Nucleation

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, BulgarianAcademy of Sciences, Sofia, Bulgaria

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OXFORD AMSTERDAM BOSTON LONDON NEW YORK PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO

Butterworth-Heinemann An imprint of Elsevier Science Linacre Home, Jordan Hill, Oxford OX2 8DP 200 Wheeler Road, Burlington, MA 01803 First published 2000 Transferred to digital printing 2003 Copyright 9 2000, Dimo Kashchiev. All fights reserved The fight ofDimo Kashehiev 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 catalogue 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 0 7506 4682 9

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Printed and bound in Great Britain by Antony Rowe Ltd, Eastbourne

Contents

ix

P refa c e

Symbols and abbreviations Part 1

4

5

6

7

8

xiii

Thermodynamics of nucleation

First-order phase transitions

3

Driving force for nucleation

9

Work for cluster formation

17

3.1 3.2 3.3 3.4

20 30 40 42

Homogeneous nucleation Heterogeneous nucleation General formulae Absence of one-dimensional nucleation

Nucleus size and nucleation work

45

4.1 4.2 4.3 4.4

45 46 50 55

General formulae Homogeneous nucleation Heterogeneous nucleation Atomistically small nuclei

Nucleation theorem

58

5.1 5.2 5.3 5.4

59 64 65 67

Phenomenological proof Thermodynamic proof Generalizations Integral form

Properties of clusters

70

6.1 6.2 6.3 6.4 6.5 6.6

70 71 73 77 78 79

Inside pressure Chemical potential Vapour pressure Solubility Melting point Specific surface energy

Equilibrium cluster size distribution

83

7.1 7.2

84 93

Equilibrium concentration of clusters Equilibrium concentration of nuclei

Density-functional approach

97

8.1 8.2 8.3 8.4 8.5

97 100 102 104 107

General considerations Gradient approximation Hard-sphere approximation Quasi-thermodynamics Quasi-thermodynamic formulation

vi

Contents

Part 2 9

Kinetics of nucleation

Master equation 9.1 General formulation 9.2 Nucleation stage 9.3 Coalescence stage 9.4 Ageing stage

113 115 115 124 130 132

10

Transition frequencies 10.1 Monomer attachment frequency 10.2 Monomer detachment frequency 10.3 Multimer attachment frequency 10.4 Multimer detachment frequency 10.5 General formulae

136 136 157 165 171 172

11

Nucleation rate

174

12

Equilibrium

178

13

Stationary nucleation 13.1 Stationary cluster size distribution 13.2 Stationary rate of nucleation 13.3 Particular cases 13.4 Concentration of supernuclei 13.5 Comparison with experiment

184 184 192 204 214 214

14

First application of the nucleation theorem

224

15

Non-stationary nucleation 15.1 Non-stationary cluster size distribution 15.2 Non-stationary rate of nucleation 15.3 Time lag of nucleation 15.4 Delay time of nucleation 15.5 Concentration of supernuclei 15.6 Suggestion 15.7 Finding the equilibrium concentration of nuclei

231 232 243 249 258 267 270 270

16

Second application of the nucleation theorem

274

17

Nucleation at variable supersaturation 17.1 Quasi-stationary cluster size distribution 17.2 Quasi-stationary rate of nucleation 17.3 Condition for quasi-stationarity

279 280 283 285

Part 3

Factors affecting nucleation

291

18

Seed size

293

19

Line energy

300

20

Strain energy

309

21

Electric field 21.1 General formulae 21.2 Nucleation on ions 21.3 Nucleation in external electric field

315 315 317 323

22

Carrier-gas pressure

330

Contents

vii

23

Solution pressure

338

24

Pre-existing clusters

346

24.1 24.2 24.3 24.4

346 353 356 362

25

Non-stationary cluster size distribution Non-stationary rate of nucleation Concentration of supernuclei Delay time of nucleation

Active centres

366

Part 4 Applications

371

26

Overall crystallization

373

26.1 26.2 26.3 26.4

373 377 383 387

27

General formulae Polynuclear mechanism Mononuclear mechanism Two-stage crystallization

Crystal growth

391

27.1 27.2 27.3

391 396 404

Continuous growth Nucleation-mediated growth Spiral growth

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 supernuclei

436

32.1 32.2 32.3

437 440 440

33

General formulae Instantaneous nucleation Progressive nucleation

Size distribution of supernuclei

446

33.1 33.2

General formulae Particular cases

447 452

34

Growth of thin films

468

35

Rupture of amphiphile bilayers

480

Appendices

489

A1 Exact formula for the non-stationary nucleation rate A2 Approximate formula for the non-stationary nucleation rate A3 Initial concentration of supernuclei in previously supersaturated systems

489 491 492

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; K~impfer 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 modem 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 t h a t . . . ' 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 I 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; Srhnel and Garside 1992; Mullin 1993; Gutzow and Schmelzer 1995; Markov 1995; Baidakov 1995; Budevski et al. 1996; Debenedetti 1996]; reviews [McDonald 1962, 1963; Feder et al. 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 Halpern 1976; Skripov 1977; Zettlemoyer (ed.) 1977, 1979; Kern et al. 1979; Stoyanov 1979; Budevski et al. 1980; Chernov 1980; Russell 1980; Gutzow 1980; Kotake and Glass 1981; Stoyanov and Kashchiev 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 nucleation, 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. I 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 A* A

p

A"

Ab Ac,n

Af Am An Ap As Az a a ae ae aef a0 B

B

P

Bz b C C*

Cn, C(n) Cn(t), C(n, t)

Co C1 Cl,e

kinetic parameter in stationary nucleation rate (13.39) area of surface of contact between nucleus and substrate (4.46) kinetic parameter in stationary nucleation rate (13.41) kinetic parameter in stationary nucleation rate (13.51) area of surface of amphiphile bilayer (35.22) area of surface of contact between n-sized cluster and old phase (10.51) area of surface of crystal face (27.14) kinetic parameter in bilayer lifetime (35.23) area of surface of contact between n-sized cluster and substrate (3.76) kinetic parameter in bilayer lifetime (35.27) area of substrate surface (3.51) area of nucleation-exclusion zone (32.11) activity of solute (2.11) shape factor for 3D nucleation (3.20) equilibrium activity of solute (2.11) factor in coefficient of light extinction (29.24) effective molecular area (3.70) molecular area (3.74) thermodynamic parameter in stationary nucleation rate (13.48) thermodynamic parameter in stationary nucleation rate (13.66) parameter characterizing the effect of nucleationexclusion zones (32.16) shape factor for 2D nucleation (3.71) concentration of solute (2.14) equilibrium concentration of nuclei (7.37) equilibrium concentration of solute (2.14) equilibrium cluster size distribution (7.3) quasi-equilibrium cluster size distribution (9.22) concentration of nucleation sites (7.3) equilibrium concentration of monomers (7.5) equilibrium concentration of monomers at Ap = 0 (7.17)

xiv Symbols and abbreviations C c* r r

p

Cz

D

D Dn Ds O

s,n

d dr

do E

E E*

Edes En

es*

Esd Es,n Ev E1 e eo

F(R, t)

F2

f y,

K fe,s

f,, f(n, t)

Lm fs

y0*

G G Gc

GC,Z

shape factor for 3D nucleation (3.18) capture number (10.36) shape factor for growth of supernucleus (26.7) shape factor for growth of monolayer supemucleus (27.16) shape factor for growth of nucleation-exclusion zone (32.2) parameter defined by eq. (21.27) vector of electric displacement (21.1) coefficient of volume diffusion of monomer (10.11) coefficient of volume diffusion of n-sized cluster (10.21) coefficient of surface diffusion of monomer (10.25) coefficient of surface diffusion of n-sized cluster (10.94) dimensionality of growth (26.7) differentially small volume (8.1) molecular diameter (8.14) vector of electric field (21.1) strength of external electric field (21.21) binding energy of nucleus (4.45) activation energy for desorption (10.29) binding energy of n-sized cluster (3.32) 'substrate' binding energy of nucleus (7.55) activation energy for surface diffusion (10.25) 'substrate' binding energy of n-sized cluster (7.31) activation energy for viscous flow (10.56) value of En at n = 1 (7.31) base of natural logarithms (7.15) charge of electron (2.27) distribution function of supernuclei (33.2) 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 off* at A/I = 0 (13.42) value Of fs at A/~ = 0 (27.6) frequency of monomer attachment to n-sized cluster (9.16) frequency of transition of n-sized cluster into m-sized cluster (9.1) frequency of monomer attachment to growth site (27.2) value off* at Ap = 0 (13.43) Gibbs free energy (1.4) radial growth rate of supernucleus (26.7) growth constant of supernucleus (26.15) growth rate of crystal (27.1) growth constant of nucleation-exclusion zone (32.12)

Symbols and abbreviations xv

a ex

Gf GI

GI(R) G2

G2(t)

gn, g(n, t) gs h h* hf hi

hp I J J j"

ga Ja, s

Js,c

j*

in, j(n, t) K

Kr Ki Kv, Kva

Ko,

K3

k kv, ko, kl Ls M m0 N

excess energy of cluster (3.3) growth rate of thin film (34.2) Gibbs free energy of old phase before cluster formation (3.1) radius-dependent factor in radial growth rate of supemucleus (33.7) Gibbs free energy of old phase after cluster formation (3.2) time-dependent factor in radial growth rate of supernucleus (33.7) frequency of monomer detachment from n-sized cluster (9.16) frequency of monomer detachment from growth site (27.3) height of cluster (3.69) height of nucleus (4.37) mean thickness of thin film (34.1) thickness of ith layer of thin film (34.1) Planck constant (10.59) impingement rate (2.9) equilibrium impingement rate (2.9) rate of nucleation (11.1) rate of non-stationary nucleation (15.1) rate of detectable nucleation (11.11) rate of non-stationary nucleation per active centre (25.10) rate of stationary nucleation per active centre (25.3) rate of quasi-stationary nucleation (17.1) rate of stationary nucleation (13.1) critical rate of stationary nucleation (31.1) net flux through nucleus size (11.5) net flux through size n (9.6) pre-exponential factor (15.121) adsorption constant (10.46) parameters in induction time (30.1), (29.53), (29.16) adsorption constant (10.43) parameters in induction time (29.20), (29.56), (29.18) Boltzmann constant (1.1) coefficient of light extinction (29.23) parameters in rate of crystal growth (27.37), (27.39), (27.31) length of path travelled by light beam in scattering medium (29.23) total number of molecules in system (1.1) molecular mass (10.1) number of supernuclei (25.1) number of nucleation-active centres (7.9)

xvi

iVex Nm Ns n n* nt n S

ns nl n2

P P Pni

Pno el

p p p* Pe Pef Pn

Q q qi R

R* R" gav

Rb Ri

Rs Rz

R0 R2 r

S

Sc Snew Snew Sold

T

Td r~

Symbols and abbreviations

extended number of supernuclei (25.16) maximum number of supernuclei (25.9) number of adsorption sites on substrate (7.8) number of molecules in cluster (3.2) number of molecules in nucleus (4.1) number of molecules in smallest detectable cluster (11.12) Gibbs excess number of molecules in cluster (3.85) Gibbs excess number of molecules in nucleus (5.13) left end of nucleus region (7.41) fight end of nucleus region (7.42) pressure of cartier gas (22.1) pressure of solution (23.2) probability for non-ingestion of nucleation-active centre (32.3) probability for non-occupation of nucleation-active centre (32.3) probability for appearance of at least one nucleus (26.37) pressure in system (2.4) pressure of vapours (2.5) pressure inside nucleus (4.3) equilibrium pressure (1.9) effective pressure (8.24) pressure inside n-sized cluster (3.6) charge of ion (21.8) KJMA exponent (26.34) kinetic index of filling of ith layer of thin film (34.4) radius of cluster (3.22) radius of nucleus (4.9) radius of smallest detectable supernucleus (33.23) average radius of detectable supernuclei (33.23) radius of biggest supernucleus (33.10) radius of ion (21.6) radius of seed (18.1) radius of nucleation-exclusion zone (32.2) molecular radius (10.1) radius of n2-sized supernucleus (33.6) position vector (8.1) supersaturation ratio (7.19) critical supersaturation ratio (31.5) entropy per molecule in new phase (5.39) entropy per molecule in nucleus (5.37) entropy per molecule in old phase (5.37) absolute temperature (1.1) absolute temperature of divergence of viscosity (10.56) absolute equilibrium (melting) temperature (2.19)

Symbols and abbreviations

To t

/av

tb

tg tic p

t R,

tl

V V*

Vex Vn V, v(n, t) Oh Us Us OO

W W* W1 X* X'* Xn, X(n) Z* Z* Zn, Z(n, t) Zn, Z(n, t) Zn,o, Zo(n) z;* z1 Z Zi

a

aa

F 7"

xvii

absolute pre-existing temperature (24.17) time (9.1) average lifetime of metaStable system (26.9) lifetime of amphiphile bilayer (35.23) growth time of n2-sized supernucleus to smallest detectable size (33.27) induction time (29.1) critical induction time (31.12) moment of appearance of supernucleus which has radius R" at time t (33.25) mean time for appearance of at least one nucleus (26.39) volume of system (1.1) volume of nucleus (4.5) extended volume (26.4) volume of n-sized cluster (3.6) volume of nucleation-exclusion zone (32.2) growth rate of n-sized cluster (9.21) growth rate of hole in amphiphile bilayer (35.22) growth constant of monolayer supernucleus (27.17) velocity of propagation of monolayer step (27.23) molecular volume (2.4) work for cluster formation (3.4) work for nucleus formation (4.2) value of W at n = 1 (7.5) stationary concentration of nuclei (13.23) derivative of X(n) with respect to n at n = n* (24.10) stationary cluster size distribution (13.3) concentration of nuclei (11.4) non-stationary concentration of nuclei (15.54) cluster size distribution (9.1) non-stationary cluster size distribution (15.2) initial cluster size distribution (9.2) pre-existing concentration of nuclei (24.10) derivative of Zo(n) with respect to n at n = n* (24.10) concentration of monomers (9.32) Zeldovich factor ( 13.33) valency of ion (2.27) fraction of crystallized volume (26.1) detectable fraction of crystallized volume (29.8) coverage of ith layer of thin film (34.1) Zeldovich coefficient (7.38) detectable relative decrement of intensity of transmitted light (29.47) gamma-function (26.19) coefficient of monomer sticking to nucleus (13.44)

xviii

Yn ')'s A* An*

ap

Aps AS* ASe AT

Arc Ave

AU

Ag Aq~ ec eo

r r

Ca r r 77 O O'

Op Ow O

oi K"

Zs

p* Pe

~.lnew ~[Anew, n

~old

Symbols and abbreviations

coefficient of monomer sticking to n-sized cluster (10.2) coefficient of monomer sticking to growth site (27.7) width of nucleus region (7.43) excess number of molecules in nucleus (5.21) underpressure (4.14) spinodal underpressure (4.23) excess entropy of nucleus (5.37) difference in molecular entropy at phase equilibrium (2.21) undercooling (2.21) critical undercooling (31.9) difference between volume of molecule in melt and crystal (10.79) supersaturation (2.1) rate of change of supersaturation (17.11) critical supersaturation (31.1) spinodal supersaturation (4.13) pre-existing supersaturation (24.17) effective specific surface energy (3.70) overvoltage (2.27) Dirac delta-function (15.71 ) dielectric constant of cluster (21.5) dielectric constant of medium (21.4) permittivity of empty space (21.4) concentration of supernuclei (11.1) concentration of detectable clusters (11.11) concentration of nucleation-active centres (33.56) maximum concentration of supernuclei (33.32) initial concentration of supernuclei (11.10) viscosity (10.54) delay time of nucleation (15.99) delay time of detectable nucleation (33.67) delay time of nucleation at pre-existing clusters (24.60) wetting angle (3.52) time constant of overall crystallization (26.18) time constant of filling of ith layer of thin film (34.4) specific edge energy of 2D cluster (3.69) specific line energy of cap-shaped cluster (19.1) molecular heat of phase transition (3.31) eigenvalue (15.9) mean surface diffusion distance (10.30) chemical potential of molecules in nucleus (4.4) equilibrium chemical potential (1.10) chemical potential of molecules in new phase (2.1) chemical potential of molecules in n-sized cluster (3.6) chemical potential of molecules in old phase (2.1)

Symbols and abbreviations

v vi ~re

P Pnew ~old O"

trb O'ef o'i trs "t"

Vd

q's r *

~s ~str

X ~P V tonm,

xix

growth exponent of supernucleus (26.15) growth exponent of 2D supernucleus in ith layer of thin film (34.6) equilibrium surface pressure of insoluble monolayer (35.8) surface pressure of insoluble monolayer (35.4) number density of molecules (8.1) number density of molecules in new phase (5.9) number density of molecules in nucleus (5.10) number density of molecules in old phase (5.8) specific surface energy of interface between new and old phase (3.18) specific surface energy of amphiphile bilayer (35.2) effective specific surface energy (4.38) specific surface energy of interface between new phase and substrate (3.53) specific surface energy of interface between substrate and old phase (3.51) time lag of nucleation (15.54) mean time of desorption (10.28) effective excess energy of cluster (3.86) effective excess energy of nucleus (4.6) effective surface energy of cluster (3.82) effective surface energy of nucleus (4.5) value of q~ at n = 1 (7.34) total surface energy of interface between cluster and old phase (3.6) total surface energy of interface between nucleus and old phase (4.12) total surface energy of interface between substrate and old phase (3.46) effective strain energy per molecule of cluster (20.6) difference between actual and pre-existing supersaturation (24.20) activity factor of nucleation (4.42) Volmer function of wetting angle (3.52) parameter of quasi-stationarity (17.8) to(n, m, t) frequency of coalescence between n-sized and m-sized clusters (9.34)

EDS HEN HON IN KJMA PN

equimolecular dividing surface heterogeneous nucleation homogeneous nucleation instantaneous nucleation Kolmogorov-Johnson-Mehl-Avrami progressive nucleation

xx

Symbols and abbreviations

ST TDE VDW 1D 2D 3D

surface of tension thermodynamic equilibrium van der Waals one-dimensional two-dimensional three-dimensional

Part 1

Thermodynamics of nucleation

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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'/V2)(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 Pcr, volume Vcr and temperature Tcr of the VDW fluid by using the relations [Guggenheim 1957; Landau and Lifshitz 1976]

Per = a'/27b'2,

Vcr = 3Mb',

Tcr = 8a'/27kb'.

