0, T„)/|[dC(n, A^„, 7"0)/dn]„=„»|,
so that with the aid of (4.6), (7.44) and (24.17) we obtain (AftB < Aft)
Co = {kT„/[dW(n, A/,„, T0)/dn]„=„,}C*
x exp [- n*(Aft/kT- A^/Wo) + **(tyi, T)lkT - <P*(Aft0, T0)/kT0]
(24.55)
where C* is the equilibrium concentration of nuclei at the working
supersaturation Aft.
This general approximate formula for Co is valid regardless of the sign of
Afj0 as long as Aft0 is sufficiently less than Aft. It resembles eqs (24.19) and
(24.36) and, like them, simplifies remarkably for EDS-defined clusters of
condensed phases in all cases of nucleation at To = T. Indeed, as then 4>*(A^0,
T0) = <t>*(Afi, T) and \dW(n, Afi0, T0)/dn]„=„, = Aft - Aft„ (see Sections 24.1
and 24.2), eq. (24.55) takes the form (j > \ln*)
Pre-existing clusters 359
io = (!*)£*«-"** (24.56)
where x 'S given by (24.21) in general and by (24.22) and (24.23) in particular.
When T0 * T, again for EDS-defined condensed-phase clusters, eq. (24.55)
leads to (A/j0 < A^i, x > '/»*)
Q, = [kTo/iAv - A»0)]C*e-»'* (24.57)
in all cases of classical HON, 3D HEN and 2D HEN in which the approximation
<2>*(A^0, To) = <tf(Afi, T) is acceptable. Here x 's defined in general by
(24.24) or (24.25) so that in the concrete case of A|i0 and A^i related to T0 and
Tby (2.23) the above formula becomes (T0 >T,x> Un*)
& = lkTo/&se(T0 - T)]C*e-*'* (24.58)
where % is expressed in terms of T0 and T by (24.26) for HON or 3D HEN
and by (24.27) for 2D HEN on foreign or own substrate.
Since the approximation (24.53) is increasingly incorrect when A^0 —> A^i
(then n* —> «*), eqs (24.56)-(24.58) cannot be used in the important % —>
0 limit. To avoid this inconvenience we can unite them into the less accurate,
but general formula (x £ 0)
f0 = C*e~"'x (24.59)
in which % is specified by (24.21 )-(24.27) in the respective cases of nucleation.
This formula is applicable regardless of the sign of A^0 and even when x is
diminished down to ^ = 0. In the concrete case of classical HON or 3D HEN
either at T0 = T or T0 * T eq. (24.59) was obtained by considering the cluster
size n as a discrete variable [Kashchiev and Kaischew 1969]. This equation
can be expected to give £o with a reasonable accuracy in the range of x
values of practical interest (e.g. between 0 and 10/«*). For instance, under
conditions characterized by x = 0.079 (calculated from eq. (24.26)) and «*
= 23, Kelton etal. [1983] found numerically that f0 = 0.17C* (see their Figs
2 and 11) for HON of crystals in vitrified Li2O-2Si02 melt. This finding for
fo agrees well with f0 = 0.16C* following from (24.59) with the above
values of n* and x- The accuracy of eq. (24.59), however, cannot always be
so good. Indeed, as shown in Appendix A3, the more accurate calculation of
fo in the case of A^i0 > 0 (presupersaturated system) leads to a formula,
(A3.7), which is considerably more complicated than (24.56) or (24.57) and,
hence, (24.59). In particular, at A^i0 = A/j and T0 = T (then x = 0), while in
accordance with (A3.9) f0 = (A*/4)C*, from (24.59) it follows that f„ = C*.
This means that in the x —> 0 limit eq. (24.59) tends to underestimate f0,
since for the width A* of the nucleus region at the working supersaturation
A^i we have A* = 2 to 40 under typical nucleation conditions.
It is important to note that eqs (24.56)-(24.59) offer a possibility for a
reliable experimental determination of the size n* and the equilibrium
concentration C* of nuclei at the working supersaturation A^i and temperature
T. This is so because, experimentally, £q is accessible to a direct measurement,
and x can be varied by means of A^0 and/or T0 at fixed A^i and T. According
to eq. (24.59), foOf) data plotted in In Q,-vs-x coordinates should lead to a
360 Nucleation: Basic Theory with Applications
linear dependence whose slope and intercept yield directly n* and C*. The
same linearity is predicted by eqs (24.56)-(24.58), but for the ratio of & and
the bracketed factors in them. Repeating the experiment at a number of fixed
Afi and T values, we can thus obtain experimental n*(A[i) and C*(A^/)
dependences and confront them with the respective theoretical ones (see
Chapter 4 and Section 7.2). This procedure does not seem to have been used
so far in nucleation experiments.
Having found £0, we now go back to eq. (24.48) in order to represent it in
the potentially more accurate form
£(0 = £o + JsU - 90 - (12/712)¾ E [(- 1)7/2] exp ( - (Vr/60)}. (24.60)
Here the quantity 0p(s) is defined by
0P = (1 -ZflX'*)e (24.61)
with 6 from (15.105) or (15.111) and with Z^*/X'* given by (24.36) in
general and by (24.37) or (24.40) in the cases considered in Sections 24.1
and 24.2. Equation (24.60) is analogous to (15.119) and passes into it at no
pre-existing clusters (then Z'0* = 0 and 6p = 0). With 0 specified by (15.105),
eq. (24.60) is identical with (24.48), but when 6 is determined from the
higher-order approximation (15.111), this equation may provide a better
description of the C,{t) dependence. It is seen also that in the t ~» «> limit C,{t)
- £o from (24.60) becomes asymptotically a linear function of time:
&)-Q = Js(t-6p). (24-62)
Comparison of this equation with (15.100) reveals that Bp is merely the
delay time of nucleation at pre-existing clusters in the system. We shall
examine 9p in the next section, but already now we see from eq. (24.61) that
the closer the size distribution Z0(n) of these clusters to the stationary cluster
size distribution X(n) corresponding to Ay and 7, the shorter 6p. In particular,
9p vanishes when Z0(«) = X(n). This makes obvious the physical meaning of
the delay time: it is a measure of the time needed for the rearrangement of
the pre-existing cluster size distribution into the corresponding stationary
one.
Figure 24.3 illustrates the effect of pre-existing clusters on the time course
of the concentration of supernuclei in the system. The curves are drawn
according to eq. (24.60) with 0p from (24.61) and ZfffX'* from (24.40).
The dashed curves correspond to n*% = 0, 1 and 3 (as indicated), and the
solid curve represents eq. (15.119) (oreq, (24.60) at «*# = «>) describing the
£(r) dependence at no pre-existing clusters. We see that the pre-existing
clusters affect the delay time, but not the stationary regime of the nucleation
process. Their presence can even eliminate completely the nucleation non-
stationarity: when their size distribution coincides with the stationary one
(then % - 0)» me £(/) dependence is linear right from the onset of the process.
In Fig. 24.4, the £(/) dependences resulting from eq. (24.60) with B? from
(24.61) and Zq*/X'* from (24.40) at n*% = 4.4 (the dashed curve) and
Pre-existing clusters 361
J?
Fig. 24.3 Time dependence of the concentration of supemuclei: solid line - eq.
(15.119) for no pre-existing clusters; dashed lines 0, 1 and 3 - eq. (24.60) for
pre-existing clusters at n*X - 0, 1 and 3, respectively.
1.5
— 0.5
t/e
Fig. 24.4 Time dependence of the concentration of supemuclei; dotted and dash-
dotted curves - numerical data of Kelton et at. [1983} for isothermal HON of
Li20 ■ 2Si02 crystals in vitrified melt at, respectively, n*% = (no pr e-existing
crystals) and n*% = 4.4; solid and dashed curves - eq. (24.60) also at n*X = and
n*% = 4.4, respectively.
362 Nucleation: Basic Theory with Applications
n*% = oo (the solid curve) are compared with the numerical £(/) dependences
obtained by Kelton et al. [1983] for HON of crystals in Li20 • 2Si02 glass
also at n*x = 4.4 (the dot-dashed curve) and n*x = •*> (the dotted curve). Due
to the somewhat higher value of ftp predicted by eqs (24.40) and (24.61), eq.
(24.60) describes less accurately the numerical data for £(/) at pre-existing
clusters in the glass.
24.4 Delay time of nucleation
The delay time ftp of nucleation at pre-existing clusters is given approximately
by eq. (24.61) which is in force for arbitrary size distribution Za(n) of these
clusters and regardless of the kind of the nucleation process (HON, HEN,
classical, atomistic, etc.). We shall now employ this equation to see what is
the influence of the pre-existing clusters on ftp when Z0(«) is specified by eq.
(24.17) or (24.18). Then, in general, the Z'a*IX'* ratio in (24.61) is given by
(24.36) and, in particular, by (24.37) or (24.40) under the conditions noted in
Sections 24.1 and 24.2. Hence, in the case of T0 = T, from (24.61) and
(24.37) it follows that (x > 1/n*) [Kashchiev 1969c]
ftp = [1 - (7tmxlP)^'x] 6 (24.63)
where^is specified by (24.21)-(24.23). Similarly, from (24.61) and (24.40)
we obtain (x>0)
ftp = (I-e^**) ft (24.64)
This formula is less accurate than (24,63) when x » Mn*, but has the
advantage to hold true in the whole range of x values down to X = 0 and to
be applicable also at Ta * T. That is why, for % in (24.64) we can use not only
eqs (24.21)-(24.23), but also eqs (24.24)-(24.27) in the respective cases of
nucleation of condensed phases.
Equation (24.64) tells us that Bp = 0 when x=0 (i-e. at T0 = 7" and A/i0 =
A^i), the reason being that the pre-existing cluster size distribution coincides
with the stationary one at the working temperature Tand supersaturation Ap.
In the other extreme, i.e. at;f = °o (no pre-existing clusters), 9p = 6 according
to both (24.63) and (24.64). We note that these equations are liable to a direct
experimental verification, since ft and ftp are directly obtainable from
experimental £(/) curves (they are the time intercepts of the asymptotic £(/)
dependences (15.100) and (24.62)), and x can be varied experimentally by
means of A^0 and/or To at fixed A/; and T. As predicted by eq. (24.64),
plotting experimental Sp(x) data in -In (1 - ftp/ft)-vs-^ coordinates should
yield a straight line whose slope is directly equal to the size n* of the EDS-
defined nucleus at T and A/i. A similar linearization of the epGf) data is
possible also when eq. (24.63) is used for their analysis, since in it ji is a
constant with respect to 70 and A^i0. With the aid of a family of 9p(x) data
obtained at different fixed T, Afj values we can thus determine experimentally
the n*(A/j) dependence and use it for verification of the respective theoretical
Pre-existing clusters 363
Gibbs-Thomson equation (see Chapter 4). Experiments aimed at obtaining
9J%) dependences in various cases of nucleation have yet to be carried out.
The effect of pre-existing clusters on the delay time is illustrated in Fig.
24.5 which depicts the 9v(x) dependence for HON of crystals in vitrified
Li20 ■ 2Si02 melt at 7= 660, 750 and 840 K (as indicated). The calculation
is done by means of eq. (24.64) with n* = 15, 23 and 39 for curves 660, 750
and 840, respectively, since this is the nucleus size at T = 660, 750 and 840
K. According to (24.26), at Te = 1300 K and Ase = 4.8* (these are the Te and
Ase values used in the study of Kelton et al. [1983]) the values of % on the
X axis in Fig. 24.5 correspond to previous undercoolings at different
temperatures T0 in the range of T < T0 < 0.77,,, i.e. to a presupersaturated
system. The circle in the figure represents the 9^6 value of 0.87 following
from the numerical data of Kelton et al. [1983] for 6 and for 9p at % = 0.19
(these authors found that 9 = 60.7 s at T= 750 K and no pre-existing clusters
and that 0p = 52.8 s again at T = 750 K, but after a long enough previous
undercooling at T0 = 800 K). As seen, eq. (24.64) predicts with an error of
some 13% the magnitude of the 9^9 ratio at the x value used in the numerical
calculation. Figure 24.5 demonstrates also that the pre-existing clusters affect
more strongly the delay time when the working temperature 7 is lower (then
«* is smaller, because the working supersaturation A/; is higher). From this
figure we can conclude as well that, since x > 0.3 when T0 > Te and T= 660,
750 or 840 K, if the U20 • 2Si02 glass is held previously at saturation or
Fig. 24.5 Dependence of the &p/& ratio on the thermodynamic parameter %: circle —
numerical finding of Kelton et al. [1983J for HON of Li20 ■ 2Si02 crystals in vitrified
melt atT = 750 K; curves 660. 750 and 840 - eq. (24.64) for the same case of HON
atT= 660, 750 and 840 K, respectively.
364 Nucleation: Basic Theory with Applications
undersaturation (i.e. at T0 £ 7*e), the pre-existing clusters exert practically no
influence on the delay time of the subsequent nucleation process at 660, 750
or 840 K.
Though general, eq. (24.61) is only an approximate formula for 0p, since
it is derived with the aid of the approximation (24.60) for £(?). Even less
accurate are, therefore, eqs (24.63) and (24.64) because of their relying on
the approximate expressions (24.37) and (24.40) for the Zy*/X'* ratio.
Fortunately, we can obtain also exact formulae for 9p, which may be utilized
for verification of the accuracy of eqs (24.61), (24.63) and (24.64). One way
to arrive at such exact formulae is to use eqs (24.44) and (24.45) in the t —>
«> limit for determination of the asymptotic £(r) dependence and to compare
this dependence with that from (24.62). Doing that yields (n* = 2, 3, . . . ,
M- 1)
M-l
9p = -(l/d%) Z (dA)(/*flB„-^ + 1£iB, +,,,-) (24.65)
/=2
when n assumes only integer values and
0P = (/*C*//S) Z (c,/A,)ar'* (24.66)
when n is considered as a continuous variable.
In these exact formulae for 6p solely the quantities d{ and c, are affected
by the initial cluster size distribution Zo(n). As these quantities are given by
eqs (24.4) and (24.9), 9p from (24.65) or (24.66) becomes equal to 9 from
(15.101) or (15.103) at Zo(ra) = 0 (no pre-existing clusters). Also, since d{ -
0 and C-, = 0 at Zq(«) = X(n), the above expressions tell us that 6p = 0 when
the pre-existing cluster size distribution coincides with the stationary one.
We note as well that (24.66) is analogous to the formula for 9p of Shizgal and
Barrett [1989].
Like (15.101) and (15.103), eqs (24.65) and (24.66) are not physically
informative, because they do not represent explicitly the dependence of 9p
on the frequencies of monomer attachment and detachment and on the
concentration of pre-existing clusters. However, such an explicit dependence
does exist and can be obtained by the method of Wu [1992a] already employed
in Section 15.4. Indeed, treating n as a continuous variable, let us start from
eq. (15.106). Performing the subsequent transformations, we see that all
expressions down to eq. (15.107b) remain unchanged. The last operation,
however, the integration of (15.107b) in accordance with (15.99), requires
accounting that now the initial condition is given by eq. (24.5) rather than by
(15.29). Hence, instead of (15.108) we get [Wu 1992a]
0P = (1//,) f [X(n)-Z„(n)]dn
| [X(n) - Zo(n)][X(n)fC(n)] dn (24.67)
Pre-existing clusters 365
where C(n), X(n) and/s are specified by (7.4) (or (12.5)), (13.18) and (13.32).
When n is allowed to assume only integer values, the integrations in (24.67)
are replaced by summations and this equation becomes [Hile 1969; Wu
1992c]
„* M-l
0P = (1A/S) [ I (X„ - Z„,„) - I (X„ - Z„0)(X„/C„)]. (24.68)
Here C„, X„ and/s are given by (12.3), (13.16) and (13.29) as functions of
f„ and g„. That is why the above two exact formulae unveil the explicit
dependence of ftp on the monomer attachment and detachment frequencies
and the concentration of pre-existing clusters. As it has to be, at Zo(n) = 0 and
Z„0 = 0 eqs (24.67) and (24.68) turn into (15.108) and (15.109), and at Z„(n)
= X(ri) and Z„ 0 = X„ they yield 8p = 0. It is clear that whereas eq. (24.68) is
more convenient for numerical calculations, eq. (24.67) is advantageous in
deriving approximate formulae for 9p of the kind of the formula for ft For
example, performing the integrations in (24.67) with Zo(n) from (24.17) or
(24.18), we could find ftp for nucleation in a previously undersaturated,
saturated or supersaturated system and compare the resulting expression
with eqs (24.61), (24.63) and (24.64). This does not seem to have been done
as yet.
Chapter 25
Active centres
In many cases of nucleation the clusters are formed on energetically preferred
sites known as nucleation-active centres (or active centres, for brevity).
Various nanoscopic structural defects, ions, impurity molecules and foreign
nanoparticles in the volume of the old phase or on the substrate surface are
examples of such active centres. In the case of nucleation on active centres
the total number of supemuclei in the system cannot exceed the number of
active centres in it. The active centres thus appear as one of the factors which
confine both the maximum number of supemuclei formed in the system and
the duration of the nucleation process. Indeed, during the process the active
centres are progressively exhausted by the supemuclei coming into being on
them so that the occupation of the last of them by a supernucleus brings
nucleation to a halt. The effect of active centres on the kinetics of nucleation
was studied, e.g. by Avrami [1939, 1940, 1941], Robins and Rhodin [1964],
Kaischew and Mutaftschiev [1965], Gutzow and Toschev [1971], Markov
and Kashchiev [1972a, 1972b, 1973], Stenzel and Bethge [1976], Trofimenko
et at. [ 1979, 1980, 1981], Stoyanov and Kashchiev [1981], Stenzel and Velfe
[ 1984], Fokin et al. [ 1997], Kozisek et al [ 1998]. In this section we shall see
what is the effect of the active centres on the time dependence of the total
number N of supemuclei in the system when the centres are of equal nucleation
activity.
We consider a supersaturated old phase which contains a total of Na active
centres each of which can be occupied by only one cluster of the new phase.
For the number N(t) of all supemuclei on the centres at time t we have
W) = £(t)V (25.1)
for HEN in the volume V of the old phase and
W) = £(')AS (25.2)
for HEN on a substrate with surface area As. The total concentration £(r) of
supemuclei is related to the nucleation rate J according to eq. (11.1). In
stationary regime we have J = Js, the stationary nucleation rate Js being
given by eq. (13.39). Since now C0 and Na are connected by eq. (7-9) or
(7.10), owing to (13.39) and (13.40) we can represent Js in the form
•A. = J*,MV (25.3)
for HEN in the volume of the old phase and
h = w.
(25.4)
Active centres 367
for HEN on a substrate. Here the quantity Jv (s_1), given by
•/..! = tf* exP (" W*/kT), (25.5)
has the physical significance of stationary nucleation rate per active centre.
Thus, using eqs (25.1)-(25.4) in (11.1), in both cases of HEN we obtain
~f- = Jv[Na-mt)], (25.6)
because at time t not all Nt active centres, but only /Va - N of them are
unoccupied by the N already formed supemuclei and, hence, able to generate
new supemuclei.
Equation (25.6) is a linear ordinary differential equation of first order
with known solution [Korn and Korn 1961]. At no pre-existing supemuclei
on the active centres the initial condition to eq. (25.6) is N(0) = 0 and the
solution of this equation reads [Avrami 1939, 1940; Robins and Rhodin
1964]
/V(/) = /Va[l-exp(-7a,st)]. (25.7)
This formula shows that in stationary HEN on active centres the number
of supemuclei in the system increases linearly with time only initially when
the exhaustion of the centres is insignificant. Indeed, for t« l//a s we have
N(t) = JasNat (25.8)
which, in view of (25.1)-(25.4), coincides with eq. (13.102). This means that
with the help of (25.3) or (25.4), from the initial slope of a given experimental
N(t) dependence we can determine the stationary nucleation rate 7S. The
effect of the active centres is manifested at longer times (t > \/Ja s) when N
slows down its rise with t and finally (r —> ■*>) assumes its limiting value /Va.
In other words, as it should be, in the considered case of occupation of each
active centre by one supernucleus the maximum number Nm of supemuclei
formed in the system is merely equal to the number N.a of active centres:
/Vm = /Va. (25.9)
In Chapter 32 we shall see, however, that Nm may not always be controlled
in such a simple way by Na.
Following the same modus operandi, we can find the N(t) dependence
also when the nucleation rate is a function of time. Analogously to (25.3) and
(25.4) J can be written down as
J(t)=J„(t)NJV (25.10)
for HEN in the volume of the old phase and as
J(t) = U<WAs (25.11)
for HEN on a substrate, Ja (s~') being the time-dependent nucleation rate per
active centre. For instance, according to (15.64), (25.3), (25.4), (25.10) and
(25.11), for non-stationary nucleation at constant supersaturation and no
pre-existing clusters on the active centres we have
368 Nucleation: Basic Theory with Applications
UD = h, 11 + 2 I (- 1)''rap (- ft/T)] (25.12)
where Jas is specified by (25.5), and the time lag T is given by (15.73), Using
eqs (25.i), (25.2), (25.10) and (25.11) in (11.1) we thus arrive again at eq.
(25.6) in which, however, /as is replaced by Ja(t):
^p-=JM[N.-N(t)]. (25.13)
The solution of this equation again under the initial condition /V(0) = 0 is of
the form [Avrami 1939, 1940; Gutzow and Toschev 1971]
(25.14)
N{t) = N, 1 - exp - JAt')dt'
This expression is a generalization of eq. (25.7) and passes into it at JJJ)
= /a, = constant (stationary nucleation). We see also that N—> /Va at t —> «>,
i.e. again Nm = N^. Employing /a(r) from (25.12) in the integral in (25.14)
and recalling that this integral can be expressed with the aid of eq. (15.114),
we find that for non-stationary nucleation at constant supersaturation and no
pre-existing clusters on the active centres the explicit N(t) dependence is as
follows:
MO = AU 1 - exp [ - 7W(t - (;r2/6)T- 2x Z [(- l)'/i2] exp (- i2t/t)}]}.
(25.15)
It is worth noting that eqs (25.7), (25.14) and (25.15) admit a general and
physically instructive presentation in the form
/V(/) = /Va(l-exp[-/Vex(/)/A'a]) (25.16)
where
va r jt(t')it-
Jo
Na(t) = N, J.(t')dt'. (25.17)
Jo
Sticking to the terminology of Avrami [1939, 1940, 1941], we may call Ncx
extended number of supernuclei, since it gives the number of supernuclei
which would have formed in the system if there were no exhaustion of active
centres by the supernuclei appearing on them (cf. eqs (25.8) and (25.17) at
ya(r) = constant = 7as). Equation (25.16) tells us that the effect of the active
centres on the N(t) dependence is controlled by A7^: while initially A^ < A^
so that A'(r) = A'CT(r), at later times A'e, » N„ and N(t) = A'a.
The circles in Fig. 25.1 display the experimental N(t) dependence of
Robins and Rhodin [1964] for HEN of Au crystals during ultra high vacuum
deposition of Au on (100)MgO substrate at 7= 654 K and impingement rate
/ = 9 x 1015 nf2s-1. As seen, this dependence is fitted well by eq. (25.7) (the
Active centres 369
3x10
,« 2x1015
1x10
1200
Fig. 25.1 Time dependence of the concentration of super nuclei on active centres:
circles - data for HEN of Au crystals on (WO)MgO substrate at T = 654 K and
impingement rate 1 = 9 x 1015 mr1 s~! [Robins and Rhodin 1964}; curve - best fit
according to eq. (25.7) for stationary nucleation.
Z
z
100
Fig. 25.2 Time dependence of the concentration of supemuctei on active centres:
circles -data of Fokin et al. j1997] for nucleation of /j-cordierite crystals on a
polished surface of cordierite glass at T - 1123 K; curve - eq. (25.15) for
nonstationary nucleation,
370 Nucleation: Basic Theory with Applications
curve) with NJAS = 3.6 x 1015 m'2 and Ja s = 0.0036 s"1. In conformity with
eq. (25.4), this means that for the stationary nucleation rate in the system
studied we have Js = 1.3 x 1013 m"2s~' under the concrete experimental
conditions. The good description of the experimental N(t) dependence by eq.
(25.7) is an evidence for stationary nucleation. The course of the N(t) function
in the case of non-stationary nucleation is illustrated in Fig. 25.2 in which
the circles represent the data of Fokin et al. [1997] for nucleation of pi-
cordierite crystals on a polished surface of cordierite glass at T = 1123 K,
and the curve is drawn according to eq. (25.15) with T= 7.88 x 104 s and Jas
= 3.74 x 10-5 s"1. As seen from Figs 25.1 and 25.2, the obvious distinction
between the stationary and the non-stationary N(t) curves is the sigmoidal
shape of the latter. Hence, the conclusion: such a shape of an experimentally
obtained N(t) curve may be an evidence for non-stationary nucleation on
active centres.
Part 4
Applications
This Page Intentionally Left Blank
Chapter 26
Overall crystallization
Overall crystallization of a melt is a complex process involving simultaneous
nucleation and growth of separate crystallites. Not surprisingly, therefore,
the theory of nucleation has found one of its most important applications in
the description of the kinetics of this process [Kolmogorov 1937; Volmer
1939; Johnson and Mehl 1939; Avrami 1939, 1940, 1941]. This application
is important, because it quantifies the effect of the nucleation rate on such
physically interesting characteristics of the resulting crystalline phase as,
e.g. the average size of the crystallites formed and their maximum number.
Moreover, this application is in fact much more general, since many of the
theoretical dependencies pertaining to overall crystallization can be used
directly also in the cases of crystallization in solutions [Chepelevetskii 1939;
Nielsen 1964], formation of droplets in vapours [Tunitskii 1941 ] and formation
of bubbles in liquids [Kashchiev and Firoozabadi 1993]. In this chapter we
shall first consider some general formulae of the Kolmogorov-Johnson-
Mehl-Avrami (KJMA) theory of overall crystallization and then employ
them in various particular cases of crystallite nucleation and growth. More
details on the subject and further developments can be found elsewhere (e.g.
Evans [1945]; Lyubov [1969, 1975]; Hopper et al. [1974]; Christian [1975];
Belenkii [1980, 1984]; Doremus [1985]; Weinberg [1985, 1991, 1992];
Weinberg and Kapral [1989]; Weinberg and Zanotto [1989a]; Furu et al.
[1990]; Orihara and Ishibashi [1992]; Shneidman and Weinberg [1993];
Weinberg et al. [1997]).
26.1 General formulae
The main quantity to be determined in the theory of overall crystallization is
the total volume Vc of crystalline phase or, equivalently, the fraction
asVJV (26.1)
of volume crystallized till time I (Vis the initial volume of the melt). Obviously,
for a closed system, the direct determination of the VC(J) function requires
finding the crystallite size distribution Zn{t) or Z(«, t) (m~3) as a solution of
the master equation (9.4) or (9.10) and using it in the expression
Vc(t) = v0V Z nZ„(t) = vl)V\ nZ(n,t)dn.
n=n*+l J„.
However, analytical formulae for the Vc(t) dependence are hardly obtainable
374 Nucleation: Basic Theory with Applications
in this way because of the formidable mathematical difficulties arising when
solving the master equation, especially at the later stages of overall
crystallization which are characterized by multiple contacts between the
crystallites. In the KJMA theory this apparent impasse is overcome by adopting
a different view on the process: it is assumed that Vc results from nucleation
of material points (crystal nuclei of radius R* = 0) at a rate /(f) (rrr3 s"1)
which then only expand irreversibly in radial direction with growth rate Git)
(m/s) (Fig. 26.1).
o
o
o
°
o
^^ o
o
o
Fig. 26.1 Overall crystallixjalion by the polynuclear mechanism corresponding to
appearance and growth of statistically many supemuclei (the circles) in the old phase.
Under this assumption, it is easy to find Vc not too late after the initial
moment t=0 at which the melt becomes undercooled and overall crystallization
begins. Then practically the whole volume V of the melt is available for
nucleation (either HON or HEN) and there are no contacts between the
growing crystallites. For that reason, at time t the volume V„(r\ t) of any
individual n-sized crystallite depends only on the earlier moment f < t of its
nucleation provided it is additionally assumed that the crystallites are
isomorphic during growth. Hence, for the crystalline volume dVc formed
between f and r" + d/ we can write
dVc = Vtf. (V(t')Vd(' (26.2)
where, according to (11.1), J(t')V it' is the number of crystallites formed
between i and f + dt'.
As we want to know Vc during the entire process of overall crystallization,
we must extend the validity of the above equation beyond the initial stage of
the process by accounting that at a sufficiently advanced time ( crystallites
can be nucleated solely in the non-crystallized volume V-Vc of the melt. This
means that, more generally, rather than by (26.2), dVc will be given by
Overall crystallization 375
dVc = Vn(f, tW)(V- Vc) dt'. (26.3)
Thus, integration of this equation under the initial condition Vc(0) = 0 with
allowing for eq. (26.1) leads to
a(t) = VM/V = 1 - exp l-V„(t)/V\ (26.4)
where the so-called extended volume Vcx [Avrami 1939,1940, 1941] is given
by
) = ,{'
Jo
Vcx(t) = V\ J(t')V„(t',t)dt'. (26.5)
Jo
Equation (26.4) is the well-known KJMA formula which was derived
rigorously by Kolmogorov [1937] in the scope of probability theory and by
Johnson and Mehl [1939] and Avrami [1939, 1940, 1941] with the help of
geometrical considerations. Not surprisingly, this equation parallels eq. (25.16).
Indeed, physically, Vex is the total crystalline volume which would have
formed in the melt till time t if there were no exhaustion of the initial melt
volume Vby the growing crystallites and no contacts between these crystallites.
For that reason, initially, Vc is practically equal to Vex: during the period at
which Vc < 0.2V, in conformity with (26.4) we have
Jo
Vc(t) = V \ J(t')V„(t', t)dt' (26.6)
Jo
with an error of less than 10% (this result follows from using the truncated
Taylor expansion exp (-Vex/V) = 1 - Vex/V). Naturally, eq. (26.6) is directly
obtainable by integration of eq. (26.2).
Since the crystallites are assumed to keep their shape during growth,
geometrically, at time t a given crystallite can be characterized by an effective
radius R which depends on the earlier moment l1 of its formation. The individual
crystallite volume V„ will then be given by
V„(t",t) = ceRd = cg
f G(t") it"
Jo
(26.7)
where d= 1, 2, 3 is the dimensionality of growth, cg (m ) is a shape factor
(e.g. cg = 4;r/3 for spherical crystallites; see also Table 26.1), t - t" is the
period of growth of the crystallite, and G(t) s dRIdt is the crystallite growth
rate.
Combining eqs (26.4), (26.5) and (26.7) yields the KJMA formula
ij{n[Cc
a(t) = 1 -exp l-cg J(t')\\ G(t")dt
dt'\ (26.8)
which shows explicitly how the evolution of the fraction of crystallized
volume is controlled by the two basic parameters of the process of overall
crystallization - the crystallite nucleation and growth rates J and G. It has to
376 Nucleation: Basic Theory with Applications
Table 26.1 Shape factor csfor ID growth of needles
with constant cross-sectional area Aq, for 2D growth of
disks or square prisms with constant thickness Hv and
for 3D growth of spheres or cubes.
Growih Shape cg
ID needle 2A0
2D disk ttH0
2D square prism AHn
3D sphere 4n/3
3D cube 8
be noted that this formula is somewhat different from that derived by
Kolmogorov [1937]: while in (26.8) the limits of the integral in the square
brackets are 0 and t - r\ in the Kolmogorov expression the respective limits
are f and t. As discussed by Belenkii [1980], this difference of (26.8) from
the rigorous Kolmogorov expression for ca[t) does not matter when G is t-
independent, but leads to a certain inaccuracy in the determination of the
time dependence of a when G is a function of t.
According to Kolmogorov [ 1937], a(t) can be interpreted as the probability
for crystallizing the melt until time t after the onset of the process of overall
crystallization. In conformity with probability theory [KornandKorn 1961],
this means that the average time /av for crystallizing the melt, i.e. the lifetime
of the melt in undercooled state, is given by [Belenkii 1980]
t„ = f tda(t)= f exp [- Vcx (t)/V] dr (26.9)
Jo Jo
where Vex is specified by (26.5) with V„ from (26.7).
Knowing the a(t) dependence, we can determine also the number N(t) of
supernucleus crystallites in the melt at time t. In doing that it is necessary to
take into account that the crystallites can be nucleated only in the continuously
decreasing volume V - Vc of the melt. At an earlier time t the melt volume
is V - Vc(t') and is nearly unchanged during the following infinitesimally
small period it'. Therefore, using eq. (25.1) in the differential form dN(f) =
[V- Vc(t')] df(/'), recalling (11.1) and (26.4) and integrating under the condition
/V(0) = 0 yields [Kolmogorov 1937]
N(t) = V f J0')U - a0')] it' = V f JO') exp [-V„(/)/V] <•''
Jo Jo
(26.10)
Since in overall crystallization the crystallites are treated as not growing
into each other upon getting in mutual contact, a certain maximum number
Wm of crystallites is produced at the end of the process. This number
characterizes the polycrystallinity of the newly formed crystalline phase and
is the limiting value of N(t) from (26.10) at t - <*■ [Kolmogorov 1937]:
Overall crystallization 377
N,
Jo
J(t)expl-V„(t)IV]dt.
(26.11)
Finally, we can calculate the average volume Vav of the crystallites at the
end of overall crystallization. This quantity is also of importance for the
properties of the resulting polycrystalline phase and can be estimated with
the help of the relation Vav = V/Nm used by Volmer [1939]. With Nm from
(26.11) it thus follows that
[f
J(t)ex?[-Vtt(t)IV]&t
(26.12)
26.2 Polynuclear mechanism
Equation (26.8) is applicable to systems which undergo phase transformation
under conditions allowing the formation of statistically many nuclei (Fig.
26.1). When this is the case, the process occurs by the so-called polynuclear
mechanism. This mechanism has two distinct manifestations known as
instantaneous and progressive nucleation, which we shall now consider
separately.
1. Instantaneous nucleation (IN)
Physically, IN corresponds to the case when all crystallites in the system are
nucleated practically at the initial moment t = 0 so that after that they only
grow irreversibly until the phase transformation is accomplished. An example
of such a process is nucleation on Na active centres at such a high nucleation
rate that right after the onset of the crystallization process the system already
contains the maximum number Nm of crystallites, which is now equal to Ns
(see Chapter 25). Mathematically, therefore, IN is characterized by a J{t)
dependence of the form
J(t) = (NmIV)o\,(t) (26.13)
where b\, is the Dirac delta-function. Employing this expression for J(t) in
(26.5) and recalling that for the product of 8^(0 with an arbitrary function
y(r', r) there holds [Korn and Korn 1961]
f WW, t) dt' = y(0,
Jo
in conformity with (26.7) we get
0,
Va(t) = NmV„(0,t) = csNt
[j>")d'"
(26.14)
This result is obvious: as in IN all /Vm crystallites are nucleated
simultaneously at t' = 0, later they have the same individual volume V„ so
378 Nucleation: Basic Theory with Applications
that the extended volume Vex is just the product of Nm and Vn. To find the
explicit Vex(0 dependence we, therefore, need a model time dependence for
the crystallite growth rate G. In many cases this dependence can be expressed
in the form
G(t) = vG?tv-1 (26.15)
implying crystallite growth according to the power law
R(t) = (Gct)v (26.16)
in which v > 0 is a number and Gc > 0 is the growth constant. Exemplary
values of v are 1/2 and 1 for growth controlled by volume diffusion and
interface transfer, respectively (e.g. Volmer [1939]; Nielsen 11964]; Lyubov
[1969,1975]; Christian [1975]; Doremus [1985]; Sohnel andGarside [1992];
Mullin [ 1993]), and Gc(mUvs~l) is obtained by kinetic considerations analogous
to those used in Sections 10.1 and 10.2 for finding the monomer attachment
and detachment frequencies /„ and gn. We note as well that when v = 1, Gc
is merely the time-independent growth rate of the crystallites (see Chapter
27).
Substituting G from (26.15) in (26.14) and performing the integration, we
find that in IN the extended volume is given by
Ve,V) = clNmG?tvd (26.17)
when the crystallites obey the growth law (26.16). From eqs (26.4) and
(26.9) it thus follows that in this case of overall crystallization we have (e.g.
Belenkii [1980])
a(t) = 1 - exp [- (r/t?)1*'] (26.18)
t„ = T(l + l/vri)t> (26.19)
where t? (s), defined by
&=(VlctNjlvd(\IGc), (26.20)
is the time constant of the process, and
T(x)= \ y"-'e-"dy (26.21)
Jo
is the gamma-function [Korn and Korn 1961], For 3D growth (d = 3) of
spheres (cg = 4^3) with time-independent growth rate (then v = 1) eq.
(26.18) passes into the formula of Kolmogorov [1937], We see that both the
time constant $ and the average time tav for crystallization shorten with
increasing the growth constant Gc and/or the concentration Nm/V of the
instantaneously nucleated crystallites. The a(t) function (26.18) is sigmoidally
shaped: this is illustrated by the solid curve IN in Fig. 26.2, which is drawn
according to eq. (26.18) with vd - 3. We note also that the usage of Jit) from
(26.13) in (26.11) and (26.12) returns the identities Wm = Nm and Vav = V/Nm
regardless of the Ve>(r) dependence, i.e. for whatever crystallite growth law.
Overall crystallization 379
0.8
0.6
0.4
0.2
PN j
An
:
/
/
,'i
/
/
/
; 5
/
/
0 12 3 4 5 6
t/fl
Fig. 26.2 Time dependence of the fraction of crystallized volume in overall
crystallization by the polynuclear mechanism: curve IN - instantaneous nucleation
according to eq. (26.18) with vd = 3; curve PN - stationary progressive nucleation
according to eq. (26.23) with vd = 3; dashed curves 1 and 5 - non-stationary
progressive nucleation according to eq. (26.30) with T/& = I and 5, respectively.
2. Progressive nucleation (PN)
Overall crystallization proceeds by PN when the crystallites are continuously
nucleated during the process. The simplest for analysis is the case when
nucleation is stationary, since then the nucleation rate is time-independent
and in (26.5) and (26.8) we have J(t) = JS1 the stationary nucleation rate /s
(nr3 s~') being given by (13.39). If in addition the crystallites grow according
to the power law (26.16), from (26.5), (26.7) and (26.15) it follows that in
this case of PN
VzM = [ceK 1 + vd)] G'/J/*'"'. (26.22)
Accordingly, using this result in (26.4), (26.9)-(26.12) yields (e.g. Belenkii
[1980])
a(t) = 1 -exp[- (//!?)' + ""]
/av = T[(2 + vd)l(\ + vd)]i
f"
7st>
Jo
N(t) = V7st> | exp (- xl + "") djr
Nm = 11(2 + vd)l(\ + vd)]Wst?°= V(]JGcf10 * vd)
vav = i/r[(2 + vd)i(\ + vd)]j,a °= (Gjjydm *"<"
(26.23)
(26.24)
(26.25)
(26.26)
(26.27)
380 Nucleation: Basic Theory with Applications
where the time constant t? is given by
t> = [(1 + vd)lceGcvdJs],n' + "*>. (26.28)
When the crystallites are spherical with radii growing linearly with time
(then cg = 47C/3, d = 3 and v= 1), eqs (26.23), (26.25) and (26.26) pass into
those obtained by Kolmogorov [1937], and eq. (26.27) parallels that given
by Volmer [1939]. According to (26.28) and (26.24), both t? and /av are
smaller (i.e. crystallization is accelerated) at higher nucleation and growth
rates of the crystallites. Also, eqs (26.26) and (26.27) are in line with the
well-known experimental fact that more and smaller (on average) crystallites
are the final product of the transformation process when nucleation is faster
and/or growth is slower. Unfortunately, although the a{t) and N(t) dependences
and the quantities /av, Nm and Vav are experimentally accessible, separately,
none of them can give information about Js, because Js is always multiplied
by Gc to some power. Physically, this merely reflects the fact that in overall
crystallization involving PN the processes of crystallite nucleation and growth
are concomitant. Yet, if we know simultaneously /av and Nm from independent
measurements, Js is readily obtainable with the aid of the formula
*s = NJVt„ (26.29)
which follows from (26.24) and (26.26). The solid curve PN in Fig. 26.2
shows that, as in the IN case, a from (26.23) is an S-shaped function of t (the
curve is drawn again at vd = 3 corresponding to growth of spheres with time-
independent rate). Understandably, despite rising later, the a(t) function for
PN saturates earlier than that for IN.
The above considerations can be extended to cover also the case of non-
stationary PN, but the mathematics becomes more challenging, because then
the nucleation rate J is a complicated function of time (see Section 15.2).
Attempts in this direction were made in a number of papers [Gutzow and
Kashchiev 1970,1971; Kashchiev 1989a; Shi and Seinfeld 1991 b; Shneidman
and Weinberg 1993; Weinberg et al. 1997]. For instance, again for spheres
with linearly growing radii, using in eq. (26.8) d= 3, v = 1,7(/) from (15.64)
and G(t) from (26.15) yields [Gutzow and Kashchiev 1970, 1971]
a(t) = 1 - exp [- (//t?)4y(//T)]. (26.30)
Here the time-dependent factor Y, a number between 0 and 1, is given by
Y(x) = 1 - 2x2Bx + ll^BOx2 - 31^/63¾3 + 1277r8/25200;t4
+ (48//) I [(- l)7is] exp (- fix) (26.31)
and has a similar form also for disk- and needle-like crystallites [Gutzow and
Kashchiev 1970]. This factor quantifies the effect of the nucleation time lag
x from (15.55) and (15.73) on the a(t) dependence. As seen, when x = 0, Y
s 1 and eq. (26.30) passes into eq. (26.23) (with vd = 3) which corresponds
to stationary PN. If T * 0, however, Y ** 0 for t« x and the whole process
Overall crystallization 381
of overall crystallization is delayed, the delay being largely determined by X,
i.e. by the nucleation non-stationarity. This delay is clearly seen in Fig. 26.2
in which the dashed curves 1 and 5 display the ait) dependence (26.30) at
T/t? = 1 and 5, respectively. Compared with the solid curve PN which
corresponds to stationary nucleation (then x = 0), the dashed curves preserve
nearly the same sigmoidal shape, but are significantly shifted towards longer
times, the shift being about 0.5X. The short-time asymptotics of the akt)
dependence (26.30) can be determined from eq. (26.6) with the help of J(t)
from (15.66) [Kashchiev 1989a]:
a(t) = 12 288(r/t?)V7tT)15'2 exp (- ith/4t). (26.32)
This formula is useful for describing the beginning of overall crystallization,
since it gives a from (26.30) with an error of less than 10% for t < 0.2T
provided XIt? < 103.
In the case of non-stationary PN the N(t) dependence and the quantities
/av, Nm and V„ are obtainable by employing a(t) from (26.30) in eqs (26.9)-
(26.12), but this has not been done so far. It is clear, however, that in this
case the N(t) curve will be shifted towards longer times like the a(t) one and
that rav will be longer than the average crystallization time corresponding to
stationary PN and given by (26.24). Since the time shift of the a(t) curve is
a fraction of x (see Fig. 26.2), tav can be approximated by the formula
rav = bpx + T[(2 + vd)l(i + vd)]ti (26.33)
which passes into (26.24) at T = 0, i.e. when nucleation is stationary. Although
the numerical factor 6p in this formula should depend on the xlt? ratio, this
dependence is relatively weak [Kashchiev 1989a] and in many cases of
practical interest the approximation bv = 0.4 to 0.6 might be sufficiently
accurate (see Chapter 29). It must be noted also that the accuracy of eq.
(26.30) depends on the accuracy of the J(t) function used in (26.8). In view
of the suggestion in Section 15.6, therefore, eqs (26.30), (26.32) and (26.33)
are likely to provide a better quantitative description if in them x is replaced
by 66/7C2, and the delay time 6 of nucleation is calculated from (15.111)
rather than from (15.105).
Summarizing, from eqs (26.18) and (26.23) we see that in the cases of
both IN and PN the a(r) dependence can be represented by the unified
KJMA formula
aS}) = 1 - exp [- (//t>)»] (26.34)
provided crystallite growth obeys the power law (26.16) and nucleation is
either progressive in stationary regime or instantaneous. The kinetic exponent
q > 0 is a number between 1 and 4: q = vd for IN and q - 1 + vd for PN. The
time constant t? is governed by the growth constant Gc and either by the
maximum number /Vm of crystallites in IN or by the stationary nucleation
rate 7S in PN (see eqs (26.20) and (26.28), respectively). Usually 7S is much
more sensitive to the experimental conditions than Gc and for that reason t?
and, thereby, overall crystallization as a whole is often controlled mainly by
382 Nucleation: Basic Theory with Application.*
the nucleation process, especially in PN. When PN is non-stationary, the a(J)
dependence is more complicated than that given above (see eq. (26.30)) and,
in addition to Js, the time lag x appears as one more nucleation parameter
controlling the overall crystallization process. As to the experimental
verification of eq. (26.34), it turns out (e.g. Christian [19751) that although
this equation can describe successfully available a(t) data, this is by far not
always the case. The latter is hardly surprising in view of the experimental
difficulties in realizing conditions corresponding to the theoretical requirements
for the validity of eq. (26.34). The straight lines in Fig. 26.3 illustrate the
applicability of this equation: the circles and the squares in this figure represent,
respectively, the a(t) data of Mikhnevich and Zaremba [ 1962] for overall
crystallization of piperine and betol melts. According to eq. (26.34), In [- In
(1 - a)] = q In t - q In tj so that represented in In [- In (1 - a)]-vs-ln t
coordinates the a(t) dependence is linear with slope q = 3 for IN and q-A
for PN if nucleation is stationary and the crystallites grow three dimensionally
(d = 3) at constant rate (v = 1). In concordance with theory, the slopes of the
best-fit straight lines in Fig, 26.3 are 3.07 ± 0.04 and 3.73 ±0.11 for piperine
and betol, respectively. Experiments with sufficiently viscous (e.g. glass-
forming and polymer) melts have shown that the Oa\i) dependence can be
affected considerably by the nucleation non-stationarity [Westman and Krishna-
Murthy 1962; Scott and Ramachandrarao 1977; Budurov el al. 1986; Spassov
and Budurov 1988; Dobreva et al. 1996],
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
In (t/min )
Fig. 26,3 Time dependence of the fraction of crystallized volume in overall
crystallization: circles - data for piperine melt at T = 345 K [Mikhnevich and
Zaremba 1962}; squares - data for betol melt at T = 283 K {Mikhnevich and
Zaremba 1962]; lines - best fit according to eq. (26.34).
Overall crystallization 383
26.3 Mononuclear mechanism
The KJMA theory is applicable to crystallization involving nucleation and
growth of statistically many crystallites (Fig. 26.1) so that Nm > 1. However,
in some cases it may be possible for the melt to crystallize completely as a
result of the appearance of only one crystalline supemucleus m it (Fig. 26.4).
Crystallization is then said to occur by the mononuclear mechanism. Inverting
the above inequality sign, we can represent the condition for the operativeness
of this mechanism as Nm < 1. When nucleation is stationary and crystallite
growth obeys the power law (26.16), the maximum number/Vm of crystallites
is given by eq. (26.26). Then, with the help of this equation (with omitted F
factor) and of eq. (26.28), it follows that the above condition takes the form
V< [caG?l(\ + vd)J^]w+vd). (26.35)
&
Fig. 26.4 Overall crystallization by the mononuclear mechanism corresponding to
appearance and growth of a single supemucleus (the circle) in the old phase.
This inequality tells us that the mononuclear mechanism is operative
when at given V and J the crystallite growth rate G is so high that after the
birth of the first supemucleus no other supernuclei have time to appear in the
melt until it is completely crystallized. The latter means that, mathematically,
the crystallite growth rate can be represented by the expression
G(t-t') = RmelMt-t'l
since Sp is equal to infinity at the moment t = t' of appearance of the crystallite
and vanishes at later times t > t' when the crystallite cannot grow any more,
because its radius is already equal to the effective radius /?me|t of the melt.
Thus, formally, employing G from the above expression in (26.8), assuming
without loss of generality that the crystallite and the melt have the same
shape (then V - cg /^,,) and using eq. (11.10) at £0 = 0, we find that
a(0 = 1 - exp [- V f J(f) df'] = 1 - exp [- VftOl- (26.36)
Jo
384 Nucleation: Basic Theory with Applications
As seen from this equation, crystallization by the mononuclear mechanism
is controlled solely by the nucleation rate J. Indeed, since growth is infinitely
fast, no time elapses from the very first nucleation event until the complete
crystallization of the melt and the growth rate G plays no role in the process.
The coincidence of the moments of formation of the first supernucleus and
of complete crystallization of the melt means that ait) from (26.36) is identical
with the probability P\{t) for appearance of at least one supernucleus until
time t. The experimentally accessible P\{t) dependence is, therefore, given
by the formula
P,(t) = 1 - exp [-V f J(t') dr"] = 1 - exp [- Vf(r)] (26.37)
Jo
which, as shown by probabilistic considerations [Bigg 1953; Skripov et al.
1970; Gutzow and Toschev 1970; Toschev e/a£ 1972;Toschev 1973a; Skripov
1977; Skripov and Koverda 1984J, is valid for any type of first-order phase
transformation.
The application of eqs (26.36) and (26.37) is again easy when nucleation
is stationary, since then J(t) = JS = constant. From eqs (26.9), (26.11), (26.12),
(26.36) and (26.37) it then follows that [Turnbull 1952; Bigg 1953; Skripov
et al. 1970; Gutzow and Toschev 1970; Toschev et al. 1972; Skripov 1972,
1977]
a(t) = P](() = 1 - exp (- JsVt) (26.38)
lav = 1, = 1/JSV (26.39)
and that, as it should be, Nm = 1 and Vav = V. Understandably, the lifetime tav
of the supersaturated old phase coincides with the mean time r, = tdPi(t)
Jo
for appearance of at least one supernucleus in it. It is worth noting as well
that the above formula for 1, is readily obtained from eq. (13.102) by setting
£" = 1/V (one supernucleus in the volume of the system) at t = t,. Also,
comparison of eq. (26.38) with the general KJMA formula (26.34) shows
that the a(() dependence in crystallization by the mononuclear mechanism is
describable by this formula with q = 1 and t?= 1/JSV. This q value corresponds
to d = 0, and the expression for r> follows formally from eq. (26.28) upon
setting d = 0 and cE = V.
In the case of non-stationary nucleation the analysis is more complicated.
Employing f(() from (15.114) in (26.36) and (26.37) leads to the relation
[Gutzow and Toschev 1970; Toschev et al. 1972]
ct(t) = P,(t) = 1 - exp [- J,V{t - (i?fo)t
- 2x Z [(- l)'/i2] exp (- (2(/f)}] (26.40)
which passes into (26.38) at T = 0. When T* 0, initially, a and Pt remain
vanishingly small because of the delayed appearance of the first supernucleus.
Overall crystallization 385
For that reason, in contrast to a and P, from (26.38) which initially are linear
with time, a and Pl from (26.40) are sigmoidally shaped functions of t. This
is illustrated in Fig. 26.5 in which the solid and dashed curves are drawn
according to eqs (26.38) and (26.40), respectively, with JsVr = 1 and 5 (as
indicated) in the latter. Experimentally, therefore, an S-shaped a(t) or P^r)
dependence may be a direct evidence for non-stationary nucleation [Toschev
et al. 1972], From Fig. 26.5 we see also that the time shift of the dashed
curves is approximately equal to the nucleation time lag T. To obtain the
short-time asymptotics of a and PL from (26.40) it is convenient to use £(t)
from(15.116) in (26.36) and (26.37) and approximate a and P| by VC,. Doing
that results in the formula
JsVt
Fig. 26.5 Time dependence of the fraction of crystallized volume in overall
crystallization by the mononuclear mechanism: solid curve - eq. (26.38) for stationary
nucleation; dashed curves 1 and 5 - eq. (26.40) for non-stationary nucleation at
JsVt = 1 and 5, respectively.
a(t) = P,(() = 8JsVr(tlK-c)m exp (- ;t2t/4t)
(26.41)
which, compared with (26.40), gives aandP, with an error of less than 15%
for/< 0.25Tprovided t< l(f/JsV.
The determination of rav or /, requires performing the integration in eq.
(26.9) with a or P, from (26.40). As the analytical solution of this problem
is not available, the explicit dependence of laT or t^ on Js and X remains
unknown. Similar to eq. (26.33), however, the relation [Gutzow and Toschev
1970]
-bmT+W,V
(26.42)
may be an acceptable approximation in many practical situations. The numerical
386 Nucleation: Basic Theory with Applications
factor bm is analogous to bp in eq. (26.33) and can be treated as a number
between 0.8 and 1.2, because it is expected to change relatively weakly with
the /,Vr product (see Fig. 26.5). Also, as already noted in Section 26.2, the
accuracy of eqs (26.40)-(26.42) is likely to be improved if in them x is
replaced by 69/ir2 with the nucleation delay time 8 given by (15.111) rather
than by (15.105).
Equations (26.36)-(26.42) find application in experiments on nucleation
in phases with a sufficiently small size (e.g. Budevski et al. [1966]; Skripov
el al. [1970]; Butorin and Skripov [1972]; Toschev et al. [1972]; Toschev
[1973]; Skripov [1972, 1977]; Skripov and Koverda [1984]; Nikolova et al.
[1989]; Kelton [1991]; Stoyanovaetal. [1994]; Baidakov [1995]). Undercooled
emulsion droplets are a typical example of such phases and their crystallization
is describable with the help of these equations (e.g. Turnbull [1952]; Bigg
[ 1953]; Turnbull and Cormia [ 1961 ]; Clausse [ 1985]; Kashchiev et al. [ 1994,
1998]). Experimentally, P, can be determined directly as the ratio between
the number of successful trials and the total number of trials (successful is a
trial in which nucleation has occurred till time ()■ The circles in Fig. 26.6
represent the />,(() data of Skripov [1977] for repeated (91 times) crystallization
at T = 383.4 K of a tin droplet with radius R = 35 fim. The experimentally
found value of t, is 35 s which, in view of (26.39), leads to JSV = 0.028 s_l
or Js = 1.6 x 10" nT3 s_1 at the temperature studied. The good description of
the experimental P\(t) dependence by the theoretical curve drawn according
to eq. (26.38) with the above J5V value is an evidence for crystallization by
the mononuclear mechanism under conditions of stationary nucleation.
0.8
0.6 -
nT
0.4 -
0.2 -
0' ■...!.... ' I
0 50 100 150 200
t(S)
Fig. 26.6 Time dependence of the probability for appearance of at least one supernucleus:
circles - data for repeated crystallization atT- 383.4 K of a tin droplet [Skripov 1977j;
curve - eq. (26.38) for stationary nucleation.
Overall crystallization 387
26.4 Two-stage crystallization
In crystallization, often the newly formed phase may not be the thermo-
dynamically most stable one when it is possible also for a metastable phase
to appear in the supersaturated old phase. This led Ostwald [1896, 1897] to
formulate the so-called Rule of Stages. According to this rule, 'if the
supersaturated state has been spontaneously removed then, instead of a solid
phase which under the given conditions is thermodynamic ally stable, a less
stable phase will be formed' [Ostwald 1897], For instance, water droplets
rather than ice can be nucleated in a supersaturated steam at temperatures
below freezing (e.g. Viisanen et al. [1993]) and precipitation of metastable
vaterite or aragonite, and not of stable calcite crystals is what can be observed
in supersaturated aqueous solutions of calcium carbonate (e.g. Sohnel and
Garside [1992]). A recent study [Sazaki et al. 1996] of the solubility of hen
egg white lysozyme indicates that metastable phases could be formed also in
protein crystallization from solutions. Formation of metastable phases is
common in the crystallization of, e.g. diamond films [Kamo et al. 1983],
metals [Tumbull 1981; Rasmussen 1982b], quasi-crystals [Shechtman et al.
1984; Kelton 1991], ice [Takahashi 1982], fatty acids [Sato 1988, 1993],
amino acids [Kitamura er a/. 1998], triglycerols [Hagemann 1988; Sato 1988,
1993] and polymers [Keller et al. 1994; Hikosaka et al. 1994, 1995].
When the conditions allow the appearance only of a metastable phase in
the supersaturated old phase, the thermodynamic accomplishment of the
overall crystallization process requires further transformation of the metastable
phase into the stable crystalline phase, for instance, by nucleation and growth
of the thermodynamically stable crystallites in the metastable rather than in
the old phase. This means that when direct formation of the stable crystalline
phase in the old phase is impossible, overall crystallization becomes a two-
stage process: the formation of the stable phase (second stage) is preceded
by the appearance of the metastable one (first stage). This two-stage process
is schematized in Fig. 26.7 in which the stable phase (the dark grey circles)
is seen to nucleate and grow at rates J& and Gs in a metastable phase (the light
grey circles) whose rates of nucleation and growth in the old phase are 7m
and Gm, respectively.
Clearly, the birth and growth of the stable phase within the metastable one
makes the kinetics of formation of the former depend on those of the latter.
Under conditions of validity of the KJMA formula (26.34), for the volume
fraction c^ of metastable phase crystallized till time t we can write
GUO = Vm(t)/V = 1 - exp [- (r/i?mf] (26.43)
where m and $m are the kinetic exponent and the time constant characterizing
the metastable phase. However, since now the stable crystalline phase is
formed in the variable volume Vm of the metastable phase, in order to determine
the volume Vs of the stable phase we cannot apply eq. (26.43) by merely
replacing Vm and V by Vs and Vm, respectively, and m and #m by the kinetic
exponent s and the time constant 1¾ of formation of the stable phase in the
388 Nuclcation: Basic Theory with Applications
Fig. 26.7 Two-stage overall crystallization: the formation of the stable crystalline
phase (the dark grey circles) is preceded by the appearance of metastable phase (the
light grey circles).
metastable one. One way to relate Vs to Vm is to use an equation analogous
to (26.43), but in a differential form, i.e. for dVs and dVm rather than for Vs
and Vm. Indeed, if we consider the small volume dVm of metastable phase
formed between /' and t' + d/', at a later time t > /' a certain part of the
metastable phase in this volume will be transformed into stable phase so that
dVm will contain in itself the small volume dVs of stable phase. Treating f as
initial moment of the crystallization process in the volume d Vm, analogously
to (26.43) we shall have
dVs = (1 - exp {- [(/ - /W)) dVm. (26.44)
This relation parallels that used by Kashchiev [1977] to couple the coverages
of the successive monolayers in polylayer growth of thin solid films (see
Chapter 34). We note as well that a formula similar to (26.44) was employed
by Vetter [1967] for describing the nucleation-mediated growth of crystals.
Integrating the r.h.s. of (26.44) from /' = 0 to f = t and its l.h.s. from Vs =
Vs(0) to Vs = Vs(t), accounting that Vs(0) = Vm(0) = 0 and dividing by Vthus
leads to [Kashchiev and Sato 1998]
ajf) = ajf) - f exp {- [(t - n/itfHdtUO/cin dC (26.45)
Jo
where a^l) = Vs(r)/V, and ajf) is given by (26.43).
Equation (26.45) describes the time dependence of the volume fraction c^
of a stable crystalline phase appearing in a metastable phase which itself is
in the course of its formation in a supersaturated parent phase. This equation
reveals that c^ depends not only on the kinetics of appearance of the stable
Overall crystallization 389
phase itself (i.e. on s and t?s), but also on the rate dajdt of formation of the
metastable phase. For KJMA kinetics involving stationary nucleation and
3D growth (d = 3) of crystallites with time-independent growth rate (v = 1)
we have m - s = 4 and, according to (26.28),
i>m = (4/ce.n,G^nl7s,m)"4 (26.46)
.>„ = (4/^0^)^, (26.47)
the subscripts m and s indicating that the shape factors cg m and cg s, the time-
independent rates Gcm and Gcs of growth and the stationary rates Jiw and Jss
of nucleation are referred to the metastable and the stable phase, respectively.
Although the numerical calculation of the as(t) dependence (26.45) is
straightforward when m, s and the t?m/t?s ratio are known, it is desirable to
have this dependence in explicit analytical form. As shown by Kashchiev
and Sato [1998], for ;/t?m > b', approximately,
0.(0 = 0,,,(0(1 - exp (- WW(tftU - *']'}) (26.48)
where b' s(m- l/m)"m is a number, and 0^(/) is specified by (26.43). For
0 < j/t?m < b', cc,(t) = 0 can be a sufficiently accurate approximation.
Curve «> in Fig. 26.8 displays the time dependence of a„, from eq. (26.43)
at m = 4. The a^t) function (26.45) is also sigmoidally shaped. This is
illustrated by the solid curves in Fig. 26.8, which represent the exact as(t)
dependence calculated numerically from eq. (26.45) at m - s = 4 and (t?m/
t?,)4 = 0.1, 1, 10 and <*■ (as indicated). We observe that the time at which (¾
starts departing appreciably from zero is controlled by the value of the ratio
Fig. 26.8 Time dependence of the volume fraction of the stable crystalline phase in
two-stage overall crystallization; solid curves - eq. (26.45) at (&„/&s)4 = 0.1, 1, 10
and °= (as indicated) and m = s = 4; dashed curves - the corresponding approximate
eq. (26.48).
390 Nucleation: Basic Theory with Applications
i?m/i?s between the time constants i?m and 1¾ for formation of the metastable
and stable phases. Namely, small i?m/^s values (i.e. large i?s because of slow
kinetics of stable-phase formation) bring about a considerable delay in the
appearance of the stable crystalline phase. This delay can be so long that the
first portions of this phase may appear only after the formation of the metastable
phase is completed (compare the positions of the 'end' of curve 00 and of the
'beginning' of curve 0.1 in Fig. 26.8). Conversely, for large i?m/i?s values
(i.e. fast stable-phase formation and, hence, small time constant i?s) c^{t) is
close to ajit) and in the &„/&* = °° limit we have as(t) = 0^,(/)- That is why
curve 00 in Fig. 26.8 represents both the 0^(/) function (26.45) at i?m/i?s = *»
and the 0^(0 function (26.43). The dashed curves in this figure illustrate the
approximation (26.48) at m = s = 4 and the indicated values of t?m/$s-
The Ojt?) curves in Fig. 26.8 reveal that the main parameter to control the
appearance of the stable crystalline phase is the i?ra/i?s ratio. This leads to the
conclusion that the general condition for formation of a long-living metastable
phase because of a delayed appearance of the stable phase in it is of the form
[Kashchiev and Sato 1998]
iVi?m> 1. (26.49)
In the concrete case of t?m and i?s specified by (26.46) and (26.47) this
inequality becomes
(^,mGc3m7s,m/cg,sGc3sJs,s)1/4 > 1. (26.50)
This expression is instructive, for it demonstrates explicitly that slower
kinetics of nucleation and growth of the stable phase inside the metastable
one (i.e. Js<s and/or Gcs sufficiently smaller than Jsm and/or Gc m, respectively)
are the reason for a protracted two-stage overall crystallization. A role in this
protraction can also be played by the shape of the growing crystallites,
which is taken into account by the factors cgs and c$m. As shown elsewhere
[Kashchiev and Sato 1998], using (26.49) allows finding the critical temperature
below which the metastable crystalline phase formed in melt crystallization
has a sufficiently long lifetime to be experimentally detectable.
Chapter 27
Crystal growth
Historically, the nucleation theory has found one of its first applications in
the field of crystal growth [ Volmer and Marder 1931; Kaischew and Stranski
1934b]. The rate of crystal growth is important both as a quantity quantifying
the growth of a given crystal face and as a parameter controlling the overall
process of formation of crystalline phases (see Chapter 26). The growth of a
crystal face depends not only on the mechanism of mass and heat transfer to,
from or across the crystal/ambient phase interface, but also on the nanoscopic
structure of this interface. In relation to crystal growth, there are three basic
types of nanostructure of the crystal surfaces and they materialize into three
distinct modes of crystal growth. First, a crystal face can be molecularly (or
atomically) rough and growth is then said to be continuous (or normal or
liquid-like) [Hertz 1882; Wilson 1900; Knudsen 1909; Frenkel 1932]. A
second possibility is the crystal face to be molecularly smooth - growth is
then nucleation-mediated [Volmer and Marder 1931; Kaischew and Stranski
1934b]. And, finally, in the presence of points of emergence of screw
dislocations the crystal face is stepped and exhibits spiral (or screw-dislocation)
growth [Frank 1949; Burton et al. 1951], Though nucleation theory is involved
in the description solely of nucleation-mediated and spiral growth, for
completeness, in this chapter we shall consider briefly not only them, but
also the continuous growth of crystals. More on the subject of crystal growth
can be found elsewhere (e.g. Volmer [1939]; Hirth and Pound [1963]; Nielsen
[1964]; Vetter [1967]; Brice [1973]; Bennema and Gilmer [1973]; Christian
[1975]; Weeks and Gilmer [1979]; Chernov [1980]; Doremus [1985]; Mullin
[1993]; van der Eerden [1993]; Sangwal [1994]; Markov [1995]; Gutzow
and Schmelzer [1995]). A vivid account of the developments of basic ideas
in the theory of crystal growth was given by Kaischew [1981],
27.1 Continuous growth
At a constant supersaturation A,u, the time-independent growth rate Gc (m/s)
of a given crystal face with a fixed area Af is defined as the velocity at which
the face advances along its normal. Hence, if we know the average time rav
of filling of a monolayer on the crystal surface by molecules arriving from
the ambient phase, we can determine Gc from the definition equality
Gc = 4/<„v (27.1)
where dG is the molecular diameter.
392 Nucleatittn: Basic Theory with Applications
In continuous growth the crystal surface is molecularly rough and preserves
its structure during the process (Fig. 27.1). For that reason every molecular
Fig. 27.1 Cross-section of a molecularly rough crystal face which advances by
continuous growth.
site on the crystal surface can be regarded as a growth site, i.e. a site at which
an arriving molecule can be incorporated into the crystal. Moreover, to a
certain approximation, it is possible to treat the growth sites as equivalent
with respect to the attachment and detachment of molecules to and from
them. This means that, on average, the time taken by a molecule to occupy
a growth site is the same for all sites on the surface and thus identical with
tm. Noting that the net flux of molecules to a given site is^ - gs, for fav we
shall, therefore, have /av = 1 /(£ - gs) where/s (s_1) and gs (s_1) are, respectively,
the frequencies of molecular attachment and detachment per site. These
frequencies depend on the concrete kinetics of attachment and detachment
and can be determined with the help of the relations
/s = flo/M„ (27.2)
gs = «oSMc,„. (27.3)
Here/(s~l) and g (s_1) are, respectively, the frequencies of monomer attachment
and detachment to and from an n-sized crystal (see Sections 10.1 and 10.2),
AC(1 is the area of the crystal surface in contact with the old phase, and Ac J
flo is the number of growth sites on this surface (the area of a growth site is
considered equal to the molecular area a0)- Thus, using the above expressions
for (av,/, and gs in eq. (27.1) leads to the formula
Gc = <«/s - gs) = 4/5(l - gift (27.4)
To proceed further we recall the general relation (10.90) between/and g,
in which now dWfdn = — A^/, because for a macroscopic ally large crystal the
effective excess energy <P is negligible with respect to the A^/ term in (10.86).
Hence, eq. (27.4) becomes
G^dofsd-e-*'*7) (27.5)
which is a general presentation of the formula given by Hertz [1882] and
Knudsen 11909] for growth from vapours and by Wilson [ 1900] and Frenkel
Crystal growth 393
[1932] for growth from the melt. Equations (27.2) and (27.5) show that at a
given supersaturation Aft, Gc is time-independent when/is proportional to
Acll (then/s does not depend on the crystal size and, thereby, on time). This
is possible in the cases of, e.g. direct impingement and interface transfer (see
eqs (10.3), (10.53) and (10.55)) provided /„ has a constant value which we
shall denote by % in order to indicate that it refers to sticking to a growth site.
It should be noted also that with ys = 1, Gc from (27.5) is the maximum rate
at which a crystal face can grow at a given supersaturation.
To find Gt as an explicit function of A^i, we must take into account the
difference in the /s(A/i) dependence when A/j is varied isothermally and
when this is done by means of T according to eq. (2.20). In the former case,
combining eqs (10.109), (27.2) and (27.5), we get
Gc = drj^^a - e-^lkT) (27.6)
where/es = a<JJAcll is the value of/s at A/( = 0, i.e. at phase equilibrium.
In the concrete case of growth from vapours A/( is given by eq. (2.8) or
(2.9). Then, for monomer attachment controlled by direct impingement, from
eqs (10.3) (used with/; = />e, % = /s, cv™nm = AC„), (13.68) and (27.6) it
follows that [Hertz 1882; Knudsen 1909]
Gc = %vMS-\) (27.7)
where /e = pJ(27an0kT)u2, and v0 = a0d0 is the molecular volume.
Similarly, for growth from solutions which is controlled by interface transfer,
from (2.14), (10.53), (10.62) (used with C= Ce), (13.69) and (27.6) we get
Gc = (rJd0)DKcCc(S-\). (27.8)
Thanks to the adsorption constant Kc, this equation takes into account that
the concentration of solute molecules in the first adsorption monolayer on
the crystal face may be different from their concentration in the bulk of the
solution. When Kc = u0, such a difference does not exist (see Section 10.1)
and (27.8) simplifies to
Gc = %aoDCc(.S -1). (27.9)
This formula could be derived with the help of/from eq. (10.60), since this
equation is valid also at Kc = v0. The appearance of D in eqs (27.8) and
(27.9) is due to the assumption behind eq. (10.53) that the activation energy
for interface transfer can be approximated by that for volume diffusion. In
reality D in (27.8) and (27.9) is an effective diffusion coefficient accounting
for the molecular motion in the immediate vicinity of the growth sites.
Equations (27.6)-(27.9) show that Gc is an increasing function of A^i or 5.
It should be noted that they are physically relevant also for Ap < 0 or S < 1.
Then Gc is negative and is the rate of evaporation or dissolution of the
crystal face. Naturally, eqs (27.5)-(27.9) are directly applicable to growth
and evaporation (or dissolution) of liquids: the liquid surface is always
molecularly rough and for it the requirements for the validity of these equations
are satisfied to the maximum. This, namely, is the reason for which the
394 Nucleation: Basic Theory with Applications
continuous growth of crystals is also called liquid-like growth.
Turning now to the case of A/i controlled by T in accordance with (2.20),
for/in (27.2) we can employ eq. (10.55) in combination with (10.58) at crs
= a, because the crystal face is the own substrate. Setting y„ = % and d\ =
fl0, from (27.2) and (27.5) we thus find that for crystal growth from the melt
under interface-transfer control
Gc = (%kTBxa0n) exp [(- X + E, + oaoWe4"""^! - e"4"'*7). (27.10)
Here the energy term Et + aa0 is approximately equal to the desorption
energy of a molecule from the crystal face and accounts for the possible
difference in the concentration of molecules in the bulk of the melt and in the
monolayer in contact with the face (see Section 10.1). The absence of such
a difference is characterized by E\ + oa0 = 0. Then (27.10) takes the following
simpler form corresponding to/from (10.64):
Gc = (7s/t7737ta„rj)e14"~mT(\ - e-^lkT). (27.11)
As seen from eqs (27.10) and (27.11), in growth from the melt Gc is a
function of Afj not only directly, but also implicitly via T and the melt
viscosity r/. Without the factor eVil'~x,lkT, eq. (27.11) is essentially the formula
of Wilson [1900] and Frenkel [1932]. In the simplest case of A^i given by
(2.23), eq. (27.11) becomes
Gc = (7st773TO0Tj) exp (- AsJk)U - exp (- Asc&T/kT)]. (27.12)
This GC(T) dependence parallels that of Gilmer [1993] and differs from the
Wilson-Frenkel GC(T) dependence by the 7-independent numerical factor
exp (-Asefk) 0.007 to 0.4 (see Section 10.1). In agreement with experiment,
eqs (27.10)-(27.12) predict a maximum of Gc with respect to T. Also, they
show that Gc is again negative when A^i < 0: then T > Te and, physically, Gc
is the melting rate of the crystal face.
Looking back at eqs (27.6) and (27.10) and recalling that e" = 1 + x for x
—> 0 [Korn and Korn 1961 ], we infer that in the range of small supeTsaturations
(A^i < 0.2kT) Gc depends linearly on A^i according to
Gc = KsAfi (27.13)
where Kg is a kinetic factor characteristic for continuous growth. This factor
can be considered as A,u-independent in growth from melts, because in the
range of small supersaturations r\(T) and exp [(E( + oa0)lkT\ are practically
T-independent, and (A/j - A)lkT ~ - Asjk. The linear Gc(A^i) dependence
(27.13) is a known experimental criterion for continuous growth. Curve CG
in Fig. 27.2 displays the Gc(A^i) function (27.6) and visualizes the linearity
between Gc and A/j for small enough A^i values.
The linear dependence of Gc on A^i in continuous growth at low
supersaturations is exemplified in Fig, 27.3 in which the squares and circles
represent experimental Gc(A^i) data [Georgieva and Nenow 1967; Nenow
and Georgieva 1968] for growth from vapours of concave faces of diphenyl
crystals at 7= 332 and 339.7 K, respectively. Being curved, the crystal faces
Crystal growth 395
CD
0.4 -
0.2
-
/CG
/ D.00D6
/ 0.0004
0.0002
MN . .]
:1
// NG
PN,-y
0.2 0.3 0.4 /
SG//NG
0.5
1.0
1.5
Au/kT
Fig. 27.2 Supersaturation dependence of the rate of crystal growth: curve CG - eq
(27.6) for continuous growth; curve SG - eq. (27.48) for spiral growth; curves NG -
eq. (27.42) for nucleation-mediated growth; curves PN and MN - eqs (27.40) and
(27.41) for polynuclear and mononuclear growth, respectively.
0.08
0.10 0.15
1 / R (unY1)
0.20
0.25
Fig. 27.3 Crystal growth rate as a function of the curvature of the crystal face in
continuous growth: squares and circles - data for growth of diphenyl from vapours at
T = 332 and 339.7 K, respectively [Georgieva and Nenow 1967; Nenow and
Georgieva 1968]: lines - best fit according to eq. (27.13).
396 Nucleation: Basic Theory with Applications
are molecularly rough and grow by the normal-growth mechanism. According
to eq. (2.8) in which the role of pe is now played by pe„ from (6.25), the
reciprocal of the radius R of curvature of the concave face is a measure of the
supersaturation Ap. Hence, in this case eq. (27.13) predicts proportionality
of Gc to VR. The straight lines in Fig. 27.3 demonstrate this proportionality
and the agreement between theory and experiment.
27.2 Nucleation-mediated growth
In the preceding section we have seen that no nucleation parameters are
needed for the theoretical description of continuous growth, the reason being
that growth sites are always available on a molecularly rough crystal face.
This is not the case, however, when the face is molecularly smooth: as
realized already by Gibbs [1928], growth can then proceed only after the
face roughens by nucleation of 2D clusters with edges having enough growth
sites on them (Fig. 27.4). Spreading of the 2D supernucleus clusters along
the crystal surface leads to the filling of the subsequent crystalline monolayers
and, thereby, to the growth of the crystal face. In analysing growth mediated
by 2D nucleation, it is convenient to distinguish between the cases of
mononuclear monolayer growth (monolayer filling by one supernucleus only,
Fig. 27.4a), polynuclear monolayer growth (monolayer filling by many
supernuclei, Fig. 27.4b) and polynuclear polylayer growth (simultaneous
filling of several successive monolayers by many supernuclei. Fig. 27.4c).
Growth in the first two cases is known also as layer-by-layer growth, and in
the last case it is simply referred to as polylayer growth.
r
O) (»> tc)
Fig. 27A Cross-section of a molecularly smooth crystal face which advances (a) by
mononuclear monolayer growth, (b) hy polynuclear monolayer growth, and (c) by
polynuclear polylayer growth.
1. Mononuclear monolayer growth
This case (Fig. 27.4a) was considered by Volmer and Marder [1931] and by
Kaischew and Stranski [1934b] as historically the first application of nucleation
theory to crystal growth. Since the filling of a crystalline monolayer by
nucleation and lateral growth of 2D clusters is in fact a process of overall
crystallization in two dimensions, for the average time of monolayer filling
r r
I—I—*-
Crystal growth 397
by a single supernucleus we can employ eq. (26.39) with V replaced by Af.
Combining (26.39) and (27.1) thus leads to [Volmer and Marder 1931;
Kaischew and Stranski 1934b]
Gc = d0AfJs. (27.14)
This general formula implies time-independent nucleation. In it Js (rrf2 s-i)
is the rate of stationary 2D HEN of clusters of monolayer thickness on their
own substrate, and Af is the area of the crystal face. Without loss of generality,
when A/i is varied isothermally, eq. (27.14) can be used in the form
Gc = d0CQArZf*e^/kTe~m,kT (27.15)
which is obtained with account of eqs (13.39)-(13.42). Here W*(A^i) is the
nucleation work for 2D HEN on own substrate (see Sections 4.3 and 4.4), C0
(m~2) is given by (7,8) or (7.9), and the pre-exponential factor d0C0Afzf*
is practically Aju-independent. With /* replaced by /* (cf. eqs (13.42) and
(13.43)), eq. (27.15) is applicable also to crystal growth from the melt when
Afi is controlled by T according to (2.20).
Analogously to the mononuclear mechanism of overall crystallization,
mononuclear monolayer growth of crystals is operative when the area Af of
the crystal face is sufficiently small. As follows from (26.35) at d - 2, the
condition for its operativeness in the case of stationary nucleation of 2D
clusters reads
At<[c'Kv2vf{\ +2v)Js2v]l/tH2l,) (27.16)
provided the cluster radius R grows according to the power law (26.16)
represented as
R{t) = {vst)v. (27.17)
Here c'g is a numerical shape factor (c'g = % for circles, c'g = 4 for squares,
etc.), and us (m,/v s"1) is the growth constant of the clusters. For linear
growth law (v = 1) us (m/s) is merely the time-independent rate of cluster
lateral growth, i.e. the velocity of advancement of the monolayer steps bordering
the 2D clusters.
2. Polynuclear monolayer growth
Given the area of the crystal face, this kind of nucleation-mediated growth
(Fig. 27.4b) has a realization when the condition (27.16) is not fulfilled, i.e.
when the rates of nucleation and growth of the 2D clusters on the crystal
surface are sufficiently high and low, respectively. The growth rate of the
crystal face is easy to find again in the case of clusters nucleated in stationary
regime and growing according to the power law (27.17). Then the average
time of monolayer filling is given by eqs (26.24) and (26.28) with d-2 and
with cg and Gc replaced by c'M and vs, respectively. Hence, from (27.1) it
follows that
Gc = {(1 + 2v)m +lv'T[(2 + 2v)/(l + 2v)]f'^0(^¾)1^2^. (27.18)
398 Nucleation: Basic Theory with Applications
In the particular case of v= 1 (cluster radius growing linearly with time) this
expression leads to the formula
Gc =0.7&d0(c'gv;Js)m (27.19)
which was derived by Todes [ 1949b], Nielsen [1964] and Hillig [1966] with
a slightly different numerical factor.
3. Polynuclear poly layer growth
A more realistic description of the growth of a crystal face by nucleation and
spreading of many 2D clusters requires allowing for the possibility of such
clusters to appear and grow on top of the already nucleated clusters of the
preceding monolayers. Then a number of successive monolayers are filled
simultaneously (Fig. 27.4c) and polynuclear polylayer growth takes place.
Since simultaneous filling of the monolayers is more efficient than their
filling one after another, in the case of polynuclear polylayer growth Gc is
expected to have a value greater than that predicted by eq. (27.18). A number
of analyses [Nielsen 1964; Vetter 1967; Borovinskii and Tsindergozen 1968;
Armstrong and Harrison 1969; Kashchiev 1977; van Leeuwen and van der
Eerden 1977; Belenkii and Lyubitov 1978; Belenkii 1980; Gilmer 1980;
Obretenov et al. 1986] shows that this is indeed so, but not at the expense of
a changed functional dependence of Gc on vs and Js. In other words, in
polylayer growth Gc is also of the form
Gc = wvd0(c'gv}vJs)m+2v) (27.20)
provided in all monolayers the 2D clusters nucleate at the same stationary
rate Js and grow laterally according to the same growth law (27.17). In this
formula the numerical factor \ffv is close to unity and its value depends on the
approximation used for the determination of Gc. For example, again in the
case of clusters growing linearly with time (v = 1), the most reliable value of
Vi is thought to be 0.97 [Gilmer 1980; Obretenov et al. 1986]. Then eq.
(27.20) leads to the expression
Gt = 0.97 d0(c'gui J s)yi (27.21)
which, as seen, corrects only numerically eq. (27.19). Comparing eqs (27.18)
and (27.20) with eq. (27.14), we see also that polynuclear (either monolayer
or polylayer) growth differs from mononuclear growth (i) in the independence
of Gc of the area Af of the crystal face, (ii) in the weaker impact of Js on Gc,
and (iii) in the dependence of Gc on u&.
Equations (27.14) and (27.20) describe crystal growth in the limiting
cases of birth and spread of only one 2D cluster or of statistically many 2D
clusters, respectively. As shown by Obretenov et al. [1989], a more general
formula for Gc, which is valid for any number of clusters on the crystal face,
can be obtained by expressing 1/GC as a sum of the reciprocals of the right-
hand sides of (27.14) and (27.20). Doing that yields
Ge =d0A{Js{l + [A./yf^'^^JJv,)2^2^. (27.22)
This interpolation formula provides a unified description of the rate of
Crystal growth 399
nucleation-mediated growth, since when the condition (27.16) is or is not
satisfied, it passes into eq. (27.14) or (27.20), respectively. In the particular
case of v= I eq. (27.22) takes the form given by Obretenov et al. [1989].
We may now ask again the question: what is the dependence of Gc on the
supersaturation Ap in nucleation-mediated crystal growth? According to eqs
(27.14),(27.20) and (27.22), we must know the Js(A^i) and us(A^/) functions
in order to find out the Gc(Ay) dependence itself. For the JK(An) function we
can use the results in Sections 13.2 and 13.3 for 2D HEN on own substrate
(then Ag= 0 and os = 6). As to the vs(Afj) function, it can be different for the
different mechanisms of attachment of molecules to the steps bordering the
monolayer clusters and for the different molecular structure of these steps. In
the often encountered case of molecularly rough steps every molecular site
at the step is a growth site and when the cluster radius R grows linearly with
time (then v = 1), us (m/s) is a 2D analogue of Gc in the case of continuous
growth. Hence, just like Gc from (27.5), vs is given by
vs = d„fs{\ - e-i»'kT) (27.23)
where now/s (s~') is the /{-independent frequency of monomer attachment
to a growth site at the monolayer step bordering the cluster. This means that
/s can again be obtained from (27.2), but with Ac„ = d„in"2 where ftn"2 is the
length of the cluster periphery {b-2(na^)]a for circles, b = Aa^1 for squares,
etc.). Hence, in order for/s to be independent of the cluster size,/in (27.2)
has to be proportional to n"2. This is so when monomer attachment is controlled,
e.g. by direct impingement or by surface diffusion (but only for R > X,) in
growth from vapours and by interface transfer in growth from solutions or
melts. Let us find us in these cases.
In the case of direct impingement, with the aid of eqs (2.8), (2.9), (10.6),
(10.7), (13.68), (27.2) and (27.23) and of the approximations dl = a0 and %
= 7S, we get the expression
"s = 4/e.,'*"'rU - e-<*<kT) = 7suMS - 1) (27.24)
whose r.h.s. coincides with that of eq. (27.7).
Similarly, for monomer attachment by surface diffusion towards circular
clusters of radius R > ^, combining (2.9), (10.31), (10.35), (10.41), (13.68),
(27.2) and (27.23) and taking into account that Ac„ = Ind^R yields
"s = 4/.^^( 1 - e-^lkT) = 2/,fl0Ve(S - 1) (27.25)
where/es = 2ysd0kjt is the doubled value of/s from (27.2) at A^i = 0 (the
factor 2 is included to allow for the contribution of the molecules diffusing
on top of the cluster towards its periphery). This is the known formula of
Burton et al. [1951] for the spreading velocity of an isolated straight step of
monolayer thickness. As seen, us from (27.25) is greater than vs from (27.24)
when As > rf0. Since the mean diffusion distance As usually satisfies this
inequality in growth from vapours, in this case the step propagation is practically
always governed by surface diffusion.
The us(A^) dependence retains the above simple form also for growth
400 Nttcleation: Basic Theory with Applications
from solutions under interface-transfer control. From eqs (2.14), (10.63),
(13.69), (27.2) and (27.23) it then follows that
vs = 4/.,s^"r(l - e-*"*7) = (%/d0)DKcCc(S - 1) (27.26)
where/es = (yJa0)DKcCe is again the value of/s from (27.2) at Afj = 0. As
already emphasized in Sections 10.1 and 27.1, here D is an effective diffusion
coefficient characterizing the random motion of the molecules in the
neighbourhood of the steps. Without the % factor, eq. (27.26) was used by
Nielsen [1964] in a pioneering analysis of polynuclear growth of crystals.
Understandably, its r.h.s. is the same as that of eq. (27.8).
Again under interface-transfer control, but in the case of growth from the
melt,/in (27.2) is given by eq. (10.66) with as = a. The us(A/<) function is
now complicated because of the relation (2.20) between Aft and T and of the
dependence of/s on the melt viscosity 7). Using eqs (10.66), (27.2), (27.23)
and the approximation v0 = aod0 results in (cf. eq. (27.10))
v„ = (y,kTI37m0rj) exp [(- X + E, + aa0)/kTie^'kT(l - e-&",tT). (27.27)
When the &^i(T) dependence has the simplest form (2.23), this expression
becomes
vs = (%kTI37m0ri) exp (-As,/*)
exp [(Ei + OB0)/*r|[i - exp (-AscATIkT)\. (27.28)
These two equations show that vs, like Gc from (27.10M27.12), has a maximum
with respect to Afj or T. This is in contrast to us from eqs (27.24)-(27.26),
which increases monotonously with Afi or S because of the isothermal variation
of the supersaturation.
We are now in position to determine the Ajj dependence of the rate Gc of
nucleation-mediated growth of crystals. Let us again consider growth first
from vapours and then from solutions and melts. In the former case surface
diffusion is the controlling transport mechanism and from eqs (2.8), (2.9),
(4.36), (10.42), (13.68) and (27.15) it follows that, classically,
G^doCo^ztfeWTe-BH*
= y*c*d0C0A,z^lcStxp(-B'HnS) (27.29)
for mononuclear growth, where C0, z, B and B' are given by eqs (7.8) or
(7.9), (13.37) (at Act = 0), (13.55) and (13.84). For polynuclear growth,
however, the Gc(A[i) dependence remains unknown, since us from (27.25)
cannot be used in eq. (27.21). This is so because at the beginning of their
growth the clusters are small and do not satisfy the condition R » As for the
validity of (27.25). Actually, due to the complicated dependence of/on R
(see eq. (10.33)), the cluster radius R is not a power function of time and can
only approximately be represented in the form of eq. (27.17) with v = 0.75
to 0.80 [Kashchiev 1978, 1981], At the advanced stages of cluster growth R
may already be greater than As, but then the linear growth can be distorted by
the overlapping of the surface-diffusion zones around the clusters. As evidenced
Crystal growth 401
by the analysis of Belenkii [1980], all this makes very hard to treat analytically
the problem of monolayer filling under surface-diffusion control when the
process occurs in the so-called regime of incomplete condensation
corresponding to appreciable desorption of molecules from the crystal face.
Let us now determine Gc for growth from solutions under interface-transfer
control. Combining eqs (2.14), (4.36), (10.63), (13.69), (27.15), (27.21) and
(27.26) and assuming that y* = %, we get
Gc = d0C0Afzf*e^kre-B""'
= /,(Wa0)C0A(DX'cCezn*"2S exp (-fi'/ln S) (27.30)
for mononuclear growth and
= (k, yJd0)DKcCcSu:,(S - 1 )2'3 exp (-B73 In 5) (27.31)
for polynuclear growth where
i, = Vl(c'ta0C0zfc*/Us)ui = Vi (^MoCoZn*"2)"3. (27.32)
Here V, = 0.97, and C0,z,n*,B and B' are given by eqs (7.8) or (7.9), (13.37)
(at Ao" = 0), (4.35), (13.55) and (13.84). Thanks to the weak dependence of
the Zeldovich factor z and the nucleus size n* on the supersaturation, the
numerical factor k] is nearly independent ofA/u or S. Also, it is close to unity
for crystal faces which are free of nucleation-active centres, as then C0 =
l/a0.
Finally, we consider the case of crystal growth from the melt when A^ is
determined by T according to eq. (2.20) and cluster nucleation and growth is
controlled by interface transfer. Again in the scope of the classical theory of
nucleation, with the aid of eqs (4.36), (10.66), (13.39), (13.40), (27.14),
(27.21) and (27.27) and the approximation y* = ys we obtain
Gc = y*bC0Afzn*"2(.kmm0ri)
exp [(- A + £, + oaoV/tric4""^ -""* (27.33)
for mononuclear growth and
Gc = (t|/sir/3fffl0i)) exp [(- X + £, + ua0)/kT\
eA*//fc7V| _ e-&u/kiy/3e - B/3A*/ (27 34)
for polynuclear growth. Here C0, z, «* and B are again given by eqs (7.8) or
(7.9), (13.37) (at Acr= 0), (4.35) and (13.55), and the factor it, is specified
by (27.32) and is practically A/i-independent. As seen from these expressions,
Gc depends on Aft both directly and implicitly through T and 7). When Aft is
related to T in the simplest way according to eq. (2.23), eqs (27.33) and
(27.34) lead to
Gc = y*fcCyifzn*"2(A:773;rDor/) exp (- Asjk)
exp [(£, + aa0)lkT\ exp (- B'/TAT) (27.35)
402 Nucleation: Basic Theory with Applications
for mononuclear growth and to
Gc = (Ai%Jfc773jra0i7) ^XP (- Asjk) exp [(£, + oa0)fkT]
x [1 - exp (- AseATIkT)]m exp (- B'/3TAT) (27.36)
for polynuclear growth, B' being given by (13.90). Equations (27.33)-(27.36)
show that, in contrast to Gc from (27.29)-(27.31), in growth from melts Gc
has a maximum with respect to Ap or T. The diminishing of Gc with decreasing
7 is due mainly to the increase of the melt viscosity r\ at lower temperatures.
Looking back at eqs (27.20), (27.23), (27.31) and (27.32) and recalling
(13.39), (13.41) and (13.42), we come to the conclusion that for the case of
isothermally varied supersaturation it is possible to generalize the Gc(Afi)
dependence for polynuclear growth of crystals as follows:
Gc = Mo/e/'^d - e-WkT)2vV + 2%-w*^)/ii+2v>kT. (27.37)
This formula is valid when the monolayer clusters nucleate in stationary
regime and grow laterally according to the power low (27.17). In it/es (s_1)
is the value at Afj = 0 of a properly defined molecular attachment frequency
per growth site at the cluster periphery, and the numerical factor kv is virtually
AjU-independent and given by
K = YMcgaoQz/e*//e,),/(]+2'l (27.38)
Here y/v ~ 1 may depend slightly on v, and C0 and z are specified by (7.8) or
(7.9) and by (13.37) with Ao= 0. Since z/*//e.s = 1, when the crystal face
is free of nucleation-active centres (then C0 = \/aQ), kv~ 1. It is important to
note that at v = 0eq. (27.37) describes mononuclear growth (cf. eq. (27.15))
if /¾ is defined as
k0^AfCozf*/ftfi. (27.39)
This expression follows from (27.38) at v = 0 by setting formally i//0 = 1 and
Cgfl0 = Af. In the scope of the classical theory of nucleation the nucleation
work W* is related to Ap by eq. (4.36) so that Gc from (27.37) becomes an
explicit function of the supersaturation:
Gc = k^ofeiSe^/kT(l - e-*/tt)2tf(l +2V5/(1 +2,>A^ (2740)
the nucleation parameter B being given by (13.55). When v>0, this equation
applies to polynuclear growth: for instance, in the particular case of v = 1 it
passes into eq. (27.31). At v = 0 eq. (27.40) describes mononuclear growth.
Indeed, then it transforms into the equation
Gc = k^c^lkTe-Bl^ (27.41)
which is equivalent to eq. (27.29) or (27.30). An important distinction between
the above two formulae is the great difference in the values of the numerical
factors kv and /cq: e.g. for a crystal face which is free of active centres, while
&y= 1, for k0 we have fc0 ~ A{Cq (because z/*//e.s ~ 1) so that k0 > 107 when
the face is, say, of area A( > I /urn2.
Crystal growth 403
Equations (27.40) and (27.41) reveal that Gc is a monotonously increasing
function of A/v when this is increased isothermally. The increase of Gc has a
threshold character, since it is almost entirely governed by the last exponential
factor which stems from the nucleation rate Js. However, the influence of the
nucleation parameter B on Gc is weaker in polynuclear growth than in
mononuclear growth. Besides, whereas at lower supersaturations mononuclear
growth is operative, at higher supersaturations polynuclear growth takes
over. In fact, it is the presence of the B containing exponential factors in
(27.40) and (27.41) that makes the Gc(Afj) dependence for nucleation-mediated
growth basically different from that for continuous growth (see eq. (27.6)).
This difference disappears only for high enough supersaturations (Aft > Bl( 1
+ 2v)) provided kv ~ 1. Physically, this is not surprising: then the nuclei are
so small (n* ~ 1) and so numerous that the crystal face becomes kinetically
rough and, thereby, able to exhibit continuous growth.
Following Obretenov el ah [1989], we can now sum the reciprocals of the
right-hand sides of eqs (27.40) and (27.41) in order to find the reciprocal of
the growth rate Gc which results from nucleation and growth of any number
of supernuclei on the crystal face. Doing that and rearranging yields (v > 0)
0, = ^0/,.^^-^
(1 + (k0/kv)[0 - e^V**]-2^^}-1. (27.42)
This unified formula for nucleation-mediated growth of crystals corresponds
to (27.22) and interpolates between (27.40) and (27.41). It gives Gc in the
scope of the classical nucleation theory and can be represented more generally
by replacing BlAy with W*/kT. Since tyK » li as required, while for
Aft —> 0 eq. (27.42) passes into eq. (27.41) for mononuclear growth, with
increasing Ay it turns into eq. (27.40) for polynuclear growth.
Finally, we note that eqs (27.37), (27.40)-(27.42) remain in force also for
growth from the melt when Ay is varied according to (2.20) provided in
them /e,,*4''"7' is replaced by the attachment frequency /s determined from
eqs (10.66) and (27.2). Thus, Gc in this case is proportional to 1/r/ and has
a maximum with respect to A^i or T. The conclusion about the transition from
mononuclear to polynuclear and then to continuous growth retains validity,
since with increasing Ay (i.e. lowering T) the crystal face again roughens
kinetically. However, continuous growth will not replace polynuclear growth
if the melt cannot be undercooled (e.g. because of its vitrification) to those
low temperatures at which the supersaturation satisfies the condition Aft > Bl
(1 + 2v). Also, it is worth noting that, as in the case of continuous growth,
the above results are equally applicable when Ay < 0. Then Gc is negative
and is the rate of evaporation, dissolution or melting of the molecularly
smooth crystal face. This process is again caused by nucleation and lateral
growth of 2D clusters on the face, but in the form of nanoscopical holes of
monolayer depth.
The solid curve NG in Fig. 27.2 depicts the Gc(Ay) dependence for
nucleation-mediated growth and illustrates the threshold character of this
kind of crystal growth. The curve is drawn according to eq. (27.42) with
404 Nucleation: Basic Theory with Applications
v = 1, /t, = 1, /(¾ = 107 and B = McT (this choice of B corresponds to /c = 20
pj/m and T = 300 K in eq. (13.55)). We see that eq. (27.42) interpolates
well between eqs (27.41) and (27.40) for mononuclear and polynuclear
growth, respectively, which are represented by the dotted and dashed curves
MN and PN in the inset in Fig. 27.2.
The circles in Fig. 27.5 display the experimental GC(S) data of Simon et
al.[l 91 A] for nucleation-mediated growth of the (110) face of paraffin C36H74
crystals in a solution of petroleum ether. It is seen that they follow the linear
dependence predicted by eq. (27.40) at v= 1 in In [GJS"\S- l)2/3]-vs-l/ln
S coordinates. In addition, the smallness of the resulting value of the intercept
led Simon et al. [1974] to the conclusion for polynuclear growth in the S
range studied. From the slope of the straight line, n* = 10 to 170 was
calculated for the number of paraffin molecules constituting the 2D nucleus
in this S range.
1/lnS
Fig. 27.5 Crystal growth rate as a Junction of the supersaturation ratio in
nucleation-mediated growth: circles - data far the (110) face of paraffin C36H74
crystals in solution of petroleum ether at temperatures between 290 and 295 K [Simon
et al. 19741; line - best fit according to eq. (27.40) with v = 1 (Gc is in ym/s).
27.3 Spiral growth
In spiral growth, molecularly, the crystal face is neither completely rough as
in continuous growth nor completely smooth as in nucleation-mediated growth.
Its roughness in terms of concentration of growth sites is intermediate between
these two extremes and is a result of the presence of the spiral steps originating
from the points of emergence of screw dislocations (Fig. 27.6). Since now
there is no need of nucleation of 2D clusters which provide growth sites at
Crystal growth 405
their edges (such sites are permanently present at the spiral steps), at small
supersaturations growth should be expected to be faster than the nucleation-
mediated growth corresponding to a perfectly smooth face. At the same time,
however, the concentration of growth sites is lower than that on a completely
rough face and if the spiral arms are not close enough to each other, the spiral
growth must be slower than the continuous growth.
Fig. 27.6 Cross-section of a stepped crystal face which advances by spiral growth
because of the emergence of a screw dislocation (the dashed line) on its surface.
Following Burton et al. [ 1951 ], let us consider a crystal face with a single
circular screw dislocation spiral on its surface. Also, let the steps forming the
spiral be equidistant and with monolayer height dG and let them spread with
time-independent velocity vs (m/s). Then each step will travel the distance d,
to the neighbouring step in front of it within the same time ds/us. This is
equivalent to a complete single rotation of the spiral or, in other words, to a
complete filling of one monolayer on the crystal surface. Hence, for the
average time of monolayer filling we shall have rav = djvs which in combination
with (27.1) leads to
Gc = doujd;. (27.43)
This relationship shows that Gc is determined by two parameters: the
interstep distance d„ and the step velocity vs. It is ds through which the
nucleation theory plays a role in the description of the spiral growth of
crystals. Indeed, the very first spiral arm (seen on the top of the growth
pyramid in Fig. 27.6) can start spreading only after its radius of curvature
becomes greater than the radius R* of the 2D nucleus. For that reason at the
moment at which this happens the second arm of the spiral is already away
from the first one at a distance proportional to R*. Far enough from the
centre of the spiral this distance is the interstep distance ds in eq. (27.43) so
that one has [Burton et al. 1951]
406 Nuclealion: Basic Theory with Applications
ds = i/R* (27.44)
where if/ = 19 is a good approximation for the proportionality factor 1//'
[Cabrera and Levine 1956; Budevski etal. 1975], This formula is, inter alia,
of experimental interest because it allows ds(A") data to be used for a direct
verification of the classical dependence (4.34) of R* on A^i.
According to (4.34) and (27.44), ds decreases with increasing A^i. This
means that for us in (27.43) we can use results for the velocity of propagation
of isolated steps (e.g. eqs (27.24)-(27.28)) only when the supersaturation is
sufficiently low. The steps are then far enough from each other and the
attachment of molecules to a given step is undisturbed by the presence of its
neighbours. We can think of the step as a sink which captures molecules
from a spatial zone extending along the step and having a characteristic
width AL of its projection on the crystal face (this projection is a strip along
the step). For instance, when surface diffusion supplies molecules for attachment
to the step, the zone is two dimensional and is a strip centred on the step and
ending at the mean diffusion distance A, on both sides of the step so that Az
= 2k,. The condition for step propagation in isolation is, therefore, d, > Az.
Due to the decrease of ds with increasing A^i, for high enough supersaturations
the interstep distance satisfies the opposite inequality, i.e. then ds < Az.
Under this condition the neighbouring steps compete for the capture of
molecules and their propagation is slower than that of isolated steps. Treating
vs in (27.43) as the velocity of an isolated step, we must therefore multiply
it with the factor Yz(ds, Az) < 1 in order to account for the competition
between neighbouring steps. By definition, Yz —» 1 in the dJA, —> <*> limit
which corresponds to isolated steps. Bearing in mind all said above, let us
now find Gc in the cases considered in Sections 27.1 and 27.2 of growth
from vapours under surface-diffusion control and from solutions and melts
under interface-transfer control.
In the former case eq. (27.25) gives vs for an isolated rough step far
enough from the spiral centre (then k, is smaller than the step radius of
curvature). Also, Az = 2k, and, as shown by Burton et al. [1951], Yz(ds, Az)
= tmh(ds/2ks). Hence, with account also of (2.8), (2.9), (4.34), (13.68) and
(27.44), eq. (27.43) leads to [Burton et al. 1951]
Gc = d&JAflK)e*"'TCL ~ ^'"Xty/A/i') tanh(AuVAu)
= 78i>o4(S - 1 )(ln 5/ln S") tanh(ln S'/ln S) (27.45)
where the characteristic supersaturation A// and supersaturation ratio S' art
defined by
A,u'= iryr'd$KftXs~7.5d$id\s (27.46)
In S' = A/j'/kT. (21 Al)
In the case of growth from solutions under the control of interface transfer,
the step can capture only molecules which are in immediate contact with it
so that the capture zone is again a strip along the step, but with width Az =
d0. We can, therefore, use the Yz factor of Burton et al. [1951] in the form
Crystal growth 407
Yz(ds, Az) = tanh(rfs/rf0) provided the step is rough enough. We note, however,
that Yz is much more complicated when the attachment of molecules to the
steps involves other transport mechanisms (e.g. volume diffusion) [Chernov
1961, 1980; Bennema 1969; Gilmer et al. 1971; Bennema and Gilmer 1973;
Ghez and Gilmer 1974; van der Eerden 1982, 1993; Sangwal 1994]. Thus,
with the aid of (2.14), (4.34), (13.69), (27.26) and (27.44), from (27.43) we
obtain
Gc = rfo/e.se4"*7^ - e-W^XAfi/An') tanhfA^'/A^)
= (yJd0)DKcCt(S - l)(ln SI In S') tanh(ln S'lln S) (27.48)
where
A/ = (ity'l4)daK= \5daK (27.49)
In S' = Ap'IkT. (27.50)
Naturally, eq. (27.49) follows directly from (27.46) upon setting 2X, = d0. It
is worth remembering that D in (27.48) is an effective diffusion coefficient
characterizing the random jumps of the molecules across the crystal/solution
interface in the vicinity of the growth sites.
It remains now to determine Gc for growth from the melt again under
interface-transfer control. As already noted above, for this kind of control we
have Az = d0 and Yz(ds, Az) = taah(ds/d0). Combining eqs (4.34), (27.27),
(27.43) and (27.44), we find that
Gc = (7skT/37m0f}) exp [(- X + E, + Oa0)lkT\
etwikT,x _ e-^'MyA/i/A/) tanh (AuVAji) (27.51)
where A/i' is again given by (27.49). This formula shows that Gc is a complicated
function of A/y when this is varied according to (2.20). Let the concentration
of the molecules in the first adsorbed monolayer on the crystal face be
essentially the same as in the bulk of the melt. Then E\ + oa$ ~ 0 and in the
particular case of A/j related to T via (2.23), eq. (27.51) results in the following
Gc(7) dependence:
Gc = (7skTI3m0i}) exp (- Asjk)
[1 - exp (- AssATIkT)](ATIAT') taah(AT'/AT). (27.52)
Here, in conformity with (27.49), the characteristic undercooling A7" = A^7
Ase is given by
AT' = 7ry/'4,/c/4Ase = 15d0K/Asc. (27.53)
Inspecting eqs (27.45), (27.48), (27.51) and (27.52), we observe that Gc is
a different function of Ap in the range of low and high supersaturations.
Indeed, since e"- 1 = x and tanh(l/i) = 1 for*-* 0 [Korn and Korn 1961],
when simultaneously A^i < 0.2kT and Ay < 0.5A^', we have
Gc = (Kg/A^OV
(27.54)
408 Nucfeation: Basic Theory with Applications
where K% is the kinetic factor appearing in eq. (27.13) and characterizing
continuous growth. Equation (27.54) represents the known parabolic growth
law of Burton et al. [1951], which is considered as a criterion for crystal
growth due to the presence of screw dislocations on the crystal face. Under
the opposite condition, i.e. for A^i > 2A//, tanh (Ap'IAfi) = Afi'/Afi [Korn and
Kom 1961] and the Gc(Afi) dependence for spiral growth passes into that for
continuous growth (cf. eqs (27.45), (27.48), (27.51) and (27.52) with eqs
(27.7), (27.8), (27.10) and (27.12), respectively).
Physically, the turnover from spiral to continuous-like growth is not
unexpected: the higher the supersaturation, the closer the steps to each other
so that, eventually, their capture zones overlap entirely. Thus, though not
necessarily completely rough (as, e.g. in growth controlled by surface diffusion),
the crystal face gets the possibility to grow at its maximum rate corresponding
to continuous growth. Both the magnitude of Gc during parabolic growth and
the transition to continuous-like growth depend on A^i', i.e. on the nucleation
parameter K - the specific edge energy of the 2D clusters of monolayer
thickness. This is of experimental interest, since analysing GC(A^) data in
accordance with (27.54) allows a determination of K. In this respect it could
be noted that the rather high values of Afi' from (27.49) suggest that, typically,
the transition from spiral to continuous-like growth may not be observable
under interface-transfer control. Indeed, with exemplary d0 = 0.3 nm, k = 10
pj/m and Aje = 3k, from (27.50) and (27.53) we calculate S'= 5 x 104 at T
= 300 K and AT' ~ 1100 K. Such high supersaturation ratios S' and
undercoolings AT' are not common in experiments. In contrast, the presence
of /Is in A/j' from (27.46) leads to considerably smaller 5' values, which
means that the departure of the Gc(A/<) dependence from the parabolic law
(27.54) is more easily observable in growth controlled by surface diffusion.