(1.2)

For this reason, with the help of the reduced pressure P' = P/Pcr, volume V' = V/Vcr and temperature T" = T/Tcr eq. (1.1) passes into the reduced VDW equation (P' + 3/V'2)(3 V' - 1) = 8T'

(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/Pcr. 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 ) d V so that using eq. (1.3) leads to G(V',T') = Gref(T' ) + MkTcr[3p'V'/8 - 9 / 8 V ' - T" In (3V' - 1)] (1.4)

where Gre f is a reference energy. From the conditions for minimum (OG/cgV) r = (cgF/,gV)r+ p = - P ( V ) + p = 0 (32G/OV2) T = (o32F/OV2)T = - ( 3 P / 3 V ) r > 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

Nucleation: Basic Theory with Applications

corresponding to such a portion of the VDW isotherm (1.3) for which (3P'/ 3V')r < 0. If the solution of eq. (1.5) falls in the range of volumes for which (OP'/OV')r> O, 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 !

,

(a) I I

~

1.0-

%

C 9

D

.o

~.

. liquid

13..

-

9g

a

0.5.

--,

:....

:

(

bd 9: .........

t.:.

J..

"'

I e

0 : '

l

i r

m

I

' ......

,

,

,

,

0.55

,

I

.."

,

i

i

i

,

,

,

I

,

I

I

I

.

1

(b)

C

IL__

I--

t~

2~ 0 . 4 5 .

i

I

.gas o

4o

liquid 0.470

0.35

0

9

t

i

I

i

1

I

I

'

'

I

,

2

V

I

'

'

l

3

,

,

,

,

.

4

!

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 T'= 0.85; dashed curves - eqs (1.3) and (1.4) at p ' - 1 and T'= 1. Nucleation-mediated firs.t-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 'fghi' defines the position of the dot-dashed line 'bfi' of gas~liquid coexistence.

First-order phase transitions

5

critical temperature (T' = 1, the dashed curve). The respective G(V') dependences at three different subcritical pressures (p' = 0.470, 0.505 and 0.540, the solid curves) and at the critical pressure (p' = 1, the dashed curve) are drawn in Fig. 1.1b according to eq. (1.4) with an arbitrarily chosen Gref = 2MkTcr. 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 P'(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 p pfs(T') and Pg,s(T') of the liquid and the gas, respectively. As seen from Fig. 1.1a, for values of p' between the spinodal pressures Pf, s(T') and Pg.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 p e ( T ) , the respective reduced pressure being p~ = pe/Pcr. 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 Pe is the equilibrium (more precisely, the phase-equilibrium) pressure of the fluid. The reduced phase-equilibrium volumes Vl,e( T ' ) of the liquid and Vg~e( T' ) of the gas are those solutions of eq. (1.5) at p' = p~ that correspond to points b and i in Fig. 1.1 a, and connecting all these points for the subcritical temperatures results in another dome-shaped curve called binodal (the dotted curve B). p In the general case when p' ~ p~, but satisfies the condition Pl;s < P" < Pg.s, the two states of the fluid in stable TDE (see points a and h and points d and j in Fig. 1.1 a, 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 metastable 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. 1.1 a), 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 - G/M of the V D W fluid and its derivative (Op/Op')T 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 = 2MkTcr. Indeed, it is seen that when T' = 0.85, the conditions for first-order phase transition are fulfilled at p' = Pe = 0.505 (~g and/.t] are the chemical potentials of the gas and the liquid, respectively): ~[.tg(p;)

[((glll]()P')T']p'=p~.

[((~]Jg/O3Pt)T']p'=pe ~

-- ]JI(P;),

7.

0.8

0.6

-

g

c

...""""

I

,

0.4

!-=L

0.2 -.

1.5

,

/

i II ,

,

.t I

,

,

I

I

%

',

i

1.0

I

(b)

i

"~

,

" ',

ga

"

,

%

\ %%

L_ g

%%%

0.5

%

ce

,

...

b L

0

I

I

'

'

I

,

I

,

=

I

1.0

0.5

p

liquid

,

i

,

,

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 T'= 0.85 (solid curves) and T'= 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 Pl derivatives are equal to the volumes occupied at Pe 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 firstorder 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 (Ola/Op')r are continuous functions of p', 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 Pe is an important thermodynamic characteristic of the fluid, for it determines the direction of the phase transition" while for p > Pe the gas-to-liquid transition takes place (the gas is then metastable (Fig. 1.1b) and pg> Pl (Fig. 1.2a)), when p < Pe the liquid-to-gas transition occurs (the metastable phase then is the liquid (Fig. 1.1b) and Pl > pg (Fig. 1.2a)). To find Pe we can use the condition for equality of the values of the two minima of G at p' = Pe, which is equivalent to the condition Pg( P~ ) = Pl( P~ ). Setting equal the right-hand sides of G from (1.4) at Pe, Vg[e and Pe, Vl,e, we obtain the exact formula P

Pe "- (Vg~e- Vl,e) -1 {(8/3)

T'ln [(3Vg,e 1)/(3Vl,e P

__

P

--

1)]

- 3(1/V~,e - 1/Vg;e)l.

(1.6)

In view of (1.3), this equation can be represented in the equivalent form

Ii;gl,e Pe -- (Vg~e- Vl~e)-1

P ' ( g ' ) 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 Pe implicitly, since Vg~ and Vl.e themselves, being the solutions of (1.5) at p' = Pe, depend on Pe and T'. However, when T' is sufficiently less than 1 (i.e. for low enough subcritical temperatures), we have Vlx << Vgx--- 8T'/3pe (the gas is nearly ideal) and Vlx = 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 t

P

P

Pe (T') = 16T' exp [-(1 + 9/4T')].

t

(1.8)

In view of (1.2), this expression can be given the form

pe(T) = (2kT/b') 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 ]-/old,e = Pnew, e -----He.

(1.10)

are, respectively, the chemical potentials of the old and the new phases at phase equilibrium, and Pe, the equilibrium chemical potential, is the abbreviated notation for either of them. H e r e ]-/old,e and/-/new, e

Chapter 2

Driving force for nucleation

In the preceding Chapter we have seen that when p' ~ Pe (but is limited between P~;s and Pg.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 p

A/I - (Gold - Gnew)/M =/-/old

-- ,//new

(2.1)

known as supersaturation. Physically, the supersaturation Ap 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 Gol d t o the minimum with lower Gibbs free energy Gnew (Pold and Ftneware, respectively, the chemical potentials of the old and the new phases at the corresponding minima). Equation (2.1) allows expressing A/I 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 Pe < P' < Pg.s (this corresponds to transition from point h to point a in Fig. 1.1). Using eq. (1.4) yields p

Go~d = Gref (T') + MkTcr[3p'Vg/8 - 9/8Vg- T ' l n (3Vg- 1)] Gnew -" Gref (T') + MkTcr[3p'Vl"/8 - 9/8V[- T'ln (3Vl'- 1)]. Here Vg (p', T') and VI' (p', T') 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 Gold and Gnew from above into (2.1) leads to the exact formula

A,tt = kTcr {T'ln [(3Vl'- 1)/(3Vg- 1)] + (9/8)(1/V1' - l/Vg) + (3/8) p'(Vg - VI')1.

(2.2)

With the help of (1.6) this formula can be given the equivalent form

At.t = kTcr{T'ln [(3Vg(e- 1)(3Vl'- 1)/(3Vg- 1)(3Vl,e- 1)] + (9/8)(1/V~'- 1/VLe+ 1/Vg~e- 1/Vg)

+ (3/8)(p'Vg - p~Vg~ + p[VI,'~ - p'V[)}

(2.3)

10 Nucleation: Basic Theory. with Applications

which shows that A/d = 0 at p' = Pe. 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 A/d < 0: the old phase is then undersaturated (its G minimum is lower than that of the new phase and/~old > 1, 3Vg.e>> 1, VI' ~ Vl,'e = Moo/Vcr (the liquid is almost incompressible) and Vgp ' - VgPep~ (8/3) T' (the gas is nearly ideal). In view of the relation 3MkTcr = 8PcrVcr, it then follows from (2.3) that, approximately, P

t

A/d = k T In (P/Pe) - vo(P - Pe) = k T In

(Vg,e/Vg) -kT(Vl,elVg,e)(Vg,elVg- l)

(2.4)

where v0 is the molecular volume. Let us now see how Ay 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/d we shall use eq. (2.1) and known expressions for the chemical potentials/dold and/dnew 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 p of the vapours is given by [Guggenheim 1957] /dold(P) =/de -t- k T In (P[Pe) /dnew(P) =/de + Oo(P- Pe)

(2.5) (2.6)

where pe(T) is the phase-equilibrium pressure (i.e. the pressure of the saturated vapours of the liquid or the solid), and/de =/dold(Pe) =/dnew(Pe) 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]) A/d(p, T) = k T In (P]Pe) - Oo(P - P e )

(2.7)

which in most cases can be approximated sufficiently accurately by [Zettlemoyer 1969] Al~(p, T ) = k T In [p/pe(T)],

(2.8)

since usually V0Pe << k T and the second summand in (2.7) is negligible.

Driving force for nucleation

11

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 fv/fv, e of the actual and the equilibrium fugacities fv and fv, e of the vapours. In condensation of molecular beams onto a substrate the experimentally controllable parameter usually is the impingement rate I of molecules (per second per m 2) rather than the pressure p. As I = p/(2xmokT) 1/2, in this case Ap is given approximately by (2.8) in the form [Sigsbee 1969; Lewis and Anderson 1978]

Ap(l, T)= kT In [I/le(T)]

(2.9)

where le = pe/(27cmokT) 1/2 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.:

Ap(p, T) = kT In

(pe/p)

-

OO(Pe-- p),

(2.1 O)

p being the actual pressure of the liquid in boiling or of the vapour in evaporation or sublimation. When v0 Pe << kT, Al~ can be approximated by

Ap(p, T)= kT In [pe(T)/p]. (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 Pold on a is of the form [Guggenheim 1957] /-/old(a) = / / e 4- kT In (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 aindependent, approximately,

12 Nucleation: Basic Theory with Applications

Pnew(a) =/~new(ae) =/~e"

(2.12)

Combining the above two expressions with eq. (2.1) yields [Walton 1969a; S6hnel and Garside 1992] A/~(a, T) - k T l n [a/a~(T)].

(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; S6hnel and Garside 1992] A/~(C, T) = kT In [C/Ce(T)].

(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 case and then, according to the condition for chemical equilibrium [Guggenheim 1957; Landau and Lifshitz 1976], ,t/old = V I ] . / 1 + V2,t/2 + . . .

+

(2.15)

Vk,tl k

where vi is the number of ith species in the molecule (i = 1, 2 . . . . . k), and pi is the chemical potential of the ith species in the solution. Using (2.11) in the form ].li(ai)- ltli,e + kT In (ai/ai,e)

leads to the following expression for/~old: ~told = ~t~ + k T l n [(al/al,e) v~ ( a z ] a z , e ) V 2

. . . (ak]ak,e)

vk ] .

Here a i is the actual activity of the ith species in the solution, a i , e and Pi,e are the equilibrium activity and chemical potential of the ith species, and ,t/e = Vl,t/1,e + V2,t/2,e +

9 9 9+

Vk,Uk,e.

Thus, with the help of eq. (2.12) and the above expression for/~ola, it follows from (2.1) that A/~(H, T)= k T l n [H/He(T)]

(2.16)

v2 Vk where H = a~ ~a~2 .. a~ k and Fie = al,v1ea2,e.., akx are, respectively, the actual and equilibrium activity products of the solute [Nielsen 1964; S6hnel and Garside 1992]. As seen, eq. (2.16) is a generalization ofeqs (2.13) and (2.14) (for sufficiently dilute solutions H and I'I e a r e the products of the corresponding concentrations of ith species, II e being the so-called solubility product). An interesting point concerning eq. (2.16) is that even when the solution is supersaturated with 9

Driving f o r c e f o r nucleation

13

respect to only one of the species in the solution, but is saturated with respect to the others (e.g. a l / a l , e > 1, a2/a2,e = a3/a3,e = 9 9 9= ak/ak,e = 1), there exists a driving force for condensation of the solute, since then A/j > 0. Moreover, sufficiently high supersaturation of the solution with regard to one (e.g. a l / a l,e >> 1) or more species can create a driving force (A/~ > 0) even if the solution is undersaturated with regard to one (e.g. a2/a2, e < 1) or more of the rest of the species. Detailed considerations concerning A/.t for crystallization from solutions can be found elsewhere [van Leeuwen 1979; van Leeuwen and Blomen 1979; S6hnel 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 A~t is given by A/~(a, T) = k T In [ a e ( T ) / a ] ~ k T In [ C e ( T ) / C ]

(2.17)

and, more generally, by A/x(H, T) = k T In [FIe(T)/H].

(2.18)

(e) Crystallization of melt or polymorphic transformation by cooling (T < Te) 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 Voidexperimentally is the temperature rather than the pressure and, for that reason, it is convenient to express Ap as a function of T. Isobarically, from thermodynamics [Guggenheim 1957], ~ ( T ) = ~e +

(2.19)

s(r') dr'

where s is the entropy per molecule (or atom), and Te is the absolute phaseequilibrium temperature (the melting point in melt crystallization). Using (2.19) twice, for/-told and ~new, and recalling (2.1) yields [Volmer 1939] AI.t(T) =

A s ( T ' ) dT'

(2.20)

where As(T) = Sold(T) - Snew(T). Obviously, the main problem in finding Abt is to know how the entropies Sold and Snew 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 AH(T) = _ Ase(T - Te) - (1/2)(dAs/d~e(T - Te)2 - (1/6)(d2As/dT2)e(T-

Te)3 - . . .

of A/I from (2.20) in the vicinity of T = Te where

mse, (dAs/dT)e

and (deAs/

14 Nucleation: Basic Theory with Applications

dT2)e are the values of As and its derivatives at T = Te. Recalling the known relationship [Guggenheim 1957]

s(r) =

fo

[Cp(

)/r']dr'

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 d A s / d T = Acp(T)/T p

d2As/dT 2 = A c p ( T ) / T - Acp(T)/T 2

where Acp - Cp,old- Cp,new, A Cp =--dAcp/dT, and Cp,oldand Cp,neware 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 Ap results in All(T) = A s e A T - (Acp,e/2Te)AT 2 + [ ( Z e A c p , e - Acp,e)/6T 2] A T 3. ( 2 . 2 1 )

Here AT = Te - T

(2.22)

is the undercooling, Ase = Sola(Te) - Snew(Te) = ~,/Te is the difference in the molecular entropies of the old and the new phase at T = Te, ~, is the latent heat (per molecule) of crystallization or polymorphic transformation, ACp,e- Cp,old(Te) -- Cp,new(Te) and Acp,e- dAcp/dT at T = Te. Equation (2.21) reduces to the widely used formula [Volmer 1939] Ap(T) = AseAT

(2.23)

when the undercooling satisfies the condition AT << 2TeAse/Acp,e - 2A/Acp,e, 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] Al.t (T) = A s e A T - (Acp,e/2Te)A T 2.