For example, with the above values of dB, srand T, from (27.46) and (27.47)
it follows that S' < 1.7 when As > I0da.
The Gc(Afj) dependence for spiral growth is illustrated in Fig. 27.2 by
curve SG which is drawn according to eq. (27.48) with Afi'lkT = 10. The
parabolic increase of Gc with the supersaturation is clearly seen. The crossing
of curves SG and NG in Fig. 27.2 means that if they refer to the same crystal
face, although in the presence of screw dislocations this face will exhibit
first spiral growth, at higher supersaturations nucleation-mediated growth
will take over because of abundant nucleation of monolayer clusters on the
terraces between the spiral steps [Weeks and Gilmer 1979],
The squares and the circles in Fig. 27.7 represent the Gc(A<p) data of
Bostanov et al. [1969] for, respectively, (100) and (111) faces of Ag in the
case of electrocrystallization in an aqueous solution of AgN03 at T = 318 K.
In this case A^i is related to the overvoltage Atp through eq. (2.27), and Gc is
obtainable with the help of the measured current density ig (A/m2) which
results from the growth of the face: Gc = v0ig/zieo where Zj = 1 is the valency
of the Ag ions. The curves in Fig. 27.7 are the best-fit parabolae drawn
according to eq. (27.54) with ATgzf ej|/£\u' = 6.74 and 15.2 mm/sV2 for the
(100) and (111) face, respectively. As A^f is related to K, the good agreement
Crystal growth 409
£
o
CD
1.5 2.0
A<p (mV)
Fig. 27.7 Crystal growth rate as a function of the overvoltage in spiral growth:
squares and circles - data of Bostanov et at. [1969] for, respectively, the (100)
and (111) face ofanAg single crystal electrode in aqueous solution ofAgNOj at
T = 318 K; curves - best fit according to eq. (27.54).
between theory and experiment allows calculating the specific edge energies
K-100and kui of the monolayer steps of Ag on the (100) and the (111) face of
Ag: Km = 25 pJ/m and K]n = 28 pJ/m [Budevski et al. 1975]. These values
are in agreement with those obtained from experiments on 2D nucleation
under the same conditions [Budevski et al. 1966; Kaischew and Budevski
1967; Bostanov et al. 1983; Budevski et al. 1996] (see Section 13.5).
Finally, we note that as in the cases of continuous and nucleation-mediated
growth, the above Gc(A^i) dependences are applicable also when Afj < 0.
Then Gc is negative and is the rate of evaporation, dissolution or melting of
the crystal face, which results from the rotation of the screw dislocation
spiral in the sense opposite to that of growth. However, in this case other
structural defects (e.g. edge dislocations) often play a dominant role in the
decay of the crystal face.
Chapter 28
Third application of the nucleation
theorem
In Section 27.2 we have seen that the dependence of the rate Gc of nucleation-
mediated crystal growth on the supersaturation Ay is practically entirely
governed by the A^i dependence of the rate of 2D HEN on own substrate.
This makes it possible to apply the nucleation theorem for a theory-independent
determination of the size n* of the monolayer nuclei appearing on the growing
crystal face. As in Chapters 14 and 16, the determination is simple only
when the supersaturation is varied isothermally The following considerations
will therefore be confined solely to this case.
Let it be independently known that the 2D clusters on the crystal face
nucleate in stationary regime and grow laterally according to the power law
(27.17). Then Gc is given by the general formula (27.37) which we rewrite
as follows:
-W* = (1 + 2v)kT In Gc - (1 + 2v)Afi - 2vWln (1 - e"A""7)
-(1 + 2v)tni(M(j;,).
Differentiating this equation with respect to AjU, employing the nucleation
theorem in the form (5.29) and treating the product kvdafes as constant with
respect to A/(, we get
n* = (1 + 2v)Wd(ln Gc)/dAji -(1+ 2v) - 2v/(e<v'kT - 1). (28.1)
This general formula shows that the A^i dependence of the size n* of the
EDS-defined 2D nucleus on the growing crystal face is obtainable from the
slope of the curve resulting from plotting available Gc(A/<) data in In Gc-vs-
Afi coordinates. The so-calculated n*(Ap) function does not rely on any
concrete theoretical model (classical, atomistic, etc.) of nucleation. However,
the determination of the absolute value of n* requires independent knowledge
of the exponent v in the growth law (27.17). We also note that the last term
in the r.h.s. of (28.1) does not introduce uncertainty in the calculated values
of n* when AfilkT> In (1 + 2v), because then it is less than unity. In the case
of mononuclear growth we have v = 0 and eq. (28.1) simplifies to (cf. eq.
(14.6))
n* = kT d(ln Gc)/dA,u - 1. (28.2)
In the case of linear growth of the cluster radius with time (then v = 1) eq.
(28.1) becomes
n* = 3kT d(ln Gc)/dAji - 3 - 2/(e^'kT - 1). (28.3)
Third application of the nucleation theorem 411
To apply eq. (28.1) to a concrete case of nucleation-mediated crystal
growth we must express A/u in terms of the corresponding experimentally
controllable parameter. With the help of (2.8), (2.9), (2.13), (2.14), (2.16),
(2.27), (10.79), (13.68) and (13.69) we find that
n* = (1 + 2v)d(ln Gc)/d(ln S) - (1 + 2v) - 2v/(S - 1) (28.4)
for growth from vapours or solutions,
0 -
CD -2
c
1
■
-
■ /
"•
-
1 ,
•
1 1 1 1 1
*/
. I . . .
> •
1 1 '
. 1 .
0
1
1 1 1 1 .
-3^
• ■ • 1
(a) :
-
(b) .
-
1.5
2.0
2.5
An/kT
3.0
3.5
Fig. 28.1 (a) Dependence of the crystal growth rate on the supersaturation in
nucleation-mediated growth: circles - Monte Carlo simulation data [Weeks and
Gilmer 1979}; line - best-fit function used for calculating the derivative in
eq. (28.}). (b) Dependence of the corresponding nucleus size on the supersaturation:
solid circles - data obtained according to eq. (28.1) with v = 1/2; open circles -
dam from Fig. 14.1b; line - Gibbs-Thomson eq. (14.18).
412 Nucleation: Basic Theory with Applications
n* = (1 + 2v)(kTlzte0)d(\n ;g)/dA<y>- (1 + 2v)
- 2v/[exp (zie^qAT) - 1] (28.5)
for electrocrystallization and
n* = (1 + 2v)(/t77Au1.)d(ln Gc)/dp -(1 + 2v)
- 2v/(exp [Auc(p - pe)/kT] - 1) (28.6)
for growth of crystals in melts (or during polymorphic transformations)
when Ap is varied by means of the pressure p at constant 7. In eq. (28.5) Gc
is substituted by the experimentally measurable current density i'E (A/m2),
since these two quantities are related by Gc = VqI^z^o-
The circles in Fig. 28.1a represent the Gc(A(i) data of Weeks and Gilmer
[1979] for direct-impingement-controlled polynuclear growth of a perfect
(100) face of simple cubic crystal. The data are obtained by a Monte Carlo
simulation at e,/kT = 4 where £[ is the energy of molecular interaction
between nearest neighbours in the monolayer. From a similar simulation
[van Leeuwen and van der Eerden 1977] it is known that the 2D clusters in
the monolayer obey the growth law (27.17) with V = 1/2. The curve in Fig.
28.1a visualizes the best-fit function In (Gc/dafcs) = - 10.300811(AplkT) +
1.72739 + 1.00258(A^ftD used for the calculation of d(ln Gc)/dA/(. With the
so-calculated derivative and v = 1/2, from eq. (28.1) we can determine in a
model-independent way n* as a function of Ap. This function is shown in
Fig. 28.1 b by the solid circles. For comparison, the open circles represent the
n*(Ap) data from Fig. 14.1b. The curve in Fig. 28.1b depicts the n*(Ap)
dependence following at e^lkT = A from the Gibbs-Thomson formula (14.18)
which corresponds to eq. (4.35) for the conditions of the simulation
'experiment'. As seen, despite that the 2D nucleus is of a few molecules
only, the classical n*(Afi) dependence is in agreement with that obtained in
a theory-independent way with the help of eq. (28.1). It is worth noting that
this finding is consistent with that resulting from the analogous analysis in
Chapter 14 of the stationary rate of 2D nucleation under the same simulation
conditions.
Chapter 29
Induction time
The possibility for realization of thermodynamically metastable states is a
characteristic feature of the first-order phase transitions. Physically, it is due
to the threshold character of the dependence of the nucleation rate on the
supersaturation. After the initial moment t = 0 of supersaturating the old
phase, a certain time I, called induction time (or induction period) may
elapse prior to the formation of an appreciable amount of the new phase.
This time is an experimental observable and is a measure of the 'ability' of
the system to remain in metastable equilibrium. The various experimental
techniques detect with a different resolution the first portions of the new
phase which nucleates and grows in the supersaturated old phase. Asa result,
under otherwise equal conditions, tt may not have the same value when
measured by different experimental techniques. This means that the theoretical
interpretation of t, data must always be done with a tt formula which corresponds
to the particular experimental method used for the measurement of tr
So far, two approaches have been most widely used for the theoretical
determination of t{. The first one is based on the assumption that the appearance
of the first supemucleus is the event that brings the system out of its metastable
equilibrium [Volmer 1939; Nielsen 1964, 1967; Hirth and Pound 1963; Kubota
1983; Sbhnel and Mullin 1988; Kashchiev et al. 1991; Sbhnel and Garside
1992; Mullin 1993]. In this case, therefore, the metastability is lost by the
mononuclear mechanism (see Sections 26.3 and 27.2). The /; formula obtained
is adequate for analysis of experimental t, data only when they are obtained
by experimental techniques which allow counting the number of supernuclei
or detecting the presence of a single supemucleus in the system. The second
approach relies on the presumption that the appearance of many supernuclei
and their overgrowth to macroscopically large sizes are responsible for the
breakdown of the metastable equilibrium [Chepelevetskii 1939; Nielsen 1964,
1969; Sbhnel and Mullin 1979, 1988; Kashchiev 1989a; Kashchiev el al.
1991; Sbhnel and Garside 1992], Accordingly, in this case the metastability
is lost by the polynuclear mechanism considered in Sections 26.2 and 27.2.
The usage of the resulting /; formula is legitimate when the t\ data analysed
are obtained by an experimental technique which is based on the detection of
the appearance of a certain sufficiently large volume of the new phase. As
the formulae for I, of these two approaches correspond to limiting cases with
respect to the number of supernuclei, they can be united into a single formula
which is valid for whatever number (one, several or statistically many)
supernuclei involved in the phase transformation [Kashchiev et al. 1991].
More about the theoretical and experimental findings concerning the induction
414 Nucleation: Basic Theory with Applications
time can be found elsewhere (e.g. van der Leeden et al. [1991]; Sdhnel and
Garside [1992]; Jakubczyk and Sangwal [1994]; Sangwal and Polak [1997]).
Let us now see how tt is related to the nucleation rate when either the
mononuclear or the polynuclear mechanism is operative. We begin with the
former. One way to define ^ in this case is to identify it with the moment of
formation of one supemucleus in the system [Volmer 1939]. Mathematically,
this definition reads
Wfri) = 1 or As£(/j)= 1 (29.1)
for HON or HEN in a volume V or for HEN on a substrate with surface area
As, respectively. This algebraic equation for /, is easy to solve when nucleation
is stationary. Then the concentration £of supernuclei (in the absence of
preexisting ones) is given by eq. (13.102) and from (29.1) it follows that [Volmer
1939; Nielsen 1964, 1967; Sbhnel and Mullin 1988]
t{ = l/JsV (29.2)
where /s is the stationary nucleation rate specified by (13.39). Here and in
what follows we present t\ only for HON or HEN in the volume of the old
phase, but it is clear that replacement of V by As makes (29.2) and all
subsequent formulae of the mononuclear approach applicable also to HEN
on a substrate. With the aid of (13.39), (13.41) and (13.42), from (29.2) we
thus find that in the cases in which the supersaturation A/i is varied isothermally
t{ can be represented by the general expression
/, = {\lzf?CvV)e{W'-WT (29.3)
in which the concentration C0 of nucleation sites is given by (7.10)-(7.12) or
(7.16). Due to the relatively weak dependence of z and /e* on A^/, the pre-
exponential factor in eq. (29.3) is virtually independent of the supersaturation.
With properly defined Ap and nucleation work W* (see Chapters 2 and 4),
eq. (29.3) represents the induction time for formation of condensed phases in
vapours, solutions, etc. provided nucleation occurs in stationary regime.
When the process is non-stationary, the determination of /, is difficult, because
fyi) is a complicated function of time (cf. eq. (15.114)). Nonetheless, for
practical purposes it may suffice to use the approximation
ti = blmx+\UsV (29.4)
where the first summand accounts for the time delay in the £(r) function
(Fig. 15.9b), and b\ m ~ 1 can be considered as independent of T, /s and V. In
the X- 0 limit (stationary nucleation) eq. (29.4) passes into (29.2), but when
r» UJ&V, /j is entirely determined by the nucleation time lag r.
Physically, the induction time can also be identified with the lifetime /av
of the supersaturated old phase or, equivalently, with the mean time t\ for
appearance of at least one supemucleus in it. Comparison of eqs (26.39) and
(26.42) with eqs (29.2) and (29.4) shows that for stationary nucleation this
definition of t\ is equivalent to that given by eq. (29.1). When nucleation is
non-stationary, however, the two definitions lead to different t\ values, but
Induction time 415
the difference is small, because the numerical factors bm and b; m in (26.42)
and (29.4) are practically equal.
Let us now determine it according to the polynuclear mechanism. Clearly,
/i can again be identified with the lifetime rav of the old phase in metastable
state. In view of eqs (26.24), (26.28) and (26.33), in the case of progressive
nucleation and power-law growth of the supernucleus particles of the new
phase we shall have
t; = T[(2 + vd)/0 + vd)][(l + vd)/clG"dJa ]"(,+'"" (29.5)
for stationary nucleation and
fi = brt + T[(2 + vd)/(l + vd)][(l + vd)lcsGldJs\m*v,') (29.6)
for non-stationary nucleation. Likewise, in the case of instantaneous nucleation
of particles growing also according to (26.16), from eqs (26.19) and (26.20)
we find that
tt = r(l + \lvd)[VlcgNjlvd(\IGc). (29.7)
The induction time from the above equations corresponds to the moment
at which the phase transformation is about half accomplished. In many cases,
however, experimental ?; data are also obtained by techniques sensitive to the
presence of a certain detectable volume Vd (or mass) of the new phase which
may be much smaller than the initial volume V. Hence, in these cases it is
more appropriate to define t\ as the time needed for the formation of the
volume Vd or, equivalently, of the detectable fraction ffd he Vd/V of V.
Mathematically, this definition of J, reads
ff('i) = 0¾ (29.8)
where the fraction a of transformed volume is given by the KJMA formula
(26.8), and ad « 1. This algebraic equation for tt is easy to solve again for
progressive nucleation in stationary regime and particle growth obeying eq.
(26.16). Then a is given by (26.23) and with account of (26.28) and usage
of In (1 - «d) = -0¾ [Korn and Kom 1961], from (29.8) it follows that
[Kashchiev et al. 1991]
», = [(1 + vd)ajcg GCV"JS]"11 + vd). (29.9)
In the particular case of spherical particles (d — 3, cg = 4^3) with radii
growing linearly with time (v= 1) this equation passes into that obtained by
Chepelevetskii [1939]. We observe also that, not surprisingly, J; from (29.5)
and (29.9) differ from each other solely by a numerical factor controlled by
0¾ (i.e. by the resolution limit of the experimental technique). When nucleation
is progressive, but non-stationary, a(r) is a complicated function (see eq.
(26.30)) and ty can be determined only approximately. It turns out that in the
case of 1 + vd - 4, similar to (29.4), the nucleation time lag r appears as a
summand in the r.h.s. of (29.9) [Kashchiev 1989a]. Generalizing this finding,
we can thus represent t\ in the form
fi = *i T+[(l + vd)adlcsG*''ji]m + K[> (29.10)
416 Nucleation: Basic Theory with Application*
where for practical purposes the factor bip can be treated as a number close
to unity. For example, in the 1 + vd = 4 case numerical analysis [Kashchiev
1989a] shows that b,v = 0.4 to 0.5, because it depends weakly on Gc, Js and
T according to 6iiP = (4ad/cgGc3JsT4)""'. Equation (29.10) parallels those
used for analysis of the effect of nucleation non-stationarity on overall
crystallization [Gutzow and Kashchiev 1970, 1971; Gutzow et al. 1985],
When x = 0 (stationary nucleation), it passes into eq. (29.9). As to the usage
of the definition (29.8) for determination of t\ in the case of instantaneous
nucleation, it is again easy under condition that the particles grow according
to the power law (26.16). In this case we can use a from (26.18) in (29.8) so
that, upon accounting for eq. (26.20) and the approximation In (1 - ad) ~
- (¾ we get [van der Leeden et al. 19911
t, = (atV/cM^a/GJ. (29.11)
Understandably, 1-, from (29.7) and (29.11) differ from each other only by an
ffd-dependent numerical factor. We note also that in the vd = 3 case eq.
(29.11) leads to formulae similar to those given by Sohnel and Mullin [1988].
Equation (29.11) is useful in experiments on the so-called seeded precipitation
(e.g. van der Leeden et al. (1991, 1992]; Verdoes et al. [1992]). In such
experiments, rather than instantaneously nucleated by the old phase itself,
small enough supernucleus crystallites (seeds) with a known concentration
NmfV are deliberately introduced from outside into the solution.
Looking back at eqs (29.2), (29.4)-(29.7), (29.9)-(29.11), we see that the
impact of the nucleation process on fj is via the maximum concentration
NmlV of supernuclei or the stationary nucleation rate /s when nucleation is
instantaneous or progressive, respectively. When metastability is lost by the
polynuclear mechanism, the role of Js is weaker than when this occurs by the
mononuclear mechanism. Also, while in the former case r, does not depend
on the volume V of the metastable old phase (or on the area As of the
substrate surface), in the latter case it does (cf. eq. (29.2) with (29.5) or
(29.9)). Analogously to Gc from (27.22), by summing the right-hand sides of
(29.2) and (29.9) it is possible to obtain a unified formula for r; which is valid
for whatever number of supernucleus particles formed in the old phase
[Kashchiev et al. 1991]:
/, = 1/7SV+ [(1 + vd) aJcgGv/j,]m*"". (29.12)
This interpolation formula implies progressive nucleation in stationary regime
and particle growth in conformity with the power law (26.16). As it should
be, under the condition [Kashchiev et al. 1991]
V<[ctGv/l(\ + vd) aiJ*d]lm+'"'> (29.13)
eq. (29.12) passes intoeq. (29.2) which is applicable to supersaturated phases
with sufficiently small volume (cf. (26.35)). Under the opposite condition
eq. (29.12) reduces to eq. (29.9) of the polynuclear mechanism.
Employing concrete Gc(Afi), Js(A[i) and t(A,i/) dependences in eqs (29.5)-
(29.7) and (29.9)-(29.12) makes it possible to answer the important question
Induction time 417
about the dependence of the induction time on the supersaturation in the case
of metastability breakdown occurring by the polynuclear mechanism. For
instance, when A^i is varied isothermally, from (13.39), (13.41), (13.42) and
(29.9) it follows that, most generally,
t, = [(1 + vd)ailc%zf*CaG:dfM*^/w'-'*'w*vd)kT (29.14)
provided nucleation is stationary. With vd = 3 this equation leads to formulae
similar to those given by Sohnel and Mullin [1988].
Inspection of eqs (29.3) and (29.14) shows that the latter turns into the
former at d = 0 if then ad and eg are formally set equal to 1 and V, respectively.
Thus, eq. (29.14) appears as a single formula applicable in the case of either
the polynuclear (d > 0) or the mononuclear (with d = 0, ad = 1 and cg = V)
mechanism. Let us use it in order to obtain the tj(A/j) dependence when the
growth constant Gc (mlrtV') is the following function of the supersaturation:
Oc = dl0"fcjle^m\ 1 " e"4"'*7), (29.15)
the factor/es (s_1) being the value at A/< = 0 of a properly defined molecular
attachment frequency per growth site. When V = 1, Gc (m/s) is merely the
growth rate of the particles and eq. (29.15) corresponds, e.g. to continuous
growth due to direct-impingement or interface-transfer control (cf. eq. (27.6))
or to growth controlled by volume diffusion through a boundary layer formed
around the particle for hydrodynamic reasons (e.g. Doremus [1985]; Mullin
[1993]). When growth is controlled by undisturbed volume diffusion, v =
1/2 and Gc (m2/s) is the respective growth constant (then /cs = 2doDCc
[Volmer 1939; Nielsen 1964]). To a certain approximation, in the v = 1 case
eq. (29.15) gives Gc also for spiral growth of crystalline particles (see eq.
(27.48)), but it does not apply when their growth is mediated by 2D nucleation,
because it does not contain the last exponential factor in eq. (27.40). Combining
(4.39), (29.14) and (29.15), we find that in the framework of the classical
nucleation theory and for the above choice of Gc the /j(A/() dependence is of
the form [Kashchiev et al. 1991]
t, = Kvde-A'"kT(l -e-4«*7)-w'"(1 + w'>exp [fi/(l + vrf)A/i2]. (29.16)
Here the nucleation parameter B is specified by (13.49), and the practically
A^i-independent factor Kvd (s) is defined by
Kvd = [(1 + vd) aJc^ftC^f:']"^^ (29.17)
where C0 (m-3) is given by (7.10)-(7.12) or (7.16). In the particular case of
particles growing in three dimensions (d = 3) with time-independent growth
rate (v= 1), with the help of the approximations zf* ~fcs and dl = v0 eqs
(29.16) and (29.17) lead to
t; = K,e~^ltT( 1 - (,-4^^)-3/4 exp (B/4AiU2j (29 , g)
K3 = (4cni/cgC0D0)"4(l//e,s). (29.19)
With cg = 4, (¾ = 10~8 to 10"4 and/es = 104 to 1010 s_1, from (7.12) and
418 Nucleation: Basic Theory with Applications
(29.19) it follows that AT3 = 1 ps to 10 jis for HON. In the case of HEN C0
is given by (7.10) or (7.11) and /C3 is expected to have higher values.
As already noted above, at d = 0 eq. (29.16) describes the /;(A^) dependence
for the mononuclear mechanism. Hence, also in the scope of the classical
theory of nucleation, for this mechanism we have (e.g. Kashchiev et al.
[1991])
t- = K0 e-^lkT exp (BIAfi2) (29.20)
the nucleation parameter B being given by (13.49). As follows from eq.
(29.17) at d = 0, (¾ = 1 and cs = V, the practically A^i-independent factor K0
is specified by
K0 = llzf*C0V (29.21)
and, in contrast to Kvd for vd > 0, is a function of the volume V of the old
phase. Naturally, eq. (29.20) is directly obtainable from (29.3) with W* and
B from (4.39) and (13.49). With z = 1 and /* = 10" to 1010 s"1, from (7.12)
and (29.21) we estimate K„V = 10~39 to 10-33 m3s for HON so that then K0
= 10"30 to 10-24 s, e.g. for V = 1 mm3. Like K3, in the case of HEN K0 is
expected to have higher values, because then C0 is determined by (7.10) or
(7.11). Comparing the above values of KQ and K3, we see that Ka is many
orders of magnitude less than /C3. This means that experimentally obtained
relatively high values of Kvlj in eq. (29.16) are a strong indication both for
the operativeness of the polynuclear mechanism and for the occurrence of
HEN in the system studied.
Employing J, from (13.50) and Gc from (29.15) in eq. (29.12), we can
find also the unified /j(A^i) dependence which is valid regardless of whether
one, several or many supernuclei bring about the loss of metastability.
Alternatively, this can be done by summing the right-hand sides of (29.16)
and (29.20), the result being [Kashchiev et al. 1991] (vd > 0)
t, = K0e-^'krexp (B/V){1 + (.KJKf,)[(\ - e^"*7)
exp (fi/V)]-""0 *vd>)- (29.22)
This unified formula gives tj in the case of progressive nucleation in
stationary regime provided the growth constant Gc depends on A^i according
to (29.15). Hence, like (29.16), eq. (29.22) is not valid when growth is
mediated by 2D nucleation. As seen, in the AjU —> 0 limit (29.22) passes into
(29.20) for the mononuclear mechanism. With increasing A/j, however, the
second summand in (29.22) increases and since KvJKq» 1, above a certain
supersaturation value the polynuclear mechanism takes over: eq. (29.22)
turns into eq. (29.16). The solid curve in Fig. 29.1 and its inset displays the
t,(&M) dependence (29.22) at /¾ = 10~25 s, Kvii/K0 = 1020, d = 3, V = 1 and B
= 300(^7^2 (according to (13.49), this B value corresponds to ac! ~ 80 mJ/m2
at T = 300 K). It is seen that this curve interpolates well between the dotted
(MN) and the dashed (PN) curves which represent, respectively, eq. (29.20)
and eq. (29.16) for the mononuclear and the polynuclear mechanism. The
curves in Fig. 29.1 demonstrate also that t; is a monotonously decreasing
Induction time 419
100
80
60
40
20
10
-
5
0
2
PN'. ' \
\\
MN-
1
^
2.3
2.5
i\
A^/kT
Fig. 29.1 Supersaturation dependence of the induction time: solid curve - eq.
(29.22); dashed (PN) and dotted (MN) curves - eqs (29.16) and (29.20) for the
polynuclear and the mononuclear mechanism, respectively.
function of Ay when this is varied isothermally. We observe as well that eqs
(29.16) and (29.20) predict much shorter induction times if they are used
outside the Ay range in which they are valid. This implies that the correct
choice of a tj(A^) formula is of crucial importance for the reliable quantitative
interpretation of t^Afi) data: full correspondence must exist between the
definition of /i( which is behind the chosen formula, and the experimental
technique employed for obtaining the t^Ay) data. To further emphasize this
point let us consider one more definition of tx and the resulting ?;(A/i)
dependence.
Suppose we can monitor optically the supersaturated old phase by measuring
the temporal evolution of the intensity /t of a monochromatic light beam
transmitted through the phase. The particles of the new phase, which nucleate
and grow in the old phase, are scatterers of light. In their absence, i.e. at the
initial moment t - 0 of supersaturating the old phase, the intensity of the
transmitted light has a given value denoted by /, 0. At a later time t particles
are already present in the old phase and /t is smaller than /l0 in accordance
with the Bouguer-Lambert law
/t(0 = /t>oexp[-*ea)Ls] (29.23)
where Z,s (m) is the length of the path travelled by the light beam through the
scattering medium, and ke (m_1) is the extinction coefficient. In the presence
of spherical particles with concentration £ (m-3) and radius R » .^, kc is
given by [Rayleigh 1899]
/re = 2;iaefl6£ (29.24)
420 Nucleatiou: Basic Theory with Applications
provided multiple scattering is negligible. Here the factor ae (m^1) depends
on the light wavelength X\ and the refractive index mr of the particles relative
to the surrounding medium:
ae = 64;r4(mr2 - 1)2 /3 (m, + 2)2l*. (29.25)
When R ~ X\, ke is a complicated function of R and X{, which is given by the
theory of Mie [1908]. In the diffraction limit (then R » X[), fce is merely
equal to the product of £ and the doubled area 2k/?2 of the particle geometrical
cross-section, i.e. (e.g. Born and Wolf [1968])
ke = 27tf?2f. (29.26)
As shown by Todes and Chekunov [1957], the expression
ke = 27meR6Q(l + acfi4) (29.27)
is a useful interpolation formula valid for all values of the R/Xt ratio. We see
that for R « X, (Rayleigh limit) and R » Xt (diffraction limit) eq. (29.27)
passes into (29.24) and (29.26), respectively.
Equation (29.27) applies to equisized spherical particles with a fixed
concentration. To extend its applicability to the process of new-phase formation,
following Todes and Chekunov [1957] and Muitjens [1996], we must take
into account that the particles have different radii and a time-dependent
concentration. This is easy to do upon realizing that the particles nucleated
at the moment t' have concentration £(r') and that at a later time t £ r' all
have the same radius R(t - t'). Therefore, the change of ke in the period
between r'and /'+ d/'can be described by eq. (29.27) in the differential form
d/te = 2natR\t - l')df(r')/[l + acR\t - t')]. (29.28)
Invoking eq. (11.1), integrating over all r'from 0 to t and allowing for the
initial condition kc = 0 at t = 0 yields
kc(t) = 2naA J(t')[R.\t - ,')l[\ + aji\t - t')]}it' (29.29)
Jo
where, according to (26.7), the particle radius R and growth rate G are
related by
R(t-tr)= f G(t")it". (29.30)
Jo
The above expression for ke(t) is an approximation to that used by Muitjens
[1996] for analysis of experimental data for HON in vapours. Employing it
in eq. (29.23) leads to the sought general formula for the time dependence of
the intensity of transmitted light:
/,(/) = /,0exp (-2jmcL, \ J(f){R6(t- rO/[l + a^it- (')]) drO-
Jo
(29.31)
Induction time 421
Let us now consider first instantaneous and then progressive nucleation.
In the former case J(t) is given by (26.13) so that performing the integration
in (29.31) is a simple matter, the result being
/,(r) = /,,o exp {- 27meLsNmR\t)/[l + aJ{\t)\V). (29.32)
Hence, when the Nm particles nucleated instantaneously in the volume V
grow according to the power law (26.16), we have
/,(r) = /,,„ exp l~2jta<.LsN,„G?vt6v/(\ + acCtf"r4v)V]. (29.33)
This expression simplifies to
/,(/) = /ti0 exp [ - IxaMNJV) G* V] (29.34)
when the condition
t<\la)!tvGc (29.35)
is satisfied. This is so not too long after the beginning of the process, as then
the particles are still small, scattering is in the Rayleigh limit and the summand
ac(Gct)4v in (29.33) is negligible with respect to unity. We note that eq.
(29.34) can be used in practice only if the particle concentration NJV is
sufficiently high to allow the departure of/, from /, 0 to be detectable at times
satisfying (29.35). When this detection is possible only at later times
/ > l/ae"4vGc, (29.36)
the particles are already so large that the unity in eq. (29.33) is negligible and
the /,(/) dependence takes the form
1,(.1) = /,,„ exp I- l7tLs(NJV) C2 V] (29.37)
corresponding to the diffraction limit.
The case of progressive nucleation is more difficult to analyse. We shall
confine the considerations to nucleation in stationary regime and growth in
accordance with (26.16). Then in (29.31) J(t') = Js and K(r-rO= G*(t- t'Y
so that we get
/,(r) = /,,„ exp [- (InaJ^Ja? + 2v)/4vGc) YM.Gft1')] (29.38)
where
Yv(x) = (1/4V) f [yd + 2l,>'47(l + v)] dy. (29.39)
Jo
Unfortunately, the integration in (29.39) cannot be carried out in a closed
form for arbitrary v > 0. Nevertheless, in the particular case of v = 1/2 this
can be done and as the result is
Yll2(x) = (V2)[x - In (1 + x)], (29.40)
in this case eq. (29.38) leads to
/,(/) = /,,0 exp {-(^Z.sJs/aeGt)[acGc2r2- In (1 +aeGcV)]), (29.41)
422 Nucleation: Basic Theory with Applications
In the Rayleigh limit, i.e. for short enough times which obey (29.35) at v --
1/2, this expression simplifies to
/,(0 = /,i0 exp [- (7T/2)aeZ.sGcV],
(29.42)
since In (1 + *) = x - jt2/2 for x -> 0 [Korn and Kom 1961],
We cannot be satisfied with finding /, only for v = 1/2, because other
values of v (e.g. v= 1) are also of physical interest. We must, therefore, look
for an approximate formula for the Yv(x) function (29.39), which could be
used for any v > 0. Obtaining such a formula is easy upon noting that Yv(x)
takes the form of the function
y;(x) = (l/4v) f v(1 + :
Jo
for x —> 0 and of the function
Y"(x) = (\IAv)
i'
viy= [1/(1 +6v)]x(1
"dy= [1/(1 +2V)]*"
for x —> <*>. Hence, one way to approximate Yv(x) is to use the relation
l/yv = VY;+ MY" which leads to
Yv(x) ■■
l7[l +6v+(l +2v)x].
(29.43)
As seen, this interpolation formula describes correctly Yv in the x—>t) and
x—»«> limits. The good quality of the approximation (29.43) is illustrated in
Fig. 29.2 in which, as indicated, the solid curves display the exact Yv (x)
Fig. 29.2 Dependence of Yv on x: solid curves 1/2 and 1 - eq. (2939) at v = 1/2
and I, respectively; dotted curves the corresponding approximate eq. (29.43).
Induction lime 423
function (29.39) at v= 1/2 and 1 (the v= 1/2 curve is drawn according to eq.
(29.40), and the v = 1 one represents Y\(x) calculated numerically from
(29.39) at v = 1). The dotted curves show the approximate YJx) dependence
(29.43) at the same v values.
Substituting Yv from (29.43) in (29.38), we thus arrive at the desired
formula for /,(/) in the case of progressive stationary nucleation of particles
which grow according to the power law (26.16) with arbitrary v> 0:
/,(() =/,,o exp {-2nacUGTJs/l+6"/[l +6v+(l + 2v) aeGc4"r4v]}.
(29.44)
This approximate formula is in fact exact in the Rayleigh limit, i.e. for times
which are short enough to satisfy (29.35). Then the last summand in the
denominator in (29.44) is negligible with respect to 1 + 6v and (29.44)
simplifies to
/,(/) =/,,„ exp (-[2jtf(l +6v)]a<lLsG^j/*'"'}. (29.45)
We see that at v = 111 this equation is identical with the exact eq. (29.42).
Equation (29.45) is of practical use only when Js is so high and, thereby, the
particle so numerous that although sufficiently small to scatter light in the
Rayleigh limit, they still cause a decrease in /,(/) which can be resolved
experimentally. When resolving this decrease requires such a long time that
it satisfies the condition (29.36) corresponding to the diffraction limit, the
particles are already large enough for 1 + 6v to be negligible in eq. (29.44).
Hence, in the diffraction limit the /,(/) dependence is of the form
/,(/) = /,,0 exp (- [2x1(1 + 2v)]Z.sGc2,V +2v}. (29.46)
Having obtained analytical expressions for the /,(t) dependence, we are in
position to define the induction time t, that can be determined by means of
the corresponding optical technique. Usually, /; is identified with the moment
at which the relative decrement ft « 1 of /, with respect to /, 0 is large
enough to be reliably detected. Mathematically, therefore, this definition
reads
1/,,0-/,(^)1//,,0 = ¾ (29.47)
and in conjunction with (29.34) and (29.45) leads to
/i = [(1 + 6v)ft/2nueZ,G6CVJS]W * "" (29.48)
for progressive nucleation and to
t, = (ftV/2raeZ.s/Vm]"6v(l/Gc) (29.49)
for instantaneous nucleation, because In (1 - ft) = -ft. Equations (29.48)
and (29.49) are valid provided t; satisfies the condition (29.35) (Rayleigh
limit). Under the opposite condition (29.36), i.e. in the diffraction limit, from
(29.37), (29.46) and (29.47) it follows that
h = [(1 + 2v)ft/27tZ.s Gc2v/J"*1 +2"> (29.50)
424 Nucleation: Basic Theory wilh Applications
for progressive nucleation and that
/, = {^VI27CL,Nm\mv{\iGc) (29.51)
for instantaneous nucleation.
Equations (29.48)-(29.51) correspond to metastability breakdown by the
polynuclear mechanism and have to be compared with eqs (29.9) and (29.11)
at d = 3. Concerning t\ for instantaneous nucleation (eqs (29.11), (29.49) and
(29.51)), we observe that both definitions of the induction time result in the
same dependence of tt on Gc, but in a different dependence on Nm. For
progressive nucleation (cf. eqs (29.9), (29.48) and (29.50)), there is a change
in the dependence of t-v already on Gc and Js. Thus, although according to
both the volumetric and the optical definitions t\ is qualitatively the same
function of Gc and Nm or Js, quantitatively, inadequate usage of some of the
formulae for t\ for a fit to /; data may lead to considerably inaccurate values
of the free parameters describing Gc and Nm or Js.
To illustrate this point let us employ in eqs (29.48) and (29.50) Js from the
classical formula (13.50) and G<. from eq. (29.15) in order to find the explicit
t[(Afj) dependence when A,u is varied isothermally, e.g. by means of p or C
in condensation of vapours or solute (see eqs (2.8), (2.9) and (2.14)). After
some rearrangement and usage of the approximation dl « u0 we get
tx = Kve"^lkT{\ - e-AM/^-fiv/u +Mexp [B/(l + 6v)A^/2] (29.52)
in the Rayleigh limit and
?i = Kve-A»m{\ - e-WkT)-2via + 2v)exp \BI{\ + 2v)A|i2J (29.53)
in the diffraction limit. Here the nucleation parameter B for HON or 3D
HEN is specified by (13.49), the practically Aju-independent kinetic factor
ATv(s) is defined as
Kv = [(1 + 6v) PJ2!mcL,zf:c„vlft"rMv) (29.54)
in the Rayleigh limit and as
Kv = 1(1 + 2v)PJ2KL,zf*C0d^fc2/]"IM2v) (29.55)
in the diffraction limit, C0 (m~3) being given by (7.10)-(7.12) or (7.16). In
the particular case of v = 1 (time-independent growth of the droplets or the
crystallites), provided zf* =/„, eqs (29.52)-(29.55) become
t, = K,e-^'kr(l - e-^1")-*11 exp (fi/7A/<2) (29.56)
Kx = (7 pd/2xacLsC0vl)w ('//«) (29.57)
in the Rayleigh limit and
t; = Kie-^'^il - e-^ltTrm exp (B/3A/12) (29.58)
AT, = (3ft/27rZ.sC0rf02)"3(l//u) (29.59)
in the diffraction limit.
Now, comparing eqs (29.18) and (29.56), we see that if instead of (29.56)
induction lime 425
we use eq. (29.18) for a fit to t^A/j) data obtained optically in the Rayleigh
limit, we shall calculate a rather different value for the free parameter B and,
thereby, for the effective specific surface energy o"el of the 3D nucleus. The
error will be quite serious if, as is sometimes done, eq. (29.20) of the
mononuclear mechanism is used for such a fit.
Equations (29.52), (29.53), (29.56) and (29.58) give t, in the scope of the
classical nucleation theory provided the growth of the particles is not mediated
by 2D nucleation. According to (13.39), (13.41), (13.42), (29.48) and (29.50),
the general presentation of t, for isothermal variation of A/( is of the form
t, = [(1 + 2v) pi/2jlacLszf*C0G^]'m*2v)e<w"-A")Wv>l<T (29.60)
in the Rayleigh limit and of the form
t, = [(1 + 6v) ^/271 Lszf*CltG^]""*6v,e(W'-Ai'>l"*t''>kT (29.61)
in the diffraction limit. These formulae are applicable to whatever kind of
stationary nucleation (classical, atomistic, etc.) and 3D growth (continuous,
spiral, nucleation-mediated, etc.) as long as this growth obeys the power law
(26.16).
Finally, we note that with f*e"1"" and/e4e^'*T replaced by/* and/s,
respectively, eqs (29.3), (29.14)-(29.22), (29.52)-(29.61) give the induction
time for appearance of crystallites in undercooled melts when Ap is varied
according to (2.20). Then the pre-exponential factors in these equations are
proportional to the melt viscosity r/ and rL exhibits a minimum with respect
to A^i or T. For example, with/* from (10.64) or (10.65) in lieu of f*^1",
for HON or 3D HEN of cap-shaped nuclei eqs (29.20) and (29.21) yield
/; = (37tv0ll'r*zn*aC0VlcT) (.^-^"""exp (BIAp.1) (29.62)
where n*„ = 1(1 - cos ew)/2i^/3(ew)] (cvl"n*mldl) is the number of
attachment sites at the nucleus surface. This formula gives t, for the mononuclear
mechanism in the scope of the classical nucleation theory when molecular
attachment is controlled by interface transfer. The result is analogous for the
polynuclear mechanism in the case of continuous growth (v = 1) of the
particles in three dimensions (d = 3): then we have (cf. eqs (27.5) and
(27.10))
/s = (%km7tv0ri) exp [(A/i - X + £, + aa0)/kT] (29.63)
so that, upon replacing/e!Aw by this expression for/^, from (29.18) and
(29.19) we get
t; = (iOilcfiaVa)wOKvar\lyjiT) exp [(A - E, - oa0)/kT]
x e-<W(\ _ e-^/kTyVt exp (£/4^2) (29.64)
We recall that this formula implies a sufficient accuracy of the approximation
zj* =/s. Similarly, in the same approximation and under the same conditions
of nucleation and growth, from (29.56M29.59) and (29.63) it follows
that
426 Nucleation: Basic Theory with Applications
t, = (ipJ27tacLJCavl)w(?,Kv0mYj<T) exp [(X - E, - 0ao)/kT]
x e-<V"»'(l - e-W^-w exp (B/7A/J2) (29.65)
when the induction time is determined optically in the Rayleigh limit and
that
t, = (3PJ27cL,CBdZ)"\3jw0r}/%kT) exp [(A - E, - aa0)lkT\
x e-^lkT(l _ e-J*ikTy2n exp (fl/3^2) (2o.66)
if this determination is in the diffraction limit. When A/y is related to T
according to (2.23) and when there is practically no difference in the
concentration of molecules in the bulk of the melt and in the adsorption layer
around the growing supernuclei (then £, + ua0 = 0), eqs (29.62), (29.64)-
(29.66) lead to the following dependences of t, on T.
t, = (.3m>BT]/y*zna,*C0VkT) exp (Asjk) exp (B'lTAT2) (29.67)
for the mononuclear mechanism (£(0, P\(t) or a(t) measuring technique),
f, = (4ad/cgC0i;())"4(3TO0n/Kir) exp (Asjk)
X [1 - exp ( - AscATIkT)Ym exp (B'I4TAT2) (29.68)
for the polynuclear mechanism (ait) measuring technique),
/j = (7ft/2^fletsC0^)"7(3OT0n/7sA:r) exp (Asjk)
x [1 - exp (- AscAT/kT)]-*"1 exp (BVTAT1) (29.69)
for the polynuclear mechanism (/,(?) measuring technique, Rayleigh limit)
and
t, = O^2nLfit)dl)mCiiw0rll7%kT) exp (Asjk)
x [1 - exp (- AstATIkT)}-™ exp (B'BTAT1) (29.70)
for the polynuclear mechanism (lt(t) measuring technique, diffraction limit).
The nucleation parameters B and B' in (29.62), (29.64)-(29.70) are given by
eqs (13.49) and (13.73), and C0 is specified by (7.10), (7.11) or (7.12). It
should be noted that, plotted in T-vs-ln t, coordinates, the ti(T) dependences
(29.67) and (29.68) represent the so-called TTT (time-temperature-
transformation) curves used for the formulation of a kinetic criterion for
vitrification of melts [Turnbull 1969; Gutzow and Kashchiev 1970, 1971;
Uhlmann 1972, 1983; Yinnon and Uhlmann 1981; Gutzow et al. 1985;
Battezzati et al. 1987; Weinberg et al. 1989, 1990; Gutzow and Schmelzer
1995].
Experiments aimed at studying the dependence of t\ on Ap are perhaps
most often done with supersaturated solutions (e.g. Nielsen [1969]; Kirkova
and Djarova [1971, 1977]; Sohnel and Mullin [1978,1979,1988]; Koutsoukos
and Kontoyannis [1984]; Kubotaetal. [1986]; Wojciechowski and Kibalczyc
[1986]; Sangwal [1989J; van der Leeden etal. [1991, 1992, 1993]; Verdoes
et al. [1992]; Symeopoulos and Koutsoukos [1992]; Sohnel and Garside
Induction time 427
[1992]; Mullin f 1993]; Jakubczyk and Sangwal [1994]; Hendriksen and Grant
[1995]; Sangwal and Polak [19971). The circles in Fig. 29.3 represent the
t,(S) data of Verdoes et al. [1992] for precipitation of CaC03 in an aqueous
solution at T = 298 K. The induction time at a given supersaturation ratio
S = n/nc = e^lkT (cf. eq. (2.16)) was determined from recordings of the
temporal evolution of the signal of a Ca-ion-selective electrode, a technique
which is consistent with the volumetric formula (29.14). In addition, it was
independently established that precipitation occurred by the polynuclear
mechanism with vd = 3 and Gt °= (S - 1 )1 8. Using these findings for vd and
Gt and the classical formula (4.39) for W* transforms (29.14) into
t, = KTV\S - I)"1-35 exp (fi'/4 In2 S) (29.71)
8.5 , ,,,,,
8.0 -
"^ 7.5 -
i
CO
5 7.0-
c
— 6.5 -
6.01 ' '
0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 / (InS)2
Fig, 29.3 Supersaturation dependence of the induction time: circles - data for
precipitation ofCaCOj crystallites in aqueous solution at T = 298 K [Verdoes et al.
1992]; line - best fit according to eq. (29.71) {t, is in seconds).
where X"(s) is a practically ^-independent factor and B'is given by (13.67).
This equation predicts a linearization of the tt(S) dependence in In [£"4(S -
1)135t;]-vs-(l/ln2 S) coordinates. The best-fit straight line in Fig. 29.3 is in
support of this prediction. With the help of the B' value of 12.2 resulting
from the slope of this straight line Verdoes et al [1992] calculated % = 37.3
mJ/m2 and n* = 3 to 14 for the number of CaC03 molecules constituting the
nucleus in the S range studied. This relatively low value of 0"cf is an indication
for 3D HEN of the CaC03 crystallites in the solution. Also, although the
nucleus is quite small, it turns out that the classical nucleation theory can
provide a reasonable description of the tt(S) dependence under the conditions
of this experiment.
Chapter 30
Fourth application of the
nucleation theorem
In Chapter 29 we have seen that in the case of progressive nucleation in
stationary regime the change of the induction time t\ with the supersaturation
A^i of the metastable old phase is largely determined by the Js(Afi) dependence.
This allows the nucleation theorem to be used for a theory-independent
determination of the nucleus size n* from experimental fj(A^) data provided
it is independently known that nucleation is stationary and growth obeys the
power law (26.16). We shall now see how this can be done with /L(A/j) data
obtained when A/; is varied isothermally as, e.g. in condensation of vapours
or solute.
Recalling eq. (13.42), we first represent eqs (29.3), (29.14), (29.60) and
(29.61) for the various definitions of t[ in the unified form
r, = iK;/Gl)w +i)e(""-«(i +WT (30 ,)
Here
q = 0, K, = l/A'V (or K, = MA'A,) (30.2)
for tt by the mononuclear mechanism,
q=vd, /4 = (1 + Ki)<Vc„A' (30.3)
for volumetrically measured f; (polynuclear mechanism),
q = 6v, K-, = (1 +6v)A/2sacLsA' (30.4)
for optically measured ?; in the Rayleigh limit (polynuclear mechanism) and
q = 2v, Kt = (1 + Iv^llKLJK' (30.5)
for optically measured t, in the diffraction limit (polynuclear mechanism).
Then, taking the logarithm of both sides of (30.1), differentiating with respect
to A^i, applying the nucleation theorem in the form of eq. (5.29) and neglecting
the possible relatively weak dependence of Kj on A^i, we obtain
n* =-(1+ q)kTd(\n t,)/d&fi - 1 - qkTd(\n GJ/d&fi. (30.6)
This is the sought general formula for n* at isothermal variation of A^. It
gives the number of molecules in the EDS-defined nucleus regardless of the
theoretical model and the kind of nucleation (HON, HEN, classical, atomistic,
etc.) if in addition to the tjiyi) data we have independent information about
the value of q and the dependence of the growth constant (or rate) Gc on Afi.
Fourth application of the nucieation theorem 429
For the mononuclear mechanism, however, knowing the Gc(Aju) function is
not necessary. Then q = 0 and the slope of the t,(Afj) curve in In rrvs-Ap
coordinates is directly related to n*:
n* = -kTd(\n fj)/dAp - 1. (30.7)
For the polynuclear mechanism the contribution of the In G^ derivative in eq.
(30.6) is rather small when growth is continuous, spiral or controlled by
volume diffusion. For instance, in the particular case of Gc from (29.15) it
follows that
kTd(\n Gc)/dA^ = (1- e-^lkTyl (30.8)
so that for Ap/kT > 2 the last summand in (30.6) is practically equal to q.
However, for nucleation-mediated growth the value of this summand can be
considerable, because then the In Gc derivative is related to the size of the 2D
nuclei on the surface of the growing crystallites (see eq. (28.1)).
To apply eq. (30.6) to particular t\{Afj) data obtained by isothermal variation
of Afj we need a concrete formula for the supersaturation. Considering as an
example the induction time for formation of droplets or crystallites in vapours
or solutions, from eqs (2.8), (2.9), (2.13), (2.14), (2.16) and (30.2)-(30.6) we
find
n* = -d(ln fi)/d(ln 5) - 1 (30.9)
for t{ by the mononuclear mechanism,
n* = -(1 + w/)d(ln 'i)/d(ln S) - 1 - vdd{\n Gc)/d(ln 5) (30.10)
for volumetrically measured /; (polynuclear mechanism),
n* = -(I + 6v)d(ln ?i)/d(ln 5)-1- 6vd(ln Cc)/d(ln 5) (30.11)
for optically measured t\ in the Rayleigh limit (polynuclear mechanism) and
n* = -{l + 2v)d(ln *i)/d(ln 5)-1- 2vd(ln Gc)/d(ln 5) (30.12)
for optically measured t-v in the diffraction limit (polynuclear mechanism).
Here the supersaturation ratio S is given by eqs (13.68) or (13.69) for
condensation of vapours or solute, respectively. In the particular case of 3D
growth (d = 3) of particles with time-independent growth rate (v = 1) we
have 1 + vd = 4 and 1 + 6 v = 7 which means that the calculation of n* from
(30.9) or (30.10) will be in considerable error if one of these equations is
used instead of (30.11) in conjunction with t](S) data obtained by an optical
technique in the Rayleigh limit. This emphasizes once again the point already
made (see Chapter 29) that adequate definitions and, thereby, formulae for t-t
are necessary for the different experimental techniques. This necessity merely
reflects the fact that ^ per se is not a fundamental physical characteristic of
the process of new-phase formation because of its dependence on the choice
and the resolution power of the method for its measurement.
Chapter 31
Metastability limit
A given one-component phase is metastable in the range of supersaturations
between Ay = 0 and Ay = Ay^ (see Chapter 1). The spinodal supersaturation
A^ij which corresponds to the spinodal point (if this exists) is the thermodynamic
limit of metastability, since for Ay > Ays there is no barrier to nucleation
(then W* - 0). In reality, however, metastability is lost at a certain critical
supersaturation Ayc < Ays at which, though not yet vanishing, W* is already
small enough for nucleation to occur at a perceptible rate. Historically, the
problem of finding A^ic is among the first addressed by the theory of nucleation
[Volmer 1939]. Since Ayc is rather easily obtainable experimentally, many
studies were carried out with the aim of predicting, measuring and/or analysing
the Afjc values in various cases of nucleation (e.g. Volmer and Flood [1934];
Volmer [1939]; Hirth and Pound [1963]; Katz and Ostermier [1967]; Andres
[1969]; Walton [1969a]; Katz [1970a, b]; Skripov [1972, 1977]; Heist and
Reiss [1973]; Ovsienko [1975]; Blander and Katz [1975]; Katz et al. [1975,
1976]; Miller [1976]; Blander [1979]; Anderson et al. [1980]; Chernov [1980];
Skripov and Koverda [1984]; Hale [1986,1988]; Dillmann and Meier [1989,
1991]; Kashchiev et al. [1991]; Mullin [1993]; Nuth III and Ferguson [1993];
lakubczyk and Sangwal [1994]; Strey etal. [1994]; Laaksonen er a/. [1994];
Baidakov [1995]; Fisk et al. [1998]). In contrast to A^is, however, Ayc is not
a fundamental characteristic of the supersaturated old phase, because it is
sensitive to the concrete kinetics of nucleation and growth of the particles of
the new phase and to the particular experimental technique employed for
detecting the onset of the process of new-phase formation. Physically, therefore,
the critical supersaturation A^c is the kinetic limit of metastability. We shall
now see how A(ic can be determined in some cases of formation of condensed
phases.
One way to define Ayc is to state that it is that value of Ay at which the
stationary nucleation rate J, takes a sufficiently high value Jsc for nucleation
to become detectable [Volmer 1939]. Mathematically, therefore, Ayc is the
solution of the algebraic equation
Js(Ayc) = JS£. (31.1)
For nucleation in the volume of the old phase or on a substrate 7S c = 1
supernucleus per second per cm3 or cm2 is often used [Volmer 1939; Hirth
and Pound 1963], but any other Jsc value is equally applicable (e.g. Dillmann
and Meier [ 1991 ]; Strey et al. [ 1994]). In the scope of the classical nucleation
theory, from (13.48) and (31.1) it thus follows that for either HON or 3D
HEN of condensed phases
Metastability limit 431
A^c = [fi/ln(A/Js>c)]l/2 (31.2)
where A and B are given by (13.40) and (13.49). As seen, Apc is proportional
to the square root of the thermodynamic parameter B and changes
logarithmically, i.e. relatively weakly, with the kinetic parameter A. The
above formula is still not the explicit solution of the problem, because A is
a function of the supersaturation (see Sections 13.2 and 13.3). That is why
the exact calculation of A^/c from (31.2) can be done only numerically upon
introducing a concrete dependence of A on A^/c (e.g. Katz [1970a, b]; Katz
et al. [1975, 1976]; Dillmann and Meier [1989, 1991]; Strey et al [1994]).
Nonetheless, the weak dependence of Apc on A allows setting A- A' and
using eq. (31.2) in the form
bpz = c&m (31.3)
for an approximate analytical calculation of A^/c with an accuracy which
may be sufficient in many cases of practical interest. The kinetic factor ck is
defined as
ck = [In (A7/S,c)]-1/2 (31.4)
where A' is given by eq. (13.42) or (13.43). If A'depends on the nucleus size
«*, it is possible to use its value at n* ~ 50 for the determination of ck from
(31.4) and from the other formulae for ck in this chapter. Also, T~Te can be
used for evaluating the viscosity r| and the other ^-dependent quantities in A'
in the cases (e.g. melt crystallization) in which A' is given by (13.43).
To exemplify the application of eq. (31.3) we shall consider first nucleation
of droplets or crystallites in vapours or solutions. In this case, from (2.8),
(2.9), (2.13), (2.14), (2.16), (13.49) and (31.3) we get (e.g. Hirth and Pound
[1963])
In Sc = ck(4c3/27)1/2t;0(o-ef/^7)3/2. (31.5)
Here ck is related to A' from (13.42), and the critical supersaturation ratio
Sc is given by
Sc = pjp* Sc = IJle (31.6)
for nucleation in vapours (cf. eq. (13.68)) and by
Se = Cc/Ce, S^ = ajae, Sc = nc/TIe (31.7)
for nucleation in solutions (cf. eq. (13.69)), pc, /c, Cc, ac and nc being the
values of p, /, C, a and n at which Js is equal to 7S c. Equation (31.5) reveals
that in a temperature range in which <% changes relatively little, the critical
supersaturation ratio decreases with increasing temperature according to the
law In Sc~rV2 [Volmer 1939]. Recalling that for HON o^ = g and for 3D
HEN <7ef < o, from (31.5) we see also that Sc can attain its maximum value
only when nucleation is homogeneous. This is the reason for which
experimentally it is very difficult to measure 5C values corresponding to
HON: it is well known that in the presence of even a trifling amount of ions,
impurity molecules, various nanoparticles, etc. which act as active centres,
432 Nucleation: Basic Theory with Applications
nucleation begins at supersaturations much below the critical supersaturation
for HON. With aeC = O", c3 = 36k (spherical nuclei) and typical A' = 1035 m"3
s-1, from eqs (12.19) (31.4) and (31.5) we find that Q3SST = 0.2 to 0.6)
In Sc = O.SvdolkT?1 = 0.5(ASTA/*r)3'2 (31.8)
for HON of condensed phases in vapours or liquid solutions at a rate of 7S c
= 1 supemucleus per cm3 per second. To estimate Sc for droplets in vapours
or for crystallites in liquid solutions at temperatures between 273 and 300 K
we can use va - 0.05 nm3 and a = 10 to 100 mj/m2 in eq. (31.8). The result
is Sc ~ 1.1 to 20. Experimentally measured Sc values in many cases of HON
of droplets in vapours are in this range (e.g. Volmer [1939]; Katz [ 1970a, b];
Katz et al. [1975, 1976]; Dillmann and Meier [1989, 1991]; Strey et al.
[1994]).
As a second example of the application of eq. (31.3) we consider nucleation
of crystallites in melts when A/; is controlled by T according to (2.23). In this
case, upon setting T- Tt in eq. (13.49), from (31.3) it follows that (e.g.
Walton [1969a])
ATJT, = ck(4e3/27)"2(to0/Ase)(cJep'«:Te)3'2 (31.9)
where the critical undercooling ATC is given by (cf. eq. (2.22))
&TC = TC-TC, (31.10)
and Tc is the value of T at which Js = J,c. Again for HON, with otf = a and
the above values of c3 and A' from eqs (12.19) (used with A = TcAsc), (31.4)
and (31.9) it follows that the critical undercooling corresponding to Js c = 1
cm"3 s"1 can be estimated from the expression
ATc/7-e = 0.5(to0/Ase)(a/tre)3'2 = 0.5 Pl^iAsJk)"2. (31.11)
With ftST = 0.4 and typical &se/k = 1 to 5 this yields &TJTt = 0.13 to 0.28.
The values of the experimentally found dimensionless undercooling ATC/Te
for crystallization of various melts are predominantly in this range [Walton
1969a; Ovsienko 1975; Skripov 1972, 1977; Chernov 1980; Skripov and
Koverda 1984],
Another way to define the critical supersaturation Ayc is to identify it with
that Afj value below which the old phase can remain in metastable equilibrium
longer than an arbitrarily chosen time. As a measure of the time spent by the
system in metastable equilibrium it is convenient to use the induction time /;,
since it corresponds to the moment of detecting the onset of the phase
transformation. According to this definition, A^ic is thus the solution of the
algebraic equation [Kashchiev et al. 1991]
r;(Auc) = /j,c (31.12)
where tic (s) is the arbitrarily chosen time mentioned above. Evidently, in
manifestation of the kinetic nature of A^ic, the usage of different /;(A^)
dependences in eq. (31.12) will result in different A^c values. This will be
seen explicitly below with the help of concrete expressions for ty from
Chapter 29.
Metasiubility limit 433
The /i(A^i) dependence for various definitions of the induction time is
given in a general form by eq. (30.1) which is valid both when hfi is changed
isothermally and when it is varied by means of T. Using this equation with
A^i neglected with respect to W* (this is equivalent to approximating A in
(13.39) by A') and recalling (4.39) and (13.49), from (31.12) we find that for
HON or 3D HEN of condensed phases, classically, A/v0., Sc and ATC are again
given by (31.3), (31.5) and (31.9), but with
ct= [In (G^t',*" IK,)]-112. (31.13)
Here q and K, are defined by eqs (30.2)-(30.5) for the different mechanisms
of phase transformation and techniques for measuring the induction time.
For the mononuclear mechanism q = 0 and with the aid of (30.2) eq. (31.13)
becomes
ck = [\n(A'Vtit)]-m (31.14)
for HON or 3D HEN in the volume of the old phase (for 3D HEN on a
substrate V should be replaced by the area Ai of the substrate surface).
Comparing eqs (31.4) and (31.14), we see that they are identical when 7S c is
chosen to be equal to 1 supernucleus formed in the volume V (or on the area
As) within time ric. The situation is different, however, for the polynuclear
mechanism which is characterized by q > 0. Then the kinetic factor c^ from
(31.13) accounts for the growth of the particles, for the resolution power of
the experimental technique and for the nature of the physical process on
which this technique is based. Indeed, with q and K\ from (30.3)-(30.5),
from eq. (31.13) it follows that for the polynuclear mechanism
ck = (In [ctA'Gl"t];vdl(\ + vd) a,]}'"2 (31.15)
for volumetrically measured A^c, Sc or A7C,
ck= {\n[27iacLsA'GTt'*6vl(\+6v)^]}m (31.16)
for optically measured A^c, Sc or A7C in the Rayleigh limit and
ct = (In [2]a^A'G^t>*lvl(\ + 2v) ft]}'1'2 (31.17)
for optically measured A^c, S,. or A7C in the diffraction limit. Equations
(31.14) and (31.15) show that while for the mononuclear mechanism ck (and,
thereby, A^ic) depends on V (or As), for the polynuclear mechanism it does
not. In optical measurements involving many supernuclei, the size of the
system may have some effect on ck via the length Z.s of the path of the light
beam through the scattering medium.
To get a feeling about the numerical difference between ct calculated
fromeq. (31.14) of the mononuclear mechanism and eqs (31.15)-(31.17) of
the polynuclear mechanism we can apply these equations to HON of spherical
particles when their growth occurs with a time-independent rate Gc (then cg
= 471/3, d = 3 and v=l). With V= 1 cm3, (ic= 1 s, (¾ = 10"*, ft, = l<r\ac
= 103 /jnT4, Ls = 1 cm, the already used A' - 1035 nr3 s'1 and an exemplary
Gc = 1 jrai/s, eqs (31.14)-(31.17) yield ct = 0.122 for the mononuclear
434 Nucleation: Basic Theory with Applications
mechanism and ck = 0.137 (volumetrically), ck = 0.127 (optically, Rayleigh
limit) and c\ *= 0.133 (optically, diffraction limit) for the polynuclear mechanism.
The value of ck for the mononuclear mechanism coincides with that following
from eq. (31.4) with A' = 1035 m"3 s"1 and JSiC = 1 cm"3 s"1. The rather small
difference between the above ck values implies that even though obtained by
various experimental techniques, the A/ic values for a given system may be
nearly the same. However, other values of the above parameters can result in
ck values which differ substantially from each other. This means that for a
rigorous analysis of given Afjc, Sc or A7C data it is necessary to use eqs
(31.3), (31.5) or (31.9) with that expression for the kinetic factor ck which
corresponds to the particular experimental technique for obtaining the data
and to the concrete mechanism (mono- or polynuclear) of phase formation.
The symbols in Fig. 31.1 represent the experimental SC(T) data of Volmer
and Flood [1934] (crosses), Katz and Ostermier [1967] (up triangles), Heist
•| l_> 1 ^-i ■ 1_> 1 I I 1 1 U.-I.,.. . ■ I 1 III!
220 240 260 280 300 320
T(K)
Fig. 31.1 Temperature dependence of the critical supersaturation: symbols - data of
Volmer and Flood [1934J (crosses), Katz and Ostermier [1967} (up triangles), Heist
and Reiss [1973] (circles), Miller [19761 (squares), Anderson et al [19801 (down
triangles) and Viisanen et al. [1993] (diamonds) for HON of water droplets in vapours
at Jsc = Iff, 10'" and 1014 ra"J s~' (as indicated); curves lCf, 101" and 10" - eq.
(31.1(1) with Ju = 106, 10'" and 10" mr3 r', respectively.
and Reiss [1973] (circles), Miller [1976] (squares), Anderson et al. [1980]
(down triangles) and Viisanen et al. [1993] (diamonds) for HON of water
droplets in vapours in the range of Jsc = 106 to 1014 nr3 s~' (as indicated).
The curves display the approximate SC(T) dependence
lnSc= (16^/3 In [(Y*vaC0pJJs,ckT)(2cj/mn0)"2}}"2v0io/kT)M2 (31.18)
which follows from eqs (13.44), (31.4) and (31.5). The calculation is done
Melaslability limit 435
with C0 from (7.53), y*, v0 and m0 from Table 3.1 and u and pc determined
from [Dillmann and Meier 1991]
a(T) = 93.6635 + 0.0091337- 0.0O027572
log \pc(T)\ = 21.426045 - 2892.3693/7- 2.892736 log 7-4.9369728 x 10~37
+ 5.506905 x 10^72 - 4.645869 x 10"973 + 3.7874 x 10-|274
where a is in mj/m2, pc is in Pa, and 7is in K. The different curves correspond
to 7W = 106, 1010 and 10M m-3 s-' (as indicated). We see that the Sc(7)
dependence predicted by eq. (31.18) without free parameters is in qualitative
agreement with the experimental one. Quantitatively, however, the error is
considerable and exceeds that due to the approximations involved in the
derivation of eq. (31.18). A thorough comparison between theory and
experiment in the case of HON of droplets in vapours was done by Dillmann
and Meier [1989, 1991],
Chapter 32
Maximum number of supernuclei
In Chapter 25 we have seen that in the presence of nucleation-active centres
the maximum number Nm of supernuclei formed in the system is equal to the
number Na of active centres. In this sense, the active centres can be considered
as a factor confining the nucleation process. On the other hand, however, in
many cases spatial limitations may intervene as another factor confining this
process. This is so because regions in which nucleation is practically arrested
always exist around the growing supernuclei. These regions, to be called
exclusion zones, are schematized in Fig. 32.1 by the grey circles. The space
-^-
° /^
( 0
( •
Vg_
O
\
\
J
o
o
o
®
o
Fig. 32.1 Free (the open circles), occupied (the solid circles) and deactivated
(the crossed circles) active centres in the presence of nucleation exclusion zones
(the grey circles).
occupied by a supernucleus itself represents necessarily such a zone [Avrami
1939. 1940]. Another example of the exclusion zone is the diffusion field
with depleted monomer concentration around a supernucleus growing in a
solution (or on a substrate in contact with vapours [Sigsbee 1971, 1972])
when growth is controlled by volume (or surface) diffusion. Growing together
with the supernuclei around which they are formed, as illustrated in Fig.
32.1, the exclusion zones can ingest free active centres and deactivate them
for nucleation. Thus, more generally, instead of eq. (25.9) we shall have
Maximum number of supemuclei 437
Nm£N,. (32.1)
Our task now is to find out how much less Nm is than Na. To that end we
shall determine^ as a function of N3, the nucleation rate /a(s~') per active
centre and the parameters of growth of the exclusion zones. Before doing
that, however, we note that /Vm is not always controlled by the combined
action of the active centres and the exclusion zones. Other important factors
in the establishment of the maximum number of supemuclei in a nucleating
system are. e.g. the exhaustion of the supersaturation in the system or the
coalescence between the growing supemuclei. As Nm is a quantity which is
relatively easily accessible experimentally, much theoretical work was devoted
to its determination in various cases of nucleation [Kolmogorov 1937; Avrami
1939, 1940; Todes 1949b; Nielsen 1964; Zinsmeister 1966, 1968, 1969,
1971; Logan 1969; Lewis 1970; Stowell 1970,1972b, 1974a, 1974b; Routledge
and Stowell 1970; Stowell and Hutchinson 1971; Markov and Kashchiev
1972a, 1972b, 1973; Venables 1973, 1994; Robinson and Robins 1974;
Venables and Price 1975; Kashchiev 1975a, 1979b; Markov 1976, 1995;
Markov and Stoycheva 1976; Shvedov et al. 1977; Lewis and Anderson
1978; Stoyanov and Kashchiev 1981; Venables etal. 1984; Obretenov 1988;
Villain et al 1992; Pimpinelli et al. 1992; Evans and Bartelt 1994; Wolf
1995; Kukushkin and Osipov 1995; Ratsch et al. 1995; Bartelt et al. 1996;
Amar and Family 1997; Jensen and Larralde 1997; Jensen et al. 1997, 1998;
Detsike/a*. 1997, 1998],
32.1 General formulae
Our analysis is restricted to the simple case of nucleation on active centres
with equal activity [Markov and Kashchiev 1972a], but its generalization to
the more realistic case of centres with different activity is straightforward
[Markov and Kashchiev 1972b, 1973]. The supemuclei are presumed to
appear only on the active centres which are randomly located in the old
phase (Fig. 32.1). An exclusion zone with effective radius Rz forms around
a given supernucleus at the moment f of appearance of the supernucleus. In
the case of volume nucleation (active centres in the volume of the old phase)
and 3D growth of the zone, similar to (26.7), at a later time t the zone volume
Vz is given by
>=<^=4r
Vz(r',0 = czfiz3 = cz Gz(t")dt"
(32.2)
where cz is a numerical shape factor (e.g. c7 = 4.T/3 for spherical zones), and
Gz(r) = dRJdt is the zone growth rate. By definition, no nucleation can occur
on those of the active centres (the crosssed circles in Fig. 32.1) which are
ingested by the growing exclusion zones. This means that at time I > 0 there
will be only NaPf(t) active centres which are still free for nucleation, Pf being
the probability that a given centre is free, i.e. neither occupied by a supernucleus
438 Nuclealion: Basic Theory with Applications
formed on it nor deactivated for nucleation as a result of its ingestion by a
zone. Hence, if the probabilities for non-occupation and non-ingestion are
Pno and Pni, respectively, for Ps we shall have Pf = Pn0Pni. Realizing that /VaPf
has the physical significance of Na - N in eq. (25.13), we can use this
equation with Na - /V replaced by NJ>[ in order to obtain the general time
dependence of the number N of supernuclei formed on active centres in the
presence of exclusion zones. Integration under the initial condition W(0) = 0
thus yields
W) = N, f Ut')Pm(S)P*i(t') it'- (32.3)
Jo
To proceed further we must know the probabilities Pno and Pni. Since the
probability Pn0 for non-occupation is equivalent to the (/Va - N)/N„ ratio in
eq. (25.16), we can write
P„0(/) = exp [- AUOWal (32.4)
where the extended number /V„, of supernuclei is specified by (25.17). As to
the probability Pni for non-ingestion, in the case of volume nucleation it can
be expressed as the ratio between the overall volume of the old phase left
outside the zones and the initial volume V of the old phase. In the scope of
the KJMA theory this ratio corresponds to 1 - a from eq. (26.4) so that for
Pni we shall have
Pni(/) = exp [- VCKM/V] (32.5)
where the extended volume Vexz of the zones is given by (26.5) with J and
Vn replaced by Va and V, from (25.10) and (32.2):
W) = N* f UW, t) dt'. (32.6)
Jo
Thus, with the help of eqs (25.17) and (32.4)-(32.6), from (32.3) it follows
that [Markov and Kashchiev 1972a]
N(t) = nA 7„((') exp J- f 7a (t")[l + N.VAt", t')IV] it" 1 At'.
(32.7)
This general formula describes the temporal evolution of the number of
supernuclei under the combined action of active centres and exclusion zones.
We see that in the absence of such zones (Vz = 0) or when N„ is sufficiently
small, eq. (32.7) turns into eq. (25.16). In the opposite limiting case of a
large enough number Na of active centres and existence of zones around the
supernuclei (Vz * 0), eq. (32.7) takes the form
W) = Wu f Ut') exp (- (NJV) f Jt(t")Vz(t", f) dr"} 6i. (32.8)
Jo Jo
Maximum number of supernuciei 439
When the zones coincide with the supernuciei themselves, we have Vz = Vn
where V„ is given by (26.7). Then, iD view of (25.10) and (26.5), eq. (32.7)
represents the N(t) dependence derived by Avrami [1939, 1940], and eq.
(32.8) is identical with the Kolmogorov formula (26.10). We note also that
N(t) from (32.7) must be used with Na - 1 instead of Na in front of V7 if Na
does not satisfy the condition NR » 1 [Markov and Kashchiev 1972a].
Having found the N(t) dependence, we can now determine the maximum
(in fact, the saturation) number Nm = N(t*>) of supernuciei formed on the Na
active centres in the system. Setting t = «> in eq. (32.7), we get
Nm=Na [ 7a(0exp|- J Ja(t')[\ + NllVz(t',t)/V]dt'ldt. (32.9)
This is the sought formula for Nm in the case of volume nucleation on
active centres with equal nucleation activity, which are subject to deactivation
by nucleation-exclusion zones. As required, when Vz = 0 or Na is a sufficiently
small number, eq. (32.9) passes into eq. (25.9). Conversely, in the presence
of zones (Vz ;£ 0) and a large enough number #a of active centres, in conformity
with (32.1), the value of the definite integral in eq. (32.9) is less than unity.
In this limiting case eq. (32.9) simplifies to the expression
Nm =** Jo Mt)exp\-WV)j'
Ja(t')Vz(t',t)dt'\dt (32.10)
which follows also from (32.8) at t = «>. As it should be, since Ja is related
to the nucleation rate J per unit volume by (25.10), eq. (32.10) coincides
with the Kolmogorov formula (26.11) in the case when Vz = Vlt, i.e. when the
supernuciei themselves play the role of exclusion zones. In this case eq.
(32.9) parallels the expression for Nm of Avrami [1939, 1940].
It must be pointed out that although pertaining to volume nucleation, i.e.
to HEN in the volume of the old phase, the above results are entirely applicable
also to surface nucleation, i.e. to HEN on a substrate with Na active centres
on the substrate surface. In this case the exclusion zones are two dimensional
and it is only necessary to replace Vand Vz in eqs (32.5)-(32.10) by, respectively,
the area As of the substrate surface and the areaA2 of a zone on the substrate.
Analogously to (32.2), Az is given by
Az{t\t) = czRl = cz\ Gz{t")dt
4f<
(32.11)
where cz is a numerical shape factor (e.g. cz - it for circular zones).
To see explicitly what is the effect of /Va, Ja and Gz on Nm we shall now
apply the general formula (32.9) to the particular cases of IN and PN in
stationary regime when the zones grow radially according to the power law
Rz = {GeJt)". (32.12)
This expression parallels (26.16) and in it v> 0 is a number expected to be
usually less than or equal to unity, and Ccz (m"vs_l) is the zone growth
440 Nucleation: Basic Theory with Applications
constant. When the zones coincide with the supemuclei themselves (then Rz
s R), we have Gcr s Gc, Gc being the growth constant of the supemuclei.
32.2 Instantaneous nucleation
Similar to J{t) from (26.13), in this case Ja(?) can be represented as
4W = Sd«, (32.13)
since each of the Na active centres in the old phase becomes occupied by a
supernucleus at the very beginning of the nucleation process. Using this
expression for Ja and recalling the property of the Dirac delta-function So,
noted in Section 26.2, from (32.9) we find that in the IN case, for both
volume and surface nucleation,
Nm = N. (32.14)
regardless of the growth law of the zones. Physically, this result is obvious:
as the supemuclei appear simultaneously on the active centres at t = 0, none
of the centres can be ingested by the zones formed around the supemuclei,
no matter how fast the zone growth may be.
32.3 Progressive nucleation
In the case of PN in stationary regime JJj) equals the time-independent
stationary nucleation rate 7a s per active site. Also, for zone growth according
to (32.12), from (32.2) and'(32.11) we have
Vl(t',t) = c1Gli(t-t'?\ Al(t',t) = czGl*(t-t')2v (32.15)
so that eq. (32.7) leads to the following N(t) dependence:
N(t) = Na\ exp \-x - (B^)1 + vd] dx. (32.16)
Jo
Here the zone dimensionality d and the numerical factor B, > 0 are given by
d = 3, Bz= [c,/V,Gc3,zV(l +3v) V7a1J]"(,+3v> (32.17)
for volume nucleation (3D zones) and by
d = 2, Bz = [c2/VaGc2zV(l +2v)A,Jl!]m+2v'> (32.18)
for surface nucleation (2D zones). Accordingly, for the maximum number of
supemuclei in the considered case of PN, from (32.9) or (32.16) we obtain
Wm = /Va f exp [-X- (fi^i)1 + *"] Ax. (32.19)
Jo
Although the integrals in eqs (32.16) and (32.19) cannot be performed in
closed form for arbitrary values of the power v in the zone growth law
Maximum number of supemuclei 441
(32.12), inspection of these equations shows that the character of the N{t)
dependence and the departure of Nm from /Va is controlled by the parameter
BL. Indeed, under the condition
Bz « 1 (32.20)
which corresponds to few active centres, slow zone growth and high nucleation
rate, eqs (32.16) and (32.19) turn into (25.7) and (25.9). Then all active
centres succeed in generating supemuclei before being ingested by the exclusion
zones and the nucleation process is confined only by the finite number of the
active centres. Under the opposite condition, i.e. when
fiz»l, (32.21)
the active centres are numerous, the zone growth is fast and the nucleation
rate is low. Then a considerable number of the centres are deactivated as a
result of ingestion by the zones and the process is confined by both the finite
number of the centres and the spatial limitations due to the zones. In this case
the Bz term in (32.16) and (32.19) is dominant and these equations take the
form
N{t) = (NJBZ) exp (- *' + "*) dx (32.22)
Jo
Nm = T[(2 + vd)l(\ + vd)\NJBz, (32.23)
r being the gamma-function defined by (26.21). Equation (32.23) reveals
that in the JVa —> ~ limit Nm is not any more a linear function of Na: with Bz
from (32.17) and (32.18), it follows that
Nm = T[(2 + 3v)/(l + 3v)][(l + 3v)V7cJl/(l + iv>(N,IJGczfm * 3">
(32.24)
for volume nucleation (d = 3) and that
Nm = T[(2 + 2v)/(l + 2v)][(l + 2v)As/cj"(l + 2v>(/VaJa,s/GCiZ)2v/" + 2v>
(32.25)
for surface nucleation (d = 2). As seen, the effect of the zones manifests itself
by weakening the linear increase of Nm with Na and making Nm depend on
Jas and Gcz. For example, when the zones grow with time-independent rate
(v- 1), we have Nm °= Na'4 and Nm °= N2'3 for volume and surface nucleation,
respectively. We note also that when the role of the zones is played by the
supemuclei themselves, we have cz = cg, Gcz = Gc and in view of (25.3) and
(32.17), as it should be, Nm from (32.24) becomes identical with /Vm from
(26.26).
Knowing the limiting dependences of Nm on /Va, eqs (25.9) and (32.23),
we can now employ them to find an interpolation formula for Wm which is
valid for all values of Na. Expressing 1/Nm as the sum of the reciprocals of
the right-hand sides of (25.9) and (32.23) and rearranging, we get
442 Nucleation: Basic Theory with Applications
Nm = /Va( 1 + Bjr\(2 + vd)/(\ + vd)]}"'. (32.26)
This Nm(Na) dependence is in fact an approximation to the exact one given
by eq. (32.19). With fiz from (32.17) and (32.18) it leads to
/V„, = /Va(l + [cz/VaG3(Z7(l + 3v) VJa3;]"(1+3v)/T[(2 + 3v)/(l + 3v)])-'
for volume nucleation (d = 3) and to
Nm = N,{\+ lc./V.G^VO + 2v)AiJll]WMv)l T[(2 + 2v)/(l + 2v)])-'
(32.28)
for surface nucleation (d = 2). As seen, for sufficiently small or large Nn (or
B,) values eqs (32.26)-(32.28) pass into the limiting formulae (25.9) or
(32.23)-(32.25), respectively.
The N(t) dependence for volume nucleation (d = 3) and linear growth of
the zone radius (v = 1) is depicted in Fig. 32.2 by the solid curves which are
..^5 _
105.-
6*1o4--^l
2»10^
"012345
w
Fig. 32.2 Time dependence of the number of supemuclei on active centres: solid
curves - eq. (32.16) for nucleation in the presence of exclusion zones: dashed curves
- eq. (25.7) for nucleation in the absence of such zones (Na values indicated).
drawn according to eq. (32.16) at /Va = 2 x 104,6x 104 and 105 (as indicated).
The corresponding Bz values of 0.38, 0.50 and 0.57 are calculated from eq.
(32.17) with cz = 47C/3 (spherical zones), V = 1 cm3, Ja s = 0.1 s~' and Gc z =
10 jUm/s. The dashed curves represent eq. (25.7) which describes the limiting
N(t) dependence in the absence of zones. We observe that with increasing Na,
due to the effect of the zones, saturation of the N(t) dependence occurs at Nm
values increasingly less than the respective /Va ones.
Figure 32.3 illustrates the decrease of the /Vm//Va ratio with Bz, i.e. with /Va.
8x10"
6x104
Z
4x10"
2x104
Maximum number of supernuclei 443
Fig. 32.3 Dependence of the Nn/N^ ratio on the zone parameter B7: circles -
computer simulation data for the case of V = 1 and d = 2 [Obretenov 1988]; solid
curve - eq. (32.19) also at V = 1 and d = 2; dashed curve - the corresponding
approximate eq. (32.26); dotted curve C - eq. (25.9) for nucleation confined by
active centres only; dotted curve C&Z - eq. (32.23) for nucleation confined by
both active centres and exclusion zones.
The solid curve displays the exact dependence (32.19) at d = 2 (surface
nucleation) and v = 1, and the dashed curve is the respective approximation
(32.26). As seen, the dashed curve follows with an error of less than 20% the
solid curve and interpolates between the dotted curves C and C&Z which
show, respectively, the dependences (25.9) and (32.23) for nucleation confined
by the centres only or by both the centres and the zones. The circles in Fig.
32.3 represent the /Vm(Bz) data obtained by Obretenov [1988] in a computer
simulation of 2D-nucleation on active centres on a substrate (d = 2) in the
presence of zones coinciding with the supernuclei themselves and growing
radially with a time-independent velocity (v= 1). We observe that these data
are described well by the exact formula (32.19).
The above N(t) and Nm(Nt) dependences are accessible to a direct
experimental verification. Following Markov and Stoycheva [1976],
Trofimenko et al. [1979, 1980, 1981] used eqs (25.7) and (32.22) in the
unified form
N(r)//Vm= [H1+ !/<?)]-
• rati/
Jo
exp (- n?) dx (32.29)
to analyse N(t) data for electrochemical nucleation under potentiostatic
conditions, i.e. at constant supersaturation. Here W s (aW/d/),=0 and Nm are
obtainable from the initial slope and the plateau of a given experimental N(t)
444 Nucleaiion: Basic Theory with Applications
dependence so that, given the q value, eq. (32.29) is a master curve which
corresponds to nucleation confined by active centres when q = 1 and by both
centres and zones when q = 1 + vd > 1. For instance, q = 2 or 3 implies v =
1/2 or 1, because in electrochemical nucleation the exclusion zones (which
may be of electrical or diffusion origin) are on the electrode surface and are
necessarily two dimensional: we have d = 2. The symbols in Fig. 32.4
represent the N(t) dependence obtained by Trofimenko et al. [1979] in
nucleation of Cu crystallites on a graphite electrode at two different
concentrations of the Cu ions in aqueous solution of CuS04 at T = 298 K,
Curves 1 and 3 are drawn according to eq. (32.29) with q = 1 and 3, respectively.
The agreement between theory and experiment led Trofimenko et al. [1979]
to the conclusion that at higher (the circles) or lower (the squares) concentrations
of Cu ions the nucleation process is limited, respectively, by both the active
centres and the exclusion zones on the electrode or by the active centres
only.
Z
Fig. 32.4 Time dependence of the number of supernuclei on active centres: symbols
— data for electrochemical nucleation of Cu crystallites on a graphite electrode in
aqueous solution ofCuS04 at T = 298 K and at two different concentrations of the Cu
ions in the solution [Trofimenko et al. 1979}; curves 1 and 3 - eq. (32.29) with q = 1
and 3, respectively.
The A'mfA'a) dependence was studied experimentally by Stenzel and Bethge
[1976] in nucleation of Au crystallites on the (100) face of an NaCl crystal
doped with Ca. In this case of deposition from vapours under ultra high
vacuum conditions the active centres are Ca ions on the substrate surface,
and the exclusion zones are due to surface diffusion of Au adatoms towards
the Au supernuclei so that d = 2. Under the assumption of parabolic growth
Maximum number of supernuclei 445
of the zone radius (then v= 1/2), in conformity with eqs (32.18) and (32.19)
the NJNa ratio is expected to decrease with the increase of the concentration
/Va/As of Ca ions on the substrate surface according to
NJN,
exp [
- (caNjAJx2] dx
(32.30)
where c0 s cfi^JU^, is a constant at constant supersaturation. The circles
in Fig. 32.5 represent the Nm(N,) data of Stenzel and Bethge [1976] for
nucleation at substrate temperature T= 473 K and impingement rate /= 3 x
1016 m"2 s"1. The JVm/JVa values in this figure are less than those reported by
Stenzel and Bethge [ 1976] because they are corrected for the presence of the
extra 43 crystallites per ,um2 of the substrate surface, which were observed
to nucleate on a Ca-ion-free substrate. The solid curve is drawn according to
eq. (32.30) with c0 = 170 nm2 and is seen to reproduce the experimentally
found drop of Nm/Na with NJAS. The dashed line in Fig. 32.5 indicates the
value of Nm in the absence of zones. The departure of the experimental
Nm(Nt) data from this line at NJAS = 1014 m~2 means that at higher
concentrations of active centres the nucleation process is confined by both
the centres and the zones on the substrate surface.
1.0
« 0.6
Z 0.4
0.2 -
10"
—•—^^^-■ -
10"
101'
Na/As(m^)
Fig. 32.5 Dependence of the N„/Na ratio on the concentration of active centres:
circles - data for nucleation in ultra high vacuum of Au crystallites on the (100) face
ofNaClatT = 473Kandl = 3xl0i6r-2 -■' (c --' -J "-'-- >n™ —
(32.30) with c0 = 170 nm2; dashed line -
exclusion zones.
. ' [Stenzel and Bethge 1976}; curve - eq.
i. (25.9) for nucleation in the absence of
Chapter 33
Size distribution of supernuclei
The comprehensive description of the kinetics of formation of a new phase
at the nucleation stage requires knowing not only the nucleation rate J(t) and
the concentration £(r) of all supernuclei in the system, but also their size
distribution function Z„(r) or Z(n, t). In the case of the Szilard model of
nucleation Z„(r) is the solution of the master equation (9.16) or (9.23). When
the number n of molecules in the clusters is treated as a continuous variable
(as it will be done in this section), the size distribution Z(fl, t) has to be found
by solving the master equation (9.25). Finding Z(n, t) in this way is, however,
a formidable mathematical problem: in Section 15.1 we have seen that even
in the simplest case of nucleation in a closed system at constant supersaturation
Z(n, t) can be determined explicitly only for n = «*, i.e. for clusters of near-
nucleus size. This determination is possible, because the 'drift' flux (the vZ
term in eq. (9.29)) is not important when the cluster size is in the nucleus
region, i.e. when nl < n < n2 where the left and right ends H] and n2 of the
nucleus region are defined by eqs (7.41) and (7.42). For n > n2, however, the
'drift' flux is not any more negligible - on the contrary, it is the dominant
component of the f[ux j(n, t) from eq. (9.29). How can we then find the size
distribution Z(n, t) of the supernuclei whose size is outside the nucleus
region, i.e. of the supernuclei which are constituted of n > n2 molecules and
are, therefore, subject to stable overgrowth?
Answering this question is our task in this chapter. It is really highly
desirable to know Z(n, t) for n > n2, because the large enough supernuclei are
experimentally detectable and their size distribution is accessible to a direct
experimental determination. Moreover, using Z(n, t) in eq. (11.13) allows
calculating the concentration ^ of all detectable supernuclei, which is also
an experimental observable. This is why a considerable interest has been
shown hitherto in the theoretical solution of the problem of the size distribution
of supernuclei during nucleation (e.g. Johnson and Mehl [1939]; Roginsky
and Todes [1940]; Todes [1940, 1949b]; Avrami [1941]; Bauer et al. [1966];
Toschev and Gutzow [1967b]; Zinsmeister [1969, 1974]; Robertson and
Pound [1973]; Kashchiev [1975b]; Borovinskii and Kruglova [1977]; Belenkii
[1980]; Stoyanov and Kashchiev [1981]; Trofimov [1983]; Kuni [1984b];
Family and Meakin [1988, 1989]; Osipov [1989, 1990a, b, 1993]; Bartelt
and Evans [ 1992]; Evans and Bartelt [ 1994]; Amar and Family [ 1995]; Ratsch
et al. [1995]; Kukushkin and Osipov [1995, 1997]; Bartelt et al. [1996];
Jensen and Larralde [1997]; Jensen et al. [1997, 1998]; Detsik et al. [1997,
1998]).
Size distribution of supernuclei 447
33.1 General formulae
Under typical nucleation conditions the supernuclei do not appear and/or
vanish as a result of non-aggregative processes. Being interested in the solution
of the master equation (9.25) only for the supernuclei of size n> n2 (to be
called stable supernuclei), we have therefore the right to set K„{t) = Ln{t) =
0 in this equation and, as discussed in Section 9.4, use it in its approximate
form (9.43). The initial condition for eq. (9.43) is given by eq. (9.2). The
only boundary condition needed for it at n = n2 is of the form
Z(n2, t)=j(n2, l)lv(ni,t) (33.1)
which follows from (9.29) upon neglecting the 'diffusion' flux - j\dUdri).
Here j(n2, t) is the flux of supernuclei at the right end n2 of the nucleus
region, and v{n2, t) is the growth rate of the «2-sized (i.e. the smallest stable)
supernucleus. While the latter is obtainable from eq. (9.30), j(n2y t) has to be
found from (9.28) with the help of the solution Z(n, t) of eq. (9.25) in the
1 < n < n2 region. However, to a first approximation [Osipov 1989, 1990a,
b, 1993], j(n2, t) can be replaced by the nucleation rate J(t) which is known
theoretically in various cases of nucleation (see Chapters 13, 15 and 17). It
is important to note that, due to (13.1) and (13.2), this approximation does
not introduce any error when the nucleation process is stationary.
Experimentally, the size of the detectable supernuclei is usually measured
in terms of their effective radius R rather than in terms of the number n of
molecules in them. That is why in eq. (9.43) and its initial and boundary
conditions (9.2) and (33.1) it is expedient to change the variable n to R.
For clusters with a fixed shape the corresponding size distribution function
F(R, t) and growth rate G(R, t) = dRidl are related to Z(n, t) and v(n, t) =
dnldt according to
F(R, t)dR = Z(n, t) dn (33.2)
G(R, t) = v(n, t) dR/dn. (33.3)
Equation (33.2) reflects the fact that just those clusters whose size is between
n and n + dn are with radius between R and R + dR. We note also that F(R,
t) is in m"" or m"3 for volume or surface nucleation, respectively. Thus,
employing (33.2) and (33.3) in eqs (9.2), (9.43) and (33.1), approximating
j(n2, t) by J{t) and restricting the analysis to nucleation at no pre-existing
supernuclei outside the nucleus region (then Z(n, 0) = 0 for n > n2), we get
(R2 < R < RM)
|WO + ^[C(J!,I)F(1!,/)] = 0 (33.4)
F(R, 0) = 0 (33.5)
F(R2, t) = J(t)IG(R2, t). (33.6)
Here R2 is the radius of the n2-sized supernucleus, and RM is the radius of the
448 Nuclealion: Basic Theory with Applications
largest possible cluster containing all M molecules available in the old phase.
In most cases it is possible to set RM = °°.
When solving eqs (33.4)-(33-6) for the unknown size distribution function
F{R, t) it is necessary to know the concrete dependence of the growth rate G
on R and t. As in Chapter 27, this dependence is obtainable by model kinetic
considerations in each particular case of interest. However, the exact analytical
solution of eqs (33.4)-(33.6) can be found in a general form [Osipov 1989,
1990a, b, 1993] when the G(R, t) function is given by
G(R, t) = G,(K)G2(I). (33.7)
This presentation of G as a product of two known functions G\ {R) > 0 and
G2(0 > 0 which depend only on R and t, respectively, covers a fairly large
class of cases of formation of new phases. For example, when the supernuclei
grow according to the power law (26.16), we have
G,(R) = vGzR' ~ "v, G2(i) = 1. (33.8)
With these Gi(R) and G2(() functions, G from (33.7) coincides with G from
(26.15).
Let us now find the size distribution F(R, i) of the stable supernuclei when
their growth rate G(R, t) is specified by eq. (33.7). In doing that, however,
instead of solving eqs (33.4)-(33.6) by mathematical methods [Osipov 1989,
1990a, b, 1993; Kukushkin and Osipov 1995, 1997], we shall employ the
physical arguments used by Roginsky and Todes [1940] (see also Todes
[1940, 1949b]; Avrami [1941]; Toschev and Gutzow [1967b]; Kashchiev
11975b]). The main idea behind these arguments is that when G is a known
function of R and t, at any time t > 0 there exists an unambiguous
correspondence between the radius R of a given supemucleus and the earlier
moment t' S t of its appearance. Indeed, since G - dR/dt, treating eq. (33.7)
as a differential equation with respect to the time dependence of R and
solving it under the initial condition R = R2at t = t\ we get
f [1/0,(/0] dfi"= f G2(r")dr". (33.9)
This formula represents implicitly the functions R(t', t) and t'(R, t) which
relate at any time ( the radius R and the moment t' of birth of a given
supemucleus when its growth is not disturbed by contacts with neighbouring
supernuclei. The supernucleus born at / = 0 is the biggest in size and its
radius Rb is obtainable from
[l/G,(R")]dK*= G2(t")dr". (33.10)
R2 Jo
In some cases eqs (33.9) and (33.10) lead to simple explicit t'(R, t) and
fi(r\ t) dependences. For instance, with the aid of G, and G2 from (33.8) we
find that
t'{R, t) = r- (1/Gc)(fi"v - R\'v) = (1/GJ[/?;'"(/) - fi"1 (33.11)
R</, t) = [ R'2'v +Gc(r-t')]v (33.12)
Size distribution of supernuclei 449
/fb(0 = («r + Gc')v (33.13)
for supernuclei growing according to the power law (26.16).
Now, let d£ = F{R, t)dR be the concentration of supernuclei with radii
from R to R + dR. It is clear from the aforesaid that these supernuclei formed
in the period between the moment t' and the earlier moment t' - dt'. As in this
period the nucleation rate is J(t'), in conformity with (11.1) we can represent
d£as d£= - Ji/)dt', the minus sign taking into account that a greater radius
R corresponds to an earlier moment f of formation. Hence, we have
F(R, t) dR = -/(0 dt'. (33.14)
In this relation, due to (33.9),
dR = -G,(fi)G2(r') dr' (33.15)
so that it takes the form [Roginsky and Todes 1940; Todes 1940, 1949b]
F(R, t) = MR, oyGmGiVW. N, R2<R<Rb(t) (33.16)
F(,R, () = 0, R > R„(t)
where the functions t'(R, t) and Rb(t) have to be determined from eqs (33.9)
and (33.10). For example, for supernuclei obeying the growth law (26.16) l'
and Rb are given by (33.11) and (33.13).
The F(R, t) function (33.16) is the sought exact solution of eqs (33.4)-
(33.6) with G(fi, t) from (33.7). This can be verified by substituting this
function in eq. (33.4), differentiating and accounting that in view of (33.9)
dt'ldt = G2{i)IG2{t') and dfldR = - l/G,(fi)G2(r'). Physically, however, eq.
(33.16) is approximate, since the boundary condition (33.6) which it obeys
is only an approximation to eq. (33.1) because of the replacement o£j(n2, t)
in the latter by J(t). It is therefore important to have also the mathematically
exact solution of eq. (33.4) again under the initial condition (33.5), but under
the exact boundary condition (33.1). Similar to F(R,t) from (33.16), when
G(R,t) is given by (33.7), this solution has the form
F(R, t) = jRl [t'(R, t)]IGl{R)G2[t'(R, t)], R2<R< fib(r)
F(R, 0 = 0, R > Rb(t)
where jn7(l)= j(n2, /) is the flux at R = R2, and the R,t dependence of f is
obtainable from (33.9). Direct substitution proves that this F(R, t) function
is the exact solution of eqs (33.1), (33.4) and (33.5).
Equation (33.16) describes the size distribution of stable supernuclei in
either HON or HEN only when the process is not affected by the spatial
limitations due to nucleation-exclusion zones or by the exhaustion of nucleation-
active centres. In order to extend the applicability of this equation to the
more advanced stage of the nucleation process when the confinements from
the centres and the zones already play a perceptible role (see Chapters 25
and 32), we have to multiply J in (33.16) by the probabilities Pn0 of non-
occupation and P„| of non-ingestion given by (32.4) and (32.5). Understandably,
450 Nucleation: Basic Theory with Applications
like 7 in (33.16), Pao and Pnt must refer to the moment f of appearance of the
fl-sized supernuclei. Thus, with the help of eqs (25.17), (32.4)-(32.6) and
(33.16), we find that for volume nucleation
F{R, t) = {J[t'(R, t)VGx{R)G2[f{R, t)]}Pno[f(R, OlWC*, 0],
R2<R<Rb(t) (33.17)
F(R, 0 = 0, R > Rb(t)
where
I
Pm(t')P„iC') = exp -J J,(t")[l +N,VI(t",t')IV\it" . (33.18)
In these expressions the R, t dependence of f is obtainable from (33.9), J and
Ju are related through eq. (25.10), and the zone volume Vz is specified by
(32.2). In the case of zones coinciding with the supernuclei themselves we
have V7 = Vn where the individual volume V„ of a supernucleus is given by
eq. (26.7) in R2 - 0 approximation. In this caseeq. (33.17) represents the size
distribution function found by Avrami [1941]. Clearly, eq. (33.17) remains
in force also for surface nucleation, i.e. for HEN on a substrate with Na
active centres, but then J and J„ are related through (25.11), and Vz and V
must be replaced, respectively, by the zone area Az from (32.11) and the area
As of the substrate surface.
In the limiting case of nucleation on active centres in the absence of zones
(then Vt = 0) the size distribution (33.17) simplifies to
F(R, 1) = {J[t'(R, t)]/G,(R)G2[t\R, J)]}
[-r
•mi j)
exp I- I J,{t")dt"
R2iRSRb(t) (33.19)
F(R, /) = 0, R > Rb(t)
where eqs (25.10) and (25.11) connect J and Ja for volume and surface
nucleation, respectively. In the opposite limiting case of confinements from
both centres and zones (then the unity in (33.18) is negligible with respect to
the Vz summand), with the aid of (25.10) eq. (33.17) takes the form
F(R, t) = (■/[/(/?, /)]/G,(fi)G2[r'(fi, r)]}
exp J- f ' J(t") Vz[t", f(R, /)] dr" I, R2 < R < Rb(t)
F(R, /) = 0, R > Rb(t). (33.20)
This expression describes the size distribution of supernuclei in volume
nucleation, but is directly applicable to surface nucleation provided in it Vz
is replaced by the zone area Az from (32.11). Setting Vz = V„, we can use eq.
(33.20) also when the nucleation-exclusion zones coincide with the supernuclei
Size distribution of supernuclei 451
themselves. When the effect of both the centres and the zones is negligible
(this is so, e.g. at the sufficiently early stage of the nucleation process), the
product P„0Pni in eq. (33.17) and the exponential factors in eqs (33.19) and
(33.20) are close to unity and these equations pass into eq. (33.16). We note
as well that it is only the nucleation rate J per unit volume or area that enters
eqs (33.16) and (33.20). The absence of Ja from these equations means that
they are applicable also to HON.
Knowing the size distribution function F(R, /), we can determine the time
dependences of a number of quantities of experimental interest. Such quantities
are, e.g. the most probable radius Rm of stable supernuclei, the value Fm of
the maximum of F, the average radius /?av of detectable stable supernuclei,
the total concentration £ (m"1 or rrr2) of detectable stable supernuclei and
the detectable nucleation rate f (irf3 s~' or m~2 s_!). By definition,
Fm(t) = F[Rm(.t), t], (33.21)
where Rm(l) is the solution of the equation
[5F(fi,0/*]fl=flln(/) =0. (33.22)
Also by definition,
f«b(0
«av(0 = ICC)]'1 RF(R,t)dR, (33.23)
and according to (11.13) and (33.2),
f(r)= F(R, t)dR. (33.24)
Here R' > R2 is the radius of the smallest detectable stable supernucleus (this
supernucleus consists of ri ^ n2 molecules), and F(R, t) is given by the
general equation (33.17) or by eqs (33.16), (33.19) and (33.20) in the respective
limiting cases.
It is worth noting that, equivalently, £ can be determined from the expression
J"«'">
Af)PW')P^f) dr- (33.25)
o
which follows from (33.24) with the aid of (33.9), (33.10), (33.15) and
(33.17). Here the PmPn[ product is given by (33.18), and t'R'(t), defined by
(cf. eq. (33.9))
| [1/G,(K")1 dR" = \ G2(t")dt", (33.26)
is the moment of appearance of those stable supernuclei that at time I have
the smallest detectable radius R'. For example, for supernucleus growth
characterized by G,(R) and G2(t) from (33.8), eq. (33.26) leads to
452 Nucfeation: Basic Theory with Applications
t'R.(t) = t-tE (33.27)
where tg is given by
rg = (1/Gc)(fi'"v - R2l/V). (33.28)
According to eq. (33.11), rg has a simple physical meaning: it is the time
needed for the growth of a supernucleus from radius R2 to radius R'. In other
words, lg is merely the growth time of the smallest detectable stable
supernucleus. As seen from eqs (33.26)-(33.28), in the case when even the
/?2-sized supernuclei are detectable (then R' = R2), we have t'R.(t) = t and rg
= 0. In this case, in the absence of pre-existing supernuclei (£0 = 0) and of
confinements from active centres (Pn0 = 1) and exclusion zones (P„i ~ 1), £
from (33.25) is identical with f from (11.10).
Finally, the time dependence of the detectable nucleation rate / is obtainable
from the expression
J'(t)={G1(t)IG2[t'R,{t)]\J[t'R.(t)}Pm[t'R\t)\PJt'R.(t)] (33.29)
which results from (11.11) and (33.25), since owing to (33.26), dt'g/dt =
G2(t)/G2[t'R-(t)]. Naturally, we could get this expression by employing
G(R, t) and F(R, t) from (33.7) and (33.17) in the general relation for f,
J'(t) = G(R', t)F(R', t), (33.30)
which follows from eqs (11.14), (11.15) (with neglected/' summand), (33.2)
and (33.3). When the growth of the detectable supernuclei is characterized
by G|(fi) and G2(t) from (33.8), thanks to (33.27), / from (33.29) simplifies
to (t > tg)
f(t) = 7(( - te)Pm(t - gpni« - rg) (33.31)
where tg is specified by eq. (33.28). As seen, compared with the nucleation
rate J, in this case the detectable nucleation rate f is merely 'shifted' to
longer times, the shift on the time axis being equal to the growth time tg of
the smallest detectable stable supernucleus. From t = 0 to t = t„ we have J'(t)
= 0, as then the stable supernuclei in the system are undetectable.
33.2 Particular cases
To exemplify the application of the results in Section 33.1 we shall now
consider several of the commonest cases of nucleation and growth. In doing
that we shall use the approximation R2 = 0, because n2 ~ n* (see eq. (7.42))
and in many practical situations the nucleus size n* is negligibly small. The
examples will be restricted to supernucleus growth characterized by G] (R)
and G2(t) from (33.8).
1 Instantaneous nucleation
In this case the time dependence of the nucleation rate J is represented
formally by the Dirac delta-function %, i.e.
Size distribution of supernuclei 453
At) = CmSoC) (33.32)
where f m = Nm/V or fm = /V„/As is the maximum concentration of all supernuclei
in volume or surface nucleation, respectively (cf. eq. (26.13)). Thus, with the
help of (33.8), (33.11) and (33.32), from eq. (33.17) we find that
F(K, f) = £„,$,[«-RbM] (33.33)
where it is taken into account that 5d(x) = 0 for x *■ 0, 5d(- x) = 5d(x), 5v(ax)
= (1/0)¾^) and 5o(xa- xj) = (ilcaa-,)5D(x-x0) [Korn and Korn 1961]
(the first of these equalities allows omitting the P„0Pni product in eq. (33.17)).
Equation (33.33) tells us that in the case of IN the size distribution of
supernuclei is a sharply peaked function. The peak is positioned at R = Rb(t)
and moves with time along the size axis according to Rb(t) = (Gcr)v, as it
follows from (33.13) at R2 = 0. The monodispersity of the size distribution
in this case is physically understandable: being formed together at t = 0, later
all supernuclei are the biggest in size, i.e. they have the same radius Rb.
Using F(R, t) from (33.33) in (33.23) and (33.24), we find that in the IN case
the average radius /?av and the total concentration £ of detectable supernuclei
are stepwise functions of time: Kav(r) = 0 and g(t) = 0 for 0 < t < te, and
«„(') = Rb(t) = (Gct)v (33.34)
<TM = (m (33.35)
for t > /g, rg = R'"v/Gc being the growth time of the smallest detectable
supernucleus (see eq. (33.28)). Also, from (33.31) and (33.32) it follows that
A0 = ?„A(<-<8). (33.36)
Thus, in the IN case the detectable rate f is the same pulse function of time
as the nucleation rate J, but the pulse is at t = t rather than at t = 0, because
tg is the moment of appearance of the smallest detectable supernuclei. The
F(R, t), fiav(r) and £(t) dependences (33.33)-(33.35) are illustrated in Figs
33.1-33.3. The size distribution is shown at times r, and t2 > t, at which the
supernuclei are, respectively, undetectable and detectable.
2. Progressive nucleation
(a) Stationary regime
In this case both 7 and Ja in (33.17) and (33.18) are time-independent and
equal to the stationary nucleation rates Js and Ja s, respectively. For simplicity,
we shall consider zone growth obeying the power law (32.12) with v value
equal to that of vin the growth law, eqs (33.7) and (33.8), of the supernuclei.
Then, from (33.8), (33.11), (33.17) and (33.18) it follows that (0 < R < Rb(t))
F(R, t) = (Js/vGc) R""-' exp K7U/GC)«V - R<")
-(B2y,.>/Gc)1+1'd(R;/v-fii"')l+vd] (33.37)
where Rb(t) = (Gct)v, Js and Ja s are related through (25.3) or (25.4), and the
numerical parameter Bz is given by eq. (32.17) or (32.18) for volume (d = 3)
or surface (d = 2) nucleation. As seen, F(R, t) has a time-dependent maximum
value
454 Nucleation: Basic Theory with Applications
W
Rb(y
R
Fig, 33.1 Size distribution of supernuclei in the case of IN according to eq. (33.33)
(at time t2 > tt the supernuclei are already detectable).
a:
a:
*■' i
f
Fig. 33.2 Time dependence of the average radius R3V of detectable supernuclei (the
solid line) and of the radius J?b of the biggest supernucleus (the dot-dashed line) in the
case of IN according to eq. (33.34).