(2.24)

This equation is very similar to the one given by Jones and Chadwick [ 1971 ]. A number of other Ap(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 Ap for Li20.2SIO2 melt for which Te = 1306 K, As e = 5.26k and Acp,e=l.56k [Kelton 1991]. The circles are the exact values computed by Kelton [1991] with the help of measured Acp(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 f o r c e f o r nucleation 1

'

i

'~

'

'1

i

~

'

~

I

'

'

'

I

'

~

~

I

iHi

'

~

J'

15

/,

/ t

4 p

3


p tpp

s "

,p"//I

2

."

o

0

O

9

2

1t

0

0.2

,~

0.4

0.6

0.8

1.0

AT/T e

Fig. 2.1 Dependence of the supersaturation on the undercooling f o r Li20.2Si02 melt: circles - experimental data [Kelton 19911; circular c o n t o u r - eq. (2.23); solid line - eq. (2.24). crystal with another modification. This means that everywhere in these equations As~, Acp,~ and A cp,~ 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: A/~(T) =-AseAT = Ase(T- Te).

(2.25)

(g) Electrochemical deposition (tp < tpe) 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 q~. The role of the chemical potentials of the old and new phases is now played by the respective electrochemical potentials ~old and ~new [Lange and Nagel 1935; Guggenheim 1957] so that rather than by eq. (2.1), Ap is determined by AlL ~- ~old -- ~new.

(2.26)

The dependence of ~n~w on tp has the form [Lange and Nagel 1935; Guggenheim 1957] ~new ((~) -" ~e q" Zieo (tp - q)e)

where zi is the valency of the solute ions, e0 is the electronic charge, and tPe and ~e (T) are the phase-equilibrium Galvani and electrochemical potentials, respectively. Since ~ola is practically ~independent, i.e.

16 Nucleation: Basic Theory with Applications

~o~d (~o) = ~old(~0o) =

~o,

using the last two equations in (2.26) leads to the approximate formula [Volmer 1939; Vetter 1967] Ap(cp) - zieoACp

(2.27)

where Aq9 - r

r

(2.28)

is the overvoltage. (h) Electrochemical dissolution (q9 > q~e) Analogously to non-electrochemical dissolution (case (d)), in this case ~old and ~n~w 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."

A/l(tp) = zie0(tp- tpe).

(2.29)

Summarizing, we see that Ap is determined always by eq. (2.1) or its appropriate generalization (e.g. eq. (2.26)). The difficulties in the determination of Ap arise from incomplete knowledge of the exact dependences of Polo and ]-/new on the respective experimentally controllable parameters. It must be emphasized that the correct determination of Ap 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 Ap 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 metastable TDE is 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 requiting 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 T' = 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.lb. 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 1.1 b shows that at the chosen values of p' and T' this energy barrier is approximately O.03MkTcror, more generally, about MAp, since Ap -- 0.03kTcr at p' = 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 n* << 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/~. This is much less than the energy barrier of about MA~t 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 D6ring 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

(a)

9

9 9

9

19

9

phase boundary

-

9 .:,',-,~I~,\o,

,

9

X

9....aV'~o oO/O 9

9"-%O.,o-~o 9 9

9

(b)

9

P

Pnew

! I I

I

,,

'\ phase boundary

I

,

Ill

i

i

'

i,

A .,.

i

_

,,,. I

i

i

_

i I

'

'

~

ill

X

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 densityfunctional 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 D6ring [1940]; Reiss [1950]; Sigsbee [1969]; Nishioka and Kusaka [1992]; Wiiemski 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 1 -" M/I old

(3.1)

Fig. 3.2 Old phase of M molecules before (state 1) and after (state 2) homogeneous formation of a cluster of n molecules. and in its final state 2 when it contains a cluster of n molecules (n = 1, 2, 9 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 socalled 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. lb). 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 welldefined 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 = 1, 2 . . . . ) G2(n) = ( M - n),Uol~ + G(n).

(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 T and pressure p and, hence, with unchanged chemical potential Pold 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 npnew which the cluster would have had if it were a part of the bulk new phase at the given T and p. This means expressing G(n) in the form (n = 1, 2. . . . )

G(n) = n//ne w + Gex(n)

(3.3)

where Gex(n), the cluster excess energy, remains to be determined in each particular case. Using the definition equality

W(n) = G2(n) - G1

(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 . . . . )

W(n) =-nA/.t + Gex(n).

(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(1) ~ 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(n) 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 and Stranski [ 1934a] and Becker and D6ring [ 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) + p V where F 2 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-/'/),//old- p V M - n + n//new, n - pnVn + ~(Vn).

(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_ n 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, Pnew,.(P,,) is the chemical potential of the molecules in it), and ~Ois the total free energy of the cluster/old phase interface. Combining the above two equations and taking into account that V = VM_,,+ V. yields Gz(n ) = ( m -

n)Pol d + n//new, n -- (Pn - - p ) V n + ~(Vn).

(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) = - ( P n - p)Vn + (J-/new,n - ~old) n -I- qb(Vn).

(3.8)

Recalling eq. (2.1), we can now compare the fight-hand sides of this equation and of eq. (3.5) to obtain finally that in the case of HON Gex(n) = qb(Vn) - (Pn -- p)Vn + (Pnew, n - Pnew) n.

(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 ]-/new,n o f the molecules in it. It shows that the cluster excess energy Gex 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 ~gG2/OVn = 0 for mechanical equilibrium of the system in state 2, with G2 from eq. (3.7) there follows the generalized Laplace equation [Toschev 1973a]

(3.10)

p. = p + dr

Since by definition [Guggenheim 1957] /~new,n(Pn) =/~n,w(P) + (l/n)

Vn(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 ~ through its derivative dr Thus, finally, we arrive at the following dependence of Gex on n (through Pn from (3.10) and Vn):

Gex(n)

= r

n) - ( P n - P) Vn +

Vn(e ) dP.

(3.12)

This relationship tells us that for the determination of the cluster excess energy Gex it is necessary to know not only the cluster total surface energy ~0, but also the equation of state, Vn = Vn(Pn, 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 G~x 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 Pn inside the cluster and equal to the molecular volume v0, i.e.

V,,(p,,) = nvo.

(3.13)

If the cluster is a gas phase, in ideal-gas approximation Vn depends on p, according to the equation of state Vn(Pn) = nkT/pn.

(3.14)

Using these two equations in the integral in (3.12) yields Gex(n) = ~(Vn) for clusters of condensed phases and

(3.15)

24

Nucleation: Basic Theory with Applications

G~x(n) = dp(V.) + nkT[ln (pn/p)- (1 - p / p . ) ]

(3.16)

for clusters of gas phases (Pn depends on ~ 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 r because of the virtually complete cancellation of the last two summands in (3.12), in the formation of gas clusters (e.g. bubbles) Gex depends in a much more complicated way on q~. The explicit determination of the G~x(n) dependence requires a concrete model for ~ 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 r as proportional to the area S. of the dividing surface (i.e. of the surface of the n-sized cluster). Physically, the proportionality factor or. (J m -z) 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. V2/3 and ~ can be written down as (P(Vn) = c.ty. V~/3

(3.17)

where c. is a numerical shape factor. The idea to employ tr. for the determination of q~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 c. = constant = c (independently of its size n, the cluster has the same shape) and or, = constant = tr (or, is identified with the specific surface energy cr of the planar new phase/old phase interface, i.e. with the limiting value of trn at n ~ oo, 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 an 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 cn and o'n eq. (3.17) simplifies to r

= ctrV 2/3

(3.18)

where c = (367r) 1/3 for spheres, c = 6 for cubes, etc. As tr 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 V. according to p , = p + 2c tr/3 V113.

(3.19)

If the cluster total surface energy ~ has to be expressed explicitly as a function of n, eq. (3.18) shows that it is necessary to know the dependence of V~ 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 (?(n)

= a a n 2/3

(3.20)

where a = co 2/3 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

213

clusters (e.g. a = (36 rcv~) 1/3 for spheres, a = ov 0 for cubes, etc.). In the case of gas-phase clusters, however, Vn can be expressed simply as a function of n only implicitly upon combining eqs (3.14) and (3.19):

n = (plkT) V n

(2col3kT)

+

Vg/3 .

(3.21)

That is why, in this case a simple representation of r 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, q~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 = (36zr) 1/3 and V,, = (4/3)zrR 3,

(3.22)

eq. (3.18) transforms into the known formula [Zettlemoyer 1969] ~(R) = 4xcrR 2.

(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]

Pn

-

-

P + 2ty/R.

(3.24)

Knowing ~0and Pn, 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 or,,. According to eqs (3.15), (3.18), (3.20) and (3.23), for clusters of condensed phases Gex(Vn) = c(y V2/3

(3.25)

Gex(n) = acrn 2/3,

(3.26)

and if the clusters are spherical, Gex(R) =

47rtrR2.

(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(Vn) = ( 1 / 3 ) c a V f/3 + pVn(1 + 2ccr/3p V213) I n (1 + 2ccr/3p V2/3)

(3.28) Gex(n) = (1/3)ccr(kT/pn)

2/3 2/3

n

+ nkT In (p#p),

(3.29)

and if the clusters are spherically shaped, Gex(R) = (47r/3)crR2 + (47r/3)pR3(1 + 2cr/pR) In (1 + 2(r/pR). (3.30) The explicit G~x(n) dependence is rather complicated and can be obtained by using eqs (3.19) and (3.21) to express Pn in (3.29) as a function of n. The above formulae for the cluster excess energy Qx are valid for sufficiently large values of n (i.e. for n --) oo). 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 Gex 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 Uold and Unew be the potential energies of a molecule in the bulk of the old and the new phase, respectively, u/be the potential energy of the ith molecule in the n-sized cluster, and U,, = ~ ui be the potential energy of the n-sized i=1

cluster as a whole. Then, it is convenient to introduce the quantities 2, > 0 and En > 0 defined by (n = 1, 2 . . . . ) E

(3.31)

Uol d -- Une w t!

gn -

?//'/old - -

Un

--

~ (Uold i=1

--

Ui)"

(3.32)

For liquid or solid clusters, to a good approximation, ~ 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 ~ 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., En 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 En, however, in HON we have U1 = UoldSOthat from eq. (3.32) it follows that E1 = 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 nA 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 $ of the cluster will be given by the formula (n = 1, 2 . . . . ) (p(n) = n A - 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 Un - nUnewin the potential energies of the cluster and these n molecules [Volmer 1939; Kaischew 1965], to a first approximation, G(n)

-

nl./ne w

=

U n -

n U n e w.

Hence, thanks to eqs (3.31)-(3.33), this formula can be rewritten as (n = 1, 2. . . . ) G(n) = npnew + ~(n)

(3.34)

which, compared with eq. (3.3), leads to (n = 1, 2 . . . . ) Gex(n) = ~0(n).

(3.35)

This equality tells us that for clusters of condensed phases identification of Gex with q~ 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. . . . ) Gex(n) = n Z - En.

(3.36)

It should be emphasized that while eq. (3.26) of the classical theory is valid for n ---) 0% eq. (3.36) holds true not only for n ~ 1, but for any number n of molecules in the cluster. Therefore, comparison of the fight-hand sides of these equations shows that for n -+ oo the cluster binding energy En is a function of n of the following general form: E n = ~n - a a n 2/3.

(3.37)

For crystal clusters, this dependence of En 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 G~x 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(Vn) = - ( A p / V o ) V n + ccrV 2/3 W(n) = - n A p

+ acrn 2/3

(3.38) (3.39)

and, if the clusters are spherical, W(R) =-(4ZCAla/3vo)R 3 + 4zrcrR 2.

(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(V,) = -[(Akt/kT) - In (1 + 2ccr/3pV2/3)l(p + 2cry/3V2/3) Vn + (1/3) ccrVf/3

(3.41)

W(n) = - n [ A p - kT In (Pn/P)] + (1/3)ca(kT/pn)2/3n2/3

(3.42)

W(R) =-(4z/3)[(Alu/kT) - In (1 + 2cr/pR)](p + 2cr/R)R 3 + (4~r/3)tyR2.

(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 Ap (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) = - ( A p - ~,)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/~ (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 Gex(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~x. 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 Tcr, Fisher [1967a, b] introduced another energy contribution in the classical expression for Gex.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 Abt in the system, the classical theory makes it possible to calculate W as a function of n or Vn if one knows only one physical p a r a m e t e r - the specific surface energy cr 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 E~, 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 W on n for HON of water droplets in vapours at T = 293 K

100 f~_ 80

W*

~176 f 20 0 -20

1

100

200

300

400

Fig. 3.3 Dependence o f the work f o r cluster formation on the cluster size: curve 'droplet'- eq. ( 3 . 3 9 ) f o r H O N o f water droplets in vapours at T = 293 K and P/Pe = 4; curve ' b u b b l e ' - eq. (3.42)for H O N of steam bubbles in water at T = 583 K and Pe/'P = 4.

and p = 4Pe and of steam bubbles in water at T = 583 K and p = (1/4)p e. The calculations are carded 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 Ap --- 1.4kT. 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

T = 259 K

T = 273 K

T = 293 K

m0 (kg) v0 (nm3) do (nm)a Pe (kPa) G (mJ/m2)

3 • 10-26 0.03 0.39 0.20 77.6

3 • 10-26 0.03 0.39 0.61 76

3 • 10-26 0.03 0.39 2.3 73

T~

1

1

1

acalculated from do = (6v0/~:)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)

v0(nm3)

pe(MPa)

o(mJ/m2)

7'n

3 x 10-26

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~t. Comparison of this energy with the energy MAp 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.

Workfor 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, ~0s,0and ~s(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 . . . . ) G1 = M/J,old+ Gs + ~s,O G2(n) = ( M - n)/~old + G(n) + Gs + ~0s(n).

(3.45) (3.46)

Fig. 3.4 Old phase of M molecules before (state 1) and after (state 2) heterogeneous formation of a cluster of n molecules on the surface of a substrate. Since the cluster Gibbs free energy G(n) 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) = - n A p + Gex(n) + ~0s(n)- q~s,0.

(3.47)

This equation says that in this case, in addition to the cluster excess energy Gex(n), it is necessary to know also the difference q~s(n) - Os,0 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 Gex(n) + ~(n) - ~s,0 is less 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(1) ~ 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 Gex 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)Pold- PVM-n + npnew,n-pnVn + q~(Vn) + as + Os(Vn) where 0s(V.) - q~s(n), and ~V.) is again the total surface energy of sized cluster of volume V., but now accounting for the presence cluster/substrate interface. As from thermodynamics [Guggenheim Gz(n) = Fz(n) + p V where V = VM_ n + V., using Fz(n) from the expression yields Gz(n ) = ( M - n)~told + nPnew,n - (Pn - p ) V n + ~)(Vn) + Gs +

the nof the 1957] above

~s(Vn). (3.48)

From the condition tgG2/3V, = 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 + Os)/dVn.

(3.49)

Substituting G1 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) = -(Pn - p ) V n + Q./new,n - Pold)n + ~)(Vn) at- ~s(Vn) - ~s,0" (3.50) We can now compare this equation for W(n) 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. f Pn Gex(n) = O(Vn)- ~ n - p)Vn + Vn(P)dP" vp

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 ~ now accounts for the presence of the cluster/substrate interface, and Pn depends on both ~ and q~sthrough their derivatives with respect to Vn (see eq. (3.49)). That is why, for the explicit determination of the Gex(n) dependence in HEN there is a need of concrete models for r 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 q~s,0= trsAs

(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

Workfor cluster formation

33

Fig. 3.5 Cross section of (a) cap-shaped, (b) lens-shaped, and (c) disk-shaped cluster of n 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, 2zrR2(1 - cos 0w), 7rR2 sin 2 0w and Vn = ~(Ow)(4/3)TrR 3.

(3.52)

As in the case of HON, we shall use the classical approximation Crn = constant = cr and cri,n = constant = cri, cri,~ and cri being, respectively, the specific surface energies of the substrate/n-sized cluster and substrate/bulk new phase interface. Then, for the total surface energy r of an n-sized cluster on the substrate and for the total surface energy q~sof the substrate with such a cluster on it we can write q~(R) = cr2zrR2(1 - c o s 0w) + o]zcR2 sin 2 0w

(3.53)

~s(R) = Ors(As- nrR2 sin 2 0w).

(3.54)

In the above relationships, for clusters of condensed phases 0w is the equilibrium wetting angle defined from 0 to zcby the Young equation [Young 1805] COS 0 w = (O" s -- O'i)/O',

(3.55)

and the numerical factor ~ has values between 0 (at 0w = 0) and 1 (at 0w = tr) (Fig. 3.6) and depends on 0w according to [Volmer 1939] ~0w) = (1/4)(2 + cos 0w)(1 - c o s 0w)2.