Size distribution of supernuclei 455
j I i . 1
o t, tg t2
t ►
Fig. 33-3 Time dependence of the concentration of detectable supernuclei in the case
of IN according to eq. (33,35).
Fm (I) = (JJvGc) <"-' = (JJvG>">) t"v-< (33.38)
at R = Rb(t)- Hence, we can represent F\R, t) from (33.37) in the equivalent
form (0 < R < Rb(t))
F(R, t)IFm = (RIRbf ~ ' exp (- JaJ[l - (R/Rb)Vv]
- W^t)1 + "\ 1 - (R/Rb)Vv], + vd) (33.39)
which is convenient for comparison of the shape of the size distribution
function at different times.
Equation (33.37) reveals that in the case of PN the size distribution of
supernuclei depends implicitly on r through Rh and is sensitive to the
confinements imposed on the process by the active centres and the exclusion
zones. At the earliest stage of the process such confinements are practically
absent, the exponential factor in (33.37) or (33.39) is close to unity and, as
it follows also from eq. (33.16), F(R, t) has the simple form (0 < R < Rh(t))
F(R, t) = (JJvGc)R"v~' = Fm(RIRb)"v-' (33.40)
for both volume and surface nucleation. The absence of Ja s in this formula
indicates that it is applicable also to HON. In the particular case of v - 1
(constant growth rate Gc of the supernuclei), according to (33.40), the differently
sized supernuclei have the same concentration: F(R, t) = JJGC = constant
[Roginsky and Todes 1940; Todes 1940].
At a later stage, if BL satisfies the condition (32.20), the nucleation process
is confined only by the finite number of the active centres in the system.
456 Nucleation: Basic Theory with Applications
Then the Bz summand in (33.37) and (33.39) is negligible and again for both
volume and surface nucleation we get (0 < R < Rb(t))
F(R, t) = (JJvGc)R«v- ' exp [- (JJGc)(Rb'/v - R,lv)]
= Fm(.RIRb)"v-' exp (- J„stU - (/?«„)'"']}. (33.41)
This expression is obtainable also from eq. (33.19) and in it Fm and Rh
depend on t as noted above. In the often considered case of v = 1 we have
[Toschev and Gutzow 1967b] (0 < R < fib(r))
F(R, t) = ys/Gc) exp [- (JJGc)(Rb - R)). (33.42)
Under conditions at which Bz » 1, according to (32.21), both the active
centres and the exclusion zones confine the nucleation process. Then the
exponential factor in (33.37) and (33.39) is controlled by the Bz term so that
(0 < R < Rb(t))
F(R, t) = (Js/vGc)R"v- ' exp [- (BJJG,)' * "'(K^ - «"V + vd\
= Fm(RIRb)uv-' exp (- (fizV)' + >"[1 - (Wfib)"T + "'J, (33.43)
the ^„,(0 and Rb(t) dependences being those given above. This formula
corresponds to eq. (33.20) and applies to volume (d = 3) or surface (d = 2)
nucleation with Bz from (32.17) or (32.18). Again in the v = 1 case, eq.
(33.43) leads to (0 < R < Rh(t))
F(R, t) = (JJGC) exp [- (BJJGj +rf(Rb - R)' +rf]. (33.44)
We recall that eqs (33.43) and (33.44) are applicable also to HON.
It is worth having explicitly the formula for the size distribution of
supernuclei in the particular case when the supernuclei themselves play the
role of exclusion zones. Then in (32.17) and (32.18) we have cz = ct, Gcz =
Gc and from eq. (33.43) it follows that [Kashchiev 1975b] (0 < R < Rb(t))
F(R,t) = (Js/vGc)R"v''txp (-[Cr/S/(1 + vd)Gc](Rb'" - R"v)' + w)
= Fm(RIRb)1,v- ' exp [- [cg<7s//(l + vd)\[\ - (R/Rb)l,v][ * *).
(33.45)
As noted above, this formula applies also to HON. When the supernuclei
grow at a constant rate Gc (then v - 1), from this formula we find that
(0 < R < Rb(r))
F(R, t) = (Js/Gc) exp [- (ceJJ4Gc)(Rh - fi)4] (33.46)
for volume nucleation (either HON or HEN) and growth of spherical (cg =
4)1/3), cubic (ce = 8), etc. supernuclei and that (0 < R < Rb(t))
F(R, t) = (JJGC) exp [- (cfJsBGc)(Rb - R?] (33.47)
for surface nucleation and growth of circular (cg = K), square (cg = 4), etc.
supernuclei. It must be kept in mind, however, that in this particular case of
zones coinciding with the supernuclei themselves, due to the mutual contacts
Size distribution ofsupemuclei 457
between the supemuclei at later times, eqs (33.45)-(33.47) can be used only
until some 30% of the initial volume Vor substrate area As is occupied by the
supemuclei [Kashchiev 1975b]. Also, eqs (33.45)-(33.47) reveal that in this
particular case, given the shape (i.e. cg) and the growth index v of the
supemuclei, their size distribution is controlled by a single parameter - the
JJGC ratio. As seen, faster nucleation and/or slower growth results in a
narrower size distribution and vice versa. Inspection of eqs (33.37), (33.41)-
(33.44) shows that this conclusion is in fact of general validity.
Curves C and C&Z in Figs 33.4 and 33.5 illustrate, respectively, the size
distribution functions (33.41) (confinements from centres only) and (33.43)
(confinements from both centres and zones) in the case of v = 1 (Fig. 33.4)
and v = 1/2 (Fig. 33.5). The dotted lines depict the corresponding size
distributions (33.40) in the absence of confinements from centres and zones.
At an early time /[ the size distributions are entirely in the undetectable size
range R < R'. We observe that in the v = 1/2 case, in which the smaller
supemuclei grow faster than the bigger ones, the maximum value Fm of the
size distribution increases linearly with Rh (cf. eq. (33.38)). Also, because of
the considerable confinements that the centres and zones impose on the
nucleation process as time goes on, at a later moment t2 the supemuclei of a
given size are less numerous than at the earlier moment ?,. The size distribution
function in stationary PN is most notable perhaps with its stepwise rise at
Fm -
unde
/C&2 /
/c
ectable detectable
/c&z /
/ /'
Rb(»1)
Rb(y
R
Fig. 33.4 Size distribution of supemuclei in the case of stationary PN at V = 1;
dotted line - eq. (33.40) for no confinements from active centres and exclusion zones;
curves C - eq. (33.41) for confinements from centres only; curves C&Z - eq. (33.43)
for confinements from both centres and zones (at time t2 > tj the supemuclei are
already detectable).
458 Nucleation: Basic Theory with Applications
Fm02>
Fmd)
undetectable
detectable
..■•' c&z/ /
/ /C
Rb(*1)
Rb(y
R
Fig. 33.5 Size distribution of supernuclei in the case of stationary PN at v = 1/2:
dotted line - eq. (33.40) for no confinements from active centres and exclusion zones;
curves C - eq. (33.41) for confinements from centres only; curves C&Z - eq. (33.43)
for confine me nts from both centres and zones (at time t2 > t{ the supernuclei are
already detectable).
Rm(t) = Rh(t) and its 'tail' towards the smaller sizes. As already mentioned
above, this 'tail' shortens by increasing the JJGC ratio and can be quite short
at high enough values of JJGC. This conclusion is of practical value, for it
tells us what should be done with the JJGC ratio when our aim in producing
a cluster size distribution is to make it as narrow as possible.
Having obtained the F(R, t) function in the considered case of PN in
stationary regime, we can now determine the corresponding Rav{t), £,'{i) and
]'{t) dependences. We first note that in this case the PmPni product (33.18)
is given by the exponential factor in (33.37) so that, in view of (33.11),
PJfiPJt) = exp [- J^t-(B7J^t)1
(33.48)
Then, from (33.13), (33.23), (33.25), (33.27), (33.28), (33.31) and (33.37)
we find that, in general,
*.v(0 = Gc
ir
rap [-J.S - (B,ya.sr')1+Vd] it'
(t - /'/exp [_ jt/ _ (Va/)1 + 1 it' (33.49)
(TO = h
r
Jo
exp [-Jvt'- (BJ^/Y
\dt'
(33.50)
Size distribution ofsupernuclei 459
J\t) = Js exp (-7,.s(t - rs) - [BJ^t - (,,)]1 + ^) (33.51)
for t > (g, and fiav(r) = 0, (7(() = 0, 7'(() = 0 for 0 < ( < (g where rg = fi'"7Gc
is the growth time of the smallest detectable supernucleus.
In the absence of confinements from centres and zones, the exponential
factor in these formulae is equal to unity and R„, % and f are simple explicit
functions of time (( ^ (g):
«„(() = R' [((/(„)" + ' - l]/(v + l)[((/(g) - 1] (33.52)
C(t) = Ut-tg) (33.53)
/(() = Js. (33.54)
Equations (33.52) and (33.53) show that the average radius Rav of the detectable
supernuclei increases gradually with time from R' at (= rg to Rb(t)/(v + 1) for
( » (g and that their total concentration ^ is a delayed linear function of
time. This behaviour of Rlv from (33.52) at v= 1 and of % from (33.53) is
seen in Figs 33.6 and 33.7, respectively. For illustration of the delay of £*,
the dash-dotted line in Fig. 33.7 represents the f(r) dependence (13.102),
resulting from (33.53) when all supernuclei are detectable (then tg = 0).
As to the detectable nucleation rate J\ it is time-independent and exactly
equal to the stationary nucleation rate Js. Experimentally, therefore,
Fig. 33.6 Time dependence of the average radius fiav of detectable supernuclei in the
case of stationary IN at v = 1: doited line - eq. (33.52) for no confinements from
active centres and exclusion zones; solid line - eq. (33.58) for confinements from
centres only. The dot-dashed line represents the time dependence of the radius Rb of
the biggest supernucleus.
460 Nucleation: Basic Theory with Applications
I
i
I
«2
Fig. 33.7 Time dependence of the concentration of detectable supernuclei in the case
of stationary PN: dotted line - eq. (33.53) for no confinements from active centres and
exclusion zones,- solid curve - eq. (33.56) for confinements from centres only. The dot-
dashed line and the dashed curve correspond to the case when all supernuclei are
detectable (then tg - 0).
eqs (33.53) and (33.54) are important, because they tell us that C,'(t) data for
nucleation in stationary regime can give the same information about Js as the
respective f(r) data which, however, are much harder to obtain, since they
require detection of all supernuclei in the system.
In the case of confinements due to active centres only (then Bz« 1), from
eqs (33.49)-(33.51) with omitted Bz term we obtain (t > rg)
«a»(0 = •/«<£ (1 - exp [- Jas(t - ts)])-'
f ''(t-t')" ex.p(-Ja.,t')dt' (33.55)
Jo
f(r) = £,{ 1 - exp [- yw(r - tg)]) (33.56)
/(t) = ysexp[-7„.s(t-tg)]. (33.57)
Here, according to (25.3) and (25.4), Js and Jas are related by 7S = Ja£a, and
fa = NJV or fa = NJAS is the concentration of active centres in volume or
surface nucleation, respectively. The integral in (33.55) can be expressed in
terms of the incomplete gamma-function for arbitrary V > 0. In the v = 1
case, however, the integration is carried out with the help of elementary
functions and we get (r > tg)
Size distribution of supernuclei 461
R„(r) = R' (//<g - UJaJ„ -(1- 1/7,,,¾) exp [- 7a.s(r - /„)]}
(l-exp[-7a,s(r-tg)]}-1. (33.58)
This equation reveals that /?av is a much more complicated function of time
than when nucleation is not confined by centres and zones (cf. eq. (33.52)).
In contrast to R„, from (33.52), R„ from (33.58) is controlled not only by Gc
(through /g), but also by the stationary nucleation rate 7as per active centre.
The solid curve in Fig. 33.6 illustrates the Rav{t) function (33.58) and shows
that as time goes on, the detectable supernuclei increase their average radius
from R' at t = /g to Rb(t) - Gc/7a s for t » /g. As to the %(t) dependence
(33.56) (the solid curve in Fig. 33.7), it parallels the £(/) dependence (25.7)
(the dashed curve in the same figure) which is valid when all supernuclei are
detectable. The growth of the supernuclei to the detectable size R' takes time
/g and this results in a delay of £*. This delay is reflected also in the detectable
nucleation rate 7' (see eq. (33.57)) which vanishes as time goes on because
of the finite number of the active centres in the system.
The limiting case of confinements from both centres and zones (then
Bz » 1) is characterized by the following Ray(t), £(t) and J'(t) functions
which we find from eqs (33.49)-(33.51) for t > rg:
R„U) = cAj ''exp [-(Bz7a,!r')1+"'] if
f ° (/-0vexp [-(fi77a/)' + vd] it' (33.59)
Jo
?(0 = 7sf"' exp[-(fiz7a/)l + '"']<i<' (33.60)
Jo
7-(/) = 7S exp {-[/yas(/ - r6)]' + *). (33.61)
These formulae apply to volume (d = 3) or surface (d - 2) nucleation with
B, from (32.17) or (32.18) and are valid also for HON (then d = 3). Qualitatively,
the above /?av(/) and £'(/) dependences are similar to those illustrated by the
solid curves in Figs 33.6 and 33.7, and the detectable rate 7' from (33.61) is
again a delayed vanishing function of time. At /g = 0 eq. (33.60) passes into
eq. (32.22) which is in force when all supernuclei are detectable. In the
particular case of zones coinciding with the supernuclei themselves, in eqs
(32.17) and (32.18) we havecz = cg and Gcz = Gc so that eqs (33.59)-(33.61)
describe this case, too, but with Bj/as replaced by [csG'dJ,K\ + vd)]m*yd).
We note also that if the Sav(() and £'(() functions are needed analytically for
arbitrary vd > 0, the integrals in (33.59) and (33.60) can be performed with
the help of the incomplete gamma-function.
(b) Non-stationary regime
The general formulae (33.16), (33.17), (33.21)-(33.25), (33.29)-(33.31) are
valid for whatever time dependence of the nucleation rate 7. In particular.
462 Nucleation: Basic Theory with Applications
they can be used in the case of non-stationary nucleation at constant
supersaturation (see Chapter 15), when the J(t) dependence is given by eq.
(15.64) or (15.118). However, due to the complexity of this J(t) dependence,
the analytical determination of the size distribution function F(R, t) and of
Sav, £ and / is a difficult mathematical problem. For that reason we shall
restrict our considerations to non-stationary volume or surface nucleation
which is not affected by the presence of nucleation-exclusion zones.
Let us first consider the process when it proceeds without any confinements
at all. In this case, from eqs (15.118), (33.8), (33.11) and (33.16) we obtain
F(R, t) = (JJvGJR""- ]
{1 + 2l (-1)' txp[-(i2lt2l6Gc8)(Rl/v - Rvv)]) (33.62)
for 0 < R < Rb(t) and F(R, t) = 0 for R > Rb(t). This formula says that in non-
stationary PN (either HON or HEN) the size distribution of supernuclei is
governed not only by Js and Gc, but also by the nucleation time lag t, since
this is related to the nucleation delay time 8 by eq. (15.105) or (15.111). The
dotted curves in Figs 33.8 and 33.9 visualize the size distribution (33.62) at
v= 1 and v= 1/2, respectively, at time r, at which none of the supernuclei
is detectable and at a later time r2 when some of them are already detectable.
In comparison with the size distribution resulting from stationary PN (the
dotted curves in Figs 33.4 and 33.5), the size distribution in non-stationary
detectable
J il I
0 WW R' Rm(t2)Rb(t2)
R
Fig. 33.8 Size distribution of supernuclei in the case of non-stationary PN at v = I:
dotted curves ~ eq. (33.62) for no confinements from active centres and exclusion
zones; solid curves - eq. (33.70) for confinements from centres only (at time t2 > tj
the supernuclei are already detectable).
Fm
unde
1 \
Size distribution of supernuclei 463
Fm<y
o Rm(V Rb('i) R| Rjypyy
R
Fig. 33.9 Size distribution of supernuclei in the case of non-stationary PN at v =
1/2: dotted curves - eq. (33.62) for no confinements from active centres and exclusion
zones; solid curves - eq. (33.70) for confinements from centres only (at time t2 > t{
the supernuclei are already detectable).
PN exhibits an important new feature: a 'tail' towards the larger sizes.
Physically, the origin of this 'tail' is clear. As the non-stationary rate J of
nucleation is vanishingly low at the onset of the process (see Fig. 15.3), the
biggest supernuclei are quite few in number. As time goes on, however, 7
increases gradually up to its stationary value Js so that more and more
supernuclei are formed, but they are already smaller in size. This 'tail',
therefore, contains information about the nucleation time lag x and delay
time 9 and can be used for determination of these quantities from experimentally
obtained size distributions of supernuclei in non-stationary PN [Toschev and
Gutzow 1967b; Koster 1984; Blanke and Koster 1985; Koster and Blank-
Bewersdorff 1987; Kelton 1991]. It should be kept in mind, however, that
this 'tail' is not an unambiguous indication for non-stationary nucleation.
Indeed, even in stationary nucleation the experimentally observed size
distribution function can have such a 'tail', since the equally sized supernuclei
cannot grow at exactly the same rate. The dispersion of the growth rate
around its mean value G in eq. (33.4) reflects the differences in the local
conditions of growth of the individual supernuclei [Trofimov 1983]. We note
also a useful approximation to F(R, l) from (33.62), which is valid with an
error of less than 1% for the bigger supernuclei, namely those with radii in
the {Rl/V - 24GC9/TC2)V < R < Rb range:
F(R, t) = (2<Uf&xv2Gc)l/2Rl/v-l(Rlhlv - Rilvy"2
X exp[-6Gc&(Rlh'v-Rl/v)]. (33.63)
464 Nucleation: Basic Theory with Applications
This formula is based on the approximation (15.66) for J(t) from (15.64).
Now, from eqs (33.21)-(33.23),(33.25) and (33.31), with the aid of (15.118),
(15.119), (33.12), (33.13), (33.27), (33.28) and (33.62) we can determine the
Rm(t), Fm(r), fiav('). (TW ancl ■/'(') dependences in unconfined non-stationary
PN. Because of the mathematical difficulties in the calculation of the position
Rw and the value Fm of the maximum of F(R, t) from (33.62) we shall not
attempt determining the first two of these dependences. Recalling that Pn0(r)
= Pm(t) = 1 when nucleation is not confined by active centres and exclusion
zones, we find that
R„(t) = Ccv {t-tg-8- (12/^)6
I [(- l)7i'2] exp [- i2tc\t- tg)/69])-]
x \ \ \i- t'VU + 21(-1)'exp (-i2;rV/60)]d/'l (33.64)
£(t) = JA'-'s-9-(l2/^9 S [(-l)''/(2]exp[-iV(r-/g)/6eil
(33.65)
J'(t) = Js (1 + 2 Z (- 1)' exp [- rV(» - (,,)/69)} (33.66)
for t > /g and that R„(t) = 0, £(t) = 0, J'(t) = 0 for 0 < t< te.
As seen, the only difference between the J(t) and £(r) dependences (15.118)
and (15.119) and the above /(/) and £"(/) dependences is that the latter are
'shifted' towards longer times with respect to the former, the 'shift' being
equal to the growth time rg of the smallest stable detectable supernucleus.
This is illustrated in Fig. 33.10 in which the dot-dashed and the dotted curves
represent % from (33.65) at rg = 0 (then all stable supernuclei are detectable)
and at rg > 0, respectively. We observe that in the (—> °° limit ^(/) becomes
a straight line (shown by double dots and dashes) with time intercept 0 or 61.
Physically, ff(s) is merely the delay time of detectable nucleation, which, for
brevity, we shall call detectable delay time. In analogy with eq. (15.100) for
the delay time 9 of nucleation, & is defined by
£(t) = Js(t- 6') at t -> ». (33.67)
Hence, with the aid of g(t) from (33.65), it follows that
6' = 6 + te (33.68)
where 9 is related to the nucleation time lag Tthrough eq. (15.105) or, more
accurately, through eq. (15.111). Equation (33.68) is similar to that of Kozisek
[1989] and is of practical interest, since & is accessible to a direct experimental
determination. According to this equation, when we measure the detectable
delay time 9, we can interpret it as the actual delay time 6 of nucleation only
Size distribution of supernuclei 465
/ i i \/ i i
o e t, tg ff t2
t -
Fig. 33.10 Time dependence of {he concentration of delectable supernuclei in the
case of non-stationary PN: dotted curve - eq. (33.65) for no confinements from active
centres and exclusion zones; solid curve - eq. (33.71) for confinements from centres
only. The dot-dashed and dashed curves correspond to the case when all supernuclei
are detectable (then tg — 0). The straight lines are the t —> °° asymptotes of Q(t) in the
absence of confinements from centres and zones and define the respective delay times.
if /g « ft. We note also that (33.68) is an approximation to the exact formula
ft' = &2 + 'g of Shneidman and Weinberg [1992a], in which 82 is the delay
time corresponding to the establishment of stationarity of the flux j(n2, t) at
the right end n2 of the nucleus region. The approximate character of eq.
(33.68) is a consequence of our replacement of the flux j(n2,t) in (33.1) by
the nucleation rate J(t) which at constant supersaturation is the flux at the
nucleus size «* (see eq. (15.1)). The nucleation rate becomes stationary
earlier than the flux j(n2, t), because n* < n2. Hence, ft < 6¼ and eq. (33.68)
underestimates ft'. Nonetheless, since typically ft2= 1.50 to 2ft [Shneidman
and Weinberg 1992a, 1992b], eq. (33.68) is an acceptable approximation in
many cases of practical significance.
Looking back at eq. (33.64), we see that the non-stationarity of the nucleation
process makes rather complicated the time dependence of the average radius
of detectable supernuclei. In the v = 1 case (supernuclei growing radially at
constant rate Gc), with the help of (15.104) and the equality [Gradshtein and
Ryzhik 1962]
Z [(-1)7^ = -7^/720,
1=1
466 Nucleation: Basic Theory with Applications
the integral in (33.64) is performable without much effort and the resulting
explicit Rav(t) dependence is of the form (( > rg)
R„(t) = R' (r//g - 1 - 6/tg - (\2dlit\) Z [(- 1)7/2]
x exp [- /V(» - ^)/601)1 ((1/2)(/2¾2 -1)- 8t/tj
- (lieiK1!,) X [(-1)712] exp [-i27C2(t - re)/6ej
+ (7e2/10/2) + (72e2M4r2)
Z [(-1)7(4] exp [-/V(/- /E)/6e]}. (33.69)
This cumbersome formula shows that Rav(t) data obtained in non-stationary
regime can give a reliable information about the growth time rg or rate Gc of
the detectable supernuclei only at longer times (t - /g » S) when, as it
should be, eq. (33.69) turns into eq. (33.52) with v= 1. Alternatively, however,
if we know rg independently, we can use (33.69) for determination of the
nucleation delay time 6 and, thereby, of the time lag r of nucleation provided
the R„(t) data are obtained at sufficiently short times (/ - rg < 9).
Finally, we shall extend the applicability of the above results to the case
of non-stationary volume or surface PN when the process is confined only
by active centres. Then F(R, /) is given by eq. (33.19) which, in view of
(15.118), (25.12), (33.8) and (33.11), leads to
F(R, /) = (JJvGc)R"v-' [ 1 + 2 Z (- 1)' exp [- (iV/6Gce) (<" - R1/v)])
xexp(-(Jas/Gc)(/?i'v- R^'-G^-iW^G^e Z [(-1)7/2]
x exp [- (i27?l6Gce)(R^ - Rilv)}\) (33.70)
for 0 < R < Rb(t) and to F(R, t) = 0 for R > Rb(t). This size distribution is
illustrated by the solid curves in Figs 33.8 and 33.9 for v = 1 and v = 1/2,
respectively. We see that its 'tail' on the right of its maximum parallels that
of the size distribution for unconfined non-stationary nucleation. This is so
for during the formation of the bigger supernuclei practically all active centres
are still available for nucleation. As to the 'tail' on the left of the maximum
of F(R, /) in Figs 33.8 and 33.9, it is analogous to that of F(R, /) from (33.41)
in the respective case of stationary PN (see the solid curves C in Figs 33.4
and 33.5), because there is again a considerable exhaustion of active centres
at the later moments of appearance of the smaller supernuclei.
Since the size distribution (33.70) is mathematically rather complex, we
shall refrain from using it in eqs (33.21)-(33.23) for analytical determination
of its maximum Fm(t), of the most probable supernucleus radius /?„,(/) and of
Size distribution of supemuclei 467
the average radius R„(t) of the detectable supemuclei. We shall obtain only
the corresponding time dependences of the total concentration % of detectable
supemuclei and of the detectable nucleation rate f. This is easy to do with
the help of eqs (25.3), (25.4), (25.10)-(25.12) and (25.15). Taking also into
account that now Pno(0 is given by the exponential factor in (33.19) and that
P^(0 = 1, from (33.25) and (33.31) we find that
CO) = ?a( 1 - exp (- J,s{t -t„- 8-(\Vi?)6 £ [(- m2]
xexp[-iV(t-tg)/6e]))( (33.71)
f(t) = Js 11 + 2 I (- 1)' exp [- iV(r - /g)/6 8]}
xexp(-/,^-/,,-6-(12/^)01 [(- 1)7(2]
x exp [-iV(t - g/68]}) (33.72)
for t > /g and that ^(/) = 0 and J'(t) = 0 for 0 < t < tg. We recall that here Js
and Jas are related by Js = /asfa and that fa = NJV or fa = Wa/As is the
concentration of active centres in volume or surface HEN, respectively. As
required, eqs (33.71) and (33.72) pass into eqs (33.56) and (33.57) in the
case of 6 = 0 which corresponds to stationary PN. Also, for t - rg « 9 the
Ja s-containing exponential term in (33.71) and (33.72) is close to unity (then
the active centres still do not confine the nucleation process) and these
equations take the form of eqs (33.65) and (33.66). We note as well that
when all supemuclei are detectable (then rg = 0), eq. (33.71) turns into eq.
(25.15) provided in the latter T is replaced by 8 from (15.105). The ^(r)
dependence (33.71) at /g = 0 and /g > 0 is depicted in Fig. 33.10 by the dashed
and solid curves, respectively. For the reasons already discussed, the 'shift'
of the solid curve towards longer times is again equal to the growth time tg
of the smallest detectable supemuclei.
Chapter 34
Growth of thin films
Nucleation theory finds a wide application in studies on the growth of thin
films. Usually, these studies are restricted to the initial stages of the process
when the substrate coverage does not exceed some 10-20%. An example in
this respect is the theory of Zinsmeister [1966, 1968, 1969, 1971] and its
developments and generalizations discussed, e.g. by Venables [1973, 1994],
Stowell [1974a], Venables and Price [1975], Lewis and Anderson [1978],
Kem et al. [1979], Stoyanov and Kashchiev [1981], Vook [1982], Venables
et al. [ 1984], Zinke- Allmang et al. [ 1992]. On the other hand, the nucleation-
mediated growth of crystals is described by the KJMA-type theory of birth
and spread of 2D clusters within the successive monolayers building up the
crystal face [Nielsen 1964; Vetter 1967; Borovinskii and Tsindergozen 1968;
Armstrong and Harrison 1969; Belenkii and Lyubitov 1978; Belenkii 1980;
Gilmer 1980] (see Section 27.2). An important feature of this theory of
polylayer growth of crystals is that it describes both the early and the advanced
stages of the process. Since the growth of a given crystal face is merely
growth of a thin solid film on its own substrate (the so-called homoepitaxial
growth), it is not surprising that the generalization of the theory of polylayer
growth of crystals for the case of growth of thin solid films on foreign
substrates (i.e. for the case of heteroepitaxial growth) allows a comprehensive
description of all stages of film formation [Aleksandrov and Entin 1975;
Kashchiev 1977; Borovinskii and Kruglova 1977; Aleksandrov 1978; Belenkii
1980; Trofimov et al. 1985], In this chapter we shall consider briefly the
theory of polylayer growth of thin films [Kashchiev 1977] and its application
to the description of their mode of growth. More on the subject can be found
elsewhere (e.g. Kashchiev etal. [1977]; Kashchiev [1978]; Lewis and Anderson
[1978]; Stoyanov and Kashchiev [1981]; Barthes and Rolland [1981]; Cadoret
and Hottier [1983]; Nieminen and Kaski [1989]; Ickert and Schneider [1990];
Bartelt and Evans [1993]; Newman and Volmer [1996]).
Our considerations pertain to the following physical picture (Fig. 34.1).
At the initial moment l = 0a molecularly smooth substrate with surface area
As is put in contact with a supersaturated old phase and growth of a thin solid
film commences. Although the substrate is free of screw dislocations, it can
have nucleation-active centres due to impurity molecules (or atoms), line
dislocations and other defects. The substrate is own or foreign when it is or
is not of the deposited material, but in either case the film grows in the same
manner. Within the first film layer 2D nuclei come into being at a rate J{(t)
(m~2s_1) and grow laterally with a radial velocity U[(t) (m/s). This process of
overall 2D crystallization results in a total area A, (/) occupied by the first
Growth of thin films 469
r
<■> <t>>
Fig. 34*1 Cross-section of thin solid film on molecularly smooth substrate:
(a) initial, and (b) stationary stage of polylayer growth.
film layer at a later time t > 0 so that the coverage of the layer at that time
is ai(f) = A\(t)lAs. Once a sufficiently large area ^4] is formed, deposition on
top of the first layer begins. This is again accomplished via nucleation and
growth of 2D crystallites, but now at rates J2(t) and v2(t\ respectively. Thus,
a second film layer of coverage a2(t) - A2(t)fAs appears on the first one,
offering its surface with total area A2(i) as a substrate for the same kind of
nucleation and growth within the third film layer. Clearly, the consecutive
filling up of film layers parallel to the substrate surface leads to a continuous
displacement of the film/old phase interface in perpendicular direction with
a speed G^t) which is the film growth rate. At the onset of the growth
process Gf can depend on t because of the changing molecular structure of
the film surface (Fig. 34.1a). However, at t -> *» this structure becomes
stationary provided the growth conditions are kept the same and, as the
influence of the substrate on the kinetics of layer filling is lost, Cf passes into
the time-independent growth rate Gc of the corresponding crystal face (Fig.
34.1b).
As seen, the main idea of the theory of polylayer growth of crystals (see
Section 27.2) is generalized with respect to accounting for the possibility for
different rates J, and v, of 2D nucleation and growth within the rth layer of
the film (i = 1, 2. 3,...) and, thereby, for differences in the kinetics of filling
of the successive film layers. In the particular case of absence of such
differences, i.e. when J\ = J2 = /3 = .. . and vx - v2 = t>3 = .. -, the theory
of polylayer growth of thin solid films passes into the theory of polylayer
growth of crystals and that allows treating the layer growth of both the
crystals and the films from a unified point of view.
The central problem in the theory is the determination of the time dependence
of the average film thickness hf (m) and of the film growth rate Gt (m/s)
which are defined by [Kashchiev 1977]
ht{t)= i hfiUt) (34.1)
470 Nucleation: Basic Theory with Applications
Gfc) = dhf(t)/dt. (34.2)
Here A,- is the thickness of the ith layer of the film, afc) = A;(t)/As is the
coverage of this layer at time t, and A,{t) is the total area occupied by the
layer at that time. When the ith layer is of monomolecular thickness, we
have hj = d0 where dQ is the molecular diameter.
Thus, the problem of finding h£t) reduces to the determination of all o-'s
on the basis of concrete model considerations and to the summation of these
quantities in conformity with eq. (34.1). A fairly general model for the filling
of the consecutive film layers is that underlying the KJMA theory of overall
crystallization (see Chapter 26). In the scope of this model, according to eq.
(26.34), for the coverage of the first layer we have
ax(f)=Ax{t)IAi=\-fmp [-(*/#,)*'] (34.3)
when the nucleation of the 2D crystallites is either progressive in stationary
regime or instantaneous, and their growth obeys the power law (27.17), Here
<?i > 0 and i?, (s) are, respectively, the kinetic index and the time constant of
filling of the first film layer. The determination of the coverage of the second,
third, etc. layers requires accounting for the possibly different kinetics of
filling of these layers, characterized by kinetic indices and time constants q-i-,
<y3, etc. and ^, #3, etc. Under the condition that no overhangs appear during
the deposition of the successive layers, the ith layer forms only on top of
layer i- 1. However, this process occurs on the increasing area At_,(/) of this
layer and the exact determination of the coverage at{t) of the ith layer is a
hard mathematical problem. To a certain approximation [Vetter 1967], an
equation analogous to (34.3) will apply, but in a differential form, i.e. for the
<±4r/<±4, _ ! ratio. Indeed, if dA, _ i is a small area occupied by layer i - 1
between the moments f and t' + dt', at a later time t > t' the ah layer will
occupy an area dA; of dA,__,. Hence, similar to (34.3), we shall have
(14,- = (1 -exp {- [(r-0/#i]*'})dA,-_, (34.4)
which, upon dividing by As and integrating under the initial condition a,-(0)
= 0, yields [Kashchiev 1977] (i = 2, 3, 4 )
a/it) = f (1 - exp {- [{t - n/WMdOi. tf)idf]dt'. (34.5)
Jo
This recursion formula shows that, due to the exclusion of overhangs, the
filling of a given layer is controlled only by the kinetic index and time
constant of the layer itself and by the rate of formation of the layer underneath.
In Section 26.4 we have already used a modification of eqs (34.4) and (34.5)
for describing the kinetics of two-stage overall crystallization (cf. eqs (26.44)
and (26.45)). In the particular case of qx = q2 = q3 = . . . and #, = 1¾ = 1¾ =
... (no difference in the kinetics of filling of the successive layers) eq. (34.5)
turns into the formula of Vetter [1967] for polylayer growth of crystals and
in this way eqs (34.3) and (34.5) describe the polylayer growth of both the
thin solid films (heteroepitaxy) and the crystals (homoepitaxy) from a unified
Growth of thin films 471
point of view. Concerning the accuracy of eq. (34.5) we note that although
the area A,■_ , of layer /-1 is dispersed among the 2D crystallites growing
in the layer, in eq. (34.5) it is treated as a compact area. This means that eq.
(34.5) is a kind of mean-field approximation with respect to the interrelation
between ff,- „ i and 0¾. Other approximate formulae for this interrelation can
also be derived for growth of both thin films [Aleksandrov and Entin 1975]
and crystals [Belenkii and Lyubitov 1978; Belenkii 1980; Gilmer 1980].
Physically, ¢, and $, are the basic phenomenological parameters
characterizing the overall filling of the successive layers of the growing thin
film when the process obeys the KJMA kinetics. In a number of cases qf and
1¾ can be expressed explicitly in terms of the quantities describing the nucleation
and growth of the 2D crystallites in the i-th layer [Kashchiev 1977, 1978].
As follows from eqs (26.18), (26.20), (26.23) and (26.28) at d = 2, when the
radius R of these crystallites increases with time according to
«(') = CM)" , (34.6)
qi and fy are given by
?,- = 2v;, t>, = (4/4^,,-)1^(1/dsj-) (34.7)
for IN of Nmi crystallites in the ith layer and by
¢,-=1 + 2v,-, t>,- = [(1 + 2v,)lc'fv?]" ./,,,-]"(l+2v'> (34.8)
for PN at stationary rate Js L. Here the numerical shape factor c'g is considered
as being the same for the 2D crystallites in all layers (e.g. c'g = TC for disks,
Cg = 4 for square prisms, etc.). From (34.7) and (34.8) we thus see that ¢, =
2 in IN and q-, = 3 in PN of crystallites with radii growing linearly with time
(then v, = 1, and the growth constant us, is in fact the time-independent
crystallite growth rate). For filling of the ith layer by the mechanism of
continuous or liquid-like growth (see Section 27.1) it can be shown [Kashchiev
1977] that
¢,-=1, t>, = 1/(fs.,-ft,) (34.9)
where/s, (s~') and gsi (s~l) are, respectively, the frequencies of molecular
attachment and detachment per growth site in the ith layer. In the absence of
detachment of molecules from the ith layer (gs, = 0) eq. (34.9) applies also
to nucleation-mediated filling of the layer in the regime of complete
condensation [Kashchiev 1978].
With the aid of a,{t) from (34.3) and (34.5) we can now determine the
time dependence of the mean thickness hc of a growing thin film whose
successive layers are characterized by KJMA kinetics of filling. Integration
by parts in (34.5) and substitution of the resulting expression for a,.in (34.1)
leads to [Kashchiev 1977]
hf(t) = h,a,(t) + \ Z A,a,._|(f')(dexp (-[(;- »')/*;]'" }/<•'') di'
JO '=2
(34.10)
where a, is specified by (34.3).
472 Nucleation: Basic Theory with Applications
In the scope of the approximation (34.5), this basic equation of the theory
of polylayer growth of thin films describes the evolution of the film thickness
during all stages of growth. If the kinetics of layer filling are different only
for a certain finite number of the first film layers deposited on the substrate,
eq. (34.10) can be transformed into an integral equation for the unknown
/if(r) function, which admits analytical solving. Hereafter, we shall confine
our considerations to the simplest case of thin film growth when the filling
of the second, third, etc. layers takes place in the same way, but differently
from the filling of the first layer. In physical terms this idealization of reality
means that the film 'feels' the presence of the substrate only by its first layer,
since it is this layer which is in immediate contact with the substrate. The
filling of the next layers is not affected by the presence of the substrate and
occurs homoepitaxially, i.e. the first layer plays the role of own substrate for
the deposition of the second, third, etc. film layers. The mathematical condition
for this simplest case of film growth is
A] * h2 = h-i = . . . = d0
<?i * «2 = «3 = ■ ■ ■ = <?
where d0 is the molecular diameter, and q and t? are the kinetic index and the
time constant of polylayer growth of the corresponding crystal face. Using
these relations and the identity
£ hial_ i = h( + (d0 - h,)at
/=2
resulting from the first of them and (34.1) simplifies essentially eq, (34.10)
to the integral equation [Kashchiev 1977]
A,0) = MiM+ f [W) + (4-fc,)a,(/')](dexp(-[(r-r')/t?]W)df'
Jo
(34.11)
in which 01,(/) is given by (34.3). When the solution hfa) of this equation is
obtained, it can be employed in (34.2) for determination of the time dependence
of the film growth rate Gf. Alternatively, Gf can be found directly by solving
the integral equation [Kashchiev 1977]
d0al(t)= f [GK0 + (4~fci)dai('')/dr']exp(-[(/-r')/t?r?)dr'
Jo
(34.12)
which follows from (34.11) after integration by parts. We note that in the
particular case ofhl = d0,qi = q and i>, = t? eqs (34.11) and (34.12) describe
the kinetics of polylayer growth of crystals [Vetter 1967], as then the substrate
is own and the filling of all layers occurs in the same way.
Growth of thin films 473
Equations (34.11) and (34.12) allow finding analytically the exact h^t)
and Gf(t) dependencies for arbitrary qx > 0 provided ¢=1. These dependencies
are obtainable only approximately if also q is an arbitrary positive number
(we recall that along with q = 1 other physically interesting values are, e.g.
q = 2 and 3). It can be shown [Kashchiev 1977] that when 0¾ is specified by
(34.3), useful approximate solutions of eqs (34.11) and (34.12) for arbitrary
qt and q values are the functions
/1,(/) = 0,/ + /1,(1 -expK«i)"]l - Gci?i exp(-x«')ajr
Jo
(34.13)
Gr(r) = Gc{l-exp [-(//!>,)'']}
+ (^,/1,/tf,)(//!?,)«'-' exp Hf/i>i)«'] (34.14)
where Gc is given by (F is the gamma-function (26.21))
Gc = 4/T(l + l/q)&. (34.15)
What is particularly notable with respect to eqs (34.13) and (34.14) is that
at q = 1 they represent the exact solutions of eqs (34.11) and (34.12) with a,
from (34.3) and that at q * 1 they describe correctly the asymptotic behaviour
of /ir and Gf both for t —> 0 and for t —» °°. The long-time asymptotics
corresponds to stationary regime of growth and from (34.13) it follows then
that the thin film grows linearly with time according to | Kashchiev 1977]
h£t) = Gc(t - »im) (34.16)
where
/,nt = T(l + l/?,)t>, - (fc,/d0)r(l + \lq)&. (34.17)
Equation (34.16) reveals that, physically, Gc in eqs (34.13) and (34.14) is
the stationary rate of film growth. As expected, this quantity depends only
on <i0, q and r? and is thus the rate of polylayer growth of crystals. The fact
that Gc from (34.15) with q and r> from (34.8) coincides with Cc from
(27.18) rather than with Gc from (27.20) reflects the approximate character
of the recursion formula (34.5). Contrary to Gc, however, the time intercept
tm of the asymptotic hfa) dependence (34.16) is sensitive to the properties of
the substrate, which are characterized by hu qx and #,.
The solid curves in Fig. 34.2 depict the hfo) dependence (34.13) at qt =
3 and q = 1 (at this q value (34.13) is the exact solution of eq. (34.11)).
According to (34.8) and (34.9), these qt and q values correspond to filling by
the mechanism of PN and of continuous growth in the first and next layers,
respectively. The calculation is done with r>]/r> = 0.25, 1 and 4 (as indicated)
and /i, = dQ. The dashed lines in Fig. 34.2 represent the corresponding ht(t)
asymptotes (34.16). As seen, at iVr? = 0.25 the first layer is filled up quickly
and after that the film grows in stationary regime. We can therefore say that
for rJ,/i>« 1 the substrate is hospitable to the deposited material. Conversely,
at f?|/f? = 4 the filling of the first layer is delayed considerably (the substrate
474 Nucleation: Basic Theory with Applications
Fig. 34.2 Time dependence of the average film thickness {the solid curves) according
to eq. (34.13) at $//# = 0.25. I and 4 (as indicated) in the case ofqt~3 and q = 1.
The dashed lines represent the corresponding asymptotes, eq. (34.16).
is inhospitable) and stationary growth occurs after a period comparable with
the time intercept /,m. Thus, ?in[ appears as a direct indicator of the differences
in the kinetics of filling of the first and next film layers. Qualitatively, the
Af(/) dependence (34.13) is in agreement with those observed in computer
[Kashchiev et al. 1977] and real [Sigsbee 1969; Cinti and Chakraverty 1972;
Reichelt et al. 1980; Hottier and Cadoret 1982] experiments on thin film
growth. Figure 34.3 illustrates the G^t) dependence (34.14) (the solid curves)
and the stationary regime of growth (the dashed line) again at qt = 3, q = 1,
hx = d0 and rJ,/rJ = 0.25, 1 and 4 (as indicated). We recall that at this q value
(34.14) is the exact solution of eq. (34.12).
Having obtained the time dependence of the mean film thickness, we can
now find how thick the film is at the moment r99 at which it reaches continuity.
Let /i99 be the mean film thickness at that moment. It is important to know
this quantity both because it is experimentally accessible and because, as we
shall see below, it contains information about the film mode of growth.
Defining r99 as the moment at which the substrate becomes 99% covered by
the first layer, we have Q!i(/99) = 0.99 so that in view of (34.3)
/,, = 4.61^. (34.18)
For example, /99/1?] = 4.6, 2.1 and 1.7 at q\ = 1, 2 and 3, respectively. In
accordance with (34.16)-(34.18), the mean thickness ft99s /1^(99) at which
the film reaches continuity is therefore given approximately by [Kashchiev
1977]
fc99 = h,+ ([4.61"'- T(l + l/g,)]/T(l + l/?))4f),/f). (34.19)
Growth of thin films 475
-1
' 0.25
lU
^—"4
i . 1 . 1 .
!^—. -+—-i 1 . 1 , 1 . 1 , 1
0 12 3 4 5 6
t/fl
Fig. 34.3 Time dependence of the film growth rate (the solid curves) according to eq.
(34.14) at &/& = 0.25, 1 and 4 (as indicated) in the case ofq, = 3 and q = I. The
dashed line indicates the stationary regime of growth.
Here the bracketed factor in front of d0 is a number close to unity, e.g. it is
equal to 0.9 at qx = q = 3 and to 3.6 at q{ = q = 1.
Equation (34.19) is easily understandable upon noting that if the substrate
is covered within time i?! by the first layer, during this period i?j/i? more
successive layers will heap up on the first one provided each of them takes
time i?for its filling. Like eqs (34.13)-(34.18), with /ij = d0, q{ = q and i?,
= i? eq. (34.19) is applicable to poly layer growth of crystals, i.e. to deposition
on own substrate. Denoting by h99own the value of h99 in this case, from
(34.19) we get
Vown = [4.6'^/T(l + Uq)]d0. (34.20)
This formula says that in deposition on an initially bare own substrate the
mean film thickness at the moment /99 is practically always the same, e.g.
h99.owJdo = 4.6, 2.4 and 1.9 monolayers at q = 1,2 and 3, respectively. This
is in sharp contrast with deposition on a foreign substrate. As seen from eq.
(34.19), h99 -» /1, at 1?,/# -» 0, which means that when the filling of the first
layer is much faster than that of the second layer, the substrate is covered by
one layer only. This corresponds to the layer mode of film growth. In the
other extreme, i.e. at sufficiently slow filling of the first layer (?>, —> «>) and/
or very rapid filling of the second, third, etc. layers (U —» 0), for t < t99 high
island-like crystallites form on the substrate and at t = /99 the film on it is
macroscopically thick (h99 —» «>). This is the island mode of growth of the
film. Hence, the conclusion: h99 or the tVt? ratio can be used as a criterion
for the film mode of growth [Kashchiev 1977, 1978]. Under isothermal
476 Nucleation: Basic Theory with Applications
conditions usually the t?,/i? ratio diminishes with increasing the supersaturation
A^i, and #, additionally depends on the properties of the substrate (e.g.
adhesion, misfit, etc.) with respect to the deposited material. Changing,
therefore, the supersaturation and/or the substrate, we can obtain various />99
values, i.e. control the film growth mode.
But how does hw depend on A;/? Equation (34.19) is the general basis for
answering this question when only the first two film layers have different
kinetics of filling and in each concrete case of layer filling it leads to a
specific dependence of /i99 on A^i [Kashchiev 1977, 1978; Kashchiev a al.
\911; Stoyanov and Kashchiev 1981]. As an example, we shall consider
KJMA kinetics of layer filling due to PN in stationary regime and growth at
a time-independent rate of the 2D crystallites in all film layers. Then q\-q
= 3 and under the assumption h\ = d0, from eqs (34.8) and (34.19) it follows
that [Kashchiev 1977]
hmldv = 1 + 0.9 (Us27s/[.sy,j)"3. (34.21)
Here Js, and us, are the rates of 2D nucleation and lateral growth of the
monolayer crystallites on the substrate, and Js and vs are the same quantities
for the crystallites within the second, third, etc. monolayers. Treating Js j and
/s as corresponding to 2D HEN on a foreign and own substrate, respectively,
from eqs (13.53), (13.54) and (34.21) we find that in the scope of the classical
nucleation theory the h^Afi) dependence is of the form (A^i > aoAcJ)
V«o = 1 + 0.9Kf(A/<) exp [fi/3(Aji - aoAo) - B/3A/(] (34.22)
where Aa and B are specified by (3.72) and (13.55). The dimensionless
factor Kf is given by
K, = [v^&^Ai&nVvli&^A^m]113, (34.23)
A, and A being the kinetic factors of 2D nucleation on the substrate and
within the second, third, etc. film monolayers, respectively. We note that
since Kf is a relatively weak function of A^i, the above dependence of fu^ on
A^i is dominated by that of the exponential factor when the supersaturation
is not too high.
Figure 34.4 displays the Aw(A/i) dependence (34.22) at Ks = 1 and Bla^a
- 30. The dashed line indicates the number hggown/d0 of monolayers which
would have been deposited on the substrate till time /99 if it were the own one
(i.e. with Ac = 0). According to eq. (34.20), hmov,n/d0 =1.9 monolayers. As
seen from Fig. 34.4, h99 is a strong function of the supersaturation in the
range of smaller A^i values. In this range the film may have a thickness of
thousands of monolayers when it reaches continuity. That means island mode
of growth. With increasing A^i, A99 diminishes steeply and above a certain
supersaturation A/(cll (indicated by the arrow in Fig. 33.4), it assumes a low
and constant value of about 2 monolayers, which is characteristic for the
layer mode of growth. In other words, the increase of A/y leads to a change
from island (A99 » /!99,0„n) to layer (hm = hmt>m) mode of growth, which
occurs at A^i = Afich.
Growth of thin films 477
100000
10000
1000
Aji /aQAc
Fig. 34.4 Supersaturation dependence of the mean thickness at which a thin film
reaches continuity: solid line - eq. (34.22); dashed line - eq. (34.20) at q = 3. The
arrow indicates the supersaturation for change between the island (for Ajj < 4MehJ
and the layer (for A/j > A/jch) modes of film growth.
It turns out that the decrease of the film thickness with increasing Ap,
which is illustrated in Fig. 34.4, is a general property of /i99 for deposition on
substrates with Ao > 0 (these substrates are energetically inhospitable, because
Ao > 0 corresponds to incomplete wetting - see eq. (3.73)). The physical
reason for this property is that at high enough supersaturations the foreign
inhospitable substrate becomes kinetically equivalent to the own hospitable
substrate (for which Ao = 0) in the sense that i?, = i?. If in addition the first
and next film monolayers are filled by the same mechanism, the mode of
film growth is then, by definition, the layer (or the continuous) one and h99
has the low and virtually Aju-independent value ^99i0wn given by eq. (34.20).
478 Nucleation: Basic Theory with Application.*
Thus, the h99(A^i) dependence contains essential information about the film
growth mode and, with the help of the definition equality [Kashchiev 1977]
M^ch) " Vown = 1, (34.24)
can be used for determination of the supersaturation A^ich for change between
the island and layer modes of film growth.
Physically, eq. (34.24) means that at A^ch, from the kinetic standpoint, the
distinction between the foreign and own substrate is practically lost, because
the difference in the thickness at which the film reaches continuity is only
one monolayer. In view of (34.19) and (34.20), eq. (34.24) implies also that
at A^ch the time constants t>, and t? are practically equal. This equality of t?!
and $ along with the complementary requirement for identical mechanism
of filling of all film layers corresponds to the condition for polylayer growth
of crystals. Clearly, in finding A/ich as the solution of eq. (34.24) we have the
possibility to take account of various mechanisms of layer filling by making
use of different /i99(A^) dependences resulting from (34.19) with different
time constants t>, and t>. For instance, in the case considered above of layer
filling by stationary PN and time-independent growth rate of the 2D crystallites
in all film monolayers, A(ich is readily obtained from (34.20), (34.22) and
(34.24) when Kc = 1 [Kashchiev 1977]:
A/ich = (1/2)[1 + (1+ 5.2B/3aoAcJ)"2]a0Aa (34.25)
This formula applies when A/<ch > fljAo. At BlatjAa - 30 it yields A^ch/a0AcJ
= 4.1 which is the value indicated in Fig. 34.4. The usage of eq. (34.24) for
determination of A^ich in other cases of thin film growth is exemplified
elsewhere [Kashchiev etal. 1977; Kashchiev 1978; Stoyanov and Kashchiev
1981], Equations (34.19), (34.20) and (34.24) were found to describe
satisfactorily the results of a Monte Carlo simulation of the transition between
the film modes of growth [Kashchiev et al. 1977].
Summarizing, we come to the conclusion that the theory of polylayer
growth of thin films makes possible the nucleation-mediated and continuous
growth of crystals and of thin films on molecularly smooth substrates to be
described quite naturally from a unified point of view. The theoretical approach
is based on accounting for the individual kinetics of layer filling and because
of that it allows detecting nanoscopic effects localized within layers even of
monomolecular thickness. For instance, if for some reasons all film layers
are filled equally fast except for the third one (then t>3 » t?! = 1¾ = t?4 = t>5
= . . .), the film will exhibit the layer-plus-island or Stranski-Krastanov
mode of growth [Kern et al. 1979; Venables et al. 1984; Zinke-Allmang et
al. 1992]: after the deposition of the first two layers high island-like crystallites
will be formed on them, because during the slow filling of the third layer a
considerable number of successive layers will have time to amass on it. The
phenomenological parameters of the theory of polylayer growth are the kinetic
indices <y, and the time constants $,- of layer filling. In some simple cases
they are easily expressible in terms of known quantities such as the flux of
molecules into the successive film layers or the rates of nucleation and
Growth of thin films 479
growth of the 2D crystallites in them. This makes the theory applicable also
to other problems of thin film growth, e.g. to those of the growth shape of the
individual island-like crystallites on the substrate [Kashchiev 1984b; Trayanov
and Kashchiev 1986] and of the oscillations of the specular beam intensity
in reflection diffraction from the surface of a growing thin film [Kashchiev
andKanter 1988],
Chapter 35
Rupture of amphiphile bilayers
Amphiphile bilayers are interesting examples of matter organized in two
dimensions. The thinnest foam films (also named Newtonian black films)
[Exerowa and Kruglyakov 1998] and the bilayer lipid membranes [Tien
19741 are well-known representatives of amphiphile bilayers. As illustrated
in Fig. 35.1, these bilayers consist of two mutually adhering monolayers of
amphiphile molecules such as the molecules of the various surface-active
agents (called surfactants) and lipids. A characteristic feature of the amphiphile
molecules is that they have hydrophobic single or double hydrocarbon tails
with length of a few nm and hydrophilic heads with cross-sectional area of
about 0.4 to 0.5 nm2. For that reason, along the bilayer normal, the molecules
in the bilayer are in head-head or tail-tail contact as, for instance, in the case
of a foam bilayer (Fig. 35.1 a) or a membrane bilayer (Fig. 3 5.1 b). In the case
of an emulsion bilayer, depending on the type of the emulsion, any kind of
contact is possible: e.g. head-head contact for oil-in-water emulsions (Fig.
35.1a) and tail-tail contact for water-in-oil emulsions (Fig. 35.1b).
ar(al)
UiUUHi
♦ ♦ ♦ ♦ oz* ♦ ♦ ♦ ♦
r(al)
water
if \U UUl
L
(a)
<b)
Fig. 35.1 Cross-section of (a) foam or emulsion bilayer, and (b) membrane or
emulsion bilayer with a hole of amphiphile vacancies (the amphiphile molecules and
vacancies are schematized by solid and open circles, respectively).
The amphiphile bilayers generally form at the boundary between two
contacting phases which can be either equivalent, as in Fig. 35.1, or different.
For instance, under certain conditions, the contact between the bubbles in
foams or the droplets in emulsions is made through amphiphile bilayers.
This implies that the stability of the bilayers with respect to rupture largely
determines the stability of the given foam or emulsion itself. Hence.
Rupture of amphiphile bilayers 481
understanding and controlling foam and emulsion stability requires knowledge
about the process of bilayer rupture. This process is of great interest also in
the case of bilayer lipid membranes, because these are a good model of the
biological membranes existing in the cells of all living tissues. The rupture
of these membranes may destroy important life functions of the cells and,
thereby, have serious consequences for the whole living organism. From a
physical point of view, the rupture of an amphiphile bilayer is a manifestation
of the thickness transitions in thin films [Kashchiev 1989b, 1990]: initially
the bilayer is with a given thickness of several nm, and after rupture it is with
zero thickness.
In this chapter we shall see that the nucleation of nanoscopical holes in an
amphiphile bilayer can be responsible for the rupture of the bilayer. Considering
foam bilayers, Derjaguin and Gutop [1962] were the first to propose this
hole-nucleation mechanism of bilayer rupture. Their idea was then used not
only in further studies of the rupture of foam bilayers [Kashchiev and Exerowa
1980, 1998; Kashchiev 1987; Prokhorov and Derjaguin 1988], but also in
analyses of the rupture of bilayer lipid membranes [Pastushenko et al. 1979;
Chizmadzhev and Abidor 1980; Kashchiev and Exerowa 1983; Kashchiev
1987]. More on the subject can be found in the review articles of, e.g. Petrov
et al. [1980], Chizmadzhev et al. [1982], Exerowa and Kashchiev [1986]
and Exerowa et al. [1992].
Restricting our considerations to one-component amphiphile bilayers (Fig.
35.1), let us first find out what is the driving force for bilayer rupture. A
given bilayer is always in normal or lateral contact with an ambient phase
(e.g. a solution) which contains amphiphile molecules building up the bilayer.
The rupture of the bilayer involves passage of the amphiphiles from the
bilayer into the ambient phase. The driving force for bilayer rupture is,
therefore, the supersaturation Ay defined by eq. (2.1) in which now yold = yh
and junew = ya, yb and ^/a being the chemical potentials of the amphiphiles in,
respectively, the bilayer and the ambient phase. Thus [Kashchiev 1987],
Ay = yh-ya. (35.1)
If y$D is the chemical potential of the amphiphiles in a reference 3D
lamellar phase constituted of infinitely large number of bilayer lamellae
identical with the bilayer itself, yb can be expressed as [Kashchiev 1987,
1989b, 1990]
^b = ^3D + fl0crb (35.2)
where a0 (m2) is the area occupied by an amphiphile molecule at the bilayer
surface, and ¢^ (J nT2) is the specific surface energy of each of the two
bilayer surfaces. The last term in eq. (35.2) is merely the free energy that,
due to the presence of the bilayer surfaces, an amphiphile molecule in the
bilayer has in excess with respect to such a molecule in the reference 3D
lamellar phase.
The determination of ya requires specification of the ambient phase. In
the often encountered case when this phase is a solution of the amphiphiles
4S2 Nucleation: Basic Theory with Applications
constituting the bilayer, according to eq. (2.11), for the chemical potential of
the amphiphile molecules in the solution we have
^ = C.m + kTIn (OCcW) (35.3)
provided the solution is sufficiently dilute. Here C (irf3) is the actual
concentration of amphiphiles in the solution, C,, 3D is their concentration at
which the reference 3D lamellar phase is in equilibrium with the solution
(i.e. neither dissolves nor grows when in contact with the solution), and jie 3D
is the value of ^a at C = Ce 3D. The ambient phase can also be an insoluble
monolayer of the amphiphiles building up the bilayer [Richter et al. 1986;
Exerowa et al. 1987]. The bilayer and the monolayer are then in lateral
contact and /va is given approximately by [Kashchiev and Exerowa 1998]
/<a = ^,10 + 0^^-^,10) (35.4)
if the monolayer compressibility is negligible (accounting for this
compressibility leads to a quadratic ;rs term in jia [Kashchiev and Exerowa 1998]). In
eq. (35.4) ns (N/m) is the actual surface pressure of the monolayer, and ^. 3D
is the rcs value at which the reference 3D lamellar phase is in equilibrium
with the monolayer in respect to growth and dissolution.
Now, since/v3D in eq. (35.2) is practically independent of C or JTS, we have
^3D~ ^e,3D- Thus, combining eqs (35.1)-(35.4) leads to
A/((C) = kT In (Ce,3D/C) + a„a-b(C) (35.5)
for a bilayer in contact with a solution of amphiphiles [Kashchiev 1987,
1990] and to
A,u(;rs) = aa(Ke,3D - ns) + a0cb(^) (35.6)
for a bilayer in contact with an insoluble monolayer of amphiphiles. With the
help of the equilibrium amphiphile concentration Ce and the equilibrium
surface pressure jre of the monolayer, which are defined by
Ce = COD exp [aoObiCJ/kT] (35.7)
nc = ^.i3D + ab(rce), (35.8)
eq. (35.5) can be given the equivalent form [Kashchiev 1987, 1990]
Afj(C) = kT In (Q/C) + a0[ab(C) - ab(Ce)], (35.9)
and eq. (35.6) becomes [Kashchiev and Exerowa 1998]
VW = 2a0(^ - s,) (35.10)
because of the relation (Jb(!Q - ab(Kt) - am(itt) - om(xc) = itc-its (am is the
specific surface energy of the monolayer). We note that eq. (35.9) can be
approximated by [Kashchiev and Exerowa 1980, 1983]
A/j(C) = kT In (CJC) (35.11)
when the last term in it is negligible, e.g. when ob is nearly C-independent.
Rupture of amphiphile bilayers 483
Also, for more concentrated amphiphile solutions the above formulae should
be used with concentrations replaced by activities.
Physically, Ce and 7¾ in eqs (35.7)-(35.11) are 2D analogues of Ce3D and
TCe 3d, since they refer to the equilibrium between the bilayer (which is a 2D
phase) and the ambient phase. Indeed, we see that at C = Ce or ;rs = 7te there
is no driving force for bilayer rupture: Ay = 0. Rupture is thermodynamically
favoured only when C < Ce or i\ < rc^, as then Ay > 0. The practical significance
of eqs (35.5), (35.6), (35.9)-(35.11) is that they reveal how it is possible to
change Ay and, hence, affect bilayer rupture by varying C or Ks which are
experimentally controllable parameters.
Knowing A/;, we can now determine the work W(n) for formation of an n-
sized hole in the bilayer, n being the number of close-packed amphiphile
vacancies in the hole. Accounting that the hole is a 2D cluster of such
vacancies and assuming that it is of bilayer depth (Fig. 35.1) and disk shape,
for Wv/t can use eq. (3.75) with a0 replaced by hq/2. Thus, in the scope of
the classical nucleation theory, it follows that [Kashchiev and Exerowa 1980,
1983]
W(n) = -nAy + (2naf,)mKnm (35.12)
where K (J m"') is the hole specific edge energy. Since an n-sized hole
appears as a result of the passage of n amphiphiles from the bilayer into the
ambient phase, physically, the first term in eq. (35.12) is the work associated
with this process. As to the second term in (35.12), in it (27za0n)1/2 is the
length of the periphery of an n-sized hole of bilayer depth so that this term
represents the work done on creating the periphery of such a hole. According
to eqs (4.1) and (4.2), eq. (35.12) leads to the following expressions for the
number n* of amphiphile vacancies in the nucleus hole and for the nucleation
work IV* [Kashchiev and Exerowa 1980, 1983]:
n* = na^llAy1 (35.13)
W* = 7m0K2l2Ay. (35.14)
As can be easily verified, n* and W* from these expressions satisfy the
nucleation theorem in the form of eq. (5.29). Substitution of Ay from (35.5),
(35.10) and (35.11) in eqs (35.12)-(35.14) yields the dependence of W, n*
and IV* on C or ^ for bilayers in contact with amphiphile solutions or
insoluble monolayers.
When the bilayer is kept at a sufficiently high supersaturation, the appearance
and lateral overgrowth of one or more supernucleus holes in it leads to its
rupture. Recalling the general formula (13.39), with the aid of eqs (35.5),
(35.10), (35.11) and (35.14) we find that the stationary rate Js(nT2 s~') of
hole nucleation is given by (C < Ce)
Js = A exp (- B/[ln (COD/C) + a0ob(C)lkT}} (35.15)
or, approximately, by
Js = A exp ]- fi/ln (Q/C)]
(35.16)
484 Nucleation: Basic Theory with Applications
for a bilayer in contact with a solution and by (zrs < ^,)
J, = A exp [- B/(7tQ - 7ts)] (35.17)
for a bilayer in contact with an insoluble monolayer. In these expressions the
thermodynamic parameter B (dimensionless or in N/m) is of the form
B = na^llikT)1 (35.18)
B = m?-l4kT (35.19)
for a bilayer contacting a solution or an insoluble monolayer, respectively.
The kinetic parameter A (m~2 s~') is specified by eq. (13.40),
A = zf*C0, (35.20)
and to a certain approximation can be treated as independent of C or ks.
Here, as usual, the Zeldovich factor z is a number between 0.01 and 1 and is
given by eq. (13.37) withn* and W* from (35.13) and (35.14). Forabilayer
free of nucleation-active centres, according to (7.8), C0= l/a0= 10,sm"2. As
to the frequency /* (s_l) of attachment of single amphiphile vacancies to the
nucleus hole, it is sensitive to the mechanism of amphiphile transport in the
bilayer. For instance, when this attachment occurs as a result of 2D diffusion
of such vacancies towards the hole periphery, in view of (10.36) taken with
yn = 1, y* is of the form [Kashchiev and Exerowa 1980]
/* = c*DvZ( (35.21)
where c* = 1 to 5 is the capture number, Z, (m~2) is the concentration of
amphiphile vacancies in the bilayer, and Dv (m2/s) is the coefficient of their
diffusion along the bilayer. When/* is governed both by vacancy diffusion
and by lateral stretching of the bilayer, eq. (35.21) takes a more complicated
form [Prokhorov and Derjaguin 1988].
Having obtained the dependence of the stationary rate Js of hole nucleation
on the experimentally controllable parameters C or 7ts, we can now determine
the average lifetime /b(s) of a supersaturated bilayer as a function of these
parameters. The determination of tb is important, for this quantity is a convenient
measure of the stability of the bilayer with respect to rupture. Indeed, from
the moment of formation of the bilayer in contact with a given ambient
phase until the random appearance of at least one supernucleus hole in it, the
bilayer remains essentially in the metastable state in which it is formed.
After the appearance of one or more supernucleus holes, however, the bilayer
ruptures because of their irreversible overgrowth to macroscopically large
sizes. Thus, the nucleation-mediated rupture of amphiphile bilayers is a 2D
analogue of overall crystallization and can occur by either the mononuclear
or the polynuclear mechanism (see Chapter 26).
If the supenucleus holes grow radially at a time-independent rate vh (m/
s) and if Ah (m2) is the area of the bilayer surface, using (26.35) with d - 2,
v= 1, cg = ;rand Gc and Vreplacedby uh and Ab, we find that the mononuclear
mechanism of bilayer rupture is operative under the condition (cf. (27.16))
Rupture of amphiphile bilayers 485
Ab<(KvllZJhm- (35.22)
In this case tb is identical with rav from eq. (26.39) with Ab in lieu of V.
Hence, with the help of eqs (35.15)-(35.17) it follows that for rupture by the
mononuclear mechanism tk is given by (C < Ce)
tb(C) = Am exp (fi/[hi (Ce,3D/C) + a(sak(C)lkT}} (35.23)
or, approximately, by [Kashchiev and Exerowa 1980, 1983]
th(C) = Am exp [fi/In (Ce/C)] (35.24)
for a bilayer in contact with a solution and by [Kashchiev and Exerowa
1998](^<^)
tb(7ts) = Am exp [Blix, - ns)] (35.25)
for a bilayer in contact with an insoluble monolayer. Since the factor Am (s)
is defined as
Am = 1/AA„, (35.26)
eqs (35.23)-(35.25) tell us that in rupture by the mononuclear mechanism
the bilayer lifetime is inversely proportional to the area Ab of the bilayer
surface.
When the condition (35.22) is fulfilled with opposite inequality sign,
bilayer rupture occurs by the polynuclear mechanism. Then th is equal to /av
from eqs (26.24) and (26.28) also with d = 2, v - 1, cg = K and Gc replaced
by vh. Employing Js from (35.15)-(35.17), we thus find that (C < Ce)
tb(C) = Ap exp (fi/3[ln (Ce3D/C) + a0ab(C)/kT]} (35.27)
or, approximately,
/„(C) = Ap exp [fi/3 In (Q/C)] (35.28)
when a bilayer contacts a solution and that (¾ < k,.)
/,,(¾) = Ap exp [fl/3(tfe - K,)} (35.29)
when the contacting phase is an insoluble monolayer. Here the factor Ap (s)
is given by
Ap=r(4/3)(3toh2A)"3 (35.30)
and its independence of Ab means that the area of the bilayer surface has no
effect on the bilayer lifetime in rupture by the polynuclear mechanism. Like
Am, to a certain approximation, the kinetic parameter Ap can be treated as
independent of C or 7C„ because A and vk are relatively weak functions of the
amphiphile concentration or the surface pressure.
As seen from eqs (35.23)-(35.25) and (35.27)-(35.29), for both the
mononuclear and the polynuclear mechanism of bilayer rupture the dependence
of/b on C or jTg has a threshold character. Namely, th diminishes monotonously
with decreasing C or ^, but remains greater than an arbitrarily chosen value
'b.c (e-g- ?b,c = 1 s) if C or 7ts is higher than a certain critical amphiphile
486 Nucleution: Basic Theory with Applications
concentration Cc or critical surface pressure nc. When C or 7^ is below this
critical concentration or pressure, however, the bilayer ruptures virtually
instantaneously. The critical concentration Cc or pressure Kc is an important
parameter not only because it is experimentally accessible, but also because
of its practical significance in assessing the stability of amphiphile bilayers
with respect to rapture by hole nucleation [Exerowa and Kashchiev 1986;
Exerowa et al. 1992]. Analogously to A/(c in (31.12), Cc and ^ can be
defined as the solutions of the algebraic equations
'b(Q = V (35.31)
'b(tfc) = 'b.c- (35.32)
With the help of (35.18), (35.24) and (35.28), from (35.31) it thus follows
that
Cc = Ce exp [- xa0K2/2(kT)2 In (tbJAJ) (35.33)
for rapture by the mononuclear mechanism [Kashchiev and Exerowa 1980,
1983] and that
Cc = Ce exp [- na0K2/6(kT)2 In (tbcIA?)] (35.34)
for rapture by the polynuclear mechanism. Likewise, using (35.19), (35.25)
and (35.29) in eq. (35.32) leads to
Kc = %. - Kff-IHkT In (tbJAm) (35.35)
when rupture occurs by the mononuclear mechanism [Kashchiev and Exerowa
1998] and to
7tc = 7te - KK'-IXlkTIn (tbcMp) (35.36)
when it occurs by the polynuclear mechanism. As seen from the above
formulae, while Cc and Kc are quite sensitive to the magnitude of the hole
specific edge energy k, they are affected relatively weakly by changes in the
kinetic parameters Am and Ap.
Finally, we can obtain a unified formula for the bilayer lifetime /b, which
is valid for whatever number of supemucleus holes causing the bilayer rapture.
Similar to the induction time t, from (29.12) and (29.22), by summing the
right-hand sides of eqs (35.23) and (35.27), eqs (35.24) and (35.28) and eqs
(35.25) and (35.29) we find that (C < Ce)
/b(C) = Am exp (fi/[ln (Ce3D/C) + ag^iQ/kT])(1 + (Ap/Am)
x exp [- 2fi/3[ln (COD/C) + a^c^iCykT]() (35.37)
or, approximately,
/„(C) = Am exp [B/ln (Ce/C)](l + (ApIAJ exp [- 28/3 In (Ce/C)])
(35.38)
for a bilayer contacting a solution and that (jrs £ 7¾)
Rupiure of amphiphile bilayers 487
<b(K) = ^m exp [B/(itc - n,)] (1 + (Ap/Am) exp [- 2fi/3(;re - jQ]) (35.39)
for a bilayer in contact with an insoluble monolayer. As it has to be, for C or
tcs values smaller than, but close enough to, Ce and Kc eqs (35.37)-(35.39)
pass into eqs (35.23)-(35.25) of the mononuclear mechanism. Conversely,
when the CJC ratio or the;% - a, difference is sufficiently large, eqs (35.37)-
(35.39) turn into eqs (35.27)-(35.29) of the polynuclear mechanism.
The tb(its) dependence (35.25) or (35.29) is in qualitative agreement with
that obtained experimentally by Richter et al. [1986] in the case of a foam
bilayer maintained in contact with an insoluble monolayer. Foam bilayers
were used also for a study of the tb(C) dependence (for reviews see, e.g.
Exerowa and Kashchiev [1986]; Exerowa et al. [1992]). The symbols in Fig.
35.2 represent the t^C) data of Nikolova et al. [1989] for a foam bilayer of
sodium dodecyl sulfate, which is in lateral contact with an aqueous solution
of the same surfactant at T= 283.15 K (the circles), 295.15 K (the squares)
and 303.15 K (the triangles). The curves display the best-fit /b(C) dependence
(35.24) with, respectively, Am = 8.5 x 10"', 2.5 x 10"7 and 8.1 x 10"6 s, B =
49, 39 and 33 and Cc = 8.8 x lO"4, 1.33 x 10"3 and 1.53 x 10"3 mol/dm3 for
the above temperatures [Nikolova et al. 1989]. Using these B values in eq.
(35.18) and the independently known a0 = 0.42 nm2 results in K= 31 pj/m
[Nikolova et al. 1989]. According to eq. (35.13), it turns out that the nucleus
C(10"4 mol/dm3)
Fig. 35.2 Bilayer lifetime as a function of amphiphile concentration: symbols - data
of Nikolova et al. [1989] for a foam bilayer of sodium dodecyl sulfate maintained in
lateral contact with aqueous solution of the same surfactant atT— 283.15 K (the
circles), 295.15 K (the squares) and 303.15 K (the triangles); curves - best fit
according tn eq. (35.24).
488 Nucleatian: Basic Theory with Applications
hole is constituted of n* = 6 to 14 surfactant vacancies in the C range studied
[Nikolova et al 1989]. Also, since Js = 1 ltbAb for rupture by the mononuclear
mechanism, with the aid of the measured tb~ 10 to 400 s (see Fig. 35.2) and
the known Ab = 0.2 mm2 we find that Js = 1.2 x 104 to 5 x 105 m"2 s"1 under
the experimental conditions. Employing the latter 7S value in (35.22) allows
checking the legitimacy of the usage of eq. (35.24) of the mononuclear
mechanism for analysis of the tb(C) data in Fig. 35.2. With the above Ab
value and an assumed vb = 10 m/s which is approximately the value reported
by Evers et al [1996], from (35.22) it follows that 1 < 3741 in confirmation
of the mononuclear mechanism. Thus, the reasonable fit between the
experimental and theoretical tb{C) dependences in Fig. 35.2 along with the
physically acceptable values of k and «* leads to the conclusion that the
rupture of the investigated foam bilayer is mediated by hole nucleation. This
conclusion is reinforced by the analysis of independently obtained data for
other bilayer characteristics which are also controlled by hole nucleation
[Exerowa and Kashchiev 1986; Exerowa et al. 1992].
Appendices
A1 Exact formula for the non-stationary nucleation rate
The rate J(t) of non-stationary nucleation can be determined exactly by
solving eqs (15.68) and (15.69) in the way followed by Rodigin and Rodigina
[1960] in finding the solution of a similar problem in the kinetics of consecutive
chemical reactions.
Let Y„ be the Laplace-Carson transform [Korn and Korn 1961]oftheflux
j„(n= 1,2, M- 1):
y„(A) = A f j„(t)e-'"it. (Al.l)
Jo
Multiplying each equation in the set (15.68) by fa'*1 At, integrating from
t = 0 to t = °» and accounting for (15.69) and (Al.l) transforms the set into
a+g2)Y,-g2Y2 = ^lC1
-/A-i+ (^+/,+^,)^-^,^, =0, (« = 2,3,...,M-2)
(A 1.2)
-^-,4-2 + (A + fru- ,)¾- , =0-
As can be verified by direct substitution, the solution of this linear algebraic
set of M - 1 equations in the M - 1 unknowns Yn is of the form [Rodigin and
Rodigina 1960] (n = 1, 2, . . ., M- 1)
Y„{X,=fif2. . .f„CiXPM_„_i{X)IPM_la). (A1.3)
Here F„(A) is a polynomial of A of degree n, which is subject to determination
from the recursion formulae
p„ = i
p\ = a+/«-,
-fM-n+lSM-*+\Pn-2, (« = 2, 3, . . . , M - 2) (A1.4)
Pm - i = (A + g2)PM _ 2 - hg2PM - 3-
We can now make use of the theorem according to which the inverse
Laplace-Carson transform (i.e. the original) of the function
AP„.„.,(A)/P«_,(A) (A1.5)
is the function [Rodigin and Rodigina 1960]
490 Nucleation: Basic Theory with Applications
£ [/V-.-iM.W-iMr)] exp (-A.-0 (A 1.6)
where ~ A, < 0 is the /th of the M - 1 simple roots of the polynomial Pm - i(A).
Since this polynomial can be written down as
M-]
P«-i(A) = n (A + A,), (A 1.7)
i=i
for the derivative /^.,(-^,) = (dP^^/dA)^.^ we have (k * i)
M-\
PJi_,(-A/)= n (A,-A,.). (A1.8)
k=\
Hence, applying the inverse Laplace-Carson transform to both sides of (A 1.3)
and accounting for (Al.l), (A1.5), (A1.6) and (A1.8), we find (k * i)
M-\ M-\
i,(t)=fifi---ACl Z [P„_„_,(-Ai)/n (At-A,-)]exp(-A,-»). (A1.9)
This equation represents the exact solution of eqs (15.68) and (15.69) for
the flux j„(t) through point n = 1, 2, . . . , M - 1 on the size axis. When ( —>
°°,jn{t) tends to the n-independent stationary nucleation rate Js, because one
of the non-positive roots - A/ of the polynomial PM_ j(A) is equal to zero.
Denoting by - A, this root, i.e. setting A, = 0, in the limit of t —> =» (then only
the first summand survives) from (A 1.9) we get (n - 1, 2, . . . , M - 1)
./, = //2- • •/nC,P„_„ _ KOVA2A3. . . AM_,. (ALIO)
We note that since Js is the same at any n, its connection with the M - 1
positive quantities A2, A3, . . . , Xm _ 1 is most simply expressed by the
equation
./. = /1/2-- ■/«- iC/AjAj. . . A„_, (Al.ll)
which results from (Al.10) at n = M - 1 (then PM_„_ , = P0 = 1).
Finally, from (A1.9) and (ALIO), for the non-stationary nucleation rate
7(t) =jn*(t) we obtain the following exact formula (k * /, «* = 1, 2, . . . , M
-1)
M-\
U.t) = J>-fJ2---f*Ci I [P„-„.-,(-A,.)/A,-
1=2
IT (A, -A,)]exp(-A,i) (A1.12)
s in
where/* =/,».
Comparing eq. (15.59) or (15.60) and eq. (A1.12), we see that the A,
the latter are nothing else but the A,-'s in the former. The equivalence between
eq. (15.59) or (15.60) and eq. (A 1.12) is easily demonstrated in the particular
Appendices 491
case of M = MeS = 4 and n* - 3, considered in Section 15.2. In this case, from
(A 1.4) we find that the polynomial PM_ ,(1) is of the form
PM-i = P}= K# + Mfi+h + gi + ft) +/2/3 + «2/3 + g2ft]
so that its roots are At = 0 and
-Xi=- (1/2)(½ + /3 + ft + ft) - [(/2 +/3 + ft + S3)2
- 4(f/3 + ftf3 + S2S3)]"2)
- A3= - (1/2)(½ +/3 + £2 + ft) + [(/2 +/3 + g2 + ft)2
- 4(//3 + ft/3+ftft)]"2}-
As seen, these .¾ and A3 coincide with Xi and A3 from (15.20) and (15.21).
Also, according to (Al. 4) and (Al. 11), PM-„*-\ =Po=l amiJs=f]fJ*C-ll
A2A3, since/3 =/*. Using these relations in (A1.12) leads again to J(t) from
(15.61).
A2 Approximate formula for the non-stationary
nucleation rate
The non-stationary nucleation rate J(t) can be determined by solving eq.
(15.70) in conjunction with eqs. (15.71) and (15.72). We shall do that in the
scope of the approximation used in Section 15.1 with respect to eqs (15.38)
and (15.39). According to this approximation, we 'shift' the boundary conditions
from n = 1 and n = M to n = n\ and n = n'j and replace (15.70)-( 15.72) by
(«( < n < n'j)
dj(n, t)/dt =f*d2j(n, t)ldn2 (A2.1)
j(n, 0) = (A7A*)/*C*(5D(n - n[) (A2.2)
dj(n, t)ldn = 0 at n = n[, dj(n, t)ldn = 0 at n = n'2 (A2.3)
where 5q is the Dirac delta-function.
Physically, eq. (A2.3) implies instantaneous establishment of steady state
for the subnuclei of size n < nf, i.e. on the left of the size region with centre
«*, width A' and left and right ends n[ and n2 defined by (15.48) and
(15.49). This size region corresponds to the nucleus region so that in it,
approximately,^/!) =/* and C(n) = C* (this approximation transforms (15.70)
into (A2.1)). Equation (A2.2) postulates that initially there is no flux in this
size region except at its left end n{ where the flux is/*X( n[) =/*(A7A*)C*
(the factor A7A* > 1 accounts that the stationary concentration X(n[) of the
n,'-sized subnuclei is higher than the equilibrium concentration C* of the
nuclei).
The solution of eqs (A2.1)-(A2.3) is known [Carslaw and Jaeger 1959]
(n[ <n< n'i):
492 Nucleation: Basic Theory with Applications
j(n, t) = (1/A') (A7A*y*C*5rj(n - n[) 6n
+ (2/A') Z cos [iiin - n{ )/A'] exp (- iVpt/A'2)
i=]
X f (A'/A*)j*C*5dn - n,')cos[i;r(n- nf)/A'] dn (A2.4)
where A'= n'i - n\. Owing to the properties of the Dirac delta-function
IKorn and Korn 1961], we have
S0(n-n\)An = 1,
I <5D(« — «f) cos [in(n - n{)/A'] d« = cosO = 1.
Hence, with the help of (7.43), (13.33), (13.35), (15.48), (15.49) and
(15.55), from (A2.4) we find that the non-stationary nucleation rate J(t) =
j{n*, t) is given by the equation
J(t) =JS[\ + 2 Z cos (i'7T/2) exp (-i'2r/4T)]
r=l
which is equivalent to eq. (15.64).
A3 Initiel concentration of supernuclei in previously
supersaturated systems
The initial concentration f0 of supernuclei is given exactly by the first integral
in (24.53) when the system is presupersaturated (then Aij0 > 0) before the
imposition of the working supersaturation A/i. Applying (24.54) to this integral,
we thus find that, approximately,
£, = X\n*, A^0, :r„)/| W(n, Aw, TB)ldn)„ ._ „, |. (A3.1)
To a good accuracy, the stationary size distribution X(n, Afi0, T0) of the
pre-existing clusters is expressed by eq. (13.21). Using this equation along
with eq. (7.4) yields
X(n, Afj0, T0) = (l/2)C0exp [- W(n, Afj0, T0)/kT0\{l - erf [A,(n - n*)]}
(A3.2)
dX(n, A/i0, Ta)l6n = - (l/2)C„exp [-W(n, A/j0, T0)lkT0]
x {WkTomdWin^o.ToVin^^U-eTtl&in- «„*)])
+ (2ftM"2) exp [-$(« - «0*)2]}. (A3.3)
Appendices 493
Combining eqs (A3.1)-(A3.3), taking into account that C* = C0 exp
[- W(n*, Afj, T)lkT\ and recalling (3.86), we arrive at the sought general
formula for the concentration of all supernuclei existing initially in a
presupersaturated system (0 < A^0 5 A^i):
6, = ZoC* exp [- n*(.Afi/kT - A^o/kTo)
+ <P*(tyi, T)lkT- <P*(Aji0. T0)/kT0]. (A3.4)
Here, for brevity,
z„ = ((2[<W(n, AA,0,T0)/d«]„ = „,/tT0}(l - erf [&(«* - «»*)]}
+ (4ft/*"2) exp [-$(„* - «„*)2])-'{l -erf [ft,(n* - n0*)]}2,
(A3.5)
the concentration C* of nuclei is specified by (7.44) with W* from (4.6), nj
= n*(A(i0,T0) is the nucleus size at A/i0 and 70, and ft s f}(Afi0,T0) is given
in conformity with (7.38) as
ft = ([- d2lV(«, A^o, 7-„)/d«2 ]n=n, /2*70 }l/2. (A3.6)
Equation (A3.4) has the structure of (24.55) and is valid for whatever
kind of nucleation (HON, HEN, classical, atomistic, etc.). It is most easily
applicable to the case of EDS-defined clusters of condensed phases when the
process occurs under the conditions considered in Sections 24.1 and 24.2,
since then the expression in the exponent in (A3.4) is equal to - n*% with %
from (24.21)-(24.27), and the dHVdn derivative in (A3.5) is equal to &fi -
A^ij). In this case we thus have
Co = ([2(AA, - Ap0)/W0]( 1 - erf[&(«* - n*a )])
+ (4&/7T1'2) exp [- 02(«* - 4)2])-1
x (1 - erf[A,(n* - nt)]}2C* exp (- n*X). (A3.7)
We observe that this equation passes into (24.56) or (24.57) when A^0 is
sufficiently less than A^i. Indeed, then ft(nj - n*) > 1 and, as erf (- x) -
- 1 and exp (-x2) = 0 for* > 1 [Korn and Korn 1961 J. the product in front
of C* in cq. (A3.7) becomes equal to the bracketed factor in front of C* in
(24.56) or (24.57). In the opposite limit of A/j0 approaching A/( (then 0 <
ft( n* - «*) < 1) eq. (A3.7) can also be simplified, but with the help of the
approximations erf (x) ~ (2/it[,2)x and exp (- x1) ~ 1 holding for x < 1 [Korn
and Korn 1961]. We thus find that in this limit
Q, = {[2(A^ - Au0)/«b][l + (2^/711¾ „0* - «*)] + W*"2}-1
x [1 + (2/y7r"2)(«o - n*)]2C* exp (- n*X). (A3.8)
In particular, when Afi0 = Afi and T0=T(then nj = n*, f}0 = /3 and % = 0),
eq. (A3.8) leads to
f0 = (!C,n/4p)C* = (l/4z)C* = (A*/4)C* (A3.9)
494 Nuclealion: Basic Theory with Applications
where we have used also (7.43) and (13.35), and C* is specified by (7.44)
with W* from (4.6).
Equation (A3.9) gives the highest possible value of the initial concentration
of supernuclei in a previously supersaturated system. Since usually the nucleus
region A* has values between 2 and 40 and the stationary concentration X*
of nuclei is half the value of C* (see eq. (13.23)), we conclude that the
previously formed supernuclei in the system can seldom outnumber more
than ten times the nuclei themselves.
References
Abraham F.F., 1969, 1. Chem. Phys. 51, 1632.
Abraham F.F., 1974a, Homogeneous Nucleation Theory, Academic, New York.
Abraham F.F., 1974b, J. Chem. Phys. 60, 246.
Abraham F.F., 1979, Phys. Rept. S3, 93.
Ahn T.M. and Tien J.K., 1976, J. Phys. Chem. Solids 37, 771.
Ahn T.M., Purushotaman S. and Tien J.K., 1976, J. Phys. Chem. Solids 37, 777.
Aleksandrov L.N., 1978, Perekhodnye Oblasti Epitaksialnykh Poluprovodnikovykh Plenok,
Nauka, Novosibirsk.
Aleksandrov L.N. and Entin I.A., 1975, Kristallografiya 20, 1140.
Aleksandrov Yu.A., Voronov G.S., Gorbunkov V.M., Delone N.B. and Ncchaev Yu.L,
1963, Puzyrkovye Kamery, Gosatomizdat, Moscow.
Alexandrov A.D. and Avramov M.Z., 1993, Colloid Polym. Sci. 271, 162.
Alexandrov A.D., Toshev B.V. and ScheludkoA., 1991, Langmuirl, 3211.
Alexandrov A.D., Toshev B.V. and Scheludko A., 1993, Colloids Surfaces A 79, 43.
Allen L.B. and Kassner J.L., 1969,7. Colloid Interface Sci. 30,81.
Amar J.G. and Family E, 1995, Phys. Rev. Lett. 74, 2066.
Amar J.G. and Family F., 1997, Surface Sci. 382, 170.
Anderson R.J., Miller R.C., Kassner J.L., Jr. and Hagen D.E., 1980, J. Atmos. Sci. 37,
2509.
Andres R.P., 1969, in: Nucleation. Ed. A.C. Zettlemoyer, Dckker, New York, p. 69.
Andres R.P. and Boudart M„ 1965, J. Chem. Phys. 42, 2057.
Anisimov M.P. and Cherevko A.G., 1982, hv. Sib. Old. Akad. Nauk SSSR (Ser. Khim.
Nauk)2, 15.
Anisimov M.P. and Vershinin S.N., 1988, in: Atmospheric Aerosols and Nucleation, Eds
P.E. Wagner and G. Vali, Springer, Berlin, p. 393.
Anisimov M.P., Cherevko A.G. and Shtein M.S., 1982, Kolloid Zh. 44, 529.
Anisimov MP, Kostrovskii V.G. and Shtein M.S., 1978, Kolloid. Zh. 40, 317.
Anisimov MP, Kostrovskii V.G. and Sbtein M.S., 1980, Kolloid. Zh. 42, 724.
Anisimov M.P.. Kostrovskii V.G., Shtein M.S. and Mikheev V.B., 1980, Kolloid. Zh. 42,
941.
Anisimov M.P., Vershinin S.N., Aksenov A.A., Sgonnov A.M. and Semin G.L., 1987,
Kolloid. Zh. 49, 842.
Armstrong R.D. and Harrison J.A., 1969, J. Eleclrochem. Soc. 116, 328.
Avrami M., 1939, J. Chem. Phys. 7, 1103.
Avrami M., 1940, J. Chem. Phys. 8, 212.
Avrami M„ 1941, J. Chem. Phys. 9, 177.
Baidakov V.G., 1994, Mezhfaznaya Granitsa Prostykh Klassicheskikh i Kvantovykh
Zhidkostei, U1F Nauka, Ekaterinburg.
Baidakov V.G., 1995, Peregrev Kriogennykh Zhidkostei, UO RAN, Ekaterinburg.
Baidakov V.G., Kaverin A.M. and Skripov V.P., 1980, Kolloid. Zh. 42, 314.
Barnard A.J., 1953, Proc. Roy. Soc. (London) A 220, 132.
Baroody E.M., 1967, J. Appl. Phys. 38, 4893.
Barrett J.C., Clement C.F. and Ford LI, 1993, J. Phys. A 26, 529.
496 Nucleation: Basic Theory with Applications
Bartelt M.C. and Evans J.W.. 1992, Phys. Rev. B 46, 12675.
Bartelt M.C. and Evans J.W., 1993, J. Phys. A 26, 2743.
Bartelt M.C, Gilnther S., Kopatzki E„ Behm R.J. and Evans J.W., 1996, Phys. Rev. B 53,
4099.
Barielt M.C, Stoldl C.R., Jenks C.J., Thiel P.A. and Evans J.W., 1999, Phys. Rev. B 59,
3125.
Barthes M.-G. and Rolland A., 1981, Thin Solid Films 76, 45.
Basu D.K., 1973, Int. ]. Heat Mass Transfer 16, 1322.
Battezzati L., Antonione C. and Riontino G., 1987, J. Non-Cryst. Solids 89, 114.
Bauer E., Green A.K., Kunz K.M. and Poppa H., 1966, in: Grundprobleme der Physik
Diinner Schichten, Eds R. Niedermayer and H. Mayer, Vandenhoek-Ruprecht, Gbttingen,
p. 135.
Bauer S.H. and Wilcox C.F, 1993, J. Chem. Phys. 97, 271.
Bealtie J.A., 1949, Chem. Rev. 44, 141.
Becker R. and Dbring W„ 1935, Arm. Phys. 24, 719.
Bedanov V.M., Vaganov V.S., Gadiyak G.V., Kodenev G.G. and Rubakhin E.A., 1988,
Khim. Fiz. 7, 555.
Belenkii V.Z., 1980, Geometriko-Veroyatnostnye Modeli Kristallizatsii, Nauka, Moscow.
Belenkii V.Z., 1984, Dokl. Akad. Nauk SSSR 278, 874.
Belenkii V.Z. and Lyubitov Yu.N., 1978, Kristallografiya 23, 1248; 1253.
Bennema P., 1969, J. Cryst. Growth 5, 29.
Bennema P. and Gilmer G.H., 1973, in: Crystal Growth: an Introduction, Ed. P. Hariman,
North-Holland, Amsterdam, p. 263.
Bienfait M., 1980, in: Current Topics in Materials Science, Vol. 4, Ed. E. Kaldis, North-
Holland, Amsterdam, p. 361.
Bigg E.K., 1953, Proc. Phys. Soc. (London) B 66, 688.
Binder K„ 1977, Adv. Colloid Interface Sci. 7, 279.
Binder K. and Stauffcr D„ 1976, Adv. Phys. 25, 343.
Blander M., 1979, Adv. Colloid Interface Sci. 10, 1.
Blander M. and Katz J.L., 1972, J. Stat. Phys. 4, 55.
Blander M. and Katz J.L., 1975, AIChE J. 21, 833.
Blanke H. and Kbstcr U., 1985, in: Rapidly Quenched Metals 5, Eds S. Steeb and H.
Warlimont, North-Holland, Amsterdam, p. 227.
Bliznakow G„ 1958, Fortschr. Mineral. 36, 149.
Bliznakow G. and Kirkova E„ 1957. Z. Phys. Chem. 206, 271.
Bohm J., 1981, Cryst. Res. Technol. 16, 869.
Born M. and Wolf E., 1968, Principles of Optics, Pergamon, Oxford.
Borovi'nskii L.A. and Kruglova T.I., 1977, in: Rost i Legirovanie Poluprovodnikovykh
Krislallov i Plenok, Part II, Ed. L.N. Aleksandrov, Nauka, Novosibirsk, p. 30.
Borovinskii L.A. and Tsindergozen A.N., 1968, Dokl. Akad. Nauk SSSR 183, 1308.
Boslanov V-, Obretenov W., Staikov G. and Budevski E., 1983, J. Electroanal. Chem.
146, 303.
Bostanov V., Roussinova R. and Budevski E., 1969, Comm. Dept. Chem. Bulg. Acad.
Scis. 2, 885.
Bostanov V., Roussinova R. and Budevski E., 1972, J. Electrochem. Soc. 119, 1346.
Boucher E.A., 1969, in: Nucleation, Ed. A.C Zettlemoyer, Dekker, New York, p. 527.
Brailsford A.D. and Wynblatt P., 1979, Acta Metall. 27, 489.
Brainin M.l. and Smolyak B.M., 1980, Zh. Fiz. Khim. 54, 962.
Brandes H., 1927, Z Phys. Chem. 126, 196.
Brice i.C, 1973, The Growth of Crystals from Liquids, North-Holland, Amsterdam.
Budevski E., Bostanov V. and Staikov G„ 1980. Anna. Rev. Mater. Sci. 10, 85.
References 497
Budevski E., Bostanov V., Vitanov 1., Stoinov 7.., Kotzeva A. and Kaischew R., 1966,
Electrochim. Acta 11, 1697.
Budevski E., Staikov G. and Boslanov V., 1975, J. Cryst. Growth 29, 316.
Budevski E.. Staikov G. and Lorenz W.J., 1996, Electrochemical Phase Formation and
Growth, VCH. Weinheim.
Budurov S„ Spassov T. and Markov T„ 1986, J. Mater. Sci. 21, 2553.
Burton W.K., Cabrera N. and Frank F.C., \95\, Philos. Trans. Roy. Soc. (London) A 243,
299.
Butorin G.T. and Skripov V.P., 1972, Kristallografiya 17, 379.
Cabrera N. and Levine M., 1956, Philos. Mag. 1, 450.
Cadoret R. and Hottier F., 1983,1. Cryst. Growth 64, 583.
Cahn J.W. and Hilliard J.E., 1958,1. Chem. Phys. 28, 258.
Cahn J.W. and Hilliard J.E., 1959, J. Chem. Phys. 31, 688.
Carlson G.A. and Levine H.S., 1975, J. Appl. Phys. 46, 1594.
Carslaw H.S. and Jaeger J.G., 1959, Conduction of Heat in Solids, Clarendon, Oxford.
Castleman A.W., Jr., 1979, Adv. Colloid Interface Sci. 10, 73.
Chakarov V.M., Alexandrov A.D., Toshev B.V. and Scheludko A., 1991, Colloids Surfaces
52, 175.
Chakraverty B.K., 1966, Surface Sci. 4, 205.
Chakraverty B.K., 1967, J. Phys. Chem. Solids 28, 2401.
Chakraverty B.K., 1970, J. Non-Cryst. Solids 3, 317.
Cheng K.J., 1984, Phys. Lett. 106A, 403.
Chepelcvetskii M.L., 1939, Zh. Fiz. Khim. 13, 561.
Chernov A.A., 1961, Uspekhi Fiz. Nauk4. 116.
Chernov A. A., 1980, in: Sovremennaya Kristallografiya. Tom 3. Obrazovanie Kristallov,
Eds B.K. Vainshtein, A.A.Chernov and L. A. Shuvalov, Nauka, Moscow, p. 7. (English
translation: Chernov A.A., 1984, in: Modem Crystallography. Vol. 3. Crystal Growth,
Eds B.K. Vainshtein, A.A. Chernov and L.A. Shuvalov, Springer, Berlin, p. 1.)
Chernov A.A. and Trusov L.I., 1969, Kristallografiya 14, 218.
Chizmadzhev Yu.A. and Abidor I.G., 1980, Bioetectrochem. bioenergetics 7, 83.
Chizmadzhev Yu.A., Chernomordik L.V., Pastushenko V.F. and Abidor I.G., 1982, in:
Biofizika Membran, Vol. 2, Ed. P.G. Kostyuk, VINITI, Moscow, p. 161.
Christian J.W., 1975, The Theory of Transformations in Metals and Alloys, Part I, Pergamon,
Oxford.
Christoffersen J., Rostrup E. and Christoffersen M.R., 1991,/ Cryst. Growth 113, 599.
Cinti R.C. and Chakraverty B.K., 1972, Surface Sci. 30, 109.
Clausse D., 1985, in: Encyclopedia of Emulsion Technology, Vol. 2, Ed. P. Becher, Dekker,
New York, p. 77.
Collins EC, 1955, Z. Elektrochem. 59, 404.
Courtney W.G., 1961, J. Chem. Phys. 35, 2249.
Courtney W.G.. 1962, J. Chem. Phys. 36, 2009.
Coutsias E.A. and Neu J.C., 1984, Physica 12D, 295.
Crank J., 1967, The Mathematics of Diffusion, Clarendon, Oxford.
Dadyburjor D.B. and Ruckcnstein E., 1977, J. Cryst. Growth 40, 279.
Debenedetti P.G., 1996, Metastable Liquids: Concepts and Principles, Princeton Univ.
Press, Princeton,
de Boer J.H., 1953, The Dynamical Character of Adsorption, Clarendon, Oxford.
Delay R. and Prigogine 1., 1966, Surface Tension and Adsorption, Longmans, London.
Delate C.F. and Meier G.E.A., 1993, J. Chem. Phys. 98, 9850.
498 Nucleation: Basic Theory with Applications
Demo P. and Kozisek Z„ 1993. Phys. Rev. B 48, 3620.
Demo P. and Kozisek Z., 1996, Thermochim. Acta 280/2S1, 101.
Derjaguin B.V. and Gutop Yu.V, 1962, Kolloid. Zh. 24, 431.
Detsik V.N., Kukushkin S.A., Osipov A.V., Kaptelov E.Yu. and Pronin I.P., 1997, Phys.
Solid State 39, 104.
Detsik V.N., Kukushkin S.A.. Osipov A.V., Kaptelov E.Yu. and Pronin I.P., 1998, J.
Cryst. Growth 1S7, 474.
Deubener J., Bruckner R. and Sternitzke M., 1993, J. Non-Crysl. Solids 163, 1.
Dillmann A. and Meier G.E.A., 1989, Chem. Phys. Lett. 160, 7l.
Dillmann A. and Meier G.E.A., 1991, J. Chem. Phys. 94, 3872.
Dobreva A., Stoyanov A., Tzuparska S. and Gutzow I., 1996, Thermochim. Acta
280/281, 127.
Doremus R.H., 1985, Rales of Phase Transformations, Academic, Orlando.
Dufour L. and Defay R., 1963, Thermodynamics of Clouds, Academic, New York.
Dunning W.J., 1969, in: Nucleation, Ed. A.C. Zettlemoyer, Dekker, New York, p. 1.
Dupre A., 1869, Theorie Mechanique de la Chaleur, Gauthier-ViHard, Paris.
Ebner C, 1986, in: Chemistry and Physics of Solid Surfaces VI, Eds R. Vansclow and R.
Howe, Springer, Berlin, p. 581.
Eggington A., Kiang C.S., Stauffer D. and Walker G.H., 1971, Phys. Rev. Lett. 26, 820.
Ehrenfest P., 1933, Comm. Leiden Univ. 20 (Suppl. b75), 628.
Einstein A., 1905, Ann. Phys. 17, 549.
Eshclby J.D., 1957, Proc. Roy. Soc. (London) A 241, 376.
Evans J.W. and Bartelt M.C., 1994, .7. Vac. Set. Technol. A 12, 1800.
Evans R„ Martini Bettolo Marconi U. and Tarazona P., 1986, J. Chem. Soc. Faraday
Trans. 11 82, 1763.
Evans U.R., 1945, Trans. Faraday Soc. 41, 365.
Evers L.J., Shulepov S.Yu. and Frens G„ 1996, Faraday Disc. 104, 335.
Exerowa D. and Kashchiev D., 1986, Contemp. Phys. 27, 429.
Exerowa D. and Kruglyakov P.M., 1998, Foam and Foam Films. Theory, Experiment,
Application, Elsevier, Amsterdam.
Exerowa D., Cohen R. and Nikolova A., 1987, Colloids Surfaces 24, 43.
Exerowa D„ Kashchiev D. and Platikanov D., 1992, Adv. Colloid Interface Sci. 40, 201.
Exerowa D., Kashchiev D,, Platikanov D. andToshev B.V., 1994, Adv. Colloid Interface
Sci. 49, 303.
Family F. and Meakin P., 1988, Phys. Rev. Lett. 61, 428.
Family F. and Meakin P., 1989. Phys. Rev. A 40, 3836.
Farkas L., 1927, Z. Phys. Chem. 125, 236.
Feder J„ Russell K.C., Lottie I. and Pound G.M., 1966, Adv. Phys. IS, 111.
Filipovich V.N., Fokin V.M., Yuritsin N.S. and Kalinina A.M., 1996, Thermochim. Acta
280/281, 205.
Fischler W., Morgan D. and Polchinski J., 1990, Phys. Rev. D 42, 4042.
Fisher J.C., Hollomon J.H. and Turnbull D., 1948, J. Appl. Phys. 19, 775.
Fisher M.E., 1967a, Physics 3, 255.
Fisher M.E., 1967b, Rept. Prog. Phys. 30, 615.
Fishman I.M., 1988, Uspekhi Fiz. Nauk 155, 329.
Fisk J. A. and Katz J.L., 1996, J. Chem. Phys. 104, 8649.
Fisk J.A., Rudek M.M., Katz J.l.., Beiersdorf D. and Uchtmann H., 1998, Amos. Res. 46,
211.
Fletcher N.H., 1958, J. Chem. Phys. 29, 572.
References 499
Flynn P.C. and Wanke S.E., 1974, J. Catalysis 34, 390; 400.
Fokin V.M., Filipovich V.N. and Kalinina A.M., 1977, Fiz. Khim. Stekla 3, 129.
Fokin V.M., Kalinina A.M. and Filipovich V.N.. 1981, J. Cryst. Growth 52, 115.
Fokin V.M., Yuritsin N.S., Filipovich V.N. and Kalinina A.M., 1997, J. Non-Cryst. Solids
219, 37.
Ford I.J., 1992a, in: Nucleation and Atmospheric Aerosols, Eds N. Fukuta and P.E. Wagner,
Deepak, Hampton, p. 39.
Ford I.J., 1992b, J. Aerosol Sci. 23, 447.
Ford I.J., 1996,1. Chem. Phys. 105, 8324.
Ford I.J., 1997a, Phys. Rev. E 56, 5615.
Ford I.J., 1997b, J. Chem. Phys. 106, 9734.
Ford I.J., Laaksonen A. and Kulmala M., 1993, J. Chem. Phys. 99, 764.
Frank F.C., 1949, Disc. Faraday Soc. 5, 48.
Frank F.C., 1974, J. Cryst. Growth 22, 233.
Frenkel J.. 1932. Phys. Z Sowjetunion 1, 498.
Frenkel J., 1939, J. Chem. Phys. 7, 538.
Frenkel J., 1955, Kinetic Theory of Liquids, Dover, New York.
Friedlander S.K., 1961, ]. Meteowl. IS, 753.
Frisch H.L. and Carlier C.C., 1971, J. Chem. Phys. 54, 4326.
Fulcher H., 1925,1. Amer. Ceram. Soc. 8, 339.
Fura T., Marthinsen K. and Nes E., 1990, Mater. Sci. Technol. 6, 1093.
Gattef E. and Dimitriev Y, 1979. Philos. Mag. B 40, 233.
Gattef E. and Dimitriev Y, 1981, Philos. Mag. B 43, 333.
Oeguzin Ya.E. and Kaganovski Yu.S., 1978, Uspekhi Fit. Nauk 125, 489.
Gelbard F. and Seinfeld J.H., 1979, / Colloid Interface Sci. 68, 173.
Georgieva A. and Nenow D., 1967, Phys. Stat. Sol. 22, 415.
Ghez R. and Gilmer G.H., 1974, J. Cryst. Growth 21, 93.
Gibbs J.W., 1928, Collected Works. Vol 1. Thermodynamics, Longmans-Green, New
York.
Gilmer G.H., 1980, J. Cryst. Growth 49, 465.
Gilmer G.H., 1993. in: Handbook of Crystal Growth, Vol. 1, Ed. D.T.J. Hurle, North-
Holland, Amsterdam, p. 583.
Gilmer G.H., Ghez R. and Cabrera N., 1971,/ Cry.tr. Growth 8, 79.
Girshick S.L., 1991, J. Chem. Phys. 94, 826.
Girshick S.L. and Chiu C.P., 1990, J. Chem. Phys. 93, 1273.
Glasstone S., 1956, Textbook of Physical Chemistry, Macmillan, London.
Glasstone S., Laidler K.J. and Eyring H., 1941, The Theory of Rale Processes, McGraw-
Hill, New York.
Golubovic L. and Feng S., 1991, Phys. Rev. A 43, 5223.
Goodrich F.C., 1964, Proc. Roy. Soc. (London) A 277, 167.
Gradshtein I.S. and Ryzhik I.M., 1962, Tablitsy Integralov, Summ, Ryadov i Proizvedenii,
Fizmatgiz, Moscow.
Graham R. and Hakcn H., 1971, Z Phys. 243, 289; 245, 141.
Granasy L., 1993a, Europhys. Lett. 24, 121.