(3.56)

The cases of 0w = 0 and 0w = zr are usually referred to as complete wetting and complete non-wetting, respectively. For clusters of gas phases the terminology is the opposite, for then 0w is in fact the angle of non-wetting: then 0w = 0 and 0w = 7r 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 i f - o" < O's - o'i--< O', as only then is I cos 0w] < 1.

34

Nucleation: Basic Theory with Applications

1.0

0.8

!

Y

-

0.6

/

0.4-

0.2-

0

0

J

30

.

60

~

I

!

90

,

I

I

120

[,

i

!

150

180

Ow (deg) Fig. 3.6 Dependence of the factor gt on the wetting angle Owaccording 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 p,, inside an n-sized cap-shaped cluster is again given by the Laplace equation (3.24). Also, substituting 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 Gex(R ) = [2o'(1 - c o s 0w) +

O'i

sin 2 0w] ~R 2

(3.57)

for clusters of condensed phases and analogously to (3.30) that Gex(R) = [2o'(1 - cos 0w) + cri sin 2 0w - (8/3)crllt(Ow)]ZcR 2 + ~(Ow)(4~/3)pR3(1 + 2a/pR) In (1 + 2cr/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 0w = zr, since then the caps are spheres and the last two equations pass into (3.27) and (3.30), respectively. Knowing Gex, Cs and q~,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] W(R) = gt(Ow)[-(4~zAp/3vo)R 3 + 47ro'R2]

(3.59)

or, in terms of the number n of molecules in the cap-shaped cluster, W(n) = - n a p + g//3(Ow)acrn2/3

(3.60)

Work for cluster formation

35

where a = (36roy02)m. 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) = ~(Ow){-(4rc/3)[(Ala/kT) - In (1 + 2r

+ 2a/R)R 3

+ (4~r/3)crR2}

(3.61)

W(n) =-n[Al.t - k T In (Pn/P)] + (1/3)lltl/3(Ow)CCr(kT/pn) 2/3n2/3. (3.62)

In the last equation c = (367r) I/3, and the dependence of Pn on n is given implicitly by the expression Pn = P + 2~[4tc~C(Ow)pJ3kTn] 1/3

(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 O's > ai - o" (then 0w < zr and, hence, ~ < 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 ~0w) 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 0w = zr (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 ~ = 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 = [~0w) + (sin 0w/sin Os)3~(Os)](4/3)zcR 3

(3.64)

where 0w and 0s are defined by [Jarvis et al. 1975; Blander 1979] cos 0w = (o'2 + o- 2 - o'2)/2o'soCOS 0 s = (a2s + a 2 -

a2)/2asai,

and ~(0w) and I//(0s) are given by (3.56). Confining the analysis only to clusters of condensed phases, for ~ and Gex we shall have Gex(R) = q~(g)

= o'2rrR2(1 - c o s 0w) + o'i2zrR2(1 - c o s 0s)(sin 0w/sin 002 and since Cs is again given by (3.54), from (3.47) it follows that (a = (36zrv02 )1/3)

36

Nucleation: Basic Theory with Applications

W(R) = [~0w) + (sin 0w/sin 0s)3~0s)][-(41rAla/3oo)R 3 + 4~ro'R2] (3.65) W(n) = - n A p + [~0w) + (sin 0w/sin

Os)31ll(Os)]l/3ao'n 2/3.

(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 o', as and o'i 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'i- as < o" < o'i + O's. (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 t h e cases of nucleation, e.g., (i) on a foreign substrate which is nearly completely wetted (then 0w~- 0), (ii) on a foreign substrate which is wetted 'better' than completely (then as > tr + cri 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 O's = cr and o'i = 0 so that 0w = 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 do, in the other two cases this model is altogether inapplicable. If R is the disk radius, considering again only clusters of condensed phases, for ~Os,~ and G~x we shall have ~s(R) = o's(As- 7rR2) Gex(R) = ~R) = o'~rR2 + O'i/rR2 + tCn21rR.

(3.67) (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 trn (J m -j) is the specific edge energy of the cluster. Since ten is a 2D analogue of O'n 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 cr's in all preceding formulae, we shall consider to,, 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, trn = constant = 1r ~r the specific edge energy of the planar lateral cluster/old phase interface, i.e. the limiting value of ten for n --~ oo. 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], ~r = trh,

(3.69)

but the actual ~r dependence may be considerably more complicated especially for h ~ do. Since the disk volume is Vn = rchR 2, 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)

= - ( A H / a e f - A(y)/gg 2 + 2mfR

W(n) = - ( A / a - aefAt~)n + btcn 1/2.

(3.70) (3.71)

Here Ao (J m -2) is an effective specific surface energy defined by zXo = ty + o'i- O's,

(3.72)

vo/h is an effective molecular area (aef equals the molecular area a0 when the cluster is of monolayer height h = do), and b = 2(Zraef)1/2. It is important to note that eq. (3.71) remains in force also for prismatically shaped clusters, but then tr is appropriately averaged over the orientations of the prism faces, and b accounts for the shape of the prism base (e.g. b = 4aef 1/2 for clusters with the shape of square prism). Combining eqs (3.55) and (3.72) leads to the formula [Kaischew 1950] aef =

Act = or(1 - cos Ow)

(3.73)

which connects Air and 0w and shows that Act > 0, Act = 0 and Aty < 0 correspond, respectively, to incomplete, complete and 'better'-than-complete wetting. Rather than through 0w, in some cases it is more convenient to express Act through the specific adhesion energy fla (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] Ao" = 2 0 " - / ~ which follows from (3.72) and the Dupr6 relation [Dupr6 1869]

cri= or+ a s - fla. Equations (3.70) and (3.71) reveal that in the case of 2D HEN, depending on the sign of Act, i.e. on the choice of the substrate, the supersaturation A~t 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 Act < 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/~ = 0 and - aefAtr is the driving force of the process), but also at undersaturation, i.e for negative A/~ values in the range of aefAG < A/./< 0. In the Atr > 0 case, i.e. for incomplete wetting, the supersaturation is effectively lowered and 2D nucleation can occur only when ACt > aefAO': then it is the alternative of 3D nucleation, e.g. of caps or lenses. When Atr= 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 = or), there exists no interface between them (o5 = 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

Nucleation: Basic Theory with Applications

of monolayer height h = do simplifies (3.70) and (3.71) to the equations [Brandes 1927]

(3.74)

W(R) =-(JzAt_t/ao)R 2 + 2zrtcR W(n) =-nAp

(3.75)

+ b KTt 1/2

which are the 2D analogues of (3.39) and (3.40). The last formula is valid not only for disks (then b = 2(7ra0)1/2), but also for prismatically shaped clusters with values of b accounting for the cluster shape: for example, b = 4 a 1/2 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/z 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 = 4pe, 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/.t from (2.8), ~ = 0.16 (this 50

W

r162

40 30 20

W" 10=

0 -10

n" 1

10

20

30

40

50

60

70

80

90

100

Fig. 3.7 Dependence o f the work f o r cluster formation on the cluster size in the case o f water droplets in vapours at T = 293 K and P/Pe = 4: curve ' s p h e r e ' - eq. (3.39) f o r HON; curve 'cap' - eq. (3.60) for H E N o f caps on a substrate with Ow = ~z/3; curve 'disk' - eq. (3.71) f o r H E N of disks on a substrate with Aty = (1/2)ty.

Work for cluster formation

39

means 0w = to/3 in eq. (3.56)), b = 2(/r, a 0 ) 1/2 , a e f = a0 = (zr/4)d2,tr = ado and zScr = cr/2 (which corresponds also to 0w = zr/3 in eq. (3.73)). The values of do, v0, crandpe 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 Gex(n) and q~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 G e x is again given by eq. (3.15) so that, in view of (3.33),

Gex(n) =

n ~ - E n.

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 U1 ~ Uold and E1 = UoldU1 ~ 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 q~s(n), analogously to (3.54), it can be represented as ~0s(n) = as(A s - An)

(3.76)

where A n = aonc(n) 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 Cs,0 from (3.51), eq. (3.47) yields the general atomistic formula for W(n) in HEN of condensed phases (n = 1, 2 . . . . ) W(n) = - ( A p - ~)n - En - O'sAn.

(3.77)

This formula is valid for any size and shape of the cluster on the substrate. In the absence of substrate, as = 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. nc = n), A n = aon and (3.77) takes the form (n = 1, 2 . . . . ) W(n) = - ( A p - ~ + a0Crs)n - E n.

(3.78)

In the particular case of 2D nucleation on own substrate, Crs = cr so that this equation becomes (n = 1, 2 . . . . ) W(n) = - ( A p - & + aocr)n- En.

(3.79)

With properly defined supersaturation A~ 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 E, 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) En 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 Gs term which ensures their passing into the classical formulae for W(n) when n ~ oo. 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: En = ( ~ - a o G - aoGi)n - b ~r 1/2.

(3.80)

In particular, for 2D clusters on own substrate Gi = 0 and (3.80) reduces to the equation En = (~ - aoG)n - b lfn 1/2

(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(G + o'i)n + b tfn 1/2 and aocrn + b ten 1/2 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-independent 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, thermodynamically, the work for cluster formation can be represented in terms of n, the cluster volume V~, the pressure p, in the cluster and the chemical potential flnew,n of the molecules in the cluster. The corresponding general formula for EDS-defined clusters reads W(n) = - ( P n - p ) V , + (Pnew,,,- Pold)n + ~s(Vn)

(3.82)

where ~s(V~) is an effective surface energy given by t~s(Vn) = r

+ 0s(Vn) - ~s,0,

(3.83)

/~new,n is determined by (3.11) with the aid of the equation of state of the cluster, and Pn and ~s are related by the Laplace-type equation

Workfor cluster formation

Pn = P + dCI)s/dVn.

41

(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 V, for which ~s, P~ and Pnew,~ 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 n - p)Vn + (,t/new,n-//old) n + (/-/s-//old)ns + qbs(Vn)

(3.85)

where ns is the Gibbs excess in the number of molecules (formally, the number of 'surface' molecules of the cluster), and ps is their chemical potential. Depending on the choice of the dividing surface, n~ can be positive, zero or negative [Ono and Kondo 1960]. 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 q~s, Pn and//new, n 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), W can be represented most generally as (n = 1, 2. . . . ) W(n) = - n A p + CP(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 energy q~(n) of the cluster. When the EDS is chosen as a dividing surface, the general expression for q~(n) is q~(n) = Gex(n) + ~Os(n)- ~Os,0

(3.87)

with Gex(n) given by (3.9) or (3.12). The concrete forms of q~(n) 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 ~(n) should include the 'surface' term (Ps-//old)ns Thus, most generally, ~(n) can be written down as 9

9 (n) = (,Us-/./old)ns + Gex(n) + ~ s ( n ) - ~s,0

(3.88)

where now Gex, ~s and ~s,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 ~(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 Effective excess energy ~(n) of EDS-defined clusters according to the classical theory of nucleation. Table 3.3

Kind of nucleation

New phase

~(n)

HON HON (spheres) 3D HEN (caps)

condensed gaseous a condensed

c G o 2/3 n 2/3

3D HEN (caps) 3D HEN (lenses)

gaseous a condensed

2D HEN

condensed

apn is

nkT In (Pn/P) + (4~r/3)I/3G(kT/pn) 2/3n2/3 i/tl/3(0w)(36tc) 1/3 GU~O3n 2/3 nkT In (Pn/P) + lltl/3(Ow)(4zc/3)l/3G(kT/pn) 2/3n2/3

[~0w) + (sin 0w/sin 0s)31/J(0s)] 1/3(36/r) 1/3 GV~/3n2/3 naefAG + b xnl/2

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), ~(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(n) in which ~(n) is given by ( n = 1,2 . . . . ) (3.89)

9 (n) = &n - E ~ - GsAn

in the cases of both HON (with O's = 0) and HEN (with Gs > 0) of condensed phases. For gas-phase clusters ~(n) is unknown, but can be represented formally as (n = 1, 2 . . . . ) (I)(/'/) = QJnew,n - / / n e w ) n -- (Pn --p)Vn + r

- GsA,

(3.90)

where again as = 0 and as > 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/1new,n, Pn and q~(n) 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 (1D) 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 A e of each of its two end surfaces and a length L = (oo/Ae)n. Let Ls and ~r (J m-l) 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 Cs,0 and Cs(n) of the substrate before and after the cluster formation we can write ~0s,0= crsAs + tCsLs ~s(n) = CrsAs + tCs(Ls- L).

(3.91) (3.92)

Fig. 3.8

Top view of cylindrically or prismatically shaped cluster of length L on a line (shown dark grey) on the surface of a substrate.

Let us now denote by tr (J m - 1) and tq,,, (J m - 1), respectively, the energy per unit length of the cluster/old phase interface and of the cluster/substrate contact line and let e, (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 ~r = constant = to, tq,,, = constant = lq and e~ = constant = e. Then, as we are considering condensed-phase clusters for which eq. (3.15) is in force, for the cluster excess energy Gex(n) and total surface energy ~ n ) we shall have Gex(n) = ~(n) = (K" + K'i)L + 2e.

(3.93)

Thus, from eqs (3.87), (3.91)-(3.93) it follows that the cluster effective excess energy is of the form 9 (n) = ndefAK" + 2e where

def= oo/m e is an effective

(3.94)

molecular diameter, and the combination Ate

44

Nucleation: Basic Theory with Applications

= tr + ~r ~r of specific line energies is analogous to Act from (3.72). For instance, Ate = 0 for 1D clusters of monomolecular thickness when they are formed at the edge of a monomolecular step on own crystalline substrate (then tCs = tr and ~q = 0). Substituting ~ ( n ) from (3.94) in (3.86), we finally find that, classically, the work for formation of 1D condensed-phase clusters is given by

W(n) = - ( A p - defAtc)n + 2e.

(3.95)

This formula shows that W(n) depends linearly on the cluster size (in the particular case of A/.t = defAtr it is even n-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 1D nucleation. Indeed, from (3.95) we see that when Ap > defA~r W(n) only decreases with increasing n, which means that work is gained when a molecule is added to the 1D cluster even if this cluster is with the smallest size ofn = 1. Comparison of eq. (3.95) with eqs (3.60) and (3.71) reveals that the reason for the absence of 1D nucleation of condensed phases is that the cluster end energy 2e, which corresponds to the effective surface energy Iltl/3(Ow)atyn2/3 of a 3D cluster and to the peripheral energy br.n 1/2 of a 2D cluster, is independent of the cluster size n. Hence, the conclusion: thermodynamically, in the scope of the classical approximations e, = constant = e and A~c, = constant = Ate we cannot speak of 1D nucleation of condensed phases. Only when e, and/or Ate, = to, + ~r - ~r are functions of n could 1D nucleus be defined and 1D nucleation be thermodynamically m e a n i n g f u l the condition for that to be the case is the dependence of e, and/or Atr on n to make W(n) from (3.95) display the maximum needed for the determination of the 1D nucleus size n* and the 1D 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 < 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 supernuclei (the clusters of size n > 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 supernucleus, 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 (dW/dn)n-n, = 0,

(4.1)

and then of W* from the definition equality (4.2)

W* - W(n*).

In some cases, for the determination of n* and W* it is mathematically more convenient to employ the formulae p* = p + (dOs/dVn)*

(4.3)

P* = Pold

(4.4)

W* = - ( p * - p ) V *

+ ~*

(4.5)

rather than eqs (4.1) and (4.2). Here, for brevity, p* - p,,,, I.t* =-/~new,n*,

46

Nucleation: Basic Theory. with Applications

V* - Vn,, (dq~s/dVn)* - (dclg~/dVn)v,,=v,, and ~s* - ~s(V*) is the nucleus effective surface energy. These formulae follow upon using W from the general formula (3.82) first in the extremum conditions (,9W/~gVn)v,,-v* = 0 and (c)W/On)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 * A l a + O*

(4.6)

which results from using W(n) from (3.86) in (4.2) and in which t/r* - rib(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] n* = 8c 3 v02o3/27A/~ 3

(4.7)

W* = 4c 3 002o'3/27A~ 2 .

(4.8)

In the particular case of spherical nuclei c = (36z01/3 and, since according to (3.13) and (3.22) n* = (4td3vo)R .3, for the nucleus radius R* and for n* and W* it follows from (4.7) and (4.8) that [Zettlemoyer 1969] R* = 2v0G/Ap n* = 327rv g o'3/3A,u 3

(4.9) (4.10)

Nucleus size and nucleation work

W* = 16~v 20"3/3A/.t2.