Granasy L., 1993b, J. Non-Cryst. Solids 162, 301.
Granasy L„ 1996a, J. Chem. Phys. 104, 5188.
Granasy L., 1996b, J. Phys. Chem. 100, 10768.
Granasy L., Egry I., Ratke L. and Herlach D.M., 1994, Scripta Metall. Mater. 30, 621; 31,
601.
Greer A.L., Evans P.V., Hamerton R.G., Shangguan D.K. and Kelton K.F., 1990,/ Cryst.
Growth 99, 38.
500 Nucleation: Basic Theory with Applications
Gretz R.D., 1966a, J. Chem. Phys. 45, 3160.
Gretz R.D., 1966b. Surface Sci. 5, 239.
Guggenheim E.A., 1957, Thermodynamics, North-Holland, Amsterdam.
Gutzow 1., 1980, Contemp. Phys. 21, 121; 243.
Gutzow I. and Kashchiev D., 1970, Comm. Dept. Chem. Bulg. Acad. Scis. 3, 645.
Gutzow I. and Kashchiev D.. 1971, in: Advances in Nucleation and Crystallization in
Glasses, Eds L.L. Hench and S.W. Freiman, Amer. Ceram. Soc, Columbus, p. 116.
Gutzow I. and Schmelzer J., 1995, The Vitreous State, Springer, Berlin.
Gutzow 1. and Toschev S„ 1968, Kristall Technik 3, 485.
Gutzow 1. and Toschev S., 1970, J. Cryst. Growth 7, 215.
Gutzow I. and Toschev S., 1971, in: Advances in Nucleation and Crystallization in
Glasses, Eds L.L. Hench and S.W. Freiman, Amer. Ccram. Soc, Columbus, p. 10.
Gutzow 1., Kashchiev D. and Avramov I., 1985, J. Non-Cryst. Solids 73, 477.
Gutzow 1., Popov E., Toschev S. and Marinov M., 1968, in: Rost Kristallov, Vol. 8, Part
2, Ed. N.N. Sheftal, Nauka, Moscow, p. 95.
Hadjiagapiou I., 1994, J. Phys. Condens. Matter 6, 5303.
Hagemann J.W., 1988, in: Crystallization and Polymorphism of Fats and Fatty Acids, Ed.
N. Garti and K. Sato, Dekker, New York, p. 9.
Hale B.N., 1986, Phys. Rev. A 33, 4156.
Hale B.N.. 1988, in: Atmospheric Aerosols and Nucleation, Eds P.E. Wagner and G. Vali,
Springer, Berlin, p. 323.
Ham F.S., 1958, J. Phys. Chem. Solids 6, 335.
Hammel J.J., 1967. J. Chem. Phys. 46, 2234.
Harrowell P. and Oxloby D.W., 1984, J. Chem. Phys. 80, 1639.
Hasegawa H. and Kozasa T., 1988, Prog. Theor. Phys. Suppl. 96, 107.
Heist R.H. and Reiss H., 1973,1. Chem. Phys. 59, 665.
Heist R.H., Janjua M. and Ahmed J., 1994, J. Phys. Chem. 98, 4443.
Hendriks E.M. and Ernst M.H., 1984, ]. Colloid Interface Sci. 97, 176.
Hendriksen B.A. and Grant D.J.W., 1995, J. Cryst. Growth 156, 252.
Hertz H„ 1882, Ann. Phys. (Leipzig) 17, 177.
Hikosaka M., Okada H„ Toda A„ Rastogi S. and Keller A., 1995, ]. Chem. Soc. Faraday
Trans. 91, 2573.
Hikosaka M., Sakurai K, Ohigashi H. and Keller A., 1994, Jpn. 1. Appl. Phys. (Part 1)
33, 214.
Hile L.R., 1969, Ph.D. Thesis, Princeton University, Princeton.
Hillig W.B., 1966.ACM Metall. 14, 1868.
Hirth J.P. and Pound G.M., 1963, Condensation and Evaporation, Pergamon, Oxford.
Hodgson A.W., 1984, Adv. Colloid Interface Sci. 21, 303.
Hoffman J.D., 1958, J. Chem. Phys. 29, 1192.
Hogan C.J., 1983, Phys. Lett. 133B, 172.
Hopper R.W.. Scherer G.W. and Uhlmann D.R., 1974, J. Non-Cryst. Solids 15, 45.
Hornbogen E_, 1969, in: Nucleation, Ed. A.C. Zettlemoyer, Dekker, New York, p. 309.
Hottier F. and Cadoret R., 1982, J. Cryst Growth 56, 304.
Hoyl J.I., 1990, Acta Metall. Mater. 38, 1405.
Huang K., 1963, Statistical Mechanics, Wiley, New York.
Hung C.H., Krasnopoler M.J. and Katz J.L., 1989, J. Chem. Phys. 90, 1856.
lckert L. and Schneider H.G., 1990, in; Advances in Epitaxy and Endotaxy, Eds H.G.
Schneider, V. Ruth and T. Kormany, Elsevier, Amsterdam, p. 229.
lsard J.O., 1977, Philos. Mag. 35. 817.
References 501
lsard J.O., James P.F. and Ramsden A.H., 1978, Phys. Chem. Glasses 19, 9.
Jakubczyk M. and Sangwal K., 1994, in: Elementary Crystal Growth, Ed. K. Sangwal,
Saan, Lublin, p. 43.
Jalaluddin A.K. and Sinha D.B., 1962, Nuovo Cimento Suppl. Ser. X 26, 234.
James P.F., 1974, Phys. Chem. Glasses 15, 95.
James P.F., 1982, in: Advances in Ceramics, Vol. 4, Eds J.H. Simmons, D.R. Uhlmann
and G.H. Beall, Amer. Ceram. Soc., Columbus, p. ].
Jarvis T.J., Donohue M.D. and Katz J.L., 1975. J. Colloid Interface Sci. 50, 359.
Jensen P., 1999, Rev. Mod. Phys., in press.
Jensen P. and Larralde H., 1997, Int. J. Mod. Phys. 11, 3635.
Jensen P., Larralde H. and Pimpinelli A., 1997, Phys. Rev. B 55, 2556.
Jensen P., Larralde H., Meunier M. and Pimpinelli A., 1998, Surface Sci. 412/413, 458.
Johnson W.A. and Mehl R.F., 1939, Trans. AIME 135, 416.
Jones D.R.H. and Chadwick G.A., 1971, Philos. Mag. 24, 995.
Kagan Yu., 1960, Zh. Fiz. Khim. 34, 92.
Kahlweit M., 1970, in: Physical Chemistry, an Advanced Treatise, Vol. 10, Ed. W. Jost,
Academic, New York, p. 719.
Kahlweit M., 1975, Adv. Colloid Interface Sci. 5, 1.
Kai S., Zimmermann W., Andoh M. and Chizumi N„ 1989, J. Phys. Soc. Jap. 58, 3449.
Kaischew R„ 1937. Ann. Phys. 30, 184.
Kaischew R„ 1950, Bull. Acad. Bulg. Scis. (Phys.) 1, 100.
Kaischew R„ 1951, Bull. Acad. Bulg. Scis. (Phys.) 2, 191.
Kaischew R., 1965, in: Rast Kristallov, Vol. 5, Ed. N.N. Sheftal, Nauka, Moscow, p. 45.
Kaischew R., 1980, Izhrani Trudove, Bulg. Acad. Scis., Sofia.
Kaischew R„ 1981, J. Cryst. Growth 51. 643.
Kaischew R. and Budevski E., 1967, Contemp. Phys. 8, 489.
Kaischew R. and Mutaftschiev B., 1962, Bull. Inst. Chim. Phys. Acad. Bulg. Scis. 2, 75.
Kaischew R. and Mutaftschiev B., 1963, Bull. Inst. Chim. Phys. Acad. Bulg. Scis. 3, 5.
Kaischew R. and Mutaftschiev B., 1965, Eleclrochim. Ada 10, 643.
Kaischew R. and Stranski I.N., 1934a, Z. Phys. Chem. B 26, 317.
Kaischew R. and Stranski I.N., 1934b, Z Phys. Chem. A 170, 295.
Kalikmanov V.I. and van Dongen M.E.H.. 1993, Europhys. Lett. 21, 645.
Kalikmanov V.I. and van Dongen M.E.H., 1995, J. Chem. Phys. 103, 4250.
Kalinina A.M., Filipovich V.N. and Fokin V.M., 1980, J. Non-Cryst. Solids 38 & 39, 723.
Kalinina A.M., Filipovich V.N. and Sycheva G.A., 1997, J. Non-Cryst. Solids 219, 80.
Kalinina A.M.. Fokin V.M. and Filipovich V.N., 1977, Fiz. Khim. Stekla 3, 122.
Kamo M., Sato Y„ Matsumolo S, and Setaka N„ 1983, J. Cryst. Growth 62, 642.
Kampfer B., 1988, Astron. Nachr. 309. 347.
Kane D. and El-Shall M.S., 1996, J. Chem. Phys. 105, 7617.
Kanne-Dannelschek I. and Stauffer D., 1981, J. Aerosol Sci. 12, 105.
KamerYu.O., 1983, Cryst. Res. Technol. 18, 1469.
Kanler Yu.O. and Neizvestnyi I.G., 1983, Poverkhnost. Fiz. Khim. Mekh. 7, 73.
Kantrowitz A., 1951, J. Chem. Phys. 19, 1097.
Kapusta J.1., 1984, Phys. Rev. D 30, 831.
Kashchiev D., 1969a, Surface Sci. 14, 209.
Kashchiev D., 1969b, Surface Sci. 18. 293.
Kashchiev D„ 1969c, Surface Sci. 18, 389.
Kashchiev D„ 1970, Surface Sci. 22, 319.
Kashchiev D., 1971, Comm. Dept. Chem. Bulg. Acad. Scis. 4, 327.
502 Nucleation: Basic Theory with Applications
Kashchiev D., 1972a, Philos. Mag. 25, 459.
Kashchiev D., 1972b, J. Cryst. Growth 13/14, 128.
Kashchiev D., 1974, Ph.D. Thesis, Bulgarian Academy of Sciences, Sofia.
Kashchiev D., 1975a, Thin Solid Films 29, 193.
Kashchiev D., 1975b, in: Rost Kristallov, Vol. 11, Eds A.A. Chernov, Kh.S. Bagdasarov,
E.I. Givargizov and P.O. Sharkhatunyan, EGU, Erevan, p. 41. (English translation:
1979, Growth of Crystals, Vol. 11, Ed. A.A. Chernov, Consultants Bureau, New York,
p. 38.)
Kashchiev D„ 1976a, Surface Sci. 55, 477.
Kashchiev D., 1976b, Comm. Depl. Chem. Bulg. Acad. Scis. 9, 52.
Kashchiev D., 1977, J. Cryst. Growth 40, 29.
Kashchiev D., 1978, Thin Solid Films 55, 399.
Kashchiev D„ 1979a, Surface Sci. 86, 14.
Kashchiev D„ 1979b, Phys. Stat. Sol. (a) 55, 369.
Kashchiev D., 1981, Phys. Stat. Sol. (a) 64, 715.
Kashchiev D., 1982, J. Chem. Phys. 76, 5098.
Kashchiev D., 1984a, Cryst. Res. Technol. 19, 1413.
Kashchiev D„ 1984b, J. Cryst. Growth 67, 559.
Kashcbiev D., 1985, Cryst. Res. Technol. 20, 723.
Kashchiev D., 1987, Colloid Polym. Sci. 265, 436.
Kashchiev D„ 1989a, in: Industrial Crystallization 87, Eds J. Ny vlt and S. Zacek, Elsevier,
Amsterdam, p. 3.
Kashchi
Kashchi
Kashchi
Kashchi
Kashchi
Kashchi
Kashchi
Kashchi
Kashch:
Kashchi
Kashchi
Kashchi
148
Kashchi
Kashchi
Kashchi
ev D., 1989b, Surface Sci. 220, 428.
ev D., 1990, Surface Sci. 225, 107.
ev D., 1996, J. Chem. Phys. 104, 8671.
ev D. and Exerowa D., 1980,7. Colloid Interface Sci. 77, 501.
ev D. and Exerowa D., 1983, Biochim. Biophys. Acta 732, 133.
iev D. and Exerowa D., 1998, J. Colloid Interface Sci. 203, 146.
D. and Firoozabadi A., 1993, J. Chem. Phys. 98, 4690.
iev D. and Kaischew R., 1969, Comm. Dept. Chem. Bulg. Acad. Scis. 2, 263.
v D. and Kantcr Yu.O., 1988, Phys. Stat. Sol. (a) 110, 61.
ev D. and Sato K., 1998, J. Chem. Phys. 109, 8530.
iev D. and van Rosmalen G.M., 1995, J. Colloid Interface Sci. 169, 214.
iev D., Clausse D. and Jolivet-Dalmazzone C, 1994, J. Colloid Interface Sci. 165,
iev D., Kaneko N. and Sato K., 1998, J. Colloid Interface Sci. 208, 167.
iev D., van der Eerden J.P. and van Leeuwen C, 1977, J. Cryst. Growth 40, 47.
iev D., Verdoes D. and van Rosmalen G.M., 1991, J. Cryst. Growth 110, 373.
Katz J.L., 1970a, J. Chem. Phys. 52, 4733.
Katz J.L., 1970b, J. Stat. Phys. 2, 137.
Katz J.L., 1992, Pure Appl. Chem. 64, 1661.
Katz J.L. and Ostermicr B.J., 1967, J. Chem. Phys. 47, 478.
Katz J.L., Fisk J.A. and Chakarov V., 1992, in: Nucleation and Atmospheric Aerosols,
Eds N. Fukuia and P.E. Wagner, Deepak, Hampton, p. II.
Katz J.L., Hung C.H. and Krasnopoler M.J., 1988, in: Atmospheric Aerosols and Nucleation,
Eds P.E. Wagner and G. Vali, Springer, Berlin, p. 356.
Katz J.L., Mirabel P., Scoppa II C.I. and Virkler T.L., 1976, J. Chem. Phys. 65, 382.
Katz J.L., Saltsburg H. and Reiss H„ 1966, J. Colloid Interface Sci. 21, 560.
Katz J.L., Scoppa 11 C.J., Kumar N.G. and Mirabel P., 1975, J. Chem. Phys. 62, 448.
Keldysh L.V., 1986, Contemp. Phys. 27, 395.
Keller A., Hikosaka M„ Rastogi S„ Toda A., Barham P.J. and Goldbeck-Wood G„ 1994.
J. Mater. Sci. 29, 2579.
References 503
Kelton K.R, 1991, Solid State Phys. 45, 75.
Kelton K.F., Greer A.L. and Thompson C.V., 1983, J. Chem. Phys. 79, 6261.
Kern R., Le Lay G. and Metois J.J., 1979, in: Current Topics in Materials Science, Vol.
3, Ed. E. Kaldis, North-Holland, Amsterdam, p. 131.
Kern R., Masson A. and Metois J.J., 197'I, Surface Sci. 27, 483.
Kiang C.S., Stauffcr D„ Walker G.H., Puri O.P., Wise J.D., Jr. and Patterson E.M., 1971,
J. Atmos. Sci. 28, 1222.
Kikuchi R„ 1969, J. Stat. Phys. 1, 351.
Kirkova E. and Djarova M„ 1971, Kristall Technikb, 601.
Kirkova E. and Djarova M., 1977, Kristall Technik 12, 763; 773.
Kirkwood J.G. and BuffF.P, 1949,1. Chem. Phys. 17, 338.
Kilamura M., Ueno S. and Sato K., 1998, in: Crystallization Processes, Ed. H. Ohtaki,
Wiley, New York, p. 99.
Klapka V., 1971, Coll. Czech. Chem. Coram. 36, 1181.
Knudsen M.. 1909, Ann. Phys. (Leipzig) 29, 179.
Kolmogorov A.N., 1937, Bull. Acad. Scis. URSS (CI. Scis. Math. Nat.) 3, 355. (See also:
Kolmogorov A.N.. 1986, Teoriya Veroyatnostei i Matematicheskaya Stalistika, Ed.
Yu.V. Prokhorov, Nauka, Moscow, p, 178.)
Kondo S„ 1956, 1. Chem. Phys. 25, 662.
Korn G.A. and Korn T.M., 1961, Mathematical Handbook for Scientists and Engineers,
McGraw-Hill. New York.
Kortzeborn R.N. and Abraham F.F., 1973, J. Chem. Phys. 58, 1529.
Koster U., 1978, Phys. Stat. Sol. (a) 48, 313.
Koster U., 1984, in: Amorphous Metals and Non-Equilibrium Processing, Ed. M. von
Allmen, Les Editions de Physique, Les Ulis, p. 175.
Koster U. and Blank-Bewersdorff M., 1987, Mater. Res. Soc. Symp. Proc. 57, 115.
Kostrovskii V.G., Mikheev V.B. and Shtein M.S., 1982, Mekhanism Obemnoi Kondensatsii
Para, Kipeniya Zhidkosti i Termodinamika Zarodyshei, SO AN SSSR, Novosibirsk.
Kotake S. and Glass I.I., 1981, Prog. Aerosp. Sci. 19, 129.
KoutsoukosP.G.andKontoyannisG.G., 1984,7. Chem. Soc. Faraday Trans. I 80, 1181.
Koverda V.P., Skripov V.P. and Bogdanov N.M., 1974, Kristallografiya 19, 613.
Kozisek 7.., 1988, Cryst. Res. Technol. 23, 1315.
Kozisek Z., 1989, Czech. J. Phys. B 39. 585.
Kozisek Z., 1990, Czech. J. Phys. B 40, 592.
Kozisek 2.., 1991, Cryst. Res. Technol. 26, 3; 11.
Kozisek Z. and Demo P., 1993a, J. Non-Cryst. Solids 152, 201.
Kozisek Z. and Demo P., 1993b, J. Cryst. Growth 132, 491.
Kozisek Z. and Demo P., 1995, J. Cryst. Growth 147, 215.
Kozisek Z„ Demo P. and Nesladek M., 1998, J. Chem. Phys. 108, 9835.
Kozlovskii M.I., 1962, Kristallografiya 7, 157.
Kozlovskii M.I., Burchakova VI. and Melentev 1.1., 1976, Eleklricheskoe Pole i
Kristullizatsiya, Shtiinlsa, Kishinev.
Krastanov L„ 1941, Meteorol. Z. 58, 37.
Krastanov L., 1947/48, Ann. Univ. Sofia Fac. Phys. Math. 44, I.
Krastanov L., 1970, Izbrani Trudove, Bulg. Acad. Scis., Sofia.
Krastanov L., Miloshev G. and Levkov L„ 1965, Bull. Inst. Geophys. Acad. Bulg. Scis.
6,5.
Kubota N., 1983, J. Cryst. Growth 62, 161.
Kubota N„ Kawakami T. and Tadaki T., 1986, J. Cryst. Growth 74, 259.
Kukushkin S.A. and Osipov A.V., 1995, Tech. Phys. 40, 615.
Kukushkin S.A. and Osipov A.V., 1996, Chem. Phys. Rept. 15, 1239.
504 Nucleation: Basic 1~heory with Applications
Kukushkin S.A. and Osipov A.V., 1997, J. Chem. Phys. 107, 3247.
Kukushkin S.A. and Osipov A.V., 1998, Uspekhi Fiz. Nauk 168, 1083.
Kukushkin S.A. and Slezov V.V., 1996, Dispersionnye Sistemy na Poverkhnosti Tverdykh
Tel: Mekhanizmy Obrazovaniya Tonkikh Plenok, Nauka, St. Petersburg.
Kuni F.M., 1984a, Kolloid. Zh. 46, 674.
Kuni F.M., 1984b, Kolloid. Zh. 46, 902.
Kuni F.M., Shchekin A.K. and Rusanov A.I., 1983, Kollod. Zh. 45, 682; 901; 1083.
La D. and Steinhardt P.J., 1989, Phys. Rev. Lett. 62, 376.
Laaksonen A., 1997, J. Chem. Phys. 106, 7268.
Laaksonen A. and McGraw R., 1996, Europhys. Lett. 35, 367.
Laaksonen A. and Oxloby D.W., 1995, J. Chem. Phys. 102, 5803.
Laaksonen A., Ford l.J. and Kulmala M., 1994, Phys. Rev. E 49, 5517.
Laaksonen A., Talanquer V. and Oxtoby D.W., 1995, Annu. Rev. Phys. Chem. 46, 489.
Landau L.D. and Lifshitz E.M., 1965, Teoriya Uprugosti, Nauka, Moscow.
Landau L.D. and Lifshitz E.M., 1976, Statisticheskaya Fizika, Part 1, Nauka, Moscow.
Landau L.D. and Lifshitz E.M., 1982, Elektrodinamika Sploshnykh Sred, Nauka, Moscow.
Landau L.D. and Lifshitz E.M., 1988, Gidrodinamika, Nauka, Moscow.
Lange E. and Nagel K„ 1935, Z. Elektrochem. 41, 575.
Laplace P., 1806, Traiti de Mechanique Celeste, Vol. 4, Courcier, Paris.
Larikov L.N. and Brick V.B., 1977, Kristall Technik 12, 1087.
Larson M.A. and Garside J., 1986,1. Crist. Growth 76, 88.
Lewis B„ 1970, Surface Sci. 21, 273; 289.
Lewis B. and Anderson J.C., 1978, Nucleation and Growth of Thin Films, Academic,
New York.
Lewis B. and Fujiwara S., 1976, J. Appl. Phys. 47, 1735.
Lewis B. and Halpern V„ 1976, J. Cryst. Growth 33, 39.
Lewis B. and Rees G.J., 1974, Philos. Mag. 29, 1253.
Lewis G.N. and Randall M., 1923, Thermodynamics and the Free Energy of Chemical
Substances, McGraw-Hill, New York.
Lifshitz I.M. and Slezov V.V., 1958, Zh. Eksper. Tear. Fiz. 35, 479.
Lifshitz l.M. and Slezov V.V., 1961, J. Phys. Chem. Solids 19, 35.
Logan R.M., 1969, Thin Solid Films 3, 59.
Lothe J. and Pound G.M., 1962, J. Chem. Phys. M, 2080.
Ludwig F.-R and Schmelzer J., 1995, Z. Phys. Chem. 192, 155.
Luijten C.C.M., 1998, Ph.D. Thesis, Eindhoven University of Technology, Eindhoven,
Lychev A.P, Rudenko Yu.S. and Cheremisin A.I., 1977, Izvestiya Vuzov. Fizika 4, 29.
Lyubov B.Ya., 1969, Kineticheskaya Teoriya Fazovykh Prevrashlenii, Metallurgiya, Moscow.
Lyubov B.Ya., 1975, Teoriya Kristallizatsii v Bolshykh Obemakh, Nauka, Moscow.
Lyubov B.Ya. and Roitburd A.L., 1958, in: Problemy Metallovedeniya i Fiziki Metallov,
Vol. 5, Eds B.Ya. Lyubov and O.P Maksimova, Metallurgizdat, Moscow, p. 91.
MacKenzic K.J.D. and Brown I.W.M., 1975, J. Thermal Anal. 7, 619; 8, 3.
Markov 1., 1976, Thin Solid Films 35, 11.
Markov I., 1995, Crystal Growth for Beginners, World Scientific, Singapore.
Markov I. and Kaischew R., 1976, Thin Solid Films 32, 163.
Markov 1. and Kashchiev D„ 1972a, J. Cryst. Growth 13/14, 131.
Markov 1. and Kashchiev D„ 1972b, J. Cryst. Growth 16, 170.
Markov 1. and Kashchiev D„ 1973, Thin Solid Films 15, 181.
Markov I. and Stoycheva E„ 1976, Thin Solid Films 35, 21.
Markworth A.J., 1973, Metall. Trans. 4, 2651.
References 505
Markworth A.J., 1986. J. Mater. Sci. Lett. 5, 161.
Masson A., Mritois J.J. and Kern R., 1971. in: Advances in Epitaxy and Endotaxy, Eds
H.G. Schneider and V. Ruth. VEB Deutscher Verlag fiir Grundstoffindustrie, Leipzig,
p. 103.
McDonald J.E., 1962, Amer. J. Phys. 30, 870.
McDonald J.E., 1963, Amer. J. Phys. 31, 31.
McGraw R. and Laaksonen A., 1996, Phys. Rev. Lett. 76, 2754.
McGraw R. and Laaksonen A., 1997. J. Chem. Phys. 106, 5284.
Mendell G. and Hiscock W.A., 1989, Phys. Rev. D 39, 1537.
Mersmann A.. 1990, J. Cryst. Growth 102, 841.
Mftois J.J., Gauch M, Masson A. and Kern R., 1972, Surface Set. 30, 43.
Metois J.J., Zanghi J.C., Eire R. and Kern R„ 1974, Thin Solid Films 22, 331.
Meyer H.J. and Stein B.J., 1980, J Cryst. Growth 49, 707.
Mie G., 1908, Ann. Phys. 25, 377.
Mikhnevich G.L. and Zaremba V.G.. 1962, in; Krislallizatsiya i Fazovye Perekhody, Ed.
N.N. Sirota, Minsk, AN BSSR, p. 214.
Milchev A., 1991, Contemp. Phys. 32, 321.
Milchev A. and Vassileva E., 1980, J. Electroanal. Chem. 107, 337.
Milchev A., Stoyanov S. and Kaischew R„ 1974, Thin Solid Films 22, 255; 267.
Milchev A., Vassileva E. and Kerlov V., 1980,1. Electroanal. Chem. 107, 323.
Miller R.C., 1976, Ph.D. Thesis, University of Missouri-Rolla, Rolla.
Miloshev G., 1963, Compt. Rend. Acad. Bulg. Scis. 16, 505; 597; 701.
Miloshev N., 1992, Atmos. Res. 28, 173.
Moelwyn-Hughes E.A., 1961, Physical Chemistry, Pergamon, London.
Monnette L.. Klein W., Zuckerman M., Khadir A. and Harris R., 1988, Phys. Rev. B 38,
11607.
Muitjens M.J.E.H., 1996, Ph.D. Thesis, Eindhoven University oCTechnology, Eindhoven.
Mullin J.W., 1993, Crystallization, Butterworth-Heinemann, Oxford.
Muralidhar R. and Ramkrishna D„ 1986, J. Colloid Interface Sci. 112, 348.
Muralidhar R. and Ramkrishna D., 1989, J. Colloid Interface Sci. 131, 503.
Mutaftschiev B.. 1993, in: Handbook of Crystal Growth, Vol. 1, Ed. D.T.J. Hurle, North-
Holland, Amsterdam, p. 187.
Nabarro F.R.N., 1940a, Proc. Phys. Soc. (London) 52, 90.
Nabarro F.R.N., 1940b, Proc. Roy. Soc. (London) A 175, 519.
Navascues G. and Mederos L., 1982, Surface Technol. 17, 79.
Navascues G. and Mederos L., 1983, J. Chem. Phys. 79, 2006.
Navascue's G andTarazona P., 1981, J. Chem. Phys. 75, 2441.
Nenow D. and Georgieva A., 1968, in: Rost Kristallov, Vol. 8, Part 2, Ed. N.N. Sheftal,
Nauka, Moscow, p. 316.
Neumann K. and Doring W„ 1940, Z Phys. Chem. A 186, 203.
Newman T.J. and Volmer A., 1996, J. Phys. A 29, 2285.
Nielsen A.E., 1964, Kinetics of Precipitation, Pergamon, Oxford.
Nielsen A.E., 1967, in: Crystal Growth, Ed. H.S. Peiser, Pergamon, Oxford, p. 419.
Nielsen A.E., 1969, Kristall Technik4, 17.
Nielsen A.E. and Sohnel 0., 1971, J. Cryst. Growth 11, 233.
Nicminen J.A. and Kaski K., 1989, Phys. Rev. A 40, 2088; 2096.
Nikolova A., Kashchiev D. and Exerowa D., 1989, Colloids Surfaces 36, 339.
Nishioka K., 1977, Phys. Rev. A 16, 2143.
Nishioka K., 1987, Phys. Rev. A 36, 4845.
Nishioka K., 1992, Metallurg. Trans. A 23, 1869.
506 Nucleation: Basic Theory wixh Applications
Nishioka K. and Fujita K., 1994, J. Chem. Phys. 100, 532.
Nishioka K. and Kusaka I., 1992, J. Chem. Phys. 96, 5370.
Nishioka K., Tomino H., Kusaka 1. and Takai T„ 1989, Phys. Rev. A 39, 772.
Nowakowski B. and Ruekenslein E., 1991, J. Colloid Interface Sci. 145, 182.
Nuth III J. A. and Ferguson R, 1993, in: Ceramic Transactions. Nucleation and Crystallization
in Liquids and Glasses, Ed. M.C. Weinberg, Amer. Ceram. Soc., Westerville, p. 23.
Obretenov W., 1988, Electrochim. Acta 33, 487.
Obretenov W., Bostanov V, Budevski E., Barradas R.G. and van der Noot T.J., 1986,
Electrochim. Acta 31, 753.
Obretenov W., Kashchiev D. and Bostanov V., 1989, J. Cryst. Growth 96, 843.
Ono S. and Kondo S„ I960, in: Handbuch der Physik, Vol. 10, Ed. S. Fliigge, Springer,
Berlin, p. 134.
Orihara H. and Ishibashi Y., 1992, J. Phys. Soc. Jap. 61, 1919.
Osipov A.V., 1989, Metallofizika 11, 3.
Osipov A.V., 1990a, Izv. Sib. Otd. Mead. Nauk SSSR ISer. Tekh. Nauk) 2, 92.
Osipov A.V., 1990b Metallofizika 12, 104.
Osipov A.V., 1993, Thin Solid Films 111, 111.
Ostwald W,, 1896, Lehrbuch der Allgemeinen Chemie, Vol. 2, Engelmann, Leipzig.
Ostwald W., 1897, Z Phys. Chem. 22, 289.
Ostwald W., 1900, Z Phys. Chem. 34, 495.
Ostwald W., 1901, Analytische Chemie. Engelmann, Leipzig.
Overbeek J.Th.G., 1952, in: Colloid Science, Vol. 1, Ed. H.R. Kruyt, Elsevier, Amsterdam,
Ch. 7.
Ovsienko D.E., 1975, in: Rost Kristallov, Vol. 11, Eds A.A. Chernov, Kh.S. Bagdasarov,
E.I. Givargizov and P.O. Sharkhatunyan, EGU, Erevan, p. II. (English translation:
1979, Growth of Crystals, Vol. 11, Ed. A.A. Chernov, Consultants Bureau, New York.)
Oxtoby D.W., 1992a, in: Fundamentals of lnhomogeneous Fluids, Ed. D. Henderson,
Dekker, New York, p. 407.
Oxtoby D.W., 1992b, J. Phys. Condens. Matter 4, 7627.
Oxtoby D.W., 1998, Ace. Chem. Res. 31, 91.
Oxtoby D.W. and Evans R., 1988, J. Chem. Phys. 89, 7521.
Oxtoby D.W. and Kashchiev D., 1994, J. Chem. Phys. 100, 7665.
Oxtoby D.W. and Laaksonen A., 1995, J. Chem. Phys. 102, 6846.
Parmar D.S. and Jalaluddin A.K., 1973, J. Phys. D 6, 1287.
Paskova R. and Gmzow 1., 1993, in: Fundamentals of Glass Science and Technology
1993, Suppl. Riv. Staz. Speriment. Vetro 23, 443.
Pastushenko V.F., Chizmadzhev Yu.A. and Arakelyan V.B., 1979, Bioelectrochem.
Bioenergetics 6, 53.
Paunov M. and Harsdorff M., 1974, Z Naturforsch. 29a, 1311.
Penkov I. and Gulzow I., 1984, J. Mater. Sci. 19, 233.
Petrov A.G., Mitov M,D. and Derzhanski A.1., 1980, in; Advances in Liquid Crystal
Research and Applications, Vol. 2, Ed. L. Bata, Pergamon, Oxford, p. 695.
Pilinis C. and Seinfeld J.H., 1987, J. Colloid Interface Sci. 115, 472.
Pimpinelli A., Villain J. and Wolf D.E., 1992, Phys. Rev. Lett. 69, 985.
Platikanov D., Nedyalkov M. and Nasteva V, 1980, J. Colloid Interface Sci. 75, 620.
Pocker D.J. and Hruska S.J., 1971,/ Vac. Sci. Technol. 8, 700.
Potapov O.V., Fokin V.M., Filipovich V.N. and Kalinina A.M., 1998, Fiz. Khim. Stekla
24, 484.
Pratsinis S.E., 1988, ]. Colloid Interface Sci. 124, 416.
References 507
Probstein R.F., 1951, J. Chem. Phys. 19, 619.
Prokhorov A.V and Derjaguin B.V., 1988, J. Colloid Interface Sci. 125, 111.
Piischl W. and Aubaucr H.P., 1980, Phys. Stat. Sol. (b) 102, 447.
Radoev B., Scheludko A. and Toshev B.V., 1986, J. Colloid Interface Sci. 113, 1.
Rasmussen D.H., 1982a, J.Cryst. Growth 56, 45.
Rasmussen D.H., 1982b, J. Cryst. Growth 56, 56.
Rasmussen D.H., Appleby M.R., Leedom O.L. and Babu S.V., 1983, ]. Cryst. Growth 64,
229.
Ratke L., 1987. J. Colloid Interface Sci. 119, 391.
Ratsch C, Smilauer P., Zangwill A. and Vvedensky D.D., 1995, Surface Sci. 329, L599.
Rayleigh, 1899, Philos. Mag. 47, 375.
Ree F.H., Ree T.S., Ree T. and Eyring H„ 1962, Adv. Chem. Phys. 4, 1.
Reichelt K., Lampert B. and Siegers H.-P., 1980, Surface Sci. 93, 159.
Reiss H., 1950, J. Chem. Phys. 18, 840.
Reiss H. and Katz J.L., 1967, J. Chem. Phys. 46, 2496.
Reiss H., Kalz J.L. and Cohen E.R., 1968, J. Chem. Phys. 48, 5553.
Reiss H., Kegel W.K. and Katz J.L., 1997, Phys. Rev. Lett. 78, 4506.
Richier L., Platikanov D. and Kretzschmar G., [9S6, Abhandl. Akad. Wiss. DDR IN, 395.
Robertson D. and Pound G.M., 1973,1. Cryst. Growth 19, 269.
Robins J.L. and Rhodin T.H., 1964, Surface Sci. 2, 346.
Robinson K„ 1951, J. Appl. Phys. 22, 1045.
Robinson V.H.E. and Robins J.L., 1974, Thin Solid Films 20, 155.
Rodigin N.M. and Rodigina E.N., 1960, Posledovafelnye Khimicheskie Reaktsii, AN
SSSR, Moscow.
Roginsky S.Z. and Todes O.M., 1940, Bull. Acad. Scis. URSS (CI. Scis. Chim.) 3, 331.
Rossi S.C.F. and Maciel W.J., 1984, Astrophys. Space Sci. 103, 143.
Routledgc K.J. and Stowell M.J., 1970, Thin Solid Films 6, 407.
Rowlands E.G. and James P.F., 1979, Phys. Chem. Glasses 20, 1.
Ruckenstein E. and Dadyburjor D.B., 1977, J. Catalysis 48, 73.
Ruckcnstein E. and Pulvermacher B., 1973, J. Catalysis 29, 224.
Rudek M.M., Fisk J.A., Chakarov V.M. and Katz J.L., 1996, J. Chem. Phys. 105, 4707.
Rundle J.B., 1989, J. Geophys. Res. 94, 2839.
Rundle J.B. and Klein W„ 1989. Phys. Rev. Lett. 63, 171.
Rusanov A.1., 1967, Fazovye Ravnovesiya i Poverkhnostnye Yavleniya, Khimiya, Leningrad.
Rusanov A.1., 1978, Phasengleichgewichte und Grenzflachenerscheidungen, Akademie-
Verlag, Berlin.
Rusanov A.I., 1979, J. Colloid Interface Sci. 68, 32.
Rusanov A.I. and Kuni F.M., 1984, J. Colloid Interface Sci. 100, 264.
Russell K.C., 1969,/ Chem. Phys. 50, 1809.
Russell K.C., 1980, Adv. Colloid Interface Sci. 13, 205.
Sampson K.J. and Ramkrishna D„ 1985, J. Colloid Interface Sci. 103, 245.
Sangwal K., 1989, J. Cryst. Growth 97, 393.
Sangwal K., 1994, in: Elementary Crystal Growth, Ed, K. Sangwal, Saan. Lublin, p. 83.
Sangwal K. and Polak W., 1997, Cryst. Res. Technol. 32, 509.
Sato K., 1988, in: Crystallization and Polymorphism of Pats and Fatly Acids,
Eds N. Garti and K. Sato, Dckker, New York, p. 227.
Sato K., 1993, ]. Phys. D 26, B77.
Sazaki G., Kurihara K„ Nakada T., Miyashita S. and Komatsu H., 1996, J. Cryst. Growth
169, 355
508 Nucleation: Basic Theory with Applications
Scharifker B. and Wehrmann C, 1985, ./. Electroanal. Chem. 185, 93.
Scheludko A., Chakarov V. and Toshev B.V., 1981, J. Colloid Interface Sci. 82, 83.
Schiffner U. and Pannhorst W., 1987, Glastechn. Ber. 60, 211; 239.
Schlesinger M.E. and Lynch DC, 1989, J. Non-Cryst. Solids 108, 237.
Schmelzer J. and Mahnke R., 1986, J. Chem. Soc. Faraday Trans. I 82, 1413.
Schmelzer J., Gutzow I. and Schmelzer J., Jr, 1996, J. Colloid Interface Sci. 178, 657.
Schottky W.F., 1962, Z Phys. Chem. (N.F.) 31, 40.
Scott M.G. and Ramachandrarao P., 1977, Mater. Sci. Eng. 29, 137.
Shablakh M„ Dissado L..A. and Hill R.M., 1983, J. Chem. Soc. Faraday Trans. II 79,
1443.
Shchekin A.K. and Warshavsky V.B., 1996, Kolloid. Zh. 58, 564.
Shchekin A.K., Rusanov A.I. and Kuni F.M., 1984, Kolloid. Zh. 46, 535.
Shcherbakov L.M. and Ryazantsev P.P., 1964, in: Issledovaniya v Oblasti
Poverkhnostnykh Sil, Ed. B.V. Derjaguin, Nauka, Moscow, p. 26.
Shechtman D., Blech 1., Gratias D. and Cahn J.W., 1984, Phys. Rev. Lett. S3, 1951.
Shen Y.C. and Oxtoby D.W., 1996, J. Chem. Phys. 105, 6517.
Shi F.G. and Seinfeld J.H., 1992, in: Nucleation and Atmospheric Aerosols, Eds N.
Fukuta and P.E. Wagner, Deepak, Hampton, p. 1.
Shi G. and Seinfeld J.H., 1990a, J. Chem. Phys. 93. 9033.
Shi G. and Seinfeld J.H., 1990b, .7. Colloid Interfuce Sci. 135, 252.
Shi G. and Seinfeld J.H., 1991a, J. Mater. Res. 6, 2091.
Shi G. and Seinfeld J.H., 1991b. J. Mater. Res. 6, 2097.
Shi G., Seinfeld J.H. and Okuyama K., 1990, Phys. Rev. A 41, 2101.
Shichtri T. and Araki Y„ 1986, J. Cryst. Growth 78, 502.
Shizgal B. and Barrett J.C., 1989, J. Chem. Phys. 91, 6505.
Shneidman V.A. and Weinberg M.C., 1991,/ Chem. Phys. 95, 9148.
Shneidman V.A. and Weinberg M.C., 1992a, J. Chem. Phys. 97, 3621.
Shneidman V.A. and Weinberg M.C., 1992b, J. Chem. Phys. 97, 3629.
Shneidman V.A. and Weinberg M.C., 1993, J. Non-Cryst. Solids 160, 89.
Shvedov E.V., Postnikov V.S. and levlev V.M., 1977, Phys. Stat. Sol. (a) 44, 423.
Sigsbee R.A., 1969, in: Nucleation, Ed. A.C. Zettlemoyer, Detcker, New York, p. 151.
Sigsbee R.A., 1971, ]. Appl. Phys. 42, 3904.
Sigsbee R.A., 1972, J. Cryst. Growth 13/14, 135.
Simon B., Grassi A. and Boistelle R„ 1974, J. Cryst. Growth 26, 77.
Sirota N.N., 1969, in: Mekhanism i Kinetika Kristallizatsii, Ed. N.N. Sirota, Nauka i
Tekhnika, Minsk, p. 6.
Skapski A.S., 1948, J. Chem. Phys. 16, 386; 389.
Skapski A.S., 1956, Acta Melall. 4, 576.
Skripov V.P., 1972, Metastabilnaya Zhidkost, Nauka, Moscow. (English translation: 1974,
Melastable Liquids, Wiley, New York.)
Skripov V.P., 1977, in: Current Topics in Materials Science, Vol. 2, Ed. E. Kaldis, Norlh-
Holland, Amsterdam, p. 327.
Skripov V.P. and Koverda V.P., 1984, Spontannaya KristalUzatsiya Pereokhlazhdennykh
Zhidkostei, Nauka, Moscow.
Skripov V.P., Butorin G.T. and Koverda V.P., 1971,/¾. Met. Metalloved. 31, 790.
Skripov V.P, Koverda V.P. and Butorin G.T., 1970, Kristallografiya 15, 1219.
Slezov V.V. and Schmelzer J., 1994, J. Phys. Chem. Solids 55, 243.
Slowinski E.J., Jr, Gates E.E. and Waring C.E., 1957, J. Phys. Chem. 61, 808.
Smolyak B.M., 1980, Ph.D. Thesis, Ural Polytechnical Institute, Sverdlovsk.
Sohnel O. and Garside J., 1988, .7. Cryst. Growth 89, 202.
Sohnel O. and Garside J.f 1992, Precipitation. Basic Principles and Industrial Application,
Butterworth-Heinemann, Oxford.
References 509
Sohnel O. and Mullin J.W., 1978. J. Cryst. Growth 44, 377.
Sohnel 0. and Mullin J.W., 1979, Kristall Technik 14, 217.
Sohnel 0. and Mullin J.W., 1988, J. Colloid Interface Sci. 123, 43.
Spassov T. and Budurov S., 1988, Cryst. Res. Technol. 23. 1225.
Stauffer D., 1976, J. Aerosol Sci. 7, 319.
Stauffer D. and Kiang C.S., 1977, Adv. Colloid Interface Sci. 7, 103.
Stefan J., 1886, Wied. Ann. Phys. 29, 655.
Steinhardt P.J., 1990, Phys. Rev. Lett. 64, 2740.
Stenzel H. and Bethge H., 1976, Thin Solid Films 32, 267.
Stenzcl H. and Velfe H.-D, 1984, Ph.D. Thesis, Martin-Luther-Universitat, Halle-Wittenberg.
Stenzel H., Velfe H.-D. and Krohn M., 1980, Kristall Technik 15, 255.
Stowed M.J., 1970, Philos. Mug. 21, 125.
Stowell M.J., 1972a, Philos. Mag. 26, 349.
Stowcll M.J., 1972b, Philos. Mag. 26, 361.
Stowell M.J., 1974a, J. Cryst. Growth 24/25, 45.
Stowell M.J., 1974b, Thin Solid Films 21, 91.
Stowell M.J. and Hutchinson T.E., 1971, Thin Solid Films 8, 41; 411.
Stoyanov S„ 1973, Thin Solid Films IS, 91.
Stoyanov S., 1979, in: Current Topics in Materials Science, Vol. 3, Ed. E. Kaldis, North-
Holland, Amsterdam, p. 421.
Stoyanov S. and Kashchiev D., 1981, in: Current Topics in Materials Science. Vol. 7, Ed.
E. Kaldis, North-Holland, Amsterdam, p. 69.
Stoyanov S., Kashchiev D. and Georgicv M., 1970, Phys. Stat. Sol. 41, 387; 395.
Stoyanova V, Kashchiev D. and Kupenova T., 1994, J. Aerosol Sci. 25, 867.
Stranski I.N., 1936, Ber. V/ien. Akad. 145, 840.
Stranski I.N. and Kaischew R., 1934, Z. Phys. Chem. B 26, 100; 312.
Strey R. and Viisanen Y., 1993, J. Chem. Phys. 99, 4693.
Strey R„ Viisanen Y. and Wagner RE., 1995, J. Chem Phys. 103, 4333.
Strey R., Wagner RE. and Viisanen Y, 1994,1. Phys. Chem. 98, 7748.
Slumm W. and Morgan J.J., 1981, Aquatic Chemistry, Wiley, New York.
Suck Salk S.H. andLutrus C.K., 1988, in: Atmospheric Aerosols and Nucleation, Eds RE.
Wagner and G. Vali, Springer, Berlin, p. 496.
Sutugin A.G. and Fuchs N.A., 1970, Aerosol Sci. 1, 287.
Sutugin A.G., Fuchs N.A. and Kotsev E.I., 1971, Aerosol Sri. 2, 361.
Swift D.L. and Friedlander S.K., 1964, J. Colloid Sci. 19, 621.
Sycheva G.A., 1997, Glass Phys. Chem. 23, 368.
Sycheva G.A., 1998a, Fiz. Khim. Stekla 24, 491.
Sycheva G.A., 1998b, Glass Phys. Chem. 24, 47.
Symeopoulos B.D. and Koutsoukos P.G., 1992, J. Chem. Soc. Faraday Trans. 88, 3063.
Takahashi T., 1982, J. Cryst. Growth 59, 441.
Talanquer V and Oxtoby D.W., 1994, J. Chem. Phys. 100, 5190.
Talanquer V and Oxtoby D.W., 1995, J. Chem. Phys. 102, 2156.
Talanquer V. and Oxtoby D.W., 1996, J. Chem. Phys. 104, 1483.
Talanquer V. and Oxtoby D.W., 1997, J. Chem. Phys. 106, 3673.
Tambour Y and Seinfeld J.H., 1980, J. Colloid Interface Sci. 74, 260.
Tammann G. and Hesse N„ 1926, Z. Anorg. Allgem. Chem. 156, 245.
Tarazona P. and Evans R„ 1983, Mol. Phys. 48, 799.
Temkin D.E.. 1977, J. Cryst. Growth 40. 173.
Temkin D.E. and Shevelev V.V., 1984, J. Cryst. Growth 66, 380.
Thomson J.J., 1886, Applications of Dynamics to Physics and Chemistry. McMillan,
London.
510 Nucleation: Basic Theory with Applications
Thomson J.J., 1906, Conduction of Electricity through Gases, Cambridge Univ. Press,
Cambridge.
Thomson W., 1870, Proc. Roy. Sot: (Edinburgh) 7, 63.
Thomson W., 1871, Philos. Mag. 42, 448.
Tien H.T., 1974, Bilayer Lipid Membranes (BLM). Theory and Practice, Dekker, New
York.
Todes O.M., 1940, Acta Physicochim. URSS 13, 617.
Todes O.M., 1946, Zh. Fiz. Khim. 20, 629.
Todes O.M., 1949a, in: Problemy Kinetiki i Kataliza, Vol. 7, Ed. S.Z. Roginsky, AN
SSSR, Moscow, p. 137.
Todes O.M., 1949b, in: Problemy Kinetiki i Kataliza, Vol. 7, Ed. S.Z. Roginsky, AN
SSSR, Moscow, p. 91.
Todes O.M. and Chekunov A.A., 1957, Kolloid. Zh. 19, 633.
Todes O.M. and Khrushchev V.V., 1947, Zh. Fiz. Khim. 21, 301.
Tohmfor G. and Volmer M., 1938, Ann. Phys. 33, 109.
Tolman R.C., 1949, J. Chem. Phys. 17, 118; 333.
Tomino H., Kusaka 1., Nishioka K. and Takai T., 1991,/ Cryst. Growth 113, 633.
Toner M., Cravalho E.G. and Karel M., 1990, J. Appl. Phys. 67, 1582.
Toschev S., 1973a, in: Crystal Growth: an Introduction, Ed. R Hartman, North-Holland,
Amsterdam, p. I.
Toschev S., 1973b, in: Crystal Growth: an Introduction, Ed. P. Hartman, North-Holland,
Amsterdam, p. 328.
Toschev S. and Gutzow I., 1967a, Phys. Stat. Sol. 21, 683.
Toschev S. and Gutzow 1., 1967b, Phys. Stat. Sol. 24, 349.
Toschev S., Milchev A. and Stoyanov S., 1972, J. Cryst. Growth 13/14, 123.
Toschev S., Paunov M. and Kaischew R., 1968, Comm. Dept. Chem. Bulg. Acad. Sets. 1,
119.
Toshev B.V., 1991, Langmuir 7, 569.
Toshev B.V., 1993, D.Sc. Thesis, Sofia University, Sofia.
Toshev B.V and Scheludko A., 1991, in: Lecture Notes in Physics. 386. Capillarity
Today, Eds G. P6U6 and A. Sanfeld, Springer, Berlin, p. 138.
Toshev B.V, Platikanov D. and Scheludko A., 1988, Langmuir 4, 489.
Trayanov A. and Kashchiev D„ 1986, J. Cryst. Growth 78, 399.
Trinkaus H. and Yoo M.H., 1987, Philos. Mag. A 55, 269.
Trofimenko V.V., Zhitnik V.P., Aleksandrova T.G. and Loshkarev Yu.M., 1980,
Elektrokhimiya 16, 1139.
Trofimenko V.V., Zhitnik VP. and Loshkarev Yu.M., 1979, Elektrokhimiya 15, 1035.
Trofimenko V.V., Zhitnik VP. and Loshkarev Yu.M., 1981, Elektrokhimiya 17, 1644.
Trofimov V.I., 1983, Fiz. Tverd. Tela 25, 1473.
Trofimov V.I., Belenkii VZ. and Osadchenko V.A., 1985, Poverkhnost. Fiz, Khim. Mekh.
8,60.
Tunitskii N.N., 1941, Zh. Fiz. Khim. 15, 1061.
Turnbull D., 1948, Trans. A1ME, 175, 774.
Turnbuil D„ 1950,1. Appl. Phys. 21, 1022.
Turnbull D., 1952,1. Chem. Phys. 20, 411.
Turnbull D., 1969, Contemp. Phys. 10, 473.
Turnbull D., 1981, Metall. Trans. A 12A, 695.
Turnbull D. and Cormia R.L., 1961, J. Chem. Phys. 34, 820.
Turnbull D. and Fisher J.C., 1949, J. Chem. Phys. 17, 71.
Uhlmann D.R., 1972, J. Non-Cryst. Solids 7, 337.
References 511
Uhlmann D.R., 1983, J. Amer. Ceram. Soc. 66, 95.
Ulhricht H., Schmelzer J., Mahnke R. and Schweitzer F., 1988, Thermodynamics of
Finite Systems and the Kinetics of First-Order Phase Transitions, Teubner, Leipzig,
linger C. and Klein W„ 1984, Phys. Rev. B 29, 2698.
van der Eerden J.P., 1982, J. Cryst. Growth 56, 174.
van der Eerden J.P., 1993, in: Handbook of Crystal Growth, Vol. 1, Ed. D.T.J- Hurle,
North-Holland, Amsterdam, p. 307.
van der Eerden J.P., Kashchiev D. and Bennema P., 1977, J. Cryst. Growth 42, 31.
van der Leeden M.C., Kashchiev D. and van Rosmalen G.M., 1992, J. Colloid Interface
Sci. 152, 338.
van der Leeden M.C., Kashchiev D. and van Rosmalen G.M., 1993,7. Cryst. Growth 130,
221.
van der Leeden M.C., Vcrdoes D., Kashchiev D. and van Rosmalen G.M., 1991, in:
Advances in Industrial Crystallization, Eds J. Garside, R.J. Davey and A.G. Jones,
Butterworth-Heinemann, Oxford, p. 31.
van Leeuwen C, 1979, J. Cryst. Growth 46, 91.
van Leeuwen C. and Blomen L.J.M.J., 1979, J. Cryst. Growth 46, 96.
van Leeuwen C. and van der Eerden J.P., 1977, Surface Sci. 64, 237.
Venables J.A., 1973, Philos. Mag. 27, 697.
Venablcs J. A., 1994, Surface Sci. 299/300, 798.
Venables J.A. and Price G.L., 1975, in: Epitaxial Growth, Part B, Ed. J.W. Matthews,
Academic, New York, p. 381.
Venables J.A., Spiller G.D.T. and Hanbiieken M., 1984, Rept. Prog. Phys. 47, 399.
Verdoes D., Kashchiev D. and van Rosmalen G.M., 1992, J. Cryst. Growth 118. 401.
Vetter K.J., 1967, Electrochemical Kinetics, Academic, New York.
Viisanen Y. and Strey R.. 1994, J. Chem. Phys. 101, 7835.
Viisanen Y. and Strey R., 1996, J. Chem. Phys. 105, 8293.
Viisanen Y, Strey R. and Rciss H., 1993, J. Chem. Phys. 99, 4680.
Viisanen Y, Strey R„ Laaksonen A. and Kulmala M., 1994, J. Chem. Phys. 100, 6062.
Villain J., Pimpinelli A. and Wolf D„ 1992, Comments Cond. Mat. Phys. 16, 1.