47

(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* = (l/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 ~0from (3.20) W* from (4.8) takes the simple general form [Gibbs 1928] W* = (1/3)r

(4.12)

where ~* - r = atTn .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 W* on Ap for HON of water droplets in vapours at T = 293 K, resulting from eqs (4.9)-(4.11) with the values of v0 and cr listed in Table 3.1. As seen, both the nucleus size and the nucleation work diminish with increasing Ap. Since detectable HON of water occurs typically when A/~ > 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/~ --~ oo, which is inconsistent with the thermodynamic requirement W*(A~s) = 0

(4.13)

for annulment of the nucleation work at the supersaturation Aps corresponding to the spinodal (Abts = Pold.s- ~new where jUold,sis 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 Ap-pe-p

>O

(the difference between the equilibrium and the actual pressures of the old phase) than of the supersaturation A~t from eq. (2.10). The formulae for n* 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 et al. 1963] P*

= Pe

exp (-voAp/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 + 2ctr/3V *1/3.

(4.15)

48

Nucleation." Basic Theory with Applications

1.0

(a)

E

c: 0.5 .g

0 100

(b)

50 --.

.Ic

,-q..

,

I

,

_

-.1....! ,

.

.

--.j I

,

,

-"

-!

(c)

60

40 .g

~:

20 0

1

1

i

,

,

i

I

2

,

,

,

,

I

.

.

3

.

.

I

4

|

|

5

A~ / kT Fig. 4.1 Supersaturation 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 eqs (4.9)-(4.11). In the range of higher supersaturations 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 n* = 8c3p*ty3/27kT(p * - p)3

(4.16)

where the p dependence of p* is given by (4.14). Recalling that in HON ~s = @, 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* = 4 c 3 O'3/27(p * - p)2,

(4.17)

p* being a function of p according to (4.14). Since in many cases pevo/kT << 1, it is worth noting that for pressures in the range of 0 < p < Pe, to a good approximation, p* = Pe. Hence, for p values in this range, n* and W* from the last two equations are given approximately by n* = 8C3peCr3/27kTAp 3

(4.18)

W* = 4c 3O'3/27Ap2.

(4.19)

In the particular case of spherical gas-phase nuclei c = (36n:) 1/3 and as from (3.14) and (3.22) n* = (4zU3)(p*/kT)R .3, for the nucleus radius R* and for n* and IV* 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* = 20"/(p* - p)

(4.20)

n* = 32/rp* o'3/3kT(p * - p)3

(4.21)

W* = 16zro'3/3(p * - p)2.

(4.22)

Here p* is a function of p (see eq. (4.14)) and can be approximated by Pe when pevolkT << 1 and 0 < p < Pe. 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* = (1/2)n*kT(1 e -zxn/kr) [Volmer 1939]. This is the analogue of the relationship W* = (1/2)n*A~ for condensed-phase nuclei and shows that, as expected in Chapter 3, W* --- n*A/j provided A/~ does not exceed too much kT. We note that when A/~ >> kT, W* = (1/2)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* = Pe and the cr and Pe 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 ~ o,,. This violates the thermodynamic requirement W*(Aps) = 0

(4.23)

which is analogous to (4.13) (Aps - Pe -Ps, Ps being the pressure of the old phase at the spinodal point, e.g. point e in Fig. 1.1a).

50 Nucleation: Basic Theory with Applications

'"

(a)

E

c.. V

.1(

n"

(b)

400

3OO E 200 100

(c)

80

I-"

60 40

N

2o 0 0.5

0.6

0.7

0.8

0.9

1.0

/ 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 n* 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]

Nucleus size and nucleation work 51 n* = Ilt(Ow)32zcv2 03/3Ap 3

(4.24)

W* = ~t(0w)16zrv~ o3/3Ap 2.

(4.25)

Comparison of these equations with (4.10) and (4.11) shows that in HEN n* and W* are diminished 1/~ times 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 0w), the smaller the nucleus on the substrate and, hence, the less the work for its formation. For example, for a hemispherical nucleus (0w = zr/2) n* and W* are twice smaller, as then ~ = 1/2 (see Fig. 3.6). In the limiting case of complete non-wetting (0w = zc so that ~ = 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 ~) and is again given by the Gibbs-Thomson equation (4.9). The relationship W* = (1/2)n*Ap 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* = Pe 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 ~, q~s,q~s,0and V~ are given by (3.51)-(3.54), from eq. (4.3) we arrive again at the Laplace equation (cf. eq. (3.24)) p* = p + 2or~R*.

(4.26)

Recalling that n* = (4zc/3)(p*/kT)R .3, 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* = 2cr/(p*- p)

(4.27)

n* = ~(0w)32zp* o'3/3kT(p * - p)3

(4.28)

in which p* depends on p according to eq. (4.14) and can be approximated by Pe if pevo]kT < < 1 and 0 < p < Pe. These two expressions are the GibbsThomson 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 r ~s, ~s,0 and Vn 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* = ~t(0w)16n:cr3/3(p * - p)2

(4.29)

where p* can be approximated by Pe when pevo]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 1/~-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 0w = ~r 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 gt = 1. In this case the substrate has no effect on n* and W*. We note, too, that the relationship W* = (1/2)n*kT (1 - e -a~'/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* = [gt(0w) + (sin 0w/sin Os)3gt(Os)]32zcv~o3/3A!a3

(4.30)

W* = [ ~ 0w) + (sin 0w/sin 0s) 3~ 0s)] 16zrvg o'3/3Ap 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 ~ 0w) + (sin 0w/sin 0s)3~0s) which depends on the two angles 0w and 0s. For that reason, the formula W* = (1/2)n*Ap which connects W*, n* and Ap 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/.t > aefA~ ) R* = K'aef/(A// - aefAO" ) n* = b2x2/4(A/~ - aefAO') 2 W* = b2K2/4(A/I - aefAt7 )

(4.32) (4.33)

for 2D nuclei on foreign substrate (b = 2(/r.aef) 1/2) and by (A/~ > 0) R * = tca0/A~u

(4.34)

n* = b2x2/4Ap 2

(4.35)

W* = b2x2/4A/~

(4.36)

for 2D nuclei on own substrate (b = 2(zra0)ln). 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 W* = n*Ap which coincides with the estimate in Chapter 3. The known 2D analogue of eq. (4.12), W* = (1/2)21rR*tr [Brandes 1927], is also evident from these formulae (2tcR*tr 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 = r4aef - ~/2 in (4.32) and (4.33) and

Nucleus size and nucleation work

53

b = 4a~/2in (4.35) and (4.36). The equations for R* and n* are the GibbsThomson 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 cr (provided tr is proportional to tr) and are less sensitive to changes of Ap. The latter distinction is illustrated in Figs 4.3a and 4.3b in which the n*(ACt) and W*(Ap) dependences

100 (a)

80 3D

60 t-

\ 2D

40

20 (b)

40 I.-

3D

2D

20

.,1r

(c) "O

"ID

t

1

.1r

t,'-"

0

0.5

1.0

1.5

2.0

,a t / kT Fig. 4.3 Supersaturation dependence o f (a) the nucleus size, (b) the nucleation work, and (c) the height o f cap-shaped (curves 3D) and disk-shaped (curves 2D) water nuclei in H E N on a substrate in vapours at T = 293 K: curves 3D - eqs (4.24) and (4.25) at Ow = re~3; curves 2 D - eqs (4.32) and (4.33) at A t r = (1/2)tr; solid-dashed curve - eq. (4.37) at Ow = to~3; circle - point o f transition f r o m 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 ~r o'd0, aef = a 0 = oo]do, I//= 0.16 and Act = 0"/2 (these tV and Ao" values correspond to 0w = rd3 in eqs (3.56) and (3.73)). The other parameter values used are given in Table 3.1. The choice of aef = a 0 in eqs (4.32) and (4.33) means that curves 2D describe heterogeneously formed disk-shaped nuclei of monomolecular height h* =do. 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 h* = (1 - cos Ow)[3oon*/4zclil(Ow)] 1/3

(4.37)

which follows from elementary geometry and in which 1 - c o s 0w and n* are specified by eqs (3.73) and (4.24). As seen from Fig. 4.3c, at a certain supersaturation Apt, given by Apt = 2(o0/d0)(1 - c o s 0w)f = 2(o0/d0)Af, the height h* of the 3D nucleus is equal to the molecular diameter do. It turns out [Toschev et al. 1968; Markov and Kaischew 1976; Kashchiev 1977] that for substrates with A f > 0 this result is of general validity, and Apt is the transition supersaturation at which a change in the mode of HEN occurs: for A/./ < A~/t nucleation is 3D, and for Ap > A]./t it is 2D. This change occurs because while for Ap < Apt the nucleus is of height h* > do which decreases with Ap, at Ap = Apt it is already with the monomolecular height h* = do. With a further increase of Ap 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 Ap > A/./t 2D nucleation takes place. The above formula for Apt is easy to obtain from the condition h*(Apt ) = d o with the help of eq. (4.37). In Fig. 4.3c we read that at the chosen T = 293 K and 0w = to/3 the transition from 3D to 2D HEN of the water droplets on the substrate occurs at Al.tt[kT = 1.4 which in view of (2.8) corresponds to p = 4pe. Inspection of the classical expressions for 3D nucleation shows that regardless of whether nucleation is homogeneous or heterogeneous, W(n), n* and W* can be represented by unified formulae if cr is replaced by an effective specific surface energy fef. The formulae for the nucleus size and the nucleation work then read:

n* = 8c302 f3ef/27 A~ 3

(4.38)

3/27 A/I 2 W * = 8 c 3 u 2 f ef

(4.39)

for nuclei of condensed phases and

3/27kT(p* - p)3 n* = 8c3p * f ef W* = 4c 3 fief3/27(p*

p)2

(4.40) (4.41)

for nuclei of gas phases (c = (36zr) 1/3 for spheres, c = 6 for cubes, etc.). The

Nucleus size and nucleation work

55

quantity crefaccounts 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 Gee is O'ef- ~0"

(4.42)

where the activity factor q~ is a number between 0 and 1. For example, for cap-shaped nuclei 7~= ~1/3(0w), and for lens-shaped ones ~ = [~0w) + (sin 0w/sin 0s)31ff(0s)] 1/3. Obviously, the ~ = 1 limit corresponds to HON, for then O'ef "- O'. The various cases of HEN are described by ~ < 1 and for them Gel < G. Since !g and O'efhave smaller values when the substrate is more active (in the sense that n* and W* are smaller), the activity factor ~or, equivalently, the effective specific surface energy O'ef 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 A/.t 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. 4.1 b illustrates one important general feature of the dependence of the nucleus size n* on the supersaturation A~t in the highsupersaturation range. It is seen that there exists a succession of Ap 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 Ap intervals the classical Gibbs-Thomson dependence of n* on Ap is replaced by n*(A~) = constant.

(4.43)

Owing to this result, it has been concluded [Stoyanov 1979; Milchev 1991] that for atomistically small crystal nuclei in HEN from vapours and electrolytic solutions W* is a broken linear function of Ap. It is clear, however, that this result is of more general validity. Indeed, from eq. (4.6) we find that W* depends linearly on Ap according to the general formula W* = - n * A p + q~*

(4.44)

provided that eq. (4.43) is satisfied and that q~* - q~(n*) is A~-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 p

+ (2n* - E*)

(4.45)

in the case of HON and to W* = - n * A l u + (2n* - E* - asA*)

(4.46)

in the case of HEN (for brevity, E* - E,, and A * - An*). In the particular case of 2D nuclei of monomolecular height A* = aon* and from eq. (4.46) (or from (3.78), (3.79) and (4.2)) it follows that W* = - n * A p

+ (2n*-E*-aoCrsn*)

(4.47)

for 2D nucleation on foreign substrate and that W* = - n * A p

+ ( 2 n * - E* - a o g n * )

(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 q~* can be represented most generally by the formula qr* = 2n* - E * - crsA*

(4.49)

in which O's = 0 and Ors > 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/j. This is so because for such nuclei q)* is a function of Ap: according to (2.1), (3.90) and (4.4) we have q~* = n * A p - (p* - p ) V *

+ d?* - asA*.

(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 *

- CrsA*)

(4.51)

where again Ors= 0 and Ors> 0 correspond to HON and HEN, respectively. As seen from this expression, if in a certain pressure range V* and the bracketed term are p-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 A~ (see Chapter 2), eqs (4.44)-(4.48) are the atomistic formulae for the A/~ dependence of W* for HON and HEN of condensed phases in vapours, solutions, melts, etc. In these equations n* and qr* (respectively, the bracketed terms which are explicit expressions for q~* 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 n*, 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 & , 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*(Ap) dependence satisfying the general thermodynamic requirement,

Nucleus size and nucleation work

57

eq. ( 4 . 1 3 ) , f o r a n n u l m e n t o f W* at t h e s u p e r s a t u r a t i o n Aps c o r r e s p o n d i n g

to

the spinodal.

Table 4.1

Work W* f o r atomistic H O N a n d H E N in various cases.

Nucleation

W*

W*

in the case of

(HON)

(HEN)

vapour condensation

- n * k T In

boiling solute condensation melt crystallization a melting a electrocrystallization

(PiPe)

-n*Ase(T-

- n * k T In (I/le) + 2n* - E* - o'sA* V*p + qb* - p ' V *

- CrsA*

- n * k T In (C/Ce) + 2n* - E* - crsA* n * A s e T - E* - ~s A *

2T e) - E*

-

-n*Ase(T-

2Te) - E* - O'sA*

-n*zieo(q~e- r

awhen Ap is given by eq. (2.23) in which

Table 4.2

- n * k T In (P/Pe) + 2n* - E* - crsA*

+ Xn* - E* V*p + qb* - p * V* - n * k T In (C/Ce) + A,n*-E* n * A s e T - E*

A S e --"

+ 2n* - E* - O'sA*

~,/Te

Work W* f o r atomistic 2 D H E N o f m o n o l a y e r nuclei on

f o r e i g n (as ~ ix) or own (ors = ix) substrate in various cases.

Nucleation in the case of

W*

vapour condensation

- n * k T In (P/Pe) + (~ - a0O's) n* - E*

evaporation or sublimation

- n * k T In (Pe/P) + (/1, - a0O's) n* - E*

- n * k T In (I/le) + (/1,- a0O's) n* - E* - n * k T In (le/l) + ( ~ -

solute condensation dissolution melt crystallization a

a0Crs) n* - E*

- n * k T In (C/Ce) + ( / ~ - a0O's) n* - E* - n * k T In (Ce/C) + (~ - a0Crs) n* - E* n*Z~seT- a0O'sn* - E*

melting a electrocrystallization

-n*AseT

electrodissolution

-n*zieo((p--Cpe) + ( i ~ - a0Crs) n* - E*

+ (2A,- a0Crs) n* - E*

-n*zieo(CPe - r

+ ( ~ - a0O's) n* - E*

awhen Ap is given by eq. (2.23) in which ZkSe= 2/Te

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* = ( 1 / 2 ) n * A p

(5.1)

W* = n * A p

(5.2)

which relate W*, n* and Ap 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 Ap. 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*/dAp = -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 Ap, which are invariants with respect to the choice of the dividing surface, with a quantity which, in contrast to n*, 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 [1993] 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 et al. [ 1980], Kostrovskii et al. [ 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 = - n A p + ~. 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: I f the work f o r cluster formation is expressed in the f o r m of eq. (3.86) and if W* and ~ are differentiable functions o f Ap, the relation dW*ldAp = - n * +

aa,,/aA ,

(5.4)

holds true, tp* - ~[n*(Ap), Ap] being the effective excess energy o f the nucleus.

To prove the nucleation theorem in the above most general form, following the original analysis [Kashchiev 1982], we find first the Ap derivative of W* from eq. (4.6): d W* /dAp = - n * - (Ap - act~*l On* )( dn */dAp ) + o c , I OA , .

(5.5)

Then, considering n as a continuous variable, from (3.86) and the condition for extremum, eq. (4.1), we find that Ap - (a~/an)n=n,

= O.

(5.6)

This equation is in fact a most general, phenomenological form of the GibbsThomson equation for one-component nucleation. Since Ot/r*/0n* = (O~/ an)n=n,, owing to (5.6) the dn*/dAp 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 qr*, 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* + Oqr*/ OAp in eq. (5.4) does not. The nucleation theorem thus relates two invariants with respect to the choice of the dividing surface, W* a n d - n * + Oq~*/OAp, 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 ~* defined by the very general, but already concrete expression qr* = n'A~~ - (p* - p ) V *

+ dp* + dp* - ~Os,0

(5.7)

where ~* = ~[n*(A//), A//] and ~* - ~s[n*(A//), 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 n s term in (3.88) owing to the fact that for the nucleus//s =//old [Toschev 1973a]. Naturally, t/r* from (5.7) is obtainable also by setting equal the fight-hand sides of (4.5) and (4.6). Using qr* from this equation allows rationalizing the 0q~/0A/~ 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 t/r* from (5.7) it becomes possible to answer the question: why does OtIr*/OAp 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 34r*/OAp with the aid of t/r* from (5.7). It is more convenient, however, to calculate first the derivative cgt/~*/t?//old, since we can profit from the three 'bulk' and three 'surface' Gibbs-Duhem relations at constant T [Guggenheim 1957]

dp= Pnewd//new

(5.8) (5.9)

d p * - - Pnewd~old

(5.10)

d p = Poldd//old

dOs,O= -nadsd//old

(5.11)

d~s* = - pa'~sd//ol d

(5.12)

d~* = -ns*d//ol d.

(5.13)

Here Pold (m-3), Pnew (m-3) and Pnew * (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 hads 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//new,n* of the molecules in the nucleus is replaced by Pold, since according to (4.4),//new, n* =//old" When the EDS is chosen as a dividing surface, by definition, ns = 0,

(5.14)

and when there is no adsorption of molecules of the old phase on the substrate, nads nads = 0

(5.15)

SO that then none of the total surface energies ~*, ~* and Cs,0 depends on Pol0" In the general case n*s, nads* and nads can also be positive or negative

Nucleation 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 ~*/0].~ol

d

= n* dA~t/d~told - V*( Pnew * - Pold) - ns* - had * s + hads. (5.16)

As by definition Pnew = n / V *

(5.17)

and since because of (2.1), (5.8) and (5.9) dA/~/dHold

=

1 - Pold/Pnew ,

(5.18)

from (5.16) we finally get 8**/

zxst=

n*(po,d/Pn*ew)(1 -

( r t s * + h a a9s

--

P*w/Pnow)/(1 -- Po, /Pnew)

- naris)/(1 -- Pold/Pnew) .

(5.19)

This expression shows that when t/r* is defined by (5.7), the role of the 3q~*/cgAp 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* s u m m a n d - when the nucleus and the bulk new phase have the same density (then Pnew * = Pnew), the second s u m m a n d - 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 cgqr*/cgAp = 0

(5.20)

is a good approximation for EDS-defined nuclei of condensed phases (then, typically, /)new * --- Pnew) in the cases of both HON and HEN if in the latter . case the substrate is so weakly adsorbing that hads -- nad s -=- 0. We thus see that the reason for the absence of the 3q~*/3Ap 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 t/y* 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 ~qr*/3Ap from (5.19) in (5.4), we can now find what is the form of the nucleation theorem in the particular case of qr* specified by (5.7). The result is dW*/dAp

where An* is given by

= - An*/(1 - Pold/Pnew)

(5.21)

62

Nucleation: Basic Theory with Applications

An* = n*(1 - Pold/Pn*ew) + n*s + nads- nads-

(5.22)

Recalling (5.18), we see that eq. (5.21) can be represented equivalently and even more simply as dW*/dlUol d = -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 ~ * 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 et al. [ 1993] and derived thermodynamically (see below) by Oxtoby and Kashchiev [ 1994] in the case . of HON for which hads = hads = 0. Here, we have generalized eq. (5.23) by . proving that it holds true also for HEN, i.e. when hads ~nad s :r 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*, Ap, Pold, 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 nads -- 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 Pl 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* = Pn*ewV* is the area ADEG, n*Pold/Pn*ew = PoldV* is the area ABFG, n* > 0 is the area DEF, nad * s = 0 has no area, and had s is the area BCF. In Fig. 5.1 the arbitrarily chosen dividing surface is positioned sufficiently away from the EDS (for this reason n* < 0) in order to emphasize that An* and all other quantities characterizing the nucleus (e.g. Pn*w, 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*(1 - Polu/Pn*w) + naus - nads

(5.24)

in the case of HEN and An* = n*(1 - Pold Ipn*w )

(5.25)

Nucleation theorem

63

Fig. 5.1 Spatial variation of the molecular density before (solid curve Pl) and after (solid curve p*) 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 had s* - hads ----0) and in HON An* is given by the same formula, eq. (5.25). In both these cases An* becomes merely equal to n*, i.e. An* = n*,

(5.26)

if the EDS-defined nucleus is much denser than the old phase (then Pnew * >> Pold)" 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 Pn~w* ~" Pnew, (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 d W * / d A p = -n*(1 - Pold/Pn*w)/(1 -- Pold/Pnew)

dW*/dPold = -n*(1 - Pold/Pn*w)

(5.27) (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 n e*w = P n e w S O that for EDS-defined nuclei of such phases eq. (5.27) leads to the nucleation theorem in its simplest form [Kashchiev 1982]: dW*/dAp = -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 t/r* 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 Pold: dW*/dPold = V*d(p- p*)/dPold + ( p - p*) dV*/dPold

+ (~9CPs*/~OV*)(dV*/d,Uoid) + ~q~s*/3Pold.

(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*/dpold = - n*(1 - Pold/Pn*ew)+ ~ s /~l.told

(5.31)

and, accounting additionally for (5.18), into

dW*/dAp = - n * ( 1 - Pold/Pn*ew)/(1 -- Pold/Pnew) + O~s*/c)Ap. (5.32) The last step is to find the derivatives of the nucleus effective surface energy CPs* which, according to (3.83), is given by ~ * = 0* + r

~s,0.

(5.33)

This is easily done with the help of the 'surface' Gibbs-Duhem equations (5.11)-(5.13):

O~l~)s~/~jIAold = -ns* - -

(5.34)

*

/'tad s + /'tad s

d s*/dAu "- (-ns* - - H a*d s +

naris)/(1

--

Pold/Pnew).

(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 q~s*. 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 q0s* 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 and Viisanen [ 1993], Viisanen et al. [ 1994] and Oxtoby and Kashchiev [ 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 ACt can be given different values by choosing different temperature T of the system. Despite the change of Ap by means of T, however, the nucleation process itself occurs at constant temperature - the same which is chosen to determine the Ap 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 fight-hand sides of the Gibbs-Duhem equations (5.8)-(5.13) [Guggenheim 1957], we shall find that (5.36)

d W* /dP old = - A n * - AS* ( d T/dP old). Here An* is given again by (5.22), and AS* is defined as * n * (1 - s oldPold[S new * Pnew * ) + A S* = Snew

s:n:

+ Sadsnad * * s-

Sadsnad s

(5.37)

where Snew and Sold are, respectively, the entropies per molecule in the nucleus and in the old phase, s* is the surface excess entropy (per molecule) , of the nucleus, and Sads and Sad s a r e 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 onecomponent 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 ~* = Ss,0 = 0 and which is thus valid only for HON. In the experimentally interesting case of changing Hold by varying T at

66

Nucleation: Basic Theory with Applications

constant pressure p of the old phase d]./ol d = -SolddT (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 d W * / d l a o l d = -n*(1 -- S n*e w ] S o l d ) .

(5.38)

This can be represented equivalently as d W * /dAl, t = - n * (1 - Snew/Sold)/(1 * - Snew]Sold),

(5.39)

since according to (2.1) and (2.19) dPnew = -SnewdT and dA/.t/d/~old = 1-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/~old and A/~ are changed by changing p at constant T is played by the entropy factors when the/~old and Ap 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 EDSdefined nucleus and in the bulk new phase are nearly the same (i.e. when * w = S n e w ) , eq. (5.39) leads again to the nucleation theorem in its simplest Sne form [Kashchiev 1982] dW*/dA/l = -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 m o d u s o p e r a n d i 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: c)W*[Olaold, i = - A n * - A S * ( d T / d l a o l d , i ) .

(5.41)

Here, analogously to (5.22) and (5.37), A n * - n* (1 AS*

* - Pold,i]Pnew,i) 4"

n*. s,, 4" n*a d s , / -

nads,i

(5.42)

* * (1 - Sold,/Pold,i/S*new,/Pnew,i) * = Snew,/rti

+ s*. * n ads, * i -- S ads,i n ads, i, s,, n*. s,t 4" S ads,i

(5.43)

the subscript i = 1, 2 , . . . , k referring the respective quantity to component i in the k-component system. It should be noted that AS* is/-independent and can be calculated if the quantities on the fight 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 nads, i = r/ads, i = 0. When nucleation is isothermal, d T = 0 and eq. (5.41) reduces to t~W* / OI.lold, i "- - A n *

(5.44)

with Ani* 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*. S,/ = 0 for, say, component i = 1, still in general n*~,i ~ 0 for the other components (i ~ 1) in the system. For example, for binary HON of condensed phases in vapours * 9 or solutions (then n a d s , i = nads, i = 0 and .Pold& << Pn~w,~), in accordance with (5.42) such a choice gives An1 = ni~ for c o m p o n e n t 1, but An~ = n~ + ns,*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 Ap or ]'/old" 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/t/ol d [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/2ol d

(5.45)

[An*+As*(dT/d/,told)]dpold

(5.46)

W*(]./old) -- W*'-I]2old

for isothermal nucleation and that W * ( l . / o l d ) - " W * ' -I-

f

~ld ~old

for non-isothermal nucleation. Here W*' - W* (~old) is the value of W* at a chosen reference value ]-/old 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] W*(Ap) = W*' +

n* dAp /.t

where Ap" is a reference supersaturation, and W*' = W*(Ap').

(5.47)

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, modelindependent 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*(,Uold) and AS*(,Uold) 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 oldphase chemical potential Hold,s or supersaturation AMs which correspond to the spinodal point (see eq. (4.13)), if we choose }s - - / ' / o l d , s or AM' = AMs, 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: W*(Mold) =

~

lloid,s

An*d/told

(5.48)

n'dAM.

(5.49)

~old

W*(AM) =

l

APs

dA/l

In the absence of spinodal, Mold,s = AVs = oo and these formulae become W*(,Uold) = f ~ An*d/tol d "llold

(5.50)

W*(AM) = f~~ n'dAM.

(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*(AM) dependence for all supersaturations from 0 to A/~s. Using (4.38) to perform the integration in (5.49) leads easily to 3/27 Al.t2 . W*( AIA ) = 4c30gty3f/27 Aj22 - 4C3O~O'ef

(5.52)

Comparing this equation with eq. (4.39) of the classical theory, we see

Nucleation theorem

69

that although both equations predict a linear dependence of W* on 1/Ap 2, their right-hand sides differ by an additive constant which equals zero in the absence of spinodal limit, as then Aps = oo. This constant depends only on T and is particularly important for Ap approaching A/~s, since by making W* vanish at A/~ = A/~s rather than at A/~ -+ o% 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 GibbsThomson equation (4.38) can be used in the whole supersaturation range from 0 to A/~s. It is worth noting that the thermodynamically consistent W*(A/~) dependence (5.52) is in qualitative agreement with the numerical W*(A/~) 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/I s term in (5.52) corresponds to the D(T) term in the W*(A/~) 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 nsized 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, n of the molecules in the cluster are not equal to the outside pressure p and the chemical potential ~new o f 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 ~ n ) in the energy balance. Let us now consider the effect of the cluster size on Pn, /ffnew,n and some other thermodynamic characteristics of homogeneously formed clusters. Unless especially noted, the considerations will be confined to onecomponent 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 ofp~ on n we need a model presentation of q~. 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 phasesPn =

P + 2c,,~,,/3 V~/3 + (1/3)d(cnG,,)/d V~/3 .

(6.1)

For clusters of condensed phases V~ and n are related in a simple way by eq. (3.13) and this allows expressing p~ from (6.1) as an explicit function of n" Pn = P + 2CnGn]3 0113nl/3 + (113 V~/3)d(cnO'n)/dn 1/3.

(6.2)

The representation of p, 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 c, = (36z01/3 = constant and V~ is conveniently expressed by eq. (3.22) as a function of the cluster radius R. Equation (6.1) then becomes p,, = p + 2Gn/R + dGn/dR

(6.3)

Properties of clusters 71 which is the known formula [Ono and Kondo 1960; Toschev 1973a; Abraham 1974a] for pn of EDS-defined spherical clusters. In the classical nucleation theory cnand (r~ are regarded as n-independent quantities whose values c and (r characterize, respectively, the specified cluster shape and the specific surface energy of the planar interface (G is independent of the choice of the dividing surface and is the limiting value of (r, for n --4 oo). In this approximation eqs (6.1) and (6.2) take the simple form Pn =

P + 2cG/3 V2/3

(6.4)

for clusters of both condensed and gas phases and Pn = P + 2cG/3 U~/3n1/3

(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 Pn >-P and that Pn ~ P when n (and, hence, V n o r R) tends to infinity. This is seen also in Fig. 6.1 a which depicts the dependence of Pn on n from eq. (6.4) (with Vn 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 = (36z01/3 (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 EDSdefined clusters of such small size is questionable.

6.2 Chemical potential The chemical potential/lnew, n of the molecules in an n-sized cluster is given by eq. (3.11). It is different from the chemical potential/1new(P) of the molecules in the bulk new phase at pressure p, because the pressure Pn inside the cluster is higher than the pressure p outside it. To find the dependence of ,t/new, n o n n we use again the capillarity approximation and consider EDS-defined clusters first of condensed phases. For them Vn can be approximated by (3.13) if they are treated as incompressible. Performing the integration in eq. (3.11) with Vn from (3.13) and using p~ from (6.2) then results in ]2new, n "- [ l n e w ( P )

+

2CnGn v2/3 /3nl/3 + ( U2/3/3)d(cnGn)/dn1/3.

(6.6)

For spherical clusters cn = (36zr) 1/3, n = (4~3v0)R 3 and (6.6) gives conveniently ]-/n as a function of the cluster radius R: ,t/new, n = [ l n e w ( P )

+

2GnUo[R + vodGn/dR.

(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

(a)

1000 t--

500 bubble I

O.

i---

I

I

I

9

,

I

!

...i

.

,

!.

.

.

.

I

,

,

.

(b)

6

bubble

4 ir

~

2

e--

:=k

~..~

droplet

0

bubble

1.0

t3 t--

(c)

droplet 0.5

1

200

400

600

800

1000

n Size dependence of (a) pressure inside cluster, (b) cluster chemical potential, and (c) cluster specific surface energy for water droplets in vapours at T = 293 K and steam bubbles in water at T = 583 K. F i g . 6.1

]-/new,,, -- ]-/new(P) -I-

kT In [1 + 2c,(r~/3p

Vf~13 -I-

(ll3p)d(c, Crn)ld Vt]/3 ] (6.8)

where Vn is a complicated function of n obtainable from eq. (3.14) with p,, from (6.1). For spherical clusters, however, it is much simpler to express V~ through the cluster radius R rather than through n. Using (3.22) in (6.8) and recalling that for this cluster shape c = (36Jr) 1/3 then leads to ~new,n --/-/new(P) +

kT In [1 + 2Crn/pR + (1/p)dcr,/dR].

(6.9)

In the scope of the classical nucleation theory with n-independent c,, and an, from eqs (6.6) and (6.8) it follows that /-/new, n -- ]-/new(P) +

2ccr 02/3 ]3n 1/3

(6.10)

Properties of clusters

73

for clusters of condensed phases [Zettlemoyer 1969] and that [/new, n = ]-/new(P) q" kT In (1 +

2cG/3p V2/3)

(6.11)

for gas-phase clusters. Accordingly, from (6.7) and (6.9) we find that if the clusters are spherical, ]Jnew, n = I/new(P) 4" 2 O v 0 / R

(6.12)

when they are liquid or solid [Zettlemoyer 1969] and [lnew,n "-[lnew(P) +

kT In (1 + 2G/pR)

(6.13)

when they are gaseous [Kaischew and Mutaftschiev 1962]. Equations (6.6)-(6.13) show that ,t/new,n(Pn ) ---+ ]anew(P) when n (and, hence, Vn or R) tends to infinity. This is understandable, since when the cluster is sufficiently large, the pressure Pn inside it is practically equal to the outside pressure p. The dependence of ]Jnew, n o n 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 Vn 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. 6.1 b 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 [lnew, n* of the molecules in the nucleus equals the chemical potential Pold 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 Ap from (2.1) and for p* and Ap from (4.14) and (2.10) leads again to eq. (4.4). In the same way it is easy to verify that [/new, n >/'/old for n < n* and that [gnew,n < I/old 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 pe(T). For that reason Pe is often called vapour pressure of the condensed phase. Likewise, a vapour pressure Pe,n(T) is necessary for a given n-sized cluster of the condensed phase to coexist with the vapours around it. Since Pe and Pe,n are related to it/e and ,t/new,n, and Pe and/-/new,n are not equal because of the finite size of the cluster, the vapour pressure Pe,n of the n-sized cluster will also differ from Pe.