Villain J., Pimpinelli A., Tang L.-H. and Wolf D., 1992, J. Phys. I France 2, 2107.
Vincent R, 1971, Proc. Roy. Soc. (London) A 321. 53.
Vogel H., 1921, Phys. Z. 22, 645.
Volmer M., 1939, Kinelik der Phasenbildung, Steinkopff, Dresden.
Volmer M. and Flood H., 1934, Z Phys. Chem. A 170, 273.
Volmer M. and Marder M., 1931, Z. Phys. Chem. A 154, 97.
Volmer M. and Weber A., 1926, Z Phys. Chem. 119, 277.
Voloshchuk V.M. and Sedunov Yu.S., 1975, Protsessy Koagulyatsii v Dispersnykh Sistemakh,
Gidrometeoizdai, Leningrad.
Volterra V. and Cooper A.R., 1985, J. Non-Cryst. Solids 74, 85.
von Smoluchowski M., 1916, Phys. Z. 17, 557; 585.
von Smoluchowski M., 1917, Z Phys. Chem. 92, 129.
Vook R.W., 1982, Int. Met. Rev. 27, 209.
Voronkov V.V., 1970, Kristallografiya 15, 13.
Wagner C, 1961, Z Elektrochem. 65, 581.
Wagner P.E. and Strey R., 1984, J. Chem. Phys. 80, 5266.
Wagner P.E., Strey R. and Viisanen Y, 1992, in: Nucleation and Atmospheric Aerosols,
Eds N. Fukuta and P.E. Wagner, Deepak, Hampton, p. 27.
Wakeshima H., 1954, J. Chem. Phys. 22, 1614.
5/2 Nucleation: Basic Theory with Applications
Walton A.G., 1969a, in: Nucleation, Ed. A.C. Zcttlemoyer, Dekker, New York, p. 225.
Wallon D., 1962, J. Chem. Phys. 37, 2182.
Walton D., 1969b, in: Nucleation, Ed. A.C. Zettlemoyer, Dekker, New York, p. 379.
Warshavsky V.B. and Shchekin A.K., 1999a, Colloids Surfaces A 148, 283.
Warshavsky V.B. and Shchekin A.K., 1999b, Kolloid. Zh. 61, 624.
Weeks J.D. and Gilmer G.H., 1979, Adv. Chem. Phys. 40, 157.
Weinberg M.C.. 1985, J. Non-Cryst. Solids 76, 253.
Weinberg M.C.. 1991,/ Non-Cryst. Solids 134, 116.
Weinberg M.C., 1992, J. Non-Cryst. Solids 142, 126.
Weinberg M.C. and Kapral R„ 1989. J. Chem. Phys. 91, 7146.
Weinberg M.C. and Zanotto E.D., 1989a, Phys. Chem. Glasses 30, 110.
Weinberg M.C. and Zanotto E.D., 1989b, J. Non-Cryst. Solids 108, 99.
Weinberg M.C, Birnie 111 D.P. and Shneidman V.A., 1997, J. Non-Cryst. Solids 219, 89.
Weinberg M.C, Uhlmann D.R. and Zanotto E.D., 1989, J. Amer. Ceram. Soc. 72, 2054.
Weinberg M.C, Zelinski B.J., Uhlmann D.R. and Zanotto E.D.. 1990, J. Non-Cryst.
Solids 123. 90.
Westman A.E.R. and Krishna-Murthy M., 1962, in: Symposium on Nucleation and
Crystallization in Glasses and Melts, Eds M.K. Reser, G. Smith and H. lnsley, Amer.
Ceram. Soc., Columbus, p. 91.
White G.M., 1969, J. Chem. Phys. 50, 4672.
Wiedersich H. and Katz J.L., 1979, Adv. Colloid Interface Sci. 10, 33.
Wilcox C.F. and Bauer S.H., 1991,/. Chem. Phys. 94, 8302.
Wilcox R.W., 1977, J. Cryst. Growth 37, 229.
Wilemski G., 1975, J. Chem. Phys. 62, 3763.
Wilemski G., 1995, J. Chem. Phys. 103, 1119.
Wilemski G. and Wyslouzil B.E., 1995, J. Chem. Phys. 103, 1127.
Wilemski G., Wyslouzil B.E., Gauthicr M. and Frish M.B., 1992, in: Nucleation and
Atmospheric Aerosols, Eds N. Fukuta and P.E. Wagner, Deepak, Hampton, p. 23.
Williams M.M.R., 1984a, J'.Colloid Interface Sci. 101, 19.
Williams M.M.R., 1984b, Physica 125A, 105.
Williams M.M.R., 1985, J. Colloid Interface Sci. 103, 516.
Wilson H.A., 1900, Philos. Mag. 50, 238.
Wojciechowski K. and Kibalczyc W„ 1986, J. Cryst. Growth 76, 379.
Wolf D.E., 1995, in: Scale Invariance, Interfaces and Non-Equilibrium Dynamics, Ed. A.
Mckane et ul., Plenum, New York, p. 215.
Wu D.T., 1992a, Mater. Res. Soc. Symp. Proc. 205, 411.
Wu D.T., 1992b, J. Chem. Phys. 97, 1922.
Wu D.T., 1992c, J. Chem. Phys. 97, 2644.
Wu D.T., 1993, J. Chem. Phys. 99, 1990.
Wynblatt P., 1976, Acta Metall. 24, 1175.
Wynblatl P. and Ahn T.M., 1975, in: Sintering and Catalysis, Ed. G.C. Kuczynski, Plenum,
New York, p. 83.
Wynblatt P. and Gjostcin N.A., 1976, Acta Metall. 24, 1165.
Wyslouzil B.E. and Wilemski G., 1995, J. Chem. Phys. 103, 1137.
Yang C.H. and Qiu H., 1986, J. Chem. Phys. 84, 416.
Yinnon H. and Uhlmann D.R., 1981, J. Non-Cryst. Solids 44, 37.
Young T„ 1805, Philos. Trans. Roy. Soc. (London) 95, 65.
Yount D.E. and Hoffman D.C., 1986, Aviat. Space Environ. Med. 57, 149.
Zeldovieh J.B., 1942, Zh. Eksper. Teor. Fiz. 12, 525. (English translation: 1943, Acta
Physicochim. URRS 18, 1.)
References 513
Zeldovich J.B. and Myshkis A.D., 1972, Elementy Prikladnoi Matematiki, Nauka, Moscow.
Zeng X.C. and Oxtoby D.W., 1991,1. Chem. Phys. 94, 4472; 95, 5940.
Zettlemoyer A.C. (Editor), 1969, Nucleation, Dekker, New York.
Zettlemoycr A.C. (Editor), 1977, Nucleation Phenomena, Adv. Colloid Interface Sci. 7.
Zeltlemoyer A.C. (Editor), 1979, Second Special Issue on Nucleation Phenomena, Adv.
Colloid Interface Sci. 10.
Zhang J. and Nancollas G.H., 1990, J. Cryst. Growth 106, 181.
Ziabicki A., 1968, J. Chem. Phys. 48, 4368; 4374.
Ziabicki A., 1986, J. Chem. Phys. 85, 3042.
Ziabicki A. and Jarecki L„ 1984. J. Chem. Phys. 80, 5751.
Zinke-Allmang M., Feldman L.C. and Grabow M.C., 1992, Surface Sci. Rept. 16, 377.
Zinsmeister G., 1966, Vacuum 16, 529.
Zinsmeister G., 1968, Thin Solid Films 2, 497.
Zinsmeister G., 1969, Thin Solid Films 4, 363.
Zinsmeister G„ 1970, Kristall Technik 5, 207.
Zinsmeister G-, 1971, Thin Solid Films 7, 51.
Zinsmeister G., 1974, Jpn. J. Appl. Phys. Suppl. 2, Pt. 1, 545.
This Page Intentionally Left Blank
Author index
Abidorl.G., 315,481
Abraham F.F., x, 22, 27, 71, 74, 83, 86, 97,
121, 185, 187, 193,231,242,244,246,
249, 265, 266, 315, 317
Ahmed J., 330, 336
AhnT.M., 135
Aksenov A.A., 59, 228
Akksandrov L.N., 468, 471
Aleksandrov Yu.A., x, 47, 49
Aleksandrova T.G., 366, 443
Alexandrov A.D., 300, 307
Allen L.B., 59, 226, 228
Amar J.G., 437, 446
Anderson J.C., x, II, 26, 56, 88, 89, 119,
132, 145-149, 161, 162,226,228,437,
468
Anderson R.J., 430, 434
Andoh M., ix
Andres R.P., 115, 118, 119, 138, 171, 187,
193, 194, 231, 260, 262, 346, 430
Anisimov M.P., 59, 226, 228
Antonione C, 426
Appleby M.R., 79, 83
Arakelyan V.B.. 315.481
Araki Y„ 315, 328
Armstrong R.D., 398, 468
AubauerH.R, 128, 134
Avrami M„ 366-368, 373, 375, 436, 437,
439, 446, 448, 450
Avramovl.,x, 14, 153, 156,247,270,416,
426
Avramov M.Z., 300
Babu S.V., 79, 83
Baidakov V.G., x, 49, 59, 79, 81, 97, 102,
140, 202, 209, 215, 226, 386, 430
Barham P.J.. 387
Barnard A.J., 86
Baroody E.M., 133
Barradas R.G., 398
Barren J.C., 88, 126, 232, 235, 239, 245,
261, 262, 265, 266, 335, 346, 348, 353,
356, 364
Bartelt M.C., 147, 437, 446, 468
Barthes M.-G., 468
Basu D.K., 315, 329
Battezzati L., 426
Bauer E., 446
Bauer S.H., 330
Beattie J.A., 332
Becker R., 18,22, 115, 121, 158, 184, 185,
192-194, 200
Bedanov V.M., 59, 215, 226, 229
Behm R.J., 437, 446
Beiersdorf D., 215,430
Belenkii V.Z., 373,376-379, 398,401,446.
468, 471
BcnnemaP., 168, 391,407
Bethge H., 366, 444, 445
Bienfait M., 202
Bigg E.K., 384, 386
Binder K., x, 115, 116, 119
Birnie III D.P., 373. 380
Blander M., ix, 27, 32, 35, 36, 49, 51, 84,
88,94, 140, 199, 209,430
Blank-Bewersdorff M„ 463
Blanke H., 463
Blech I., 387
Bliznakow G., 140
Blomen L.J.M.J., 13
Bogdanov N.M., 256, 265-267, 273, 276,
277
Bohm J., 10, 13, 14
Boistelle R„ 404
Born M„ 420
Borovinskii L.A., 398, 446, 468
Bostanov V., x, 220, 221. 286, 386, 398,
399, 403, 406, 408, 409
Boucher E.A., 315, 322
BoudartM., 115, 118, 119, 171,231,260,
262, 346
Brailsford A.D., 135
BraininM.L, 315
Brandes H., 38, 52
Brice J.C., 391
Brick V.B., 270
Brown I.W.M., 315
516 Author index
Bruckner R„ 256
Budevski E. x, 202, 220, 221, 286, 386,
398, 406, 408, 409
Budurov S., 382
Buff F.R, 79, 80
Burchakova V.I., 315, 329
Burton W.K., 146, 391, 399, 405, 406, 408
Butorin G.T., 213, 218, 219, 289, 384, 386
Cabrera N., 146, 391, 399, 405, 406-408
Cadoret R., 468, 474
Cahn J.W., 18, 69, 97. 100-102, 112, 387
Carlicr C.C., 231,262,346
Carlson G.A., 258, 287
CarslawH.S., 142,167,241,246,264,491
Castleman A.W., Jr., 315, 317, 318, 322
Chadwick G.A.. 14
Chakarov V.M., 215, 300, 307. 330, 336
Cbakraverty B.K., ix, 135, 231, 237, 474
Chekunov A.A., 420
Cheng K.J., 315, 324, 325
Chepelevetskii M.L., 373, 413, 415
Cheremisin A.1., 315
Cherevko A.G., 59, 226, 228
Chernomordik L.V., 315, 481
Chernov A.A., x, 315, 391, 407, 430, 432
Chiu C.P., 86, 88, 204
Chizmadzhev Yu.A.. 3t5. 481
Chizumi N., ix
Christian J.W., x, 89, 309-311, 373, 378,
382, 391
Christoffersen J., 183
Christoffersen M.R., 183
Cinti R.C., 474
Clausse D., 386
Clement C.F., 335
Cohen E.R., S3, 84
Cohen R., 482
Collins EC, 231, 246, 249, 265, 266
Cooper A.R., 231, 249
Cormia R.L., 386
Courtney W.G., 88, 231
Coutsias E.A., 135
Crank J.. 142, 167
Cravalho EG., 14, 153
Dadyburjor D.B., 126, 135
Debenedetti P.G.. x
deBoer J.H., 149, 150
Defay R„ x, 79, 81, 88
Delale C.F., 28
Delone N.B., x, 47, 49
Demo P., 115, 125,232, 237, 242-244,246,
279, 288, 289, 366
Derjaguin B.V., 481,484
Derzhanski A.I., 481
Detsik V.N., 437, 446
Deubener J., 256
Dillmann A., 28, 79, 430-432, 434
Dimitriev Y„ 315, 329
DissadoL.A., 315, 329
Djarova M., 426
Dobrcva A., 382
Donohue M.D., 35
Doremus R.H., x, 373, 378, 391, 417
DbringW., 18, 20, 22. 115, 121, 158, 184,
185, 192-194,200
Dufour L., x, 88
Dunning W.J., 24
Dupre" A., 37
Ebner C, 202
Eggington A., 28
Egry 1., 97
Ehrenfest P., 6
Einstein A., 143
El-Shall M.S., 330, 336
Eruin I.A., 468, 471
Ernst M.H., 133
ErreR.. 133, 168
Eshelby J.D., 311
Evans J.W., 147, 437, 446, 468
Evans P.V., 115
Evans R„ 69, 97, 100, 102-104, 108, 109,
112
Evans U.R., 373
Evers L.J., 488
Exerowa D., ix, 301, 315, 386, 480-488
Eyring H„ 119, 122, 154, 160, 196, 232,
233
Family F, 437, 446
FarkasL., 18,22, 115, 118, 121, 124, 158,
178, 184, 192, 200
Feder J., x, 83, 88, 249
Feldman L.C., x, 119, 468, 478
Feng S., ix
Ferguson F., 430
Author index 517
Filipovich V.N., 256, 261, 269, 270, 366,
369, 370
Firoozabadi A., 373
Fischler W., ix
Fisher J.C., 154, 160, 309, 311, 346
Fisher M.E., 28
Fishman I.M., ix, x
Fisk J.A., 215, 330, 336, 430
Fletcher N.H., 293,294, 296
Flood H., 430, 434
FlynnP.C, 135
Fokin V.M., 256, 261, 269, 270, 366, 369,
370
Ford I.J., 28, 59, 83, 84, 330, 335, 430
Frank F.C., 42, 146, 391, 399, 405, 406,
408
Frenkel J., x, 28, 83, 84, 88, 93, 128, 138,
152, 153,346,391,392,394
Frens G„ 488
Friedlander S.K., 133
Frisch H.L., 231, 262, 346
Frish M.B., 330, 336
FuchsN.A., 132
Fujiia K., 115
Fujiwara S., 147
Fulchcr H., 153
Full! T., 373
Gadiyak G.V., 59, 215, 226, 228
Garside j., x, 12, 13, 79, 89, 378, 387.413,
414, 426
Gates E.E., 333
GattefE., 315, 329
GauchM., 168
Gaulhier M., 330, 336
Geguzin Ya.E., 148
Gelbard F, 135
Georgiev M„ 182, 315
Georgieva A., 394, 395
Ghez R„ 407
Gibbs J.W., ix, x, 18,20, 22, 35,47,49,58,
79,80, 111,301,396
Gilmer G.H., 228, 229, 391, 394, 398,407,
408,411.412,468,471
Girshick S.L., 86, 88, 204
Gjostein N.A., 135
Glass 1.1., x
Glasstone S., 7, 87, 154, 160, 196, 332
Goldbeck-Wood G., 387
Goluhovic L., ix
Goodrich FC, 88, 126
Gorbunkov V.M., x, 47, 49
Grabow M.C., x, 119, 468, 472
Gradshtein I.S., 261,465
Graham R., 119
Granasy L., 97
Grant D.J.W., 427
Grass! A., 404
Gratias D., 387
Green A.K., 446
Greer A.L., 115, 153, 161, 196,231,232,
244, 246, 247, 249. 265, 266, 346, 351,
359, 361-363
Gretz R.D., 300, 301, 305, 307
Guggenheim E.A., 3, 7, 10-15, 20, 22, 23,
32, 60, 65, 66, 73, 83-85, 98, 102, 106,
315,331, 338, 343
Gunther S., 437, 446
Gutop Yu.V., 481
Gutzow 1., x, 14, 79, 81, 153, 156, 208,
247, 251, 256, 261, 265-267, 270, 271,
309, 366, 368, 380, 382, 384, 385, 391,
416.426,446,448,456,463
Hadjiagapiou 1.,79,81, 109
Hagcmann J.W., 387
Hagen D.E., 430, 434
HakenH., 119
Hale B.N., 430
HalpemV.x, 129, 147, 149, 161, 162, 178
Ham F.S., 144
Hamerton R.G., 115
Hammel J.J., 270
Hanbucken M., x, 88, 119, 132, 437, 468,
478
Harris R., 102
Harrison J.A., 398, 468
Harrowell P., 97
Harsdorff M, 228
Hasegawa H., ix
Heist R.H.. 330, 336, 430, 434
Hendriks E.M., 133
Hendriksen B.A., 427
Herlach D.M., 97
Hertz H., 391-393
Hesse N., 153
Hikosaka M., 387
Hile L.R., 231, 262, 263, 346, 365
HillR.M., 315, 329
Hilliard J.E., 18,69,97, 100-102, 112
518 Author index
Hillig W.B., 398
Hinh J.P., x, 12, 32, 37, 49, 51, 52, 89, 93,
94, 138, 140, 145, 158, 196, 200-202,
204, 206, 209, 215, 315. 317, 319, 322,
391,413,430,431
Hiscock W.A., ix
Hodgson A.W., x
Hoffman D.C., ix
Hoffman J.D., 14
Hogan C.J., ix
Hollomon J.H., 309, 311, 346
Hopper R.W., 373
Hornbogen E., 309, 310
Hottier F„ 468, 474
HoytJJ., 97, 102
Hruska S.J., 147
Huang K., 7, 119
Hung C.H., 215
Huichinson T.E., 437
lckert L., x, 468
levlev V.M., 437
IsardJ.O., 315, 324-326, 328
Ishibashi Y., 373
Jaeger J.O., 142, 167, 241, 246, 264, 491
Jakubczyk M., x, 414, 427, 430
Jalaluddin A.K., 315,329
James PR, x, 153,247, 256, 261, 269,270,
315,325,326, 328
Janjua M., 330, 336
Jarecki L., 88
Jarvis T.J., 35
Jenks C.J., 147
Jensen P., 148, 168,437,446
Johnson W.A., 373, 375, 446
Jolivet-Dalmazzone C, 386
Jones D.R.H., 14
Kagan Yu., 140
Kaganovski Yu.S., 148
KahlweitM., x, 135, 174
Kai S„ ix
Kaisehew R., x, 18, 21, 22, 26, 27, 32, 35-
37, 40, 51, 54, 73, 89, 115, 116, 121,
140,158, 180, 181,184, 192,200,203,
206, 209, 226, 228, 293, 346, 349, 358,
359, 366, 386, 391, 396, 397, 409
Kalikmanov V.I., 28
Kalinina A.M., 256, 261, 269, 270, 366,
369, 370
Kamo M., 387
Kampfer B., ix
Kane D., 330, 336
Kaneko N., 386
Kanne-Dannetschek 1., 231
KanterYU.O., 315, 329, 479
Kantrowitz A.,231, 246
Kapral R„ 373
Kaptelov E.Yu., 437, 446
Kapusta J.I., ix
Karel M., 14, 153
Kashchiev D., ix, x, 14, 54, 58, 59, 61, 62,
64-67, 115,116,118-123, 125, 127-130,
133, 147, 148, 151, 153, 156, 164, 166-
168, 171, 172, 174, 179-182, 191,224-
227, 231, 237, 240, 241, 245, 334, 338,
339, 343, 346-350, 354, 357-359, 362,
366, 373, 380, 381, 386, 388-390, 398-
400, 403, 413-418, 426, 427, 430, 432,
437-439, 446, 448, 456, 457, 468-479,
481-488
Kaski K., 468
Kassner J.L., 59, 226, 228
Kassner J.L., Jr., 430, 434
KatzJ.L.. ix, 28, 83, 84, 88, 115, 118, 120,
140, 171, 199, 209, 215, 330, 336, 430-
432, 434
Kaverin A.M., 59, 226
Kawakami X. 426
Kegel W.K., 28, 84
Keldysh L.V., ix
Keller A., 387
Kelton K.F., x, 14-16, 89, 97, 115, 153,
161, 183, 196,215,231,232,244,246,
247, 249, 256, 261, 265, 266, 270, 346,
351, 359, 361-363, 386, 387, 463
Kern R., x, 133, 148, 168, 183, 202, 468,
478
Kertov V., 219
KhadirA., 102
Khrushchev V.V., 135
Kiang C.S., 28
Kibalczyc W., 426
Kikuchi R„ 84
Kirkova E„ 140, 426
Kirkwood J.O., 79, 80
Kitamura M., 387
Klapka V., 286
Author index 519
Klein W.,ix, 102
KnudsenM., 391-393
Kodenev G.O., 59, 215, 226, 229
Kolmogorov A.N., 373, 375, 376, 378,380,
437, 439
Komalsu H., 387
Rondo S„ 20, 24, 41, 71, 74, 75, 79-81,
100, 104-107, 109, 111
Kontoyannis G.G., 426
Kopatzki E., 437, 446
Korn G.A., 99, 101, 142, 149, 185, 192,
232-234, 238-240, 248, 281-283, 367,
376-378, 394, 407, 408, 415, 422, 453,
489, 492, 493
Korn T.M., 99, 101, 142, 149, 185, 192,
232-234, 238-240, 248, 281-283, 367,
376-378, 394, 407, 408, 415, 422, 453,
489, 492, 493
Kortzeborn R.N., 315,317
Roster U„ 247, 256, 271, 463
Rostrovskii V.G., 58, 59, 226, 228
Kotake S., x
Kolsev E.I., 132
Rolzeva A., 386, 409
Routsoukos P.G., 426
Koverda V.P., x, 200, 212, 215, 218, 219,
256, 257, 265-267, 273, 276, 277, 384,
386, 430, 432
Kozasa T., ix
Kozisek Z„ 115, 125, 153, 161, 232, 237,
242-244, 246, 249, 279, 288, 289, 366,
464
Rozlovskii M.I., 315, 329
Krasnopoler M.J., 215
Krastanov L., x, 32, 293, 296
Kretzschmar G., 482, 487
Krishna-Murthy M., 382
Rrohn M„ 228
Kruglova T.I., 446, 468
Rruglyakov P.M., 480
RubotaN., 413, 426
Kukushkin S.A., x, 135, 437, 446, 448
Rulmala M., 28, 58, 65, 67, 227, 228, 430
Rumar N.G., 430-432
Runi F.M., 315, 317, 318, 446
Kunz K.M., 446
Rupenova T„ 267, 386
Kurihara K., 387
Kusaka I., 20, 69, 79, 97, 102, 112
La D., ix
Laaksoncn A., x, 28. 58, 59, 65, 67, 69, 79,
97, 108, 227, 228, 330, 335, 430
LaidlerR.J., 154, 160, 196
Lampert B., 474
Landau L.D., 3, 12, 84, 86, 102, 103, 105,
121, 134,309,315,316, 324,325
Lange E., 15
Laplace P., 25
Larikov L.N., 270
Larralde H., 437, 446
Larson M.A., 79
Leedom G.L., 79, 83
Le Lay G., x, 133, 148, 168, 183, 202,468,
478
Levine H.S., 258, 287
Levine M., 406
Levkov L., 293
Lewis B., x, 11, 26, 56, 88, 89, 119, 129,
132, 145-149, 161, 162, 168, 178, 226,
228, 437, 468
Lewis G.N., 338, 339
LifshitzE.M., 3, 12, 84, 86, 102, 103, 105,
121, 134, 309, 315, 316, 324, 325
Lifshitz I.M., 135
Logan R.M., 137
Lorenz W.J., x, 202, 409
Loshkarev Yu.M., 366, 443, 444
Lothe J., x, 28, 83, 88, 249
Ludwig F.-R, 125
Luijten C.C.M., 59, 228, 330, 333, 336
Lutrus C.R., 83
Lychev A.P., 315
Lynch D.C., 278
Lyubitov Yu.N., 398, 468, 471
Lyubov B.Ya., x, 231, 265, 309, 373, 378
Maciel W.J., ix
MacRenzie K.J.D., 315
MahnkeR., 79, 310, 311
MarderM., 391, 396, 397
Marinov M., 256, 265, 267, 270
Markov 1., x, 42, 54, 366, 391, 437-439,
443
Markov T., 382
Markwotth A.J., 135
Martbinsen R., 373
Martini Bettolo Marconi U., 102
Masson A., 148, 168
Matsumoto S., 387
McDonald J.E., x, 138, 161, 194, 196
McGraw R„ 59, 69, 79, 108
520 Author index
Meakin P., 446
Mederos L., 300
Mehl R.F., 373, 375, 446
Meier G.E.A., 28, 79, 430-432, 434
Melentev 1.1.,315, 329
Mendeli G., ix
Mersmann A., 183
MetoisJ.J.,x, 133, 148, 168, 183,202,468,
478
Meunier M., 437, 446
Meyer H.J., 228
Mie G., 420
Mikhcev V.B., 58, 59, 226, 228
Mikhnevich G.L., 382
Milchev A., x, 26, 40, 55, 89, 203, 206,
209, 219-221, 226, 228, 384-386
Miller R.C., 430, 434
Miloshev G., 293
Miloshev N., 232, 246, 249, 265, 266
Mirabel P., 430-432
Mitov M.D., 484
Miyashita S., 387
Moelwyn-Hnghes E.A.. 87, 137, 143, 166
Monnetle L., 102
Morgan D., ix
Morgan J.J., 338
Muitjens M.J.E.H., 330, 336, 420
Mullin J.W., x, 378, 391, 413, 414, 416,
417,426,427,430
Muralidhar R„ 133
Mutaftschiev B., x, 21, 22, 27, 28, 35, 51,
73,83, 116,293,366
Myshkis A.D., 358
Nabarro F.R.N., 311
Nagel K., 15
Nakada T., 387
Nancollas G.H., 42
Nasteva V., 301
Navascues G., 300
Nechaev Yu.l., x, 47, 49
Nedyalkov M., 301
Neizvestnyi I.G., 315, 329
Nenow D., 394, 395
Nes E„ 373
Nesladek M., 366
NeuJ.C, 135
Neumann K„ 20, 142
Newman T.J., 468
Nielsen A.E., x, 12, 58, 59, 89, 124, 142,
183, 189, 226, 231, 373, 378, 391, 398,
400, 413, 414, 417, 426, 437, 468
Nieminen J.A., 468
Nikolova A., 386, 482, 487, 488
Nishioka K., 20, 69, 79, 81, 97, 102, 112,
115
Nowakowski B., 248
Nuth III J.A., 430
Obretenov W„ 220,221, 398, 399,403,409,
443
Ohigashi H„ 387
Okada H„ 387
Okuyama K., 232, 237
Ono S., 20, 24, 41, 71, 74, 75, 79-81, 100,
104-107, 109, 111
Orihara H., 373
Osadchenko V.A., 468
Osipov A.V., x, 135, 437, 446-448
Ostermier B.J., 430, 434
Ostwald W„ 77, 133, 387
Overbeek J.Th.G., 132, 142, 143, 166
Ovsienko D.E., 430, 432
Oxtoby D.W., x, 58, 62, 64, 65. 67,69, 97,
100, 102-104, 108, 109, 112. 227, 300,
330, 335
Pannhorst W„ 256, 270
ParmarD.S., 315, 329
PaskovaR., 208,271
Pastushenko V.F., 315,481
Patterson E.M., 28
Paunov M„ 54, 228
Penkov 1., 261, 267, 270, 271
Petrov A.G.,481
Pilinis C, 133
Pimpinelli A., 437, 446
Platikanov D., ix, 300, 301, 481, 482,486-
488
Pocker D.J., 147
Polak W., 414, 427
Polchinski J., ix
Popov E„ 256, 265, 267, 270
Poppa H., 446
Poslnikov V.S., 437
Potapov O.V., 270
Pound G.M., x, 12, 28, 32, 37, 49, 51, 52,
83, 88, 89, 93, 94, 138, 140, 145, 158,
196, 200-202, 204, 206, 209, 215, 249,
315,317,319,322,391,413,430,431,
446
Pratsinis S.E., 132
Author index 521
Price G.L., x, 56, 88, 119, 132, 147, 167,
170, 437, 468
Prigogine I., x, 79, 81
Probstein R.R, 231
Prokhorov A.V, 481,484
Pronin I .P., 437
Pulvermacher B„ 133, 166, 168
Puri OP, 28
Purushotaman S., 135
PiischlW., 128, 134
QiuH., 182, 183, 194
Radoev B., 84
Ramachandrarao P., 382
Ramkrishna D., 133
Ramsden A.H., 315, 325, 326, 328
Randall M„ 338, 339
Rasmussen D.H., 79, 83, 387
Rastogi S., 387
RatkeL., 97, 133, 135
Ratsch C, 437, 446
Rayleigh, 419
ReeF.H., 119, 122,232,233
ReeT., 119, 122,232,233
RceT.S., 119, 122,232,233
ReesGJ., 147, 148
Reichelt K., 474
Reiss H., 20, 28, 58, 62, 83, 84, 115, 118,
120, 171, 215, 217, 218. 227, 229, 230,
330, 387, 430, 434
Rhodin T.H., 366-369
Richter L., 482, 487
Riontino G., 426
Robertson D„ 446
Robins J.L., 228, 366-369, 437
Robinson K., 311
Robinson V.H.E., 228, 437
Rodigin N.M., 489
Rodigina E.N., 489
Roginsky S.Z., 446, 448, 449, 455
Roitburd A.L., 231,262
Rolland A., 468
Rossi S.C.F., ix
Roslrup E.. 183
Roussinova R., 287, 408, 409
Routledge K.J., 437
Rowlands E.G., 153
Rubakhin E.A., 59, 215, 226, 229
Ruckenstein E„ 126, 133, 135, 166, 168,
248
RudekM.M., 215, 430
Rudenko Yu.S., 315
Rundle J.B., ix
Rusanov A.I., x, 20, 22, 36, 79, 111, 315,
317,318,333.339
Russell K.C., x, 32, 52, 83, 88, 249, 309,
311,312,315,317,318, 322
Ryazantsev P.P., 301
Ryzhikl.M., 261,465
Sakurai K„ 387
SaltsburgH., 115, 118, 120, 171
Sampson K.J., 133
Sangwal K„ x, 183, 391, 407, 414, 426,
427, 430
Sato K., 386-390
Sato Y., 387
Sazaki G., 387
Scharifker B.. 228
Scheludko A., 84, 300. 307
SchererG.W., 373
Schiffner U.. 256, 270
Schlesinger M.E., 270
SchmelzerJ., x, 14,79,81, 125, 126,248,
270,309-311,391,426
SchmelzerJ., Jr., 79, 81
Schneider H.G., x, 468
Schottky W.F., 286
Schweitzer F., 310, 311
Scoppa II C.J., 430-432
Scott M.G., 382
Scdunov Yu.S., 117, 132, 133
Seinfeld J.H.. 115, 131-133, 135,232,237,
249, 380
Semin G.L., 59, 228
Setaka N, 387
Sgonnov A.M., 59, 228
ShablakhM., 315, 329
Shangguan D.K., 115
Shchekin A.K., 315, 317, 318, 324
Shcherbakov L.M., 301
Shechtman D., 387
Shen Y.C., 97
Shevelev V.V., 115
Shi F.G.. 232
ShiG., 115, 131, 132,232,237,249,380
ShichiriT., 315, 328
Shizgal B., 88, 126, 232, 235, 239, 245,
261, 262, 265, 266, 346, 348, 353, 356,
364
Shneidman V.A., 232, 237, 262-266, 373,
380, 465
522 Author index
Shtein M.S., 58, 59, 226, 228
Shulepov S.Yii,, 488
Shvedov E.V., 437
Siegers H.-P., 474
Sigsbee R.A., 11, 20, 115, 145-147, 151,
194, 227, 436, 474
Sinon B., 404
SinhaD.B., 315,329
Simla N.N., 315
Skapski A.S., 183
Skripov V.P., x, 49, 51, 59, 140, 200, 202,
209, 212, 213, 215, 218, 219, 222, 223,
226, 256, 257, 265-267, 273, 276, 277,
289, 384, 386, 430, 432
Slezov V.V., 125, 135,248
Slowinski E.J., Jr., 333
Smilauer P., 437, 446
SmolyakB.M., 315, 327
Sonel 0., x, 12, 13, 79, 89, 183, 378, 387,
413,414,416,417,426
SpassovT., 382
Spiller G.D.T., x, 88, 119, 132, 437, 468,
478
Staikov G., x, 202, 220, 221, 406, 409
Stauffer D„ x, 28, 115, 227, 231
Stefan J., 183
Stein B.J., 228
Steinhardl P.J., ix
Stenzel H., 228, 366, 444, 445
Sternitzke M., 256
Stoinov Z., 386, 409
StoldlC.R., 147
Stowell M.J., x, 147, 149, 437, 468
Stoyanov A„ 382
Stoyanov S„ x, 26, 40, 55, 56, 89, 119,
133, 147, 148, 161, 162, 182,203,206,
209, 226, 315, 366,384-386, 437, 446,
468, 476, 478
Stoyanova V., 267, 386
Stoycheva E„ 437, 443
Stranski I.N., 18, 22,26, 36, 115, 121, 140,
158, 181, 184, 192, 200, 228, 391, 396,
397
Strey R„ 58, 62, 65, 67, 215, 217, 218,
227-230, 330, 336, 387, 430-432, 434
Stumm W., 338
Suck Salk S.H., 83
Surugin A.G., 132
Swift D.L., 133
Sycheva G.A., 256, 270
Symeopoulos B.D., 426
Tadaki T., 426,
Takahashi T., 387
Takai T., 69, 79, 97, 102, 112
Talanquer V., x, 97, 300
Tambour Y., 133
Tammann G., 153
Tang L.-H., 437
Tarazona P., 102, 300
TemkinD.E., 115, 147
Thiel P.A.. 147
Thompson C.V., 153, 161, 196, 231, 232,
244, 246, 247, 249, 265, 266, 346, 351,
359, 361-363
Thomson I.J., 78, 315, 317
Thomson W., 74
Tien H.T., 480
TienJ.K., 135
Toda A., 387
Todes O.M., 133-135, 398, 420, 437, 446,
448, 449, 455
TohmforG., 315,317
Tolman R.C., 79, 80
Tomino H., 69, 79, 97, 102, 112
Toner M„ 14, 153
Toschev S., x, 10, 20-23, 41, 46, 54, 60,
63, 71, 116, 153, 181, 247, 251, 255,
256, 265-267, 270, 366, 368, 384-386,
446, 448, 456, 463
Toshev B.V., 84, 300, 301, 307
Trayanov A., 479
Trinkaus H., 232, 237, 279
Trofimenko V.V., 366, 443, 444
Trofimov V.I., 446, 463, 468
TrusovL.l., 315
Tsindergozen A.N., 398, 468
Tunitskii N.N., 125, 126, 279, 286, 373
Tumbull D., 154, 160, 183, 231, 309, 311,
346, 384, 386, 387, 426
Tzuparska S„ 382
UchtmannH., 215, 430
Ueno S„ 387
Uhlmann D.R., 373, 426
UlbrichtH., 310, 311
Unger C, 102
Vaganov V.S., 59, 215, 226, 228
van der Eerden J.P., 168, 391, 398, 407,
412,468,474,476,478
Author index 523
van der Leeden M.C., 414, 416, 426
van der Noot T.J., 398
van Dongen M.E.H., 28
van Leeuwen C, 13, 398, 412, 468, 474,
476,478
van Rosmalen O.M., 338, 339, 343, 413-
418,426,427,430,432
VassilevaE., 219-221, 228
Velfe H.-D., 228, 366
Venables J.A., x, 56, 88, 119, 132, 147,
167, 170,437,468,478
Verdoes D„ 413-418, 426, 427, 430, 432
Vershinin S.N., 59, 226, 228
Vetter K.J., x, 16, 146, 147, 206, 208, 388,
391, 398,468,470,472
Viisanen Y., 58, 62, 65, 67, 215, 217, 218,
227-230, 330, 336, 387, 430-432, 434
Villain J., 437
Vincent R„ 132, 170
Virklcr T.L., 430-432
Vitanov T., 386, 409
VogelH., 153
Volmer A., 468
VolmerM., ix, x, 13, 14, 16, 18,22,26,27,
33, 36, 46, 49, 138, 140, 142, 153, 159,
160, 161, 181, 197, 200-202, 204-207,
209, 214, 315, 317, 319, 321, 322, 373,
377, 378, 380, 391, 396, 397, 413, 414,
417,430-432,434
Voloshchuk V.M., 117, 132, 133
Volterra V, 231, 249
von Smoluchowski M., 133, 143, 166, 170
Vook R.W., 468
Voronkov V.V., 42
Voronov O.S., x, 47, 49
Vvedenski D.D., 437, 446
Wagner C, 135
Wagner P.E., 215, 217, 218, 228-230, 330,
336, 430-432
Wakeshima H., 131,249
Walker G.H., 28
Walton A.G., 12, 24, 183, 231, 246, 249,
430, 432
Walton D., 26, 40, 55, 83, 88, 89, 95, 203,
207
Wanke S.E., 135
Waring C.E., 333
Warshavsky V.B., 315, 317, 324
Weber A., ix, 18, 22, 181, 197, 200
Weeks J.D., 228, 229, 391, 408, 411, 412
Wehrmann C, 228
Weinberg M.C., 232. 237, 262-266, 271,
373, 380, 426, 465
Westman A.E.R., 382
White G.M., 194, 196
Wiedersich H., ix
Wilcox C.F., 330
Wilcox R.W., 143
Wilemski G., 20, 84, 86, 87,115, 227, 330,
336
Williams M.M.R., 117, 133
Wilson H.A., 391,392, 394
Wise J.D., Jr., 28
Wojciechowski K., 426
Wolf D.E., 437
Wolf" E., 420
Wu D.T., 115, 126, 232, 262-265,346,364,
365
WynblattP., 135
Wyslouzil B.E., 20, 115, 330, 336
YangC.H., 182, 183, 194
Yinnon H., 426
Yoo M.H., 232, 237, 279
Young T., 33
Yount D.E., ix
Yuritsin N.S., 256, 270, 366, 369, 370
Zanghi J.C., 133, 168
ZangwiU A., 437, 446
ZanottoE.D., 271, 373,426
Zaremba V.G., 382
Zeldovich J.B., 92, 93, 115, 121, 125-129,
163, 188, 189, 193, 194, 231, 249, 271,
358
Zelinski B.J., 426
ZengX.C, 97, 103, 109
Zettlemoyer A.C., x, 10, 21, 25,27, 28, 32-
34, 36, 37, 46, 47, 50, 52, 73, 83, 89,
93, 94, 116, 121, 124, 138, 153, 194,
196, 200, 201, 204-207, 215
Zhang J., 142
Zhitnik V.P., 366, 443, 444
Ziabicki A., 88, 116, 346
Zimmermann W., ix
Zinke-Allmang M„ x, 119, 468, 478
Zinsmeister G., x, 119, 126, 129, 132, 148,
161, 162, 178, 180, 181, 185, 187,437,
446, 468
Zuckerman M., 102
This Page Intentionally Left Blank
Subject index
Active centres for nucleation, 86,366,436,
444, 445, 449
Activity factor of nucleation, 55, 295, 305
Adsorption equilibrium, 148
AgNO-,, 221,408,409
Amphiphile bilayers, 480
Argon, 217, 218
Atomistic theory of nucleation, 26, 39, 42,
55-57, 89, 94, 95, 202, 205-209, 216,
220, 250, 251, 253-255, 257, 272
Average:
radius of detectable supernuclei, 451,
453, 454, 458-461, 464, 466
thickness of thin film, 469, 471-474
time for phase transformation, 376, 378,
379,381,384,385
Benzene, 183, 338
Benzole, 222, 223
Betol, 382
Bilayer lipid membranes, 480
Bilayers, rupture of, 480
Binodal, 5
CaCO,, 338, 345, 427
Calcium, 444, 445
Cap-shaped clusters, 33,38,50,53,55,138,
145, 147, 150, 155, 164, 166, 169, 250,
251,253-257,300
Capillarity approximation, 24
Capture number, 147
Carbon, 219, 220, 444
Carrier gas, 217, 218, 330
Classical theory of nucleation, 22, 31
Cluster:
approach, 18
binding energy, 26, 40
chemical potential, 71
density profile, 98
excess energy, 21
inside pressure, 22, 23, 32, 40, 70
melting point, 78
solubility, 77
specific surface energy, 24, 79, 111
'surface' binding energy, 89
vapour pressure, 73
Coagulation, 133
Coalescence, 123, 130, 132, 166, 168
Concentration of :
detectable supernuclei, 176, 451, 453,
455, 458-461, 464, 465, 467
nucleation sites, 84, 86, 87, 94, 273
nuclei:
equilibrium, 93, 270
non-stationary, 236, 241, 242, 270,
349,351, 352
quasi-stationary, 282
stationary, 189
supernuclei, 174, 175:
in non-stationary nucleation, 267,
270,356
in stationary nucleation, 214
initial, 356, 358-360, 492
Copper, 444
Cordierite, 369, 370
Coverage of film layer, 469, 470
Critical amphiphile concentration, 485,486
Crystal growth:
continuous, 391
nucleation-medialed, 396
spiral, 404
Crystallization:
overall, 373, 396, 397, 468
two-stage, 387
CuS04, 44
Delay time of:
detectable nucleation, 464
nucleation, 258, 270,271, 274,360,362,
461
Density functional:
approach, 18, 97
theory of nucleation, 97, 100, 102, 107
Detailed balance, 163, 164, 171
Diethyl ether, 222, 223
526 Subject index
Diffraction limit, 420, 421, 423-426, 428,
433, 434
Diffusion equation, 142, 146, 167
Diphenyl, 394, 395
Dircct-impingementconlrol, 137,158, 166,
198,204,205,250, 334
Disk-shaped clusters, 33, 36, 52, 139, 145,
147,155, 162, 166, 169, 170,251,254,
257
Edge energy, 36, 220, 221, 409, 483, 487
Effective:
excess energy, 41
pressure, 105, 108, 109, 111
surface energy, 40
Effect of:
■ active centres, 366, 436, 449, 452
carrier-gas pressure, 330
cluster size, 70
electric field, 315
line energy, 300
preexisting clusters, 346
seed size, 293
solution pressure, 338
strain energy, 309
Electric field, 315
Electrostatic energy, 316, 318, 324
Equation of:
Bouguer-Lambert, 419
Dupre, 37
Gibbs-Thomson, 46-55, 59, 183, 228,
277, 296, 304, 312, 319, 326, 333,
339.411,483
Laplace, 23-25, 32, 35, 40, 70
Stefan-Skapski-Turnbull, 183
Slokes-Einstein, 143, 153
Tunilskii, 125, 184,232,347
van der Waals, 3
Vogel-Fulcher, 153, 200
von Smoluchowski, 132
Young, 33, 301
Zeldovich, 128, 187, 237, 347
Equilibrium:
chemical potential, 8
monomer concentration, 85, 138, 148,
153
Equimolecular dividing surface, 20,41,60,
62,80
Ethanol, 183
Evaporation control, 140, 158, 199, 257
Exclusion zones for nucleation, 436, 449
Experimental data, 15, 217, 228, 247, 265,
269, 273, 276, 289, 368, 382, 386, 394,
404, 408, 427, 434, 443, 487
Extended:
number of supernuclei, 368, 438
volume, 375, 377, 379, 438
Extinction coefficient, 419
Flux through cluster size, 120, 122, 125,
128
Fraction of transformed volume, 373, 387,
388,415
Frequency of:
monomer attachment, 136, 172, 392
monomer detachment, 157, 172, 392
multimer attachment, 130, 165
multimer detachment, 131, 171
Gallium, 218
Glass-transition temperature, 212,256, 267
Gold, 368, 444
Growth:
coalescence, 168
mode of thin films, 475
site, 392
time of smallest detectable supernucleus,
452
Growth rate of:
clusters, 127, 129, 169
crystals, 391,410, 417
exclusion zones, 439
supernuclei, 378, 397, 399, 400, 417,
448
thin films, 469, 472-474
Ice, 78, 79, 156, 161, 199, 211, 218, 256,
266, 273, 276, 289
Impingement rale, 11
Induction time, 413, 428, 432
Insoluble monolayers, 482
Interface-transfer control, 151, 160, 198,
253-257, 342
Ions, 315, 317, 444
Kinetic factor of nucleation, 197, 205-208,
224, 227
Subject index 527
KJMA theory, 373, 438, 480
Law of Mass Action, 84-86, 90, 203
Lens-shaped clusters, 35, 52
Lifetime of amphiphile bilayer, 484
Line energy, 300
Li02.2Si02, 14, 242, 244, 269, 288, 359,
361-363
Master equation:
general, 115-123
of coalescence, 131-133
of nuclealion, 125-128, 130
of Ostwald ripening, 133-135
Maximum number of supemuclei, 367,376-
380, 436
Mean time for appearance of at least one
supernucleus, 384-386
Metastability limit, 430
MgO, 368
Mobility coalescence, 166
Mononuclear mechanism, 383, 396, 410,
413, 414, 418,425,426, 428,433, 484-
488"
NaCI. 444
Naphthalene, 338
NaP03, 266, 267
n-butanol, 217, 229
Nucleation:
at pre-existing clusters, 346, 492
at variable supersaturation, 279
atomistic, 26, 39, 42, 46, 55. 202, 205-
209, 216, 220, 250, 251, 253-255,
257, 272
classical, 22, 31, 40, 45, 46, 50, 200,
204-209,215-219,221-223,250-258,
272, 273
electrochemical, 15, 57, 206, 208,219-
222, 226, 275, 286, 412, 443
heterogeneous, 30, 50,57,200-202,215,
216, 251, 254. 256, 293, 300, 317,
366, 396, 418, 431, 436, 446, 468
homogeneous, 20,46, 57, 97, 181, 182,
200-202, 215, 216, 309, 323, 431,
446
in external electric field, 315, 323
in melts, 13, 57, 65, 66, 153-157, 160,
161,197-201,205-208,211,215,216,
218, 219, 226, 227, 250, 255-257,
265-267, 269, 272, 273, 275-278,
287, 360-364, 425, 426, 432
in solutions, II, 57, 77, 141-145, 151-
155, 158-160, 197, 204-207, 226,
249, 253-255, 272, 275, 285, 309,
338, 404, 411, 426, 427, 429, 431
in vapours, 10, 30, 57, 87, 89, 94, 137-
140, 145-151, 157-162, 164-168,
181-183, 198, 204-207, 210, 217,
218, 226, 229, 230, 250-252, 272,
275, 285, 330, 368, 411, 429, 431,
434, 444
instantaneous, 377, 415, 416, 421, 423,
424,440,452,471
mode, 54
non-stationary, 231, 274, 346,367, 380,
384,414,415,461,489,491
of bubbles, 11, 24, 25, 27-30, 34, 35,
42, 47-52, 54-57, 88, 94, 140, 158,
198-200, 202, 209, 212, 213, 217,
222, 225, 257, 272, 275, 286
of holes, 481
onions, 315, 317
on seeds, 293
progressive, 377, 379, 397, 415, 416,
418,421,440,453,471
stationary, 184
ID, 42
2D, 33, 36-40,42, 52-54, 56-58, 86, 88,
89, 92, 94, 95, 139, 155, 162, 166,
169, 182, 194, 201, 206, 215, 216,
221, 228, 229, 251, 254, 257, 272,
351,396,410,468,483
3D, 33, 38, 42, 52-54, 58, 86, 88, 89,
92,94,138,145,155, 164,166,168,
194, 200-206, 209, 215, 219, 222,
230, 250-258, 266, 267, 272, 273,
277, 293, 300, 333, 339, 350, 417,
419, 424-427, 430
Nucleation process, ix, 5, 18
Nucleation rate, 174:
detectable, 176,452,453,458-461,464,
467
non-stationary, 231,243, 270,353,367,
368,489,491
per active centre, 367, 368
quasi-stationary, 279, 283
stationary, 184, 192,204, 214,224,274,
296, 307, 312, 321, 326, 333, 339,
343, 367, 483
Nucleation theorem, 58, 109:
528 Subject index
application of, 224, 274, 334, 340, 410,
428
Nucleation work, 29, 30, 38, 39, 45, 58,
99,108,110,181,183, 294,305, 311,
321,326, 333,339,483
Nucleus, 29, 30, 45, 46, 99, 159, 181:
binding energy, 56, 95
density profile, 62, 63, 99, 101, 103,
108
effective excess energy, 46, 56, 59
effective surface energy, 45, 64
excess number of molecules in, 62, 110,
224, 274
region, 93, 190, 191, 194, 249, 280
size, 29, 38, 45, 58, 159, 161, 165, 181,
183, 216-224, 273, 274, 296, 304,
311, 319, 326, 333, 339, 404, 410,
427, 428, 483, 488
'surface' binding energy, 95
Number of supcrnuclci, 366,376,378, 379,
438,440,441,443,444
Numerical data, 228, 236, 242, 244, 246,
265,288,360,363,411,443
Newtonian black, foam Films, 480
Ostwald:
ripening, 123, 133
Rule of Stages, 387
Overvoltage, 16
Paraffin C36H74, 404
Petroleum ether, 404
Phase:
equilibrium, 5
transition of first order, 5:
ageing stage, 123, 132
coalescence stage, 123, 130
nucleation stage, 123, 124
Piperine, 382
Polynuclear mechanism, 377, 396-399,401-
404, 410, 415-419, 421-428, 433, 434,
468, 484-487
Pores, 343
Pressure tensor, 98, 105-107
Probability far:
appearance of at least one supernucleus,
384-386
non-ingestion, 438, 449, 450, 458
non-occupation, 438, 449, 450, 458
Quasi-thermodynamics, 104, 107
Rayleigh limil, 419-426, 428, 433, 434
Seeds, 293, 416
Silicon, 247, 271
Silver, 219-222, 408
Size distribution:
equilibrium, 83, 163,164, 178,179,182
non-stationary, 232, 346
of supernuclei, 446
quasi-equilibrium, 164, 179
quasi-stationary, 280
stationary, 184
Sodium dodccyl sulfate, 487
Solubility product, 12
Spinodal, 5
Spinodal pressure, 5, 49
Sticking coefficient, 140, 266
Strain energy, 309
Subnuclei, 45
Supernuclei, 45
Supersaturation, 9, 161, 331, 338,481,482:
critical, 211,430
for change of film mode of growth, 476
for 3D-2D nucleation transition, 54
spinodal, 47, 68, 430
Supersaturation ratio, 87, 204:
critical, 431
Surface energy:
effective specific, 54
specific, 24, 32,33,79, 111
total, 22, 31, 111
Surface-diffusion control, 145, 161, 162,
164-168, 170, 207, 251, 252, 334, 399-
401,406,408,484
Surface nucleation, 439-445,450,453,455,
456, 460-462, 467
Surface of tension, 79-81
Surface pressure, 482:
critical, 486
Szilard model of nucleation, 115,118,124,
125
Thermodynamic equilibrium, 3
Time lag of nucleation, 241, 242, 245-247,
249, 259, 261, 265, 270, 271, 282-289,
312, 327, 334, 341-343, 380, 381, 384-
386,414-416,462
Subject index 529
Time-temperature-transt'ormation (TTT)
curve, 426
Tin, 386
Transmitted light, intensity of, 419-423
Undercooling, 14:
critical, 432
Underpressure, 47, 209
Volume-diffusion control, 141, 158, 159,
166, 253, 327, 342
Volume nucleation, 439^142,450,453,455,
456, 460-462, 467
Water, 29, 30, 38,47-50, 53, 54, 71-73, 75,
76, 78, 79, 81, 91, 95, 96, 139, 150,
159, 165, 183, 190, 191, 195, 199,200,
209-213, 217-219, 229, 230, 236, 237,
242, 244, 251, 252, 256-258, 265-267,
273, 276-278, 289, 297-299, 302-308,
318-320, 322, 323, 327-329, 332, 333,
335-337, 434, 435
Wetting angle, 33-35, 300-306, 294-299
Work for cluster formation, 21,40, 41, 164,
179-181, 309, 316, 483:
in HEN, 31, 32, 34-40, 294, 303, 304,
318,319
in HON, 21,22, 27-29,98, 99, 107, 108,
181-183, 311, 325
Xenon, 217, 218
Zeldovich factor, 194-196
2Na2O.Ca0.3Si02, 269, 270
This Page Intentionally Left Blank