74 Nucleation: Basic Theory with Applications

To find Pe,n w e 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 /-/old(Pe,n) "-]-/e d- kT In (Pe,n[Pe)

(6.14)

~new,n(Pn) ----/-/new(fie,n) q- 2cry v 2/3/3n 1/3

(6.15)

and for the chemical potentials of the molecules in the vapours around the nsized cluster and of the molecules inside the cluster, respectively. Equation (6.15) expresses/-/new,n in the scope of the classical nucleation theory and in it ~n~w(Pe,~) is given by eq. (2.6). Coexistence between the cluster and its vapours is possible only when//new,n(Pn) "-/t/old(fie,n) SO that from (2.6), (6.14) and (6.15) it follows that Pe,n = Pe exp [ 2 c t 7 0 2 / 3 / 3 k T n 1/3 + Uo(Pe,n- pe)/kT]

(6.16)

or, to a good approximation, Pe,n = Pe exp (2ccr v 2/3/3kTnl/3).

(6.17)

For spherical clusters (6.17) takes the form of the well-known Thomson (Lord Kelvin) equation [Thomson 1870, 1871 ] Pe,n = Pe exp (2crvo/kTR).

(6.18)

The approximation of the classical theory for constant tYn can be relaxed by using eq. (6.6) in lieu of (6.10) throughout the above derivation. Equation (6.17) is then replaced by Pe,n = Pe exp {( u~/3/3kT)[2CntYn/n 1/3 + d(cntYn)/dnl/3]},

(6.19)

and eq. (6.18) becomes [Ono and Kondo 1960; Abraham 1974a] Pe.,, = Pe exp [(vo/kT)(2cr,,/R + dcr,,/dR)].

(6.20)

The above formulae apply to EDS-defined condensed-phase clusters surrounded by their vapours. Similarly, it is possible to speak of the vapour p r e s s u r e Pe,n of gas-phase clusters embedded into bulk condensed (liquid or solid) phases. Physically, Pe,n 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, Pe,n should not be confused with Pn which is the pressure inside the gaseous cluster necessary only for the cluster's mechanical equilibrium. However, despite their different physical nature, Pe,n 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 c~, cr~ = 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 Pea, of spherical gaseous clusters in the c,,cr, = constant approximation of the classical theory we use eq. (6.11) in the form ]-/new, n(Pe.n) = ]-/new(Pe.n -- 2C~[3 V113)

+ kT In [1 + 2ccr/3(pe,n - 2ccr/3 V2/3) V2/3].

(6.21)

This expression takes into account that when the pressure inside the cluster is Pe,n, in conformity with (6.4), the outside pressure is Pe,n - 2ccr/3 V2/3. With this outside pressure, from (2.6) it follows that we can use the formula flold(Pe,n

--

2ccr/3 V2/3) = Ire + vo(Pe,,- 2ccr/3 V,u3

-Pe)

(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, Pe,n can be found by setting equal the fight-hand sides of (6.21) and (6.22) noting that due to (2.5) ]-/new(Pe,n- 2ccr/3 V~/3) = Ire + kT In [(Pe,~ - 2ccr/3 V~/3)/pe],

since now the gas phase is the new one. The result is Pe,n = P e

exp [ - (vo/kT)(2ccr/3

V 113 +

Pe

-

Pe,n)]

(6.23)

or, to a good accuracy, Pe,,, = Pe exp (- 2ctrvd3kTV1/3).

(6.24)

In these formulae Vn is a complicated function of n obtainable from (3.21) with p replaced by Pe,n - 2ccr/3 V2/3. 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] Pe,n = Pe exp (- 2Ovo/kTR).

(6.25)

If we want to avoid the approximation of the classical nucleation theory for n-independent Cn and on, 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 = P e Pe,n =

exp {- (Vo/3kT)[2cnCrn/ V1/3 + d(cn~n)]d V1/3 ] }

Pe exp [- (vo/kT)(2tYn/R + dan~dR)]

(6.26) (6.27)

for EDS-defined gaseous clusters with arbitrary and with spherical [Ono and Kondo 1960] shape, respectively. The dependence of p~,,, on n from eqs (6.17) and (6.24) is illustrated in Fig. 6.2a for spherical water droplets at T = 293 K and steam bubbles at T = 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, Pe,n ---)Pe for n --> ,~, but while Pe,n >-Pe for the droplets,

76

Nucleation: Basic Theory with Applications

c-

0 15

o

lo l--

0

g

5 0 1.0

-

(c)

f e-.

0.5

h-0

1

200

400

600

800

1000

Fig. 6.2 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 for the 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 between Pe (which is the vapour pressure over the planar surface) a n d Pe,n.

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 C~,, 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 Pe,n, 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,n of an n-sized cluster of condensed phase we can rewrite eqs (6.17) and (6.19) with Pe,n and Pe replaced by Ce, n and Ce, respectively: Ce, n = C e

exp (2cGv~/3/3kTn 1/3)

(6.28)

Ce,,~ = Ce exp {( U 2/3 ]3kT)[2CntTn/n 1/3 + d(cntTn)/dn 1/3] }.

(6.29)

For spherically shaped clusters these dependences become (cf. eqs (6.18) and (6.20)) Ce,n = Ce exp (2OVo/kTR)

(6.30)

Ce,,, = Ce exp [(vo/kT)(2Gn/R + dG,/dR)].

(6.31)

Equations (6.29) and (6.31) account for the dependence of the cluster specific surface energy cr~ on the size of the EDS-defined cluster and, in the c,,cr, = 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, n 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 Vo (nm3)

do (nm)~

Ce (m-3)

O" (mJ/m2)

D (~tm2/s)

0.05

0.46

1023

1 O0

1000

acalculated from do = (6vo/~)1/3

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,n 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 Pe,n and Ce,,, Te,~ can be found from the condition for equality of the chemical potentials it/new, n and ~told 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/~new,~ and/~oJd we have the expressions ]Jnew,n(Te,n) = ~e(P) at- Snew(Te)(Ze- Te,n) at- 2cGv2/a/3nl/3 ~old(Te,n) = ]Je(P) + Sold(Te)(Te- Te,n)

(6.32) (6.33)

which are restricted by the approximation of the classical theory for nindependent c~ and tr, and by the approximations Snew, n(T) = constant = Snew(Te)and Sold(T) = constant = So]d(Te) for the molecular entropies Snew,n and Sold of the crystallite and the melt in the temperature range between Te,n and Te. Setting equal the fight-hand sides of the above equations we thus find that Te,n = Te - 2cG v 2/3/3Ase nil3

(6.34)

where Ase- So]d(Te) - Snew(Te) 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] T~,, = T e - 2ov0/As~.

(6.35)

The approximation for constant Cn and or, 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,n = Te - ( 02/3/3 Ase)[2CnGn/n 1/3 + d(cnGn)/dn 1/3]

(6.36)

when the clusters are arbitrarily shaped and Te, n -" T e -(vo/Ase)(2Crn/R + dGn/dR)

(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,n on n, calculated from eq. (6.34) with c = (367r)1/3 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 Vo (nm 3)

do (nm) a

Te (K)

Ase/k

cr (mJ/m 2)

0(230 K) (mPa.s) b

77(240K) (mPa.s) b

7n

0.033

0.4

273.15

2.6

26

43

16

1

acalculated from do = (6Vo/tr)1/3 bcalculated 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, bin, Pe,n, Ce,n and Te,n for a cluster of specified shape (Cn = constant) depend on n not only directly, but also through the n-dependence of the cluster specific surface energy G,,. If the effect of the cluster size on crn manifests itself more strongly for smaller values of n, it might be expected that the approximation Gn = constant = cr 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 ty, is an important problem and this may explain the interest in this problem shown first by Gibbs [1928] and then by many others [Tolman 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; S6hnel 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 o-n 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 tyn 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 or, 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 O'sT,n)/d(ln RST) = (280/RsT)/(1 + 280/RsT).

(6.38)

80 Nucleation: Basic Theory with Applications

Here the subscript 'ST' indicates that crn is referred to the ST, and S0 is defined by 60 = R - RST

(6.39)

where R and RST are the radii of the EDS- and the ST-defined clusters, respectively. The Gibbs parameter ~ is thus the distance between the EDS and the ST. Gibbs [1928] solved eq. (6.38) by assuming (i) that ~0 is constant and (ii) that RST >> 2~0. He thereby found that O'ST,n = Cr exp (- 26o/RsT)

(6.40)

where cr (independent of the choice of the dividing surface) is the specific surface energy of the planar interface (i.e. the value of CrST,,,at RST = ~). 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 80 is represented as CrsT,n = cr/(1 + 260/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 1960] that d(ln O'sT,n)/d(ln RST) = d(ln O'n)/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 o'n in this case). This means that with the help of (6.39) eq. (6.38) can be rewritten as d(ln Crn)/d(ln R ) = (260/R)/(1 + ~ / R ) .

(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 EDSdefined clusters: O "n " -

O"

exp (- 2S0/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 >> ~o. The result is an= cr/(1 + ~0/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 V, of the EDS-defined cluster: cr~ = or~(1 + C6o/3 V~/3 )2.

(6.45)

Here c is the cluster shape factor equal to (36t01/3 for spheres, 6 for cubes, etc.

Properties of clusters 81

In accordance with (3.13), for condensed-phase clusters the explicit ndependence of cr,, from (6.45) is given by ty,, = or~(1 + Ct~o/3O~/3 n 1/3) 2.

(6.46)

For gaseous clusters, however, the dependence of o'n 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 = (p/kT)V,, + (2ccr/3kT) V1/3[1 + (Ct~o/3V, l,/3)/(1 + Ct~o/3Vnl/3) 3] (6.47)

which follows from (3.14) upon using Pn from (6.1) with c,, = constant = c and dcr,,/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 do: tS0 > 0 leads to a lower specific surface energy for the smaller clusters, and d;0 < 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 60. 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 60 between them. Under such circumstances it remains only to resort to theoretical arguments and/or to determinations of ~ with the aid of numerical calculations. On such grounds it is believed that, typically, I~lis of the order of the molecular diameter do, but the question about the sign of 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] S01is 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, d;0 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 ~ as an R, Rsz-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 for both R ~ oo and RST ~ co. Recently, Hadjiagapiou [ 1994] has shown that 60 may decrease linearly with R, and Schmelzer et al. [ 1996] have demonstrated that different tS0(R,RsT) functions can affect essentially the size dependence of the cluster specific surface energy. Figure 6.1c illustrates the n-dependence of o'~ from eqs (6.45) and (6.46) for spherical EDS-defined steam bubbles and water droplets, respectively. For the bubbles T = 583 K and p = 0.1 MPa, and n is determined with the help of (6.47). The parameter values used are c = (36/r) 1/3, S 0 = 0.1 nm and those listed in Tables 3.1 and 3.2. As seen, ty~ does not differ more than 10% from its limiting value ty for n = oo (i.e. for the planar interface) when n > 1000 molecules. In the range of n < 100 molecules the change of ty, with respect to ty is already greater. However, in this range of n values the predicted effect cannot be regarded without reservation, because then the ~0(R, RST) = 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 n-dependence of o'n, 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 O'n with n we shall confine all further considerations within the approximation an = constant = cr 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 et al. [ 1983]; Suck Salk and Lutrus [ 1988]; Mutaftschiev [ 1993]; Ford [ 1997a]). Nevertheless, so far there is no generally agreed formula for C(n), 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(n). 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(n) has recently been presented by Wilemski [1995]. In principle, the problem of finding C(n) is sufficiently hard both physically and mathematically so that we shall confine our considerations only to a derivation of a formula for C(n) 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(n), 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 n-mer, formed of n monomers (i.e. single molecules) in accordance with the reversible 'reaction' (n = 1, 2 . . . . ) n [C~] ~ [Cn]

(7.1)

where [C1] and [Cn] are the 'chemical' formulae of the monomer and the nsized 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

npo~d = P,,,

(7.2)

for any n = 1, 2 . . . . . Here p~ is the chemical potential of the n-mer considered as a separate 'macromolecule', and it has been accounted that in equilibrium, by definition, P l = Pold. The problem is, therefore, to express p~ in terms of the concentration C(n) of the n-mers, i.e. of the n-sized clusters. To do that we consider first HON and denote by Co (m -3 or m -2) 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 size 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/~,, 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) = Co) by the formula [Guggenheim 1957] l.tn = G(n) + kT In [C(n)/Co].

(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 t~n = 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/~ 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) = Co exp [ - W(n)/kT].

(7.4)

It follows from here that the monomer concentration C(1) is related to Co by C1 - Co exp ( - Wl/kT)

(7.5)

so that C(n) from (7.4) can be expressed alternatively as (n = 1, 2 . . . . ) C(n) = Cl exp {- [W(n)- W1]/kT}

(7.6)

where, for brevity, C1 - C(1) and W1 - W(1). 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 (Cl/Co)"-lexp [(n - 1)W1/ kT], which equals unity by virtue of (7.5), changes (7.6) into (n = 1, 2 . . . . ) C(n) = Co(Cl/Co) n exp {- [W(n) - nW1]/kT}.

(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(n) - q~s,0.The difference q~s(n)- Cs,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 Co, C1 and W1. Equation (7.5) is thus a general condition which must be obeyed by the limiting value W1 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)/Co 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(1) = Cl 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 W1 = 0. It is clear, however, that neglecting W1 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 Co 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 Co has the following form:

Co = Ns/As (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 Ns = As/ao and Co = l/ao = 1019 m-2),

Co = Na/As (rn-2)

(7.9)

for 3D or 2D HEN on a substrate with Na active centres (e.g. impurity molecules, structural point defects, etc.) which are always less than Ns, Co = Na/V

(m -3)

(7.10)

for 3D HEN in the volume V of an old phase in which there are Na active centres (e.g. impurity molecules or ions),

Co = NaMs/V (m -3)

(7.11 )

for 3D HEN in an old phase containing Ms seeds each of them with Na 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 Co 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 nucleation and, analogously to (7.10), Co = M~ V = 1/Vo = 10 28 to 10 29 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 Co 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 C1 = p / k T

(7.13)

and also, according to (2.8) and (3.44), W1 = ~ - k T In (piPe),

(7.14)

because E1 = 0. For the equilibrium (or saturation) pressure Pe we have the integrated Clapeyron-Clausius formula [Glasstone 1956] Pe = P0 e -~vkr

(7.15)

where p0(Pa) is a pre-exponential factor. With C1 from (7.13) and W1 from (7.14) eq. (7.5) yields Co = (Pe]kT) e ~kr = po/kT.

(7.16)

This formula gives Co in the case of HON in vapours. It is indeed close to eq. (7.12), since often P0 can be approximated as P0 = kT/vo [Moelwyn-Hughes 1961]. It must be emphasized that, in fact, knowing Co for HON in gaseous phases is not necessary (provided, of course, W1 is known), since in this case the convenient formula for C(n) is eq. (7.6) with C1 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(n) =

Cl, e

exp [n In S - (aG/kT)(n 2 / 3 - 1)].

(7.17)

Here Cl,e, given in ideal-gas approximation by Cl, e = pe/kT,

(7.18)

is the concentration of the molecules in the gas phase at the equilibrium (or saturation) pressure Pe, and S =- pipe = C1]Cl,e

(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(1) = C1 with C1 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 1

exp [n In S - (acr/kT)n 2/3]

(7.20)

which, as seen, follows from (3.39) and (7.6) after neglecting W1 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 and Chiu 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(n) appears to be (7.4) when Co is known independently, and (7.6) when we have such a knowledge of C1. If both Co and C1 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 Ap 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) = Co exp [(Ap/kT)n - ( a t y e f / k T ) n 2/3]

(7.21)

C(n) = C1 exp [(Ap/kT)(n- 1 ) - ( a t T e f / k T ) ( n 2 / 3 - 1)]

(7.22)

for either HON or 3D HEN of condensed phases (tref = tr for HON, and tref = ~o" < tr for HEN of caps, lenses, etc. - see eq. (4.42)),

C(n) = Co exp {[(Ap

-

aefA(y)/kT]n

- (brdkT)n 1/2}

(7.23)

for 2D HEN of condensed phases on foreign (Aa ~ 0) or own (Ac = 0, a0) substrate and

C(n) = Co (P/pn)n exp{ (Ap/kT)n - [C(Yef[3Pn 2/3 (kT) 1/3 ]n 2/3 }

aef =

(7.24)

for either HON (Cref= or) or HEN (O'ee"" t/tO"< O') of gas phases (the dependence of Pn on n is given by (3.63) for spherical (gt = 1) or cap-shaped (gt < 1) gaseous clusters).

Equilibrium cluster size distribution

89

The respective atomistic formulae for clusters of condensed phases are (n =1,2 .... ) C(n) = Co exp {[(Ap - &)n + En]/kT}

(7.25)

C(n) = C1 exp { [ ( A ~ - 2 ) ( n - 1) + En]/kT}

(7.26)

in the case of HON (then E1 = 0), C(n) = Co exp {[(A/.t - X,)n + E n + crsAn]/kT }

(7.27)

C(n) - Co ( C l / C o ) n exp {[Es, n +

(7.28)

tYs(A n -

nao)]/kT}

in the general case of HEN (then E1 ~ 0) of arbitrarily shaped clusters (for them A n ~ nao) on a foreign substrate and C(n) = Co exp {[(Ap - & + aoCrs)n + En]/kT}

(7.29)

C(n) = Co (CI/Co) n exp (Es,n/kT)

(7.30)

in the particular case of 2D HEN of monolayer clusters (for them An = nao) on a foreign (as ~ ty) or own (as = or) substrate. In writing eqs (7.28) and (7.30) it is taken into account that A1 equals the molecular area a0, and the quantity Es,n > 0 is defined by (n = 1, 2 , . . . ) Es,, - FEn - nE1 = nul - Un.

(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 E~ from (3.32) and shows that Es,~ differs from E~ only by the chosen zero of the potential energy: Uoldfor En and Ul for Es,,,. We see also that Es,1 = 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 Co, C1, O'efand A~t 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 Ap from (2.8), O'ef "" O" and C1/CI,e = P/Pe, 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 Ap 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; S6hnel and Garside 1992]. With Ap 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 C1 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)

=

C1, e

Sn

exp {- [(n - 1)~,- En]/kT},

(7.32)

which follows from (7.26) with Ap from (2.8) and S from (7.19). Recalling

90

Nucleation: Basic Theory with Applications

that in HON E1 = 0, we observe that eq. (7.32) is a complete analogue of the self-consistent classical equation (7.17), because nA,- En is just the atomistic representation of the classical total surface energy aO'n 2/3 of the n-sized 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 supersaturation 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) = Co exp [- tI~(n)/kT] e n~/kr

(7.33)

C(n) = C1 exp { - [ t / J ( n ) - ~l]/kT} e Cn-1)a~/kr

(7.34)

C(n) = CO(C1/Co)n exp { [ n ~ l - tIJ(n)]/kT}.

(7.35)

In these most general formulae for C(n), in which ~1 = t/~(1), t/~(n) is given thermodynamically 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 9 (n) is practically Ap-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 A/~ and decreasing ~(n) 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 ~ l - tI~(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/u, 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/J < 0). From eq. (7.33) we thus see that since typically ~(n) is a positive quantity which increases more or less steadily with increasing n, when A~u < 0, most generally, C(n) diminishes with n and vanishes in the n --~ ~ limit (in 2D HEN this is so only in the A/u <_ aefAt~ range). On the contrary, when A/~ > 0 and the old phase is supersaturated, C(n) ~ ~ in the limit of n ~ ~. This behaviour of C(n) reflects the fact that when A/~ < 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(n) with increasing n when Ap > 0, i.e. when the old phase is in metastable equilibrium, implies that C(n), 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

1025

10 20

I

E

c.. ~0

1 015

10

9

.-'"

1.5

10

nli i2

105

"/~

. . . . . . 20

40

~ .... 60

~... 80

~ . . . , . .. 100

120

140

n

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 A ~ / k T = In S = - 1.5, 0 and 1.5 (as indicated). The values of o0, cr and Pe 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 (Ap < 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 108 droplets of, say, 12 water molecules in vapours with volume of 1 m 3 at A p / k T = - 1.5 and 0, respectively. Increasing Ap thus has a strongly stimulating effect on the droplet formation: Ap 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 m 3 at A p / k T = 1.5. However, at this Ap value the course of the C(n) curve is fundamentally different: the curve exhibits a minimum at n = n* and diverges for n ~ oo. 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 A p / k T = 1.5. Since the water vapours are in metastable equilibrium when Ap > 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 Ap > 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 Ap > 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/~ > 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 n* by the truncated Taylor series

W(n) = W* + (1/2)(d2W/dn2)n-n,(n- n*) 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 [flE(n- n*) 2]

(7.37)

provided In - n*l < 1~ft. Here C* - C(n*) = Co exp (- W*/kT) is the equilibrium concentration of nuclei, and the numerical factor fl > 0 is given by fl = [(-- dEW/dnE)n=n,/2kT] 1/2.

(7.38)

Geometrically, fl characterizes the curvature of the W(n) curve at n - n*" a greater fl value corresponds to a sharper maximum of W(n) at the nucleus size. To estimate fl we can use the classical formulae for W(n). For example, in the cases of both HON (o'ef -- o') and 3D HEN (o'ef < or) of condensed phases, from (3.39), (3.60) or (3.66) and from (4.38) and (4.39) it follows that (A/I > 0)

r2 = atTef/9kTn,4/3 = 9Ap4/16kTa3 0.ef 3 = W./3kTn.2 = Alu/6kTn*

(7.39)

Similarly, for 2D HEN of condensed phases on foreign (Ao ~ 0) or own (Ac = 0) substrates, with the aid of (3.71), (4.32) and (4.33) we find that, classically, (A/z > aefAff )

r2 = b ~ 8 k T n , 3 / 2 = (A/~ - aefAty)3/kTb2K 2 = W*/4kTn .2

= (Aft - aefA6)/4kTn*.

(7.40)

These relations tell us that fl is an increasing function of Ag and that, typically, fl = 0.01 to 1, since in most cases of nucleation Atz/kT = 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(n) 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 Alt/kT = 1.5 (in this case n* = 53 and hence, according to (7.39), fl = 0.07). As seen, eq. (7.37) approximates well the corresponding C(n) dependence from (7.17) solely in the nucleus region (known also as critical region) around n = n*. This region extends from n = n l to n = n2 which are defined by

so that its width

A*

-

n2

-

n1=

n*

n2=

n* + zcl/2/2fl

(7.41)

- ~1/2]2fl

(7.42)

n 1 is given by A * - /~,l/2/fl.

(7.43)

This formula reveals the physical significance of fl: 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(n) derived above in order to determine the equilibrium concentration C* (m -3 or m -2) 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* = Co exp (- W*/kT) C* = C 1

C* -

exp [- (W*

-

(7.44) (7.45)

WI)/kT]

Co(C1/Co)n* exp [- (W*

-

n*Wl)/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 W1 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* = Co exp (- tI>*/kT) e n*Au/kr

(7.47)

C* = C1 exp [- (q~* - q~j)/kT] e~n*-l)~/kr

(7.48)

C* = Co(C1/Co) n* exp [(n*q~l - q~*)/kT]

(7.49)

where t/r* 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 qr* are functions of Ap. With concrete expressions for Co (or Cj), W* (or qr*), W1 (or ~1) and n*, eqs (7.44)-(7.49) apply to all cases of nucleation. The general thermodynamic formulae for W* and qr*, valid for arbitrarily chosen Gibbs dividing surface, are eq. (4.5) (with @s* from (5.33)) and eq. (5.7). In the scope of the classical nucleation theory these formulae result in specific Ap dependences of W*, n* and qr,. 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/~ > 0) 3/27kTAIa2) C* = Co exp ( - 4 c 3 V2aef

(7.50)

in the cases of both HON (O'ef --- O') and 3D HEN (O'ef < O') of condensed phases, that [Hirth and Pound 1963; Zettlemoyer 1969] (Ap > aefA(y ) C* = Co exp [- b 2x 2 / 4 k T ( A p - aerate)]

(7.51)

for 2D HEN of condensed phases on foreign (Ao ;~ 0) or own (Aa = 0, aef = a0) substrate and that [Hirth and Pound 1963; Blander 1979] (0 < p < Pe) 3/27kT(p* C* = Co exp [- 4 c 3 Gee

p)2]

(7.52)

for either HON (O'ef -" O') or 3D HEN (O'ef < O') of gaseous phases, p* - p being related to Ap 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 Co in (7.50) in the form C O = (pe/kT) exp (c u 213 cr/kT),

(7.53)

because, classically, W1 = - Ala + c v~/3 cr (see eq. (3.39)). This expression for Co is only an approximation to (7.16), for it results from using c v 2/3 cr instead of ~, (cf. eqs (3.20) and (3.33)) in the exact formula W1 = - A/.t + ~, which follows from the atomistic equation (3.44) with E1 = 0. In addition to the classical formulae for W* and t/r, 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 n*, e.g.

Equilibrium cluster size distribution 95 C* = Co exp {[(A~ - ~)n* + E*]/kT}

(7.54)

in the case of HON of condensed phases and

C* = Co (C1/Co) n* exp (Es*/kT)

(7.55)

in the case of 2D HEN of monolayers of condensed phases (E* - En, and E* Es,n, 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 t/~ contains the information about the kind of nucleation (see eqs (4.49) and (4.50)). Since in (7.47)-(7.49) n* and t/r* are unknown functions of Ap, these equations become practically working formulae in the n* ~ 1 limit when in certain A/~ ranges both n* and t/r* remain constant with respect to A/~ (see Section 4.4). As this can be the case with n* and t/r* 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 A/.t controlled by two unknown parameters: n* and Co exp (- qr*/kT). The C*(A/~) 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 tree, ~r and Ate. Figure 7.2 depicts C* from eq. (7.50) as a function of Al.t/kT = In S in the cases of HON of spherical (gt = 1, o-el "- O-) and HEN of hemispherical (gt = 1/2, o-ef = (1/2) 1/3o-) water droplets in vapours at T = 293 K. The numbers at the symbols on the C*(A/~) curves indicate the number n* of water molecules in the nucleus droplet (according to eq. (4.38)) at the corresponding supersaturation. The values of o0, o- and Pe used in the calculations are those given in Table 3.1, for HON Co = 2.6 • 1027 m -3 is evaluated from (7.53), and for HEN it is assumed that Co = 1019 m -e 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 A/~. Indeed, if in HON the volume of the vapours is 1 m 3, we observe that while at Ap/kT = 1.2 the number of nucleus droplets in the vapours is 5, at AI~/kT = 1.4 it is already 5 • 10 7. Similarly, in the case of HEN on a substrate with surface area of 1 m e, at the same AI~/kT values the respective number of nucleus droplets on the substrate is 5 • 105 and 10 9. The physical reason for this strong impact of A~ on C* is the diminishing of both n* and W* with increasing A/I (in HON n* = 103 and 53 molecules and W*/kT = 62 and 40 at the above At~/kT 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 m 3, and the latter on a substrate with surface area of 1 m 2, the substrate being, e.g. a wall of the container. As already noted above, at =

96 Nucleation: Basic Theory with Applications 10 25

10 2o

1015 -

b

1010

40

-

20

o/i,o

105

1

0

1

2

3

4

,a~ / kT Fig. 7.2

Supersaturation dependence of the equilibrium concentration of nuclei:

curves H O N and H E N - 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. A p / k T = 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 A p / k T = 2, however, the

number (-- 10 j8) 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 Ap 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 Co 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 cr be treated as independent on the cluster size n? Fourth, what in fact is the physical meaning of cr for small clusters when, geometrically, they may have fewer 'bulk' than 'surface' molecules? And, of principle importance, why is W* ~ 0 at A/~ = A/~s, 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], Harrowell and Oxtoby [1984], Oxtoby and Evans [1988], Nishioka et al. [ 1989], Hoyt [ 1990], Tomino et al. [ 1991 ], Zeng and Oxtoby [ 1991 ], Nishioka [ 1992], Granasy [ 1993a, b, 1996a, b], Granasy et al. [ 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 onecomponent HON under isobaric-isothermal conditions, we shall now outline the density-functional theory of nucleation which is based on the densityfunctional 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, z). The existence of spatial inhomogeneity of the molecular density/9 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 Pik(r) (i, k = x, y, z), and the Helmholtz free energy f per molecule becomes locally different and a function of r, p and its derivatives, which we shall denote for short as f(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 F 2 of the system in state 2 will be given by F2{p} = f f(r)p(r) dr, ,Iv

(8.1)

since p(r)dr is the number of molecules in the differentially small volume dr - dx dy dz 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 = Fe + pV

(8.2)

of the system in state 2 become functions of the function p(r), i.e. functionals of the molecular density. We shall denote these functionals as Fz{p} and G2 {P} in distinction with the usual notations F2(p) and G2(p) for/72 and G2 when these are functions of the variable (and not of the function)/9. The density-functional theory operates with the functionals Fz{p} and Gz{P} which thus give the name of this theory. These functionals are non-local, because f 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{R} to form a density fluctuation (not a cluster) having a density profile p(r). Rather than a function of the cluster size n, now W is a functional of p, since it is again given by eq. (3.4) in which, according to (8.1) and (8.2), G2{P} = ~v [f(r)p(r) + p] dr.

(8.3)

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} = [ {[f(r) - Pold]p(r) + P]} dr dv

(8.4)

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 12ne w 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 or. 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 f 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 (b'W is the variation of W with respect to p)

(b3V{p})p=p. = O.

(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 thef(r) function. Without losing the essential physics, the simplest to assume is that besides of r, f i s a function only of p and its first derivatives p~" - c?p/ o3/(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 - l.tolaP + P) - ~. cgpi'cgi(fp -/-told P + P)

p= p*

= 0. (8.7)

Upon performing the differentiation this equation takes the form (Pi'," -~2p/ &'cgk with i, k = x, y, z)

32 c92 -~p ( f P - ~i o3pigi (fp) - ~i Pi'cgpi'cgp (fp)

,,

- E E Pik ~

i k

O~piz O~pk

(fp)

] p=p,

= #o~a.

(8.8)

Equation (8.8) plays the role of the Gibbs-Thomson equation in the densityfunctional 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 n*) upon

100 Nucleation: 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 densityfunctional theory depend crucially on the modelling of the f(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 f 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 [1958, 1959] which accounts for the dependence o f f on/9 and its first derivatives Pi" and the hard-sphere approximation of Oxtoby and Evans [1988] which represents f 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 [1960]. Let us now consider briefly the gradient and the hard-sphere approximations. The quasithermodynamic approximation has not been used so far in the theory of nucleation 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 nucleation. 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 nucleation in two-component incompressible fluids [Cahn and Hilliard 1959]. The assumption is that f is a function only of p and its first derivatives Pi' and can be approximated as

f(r) = fu [p(r)] + K[p(r)] [Vp(r)] 2 .

(8.9)

Here fu 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 = (tgp/o3x) 2 + (o3p/cgy) 2 + (tgp/&) 2 is the squared gradient of p. Equation (8.9) is a truncated expansion of f i n gradients of/9, and the gradient term describes the departure of the actual energy f at point r from the energy fu 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 fir) 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 f from (8.9) in (8.8) leads to the equation of Cahn and Hilliard [1959]

a( fu*p,)/ap,

- [ a ( K * p * ) / a p , ] ( V p , ) 2 - 2 K * p * ( V 2 p *) =/Jold

(8.10)

where f * -fu(P*), K* =- K ( p * ) and 72t9* = o32p*/o3x 2 + a2p*/Oy 2 + a2p*/Oz 2. 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 v Z p * = (2/r)(dp*/dr) + dZp*/dr2 [Korn and Korn 1961] so that in this case (8.10) reads 2K*p*(d2p*/dr 2) + (4K*p*lr)(dp*ldr) + [d(K*p*)ldp*](dp*ldr) 2

= d( fu* P*)/dP* -/./old,

(8.11)

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*(oo) = Pold, (dp*/dr)r=O = O.

(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 = 4n'r2dr 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/-toldP- P in (8.4) should be replaced by foldffold (fold 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 Hilliard [1959]. Cahn and Hilliard [1959] analysed the properties of p* and W* 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 Ap the nucleus fluctuation resembles the classical nucleus cluster in that (i) the density P*w at its centre approaches the density/9new of the bulk new phase, (ii) the specific surface energy cr 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 Aps: the density Pn*ewat the nucleus centre tends to the spinodal density Pold,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 Ap = Aps. 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 Ap = oo. 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/9 with r. Following Tarazona and Evans [1983] and Evans et al. [1986], Oxtoby and Evans [1988] expressed f analogously to the molecular Helmholtz free energy fvow 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) we find thatfvow = F/M is given by [Landau and Lifshitz 1976] fVDW -" fref + kT In [b'pu/(1 - b'pu)] - a'pu

(8.13)

where fref is a p,-independent reference energy. The b' term in this formula accounts for the harsh repulsion between the molecules at distances smaller than the molecular diameter do. This term becomes important when the density Pu of the VDW fluid approaches its upper limit 1/b" 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 do and is determined by the expression [Landau and Lifshitz 1976] a' = -(1/2) Ja0 u(r) dr.

(8.14)

Here u(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 = 47~r2 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), f can be represented as

f(r)

-- fh [p(•)]

+ (1/2) f p(r')Ua(Ir- r'[) dr'

(8.15)

where the integration is over the volume of the system. In the abov.e approximationfh 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 fh(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 Ir- r'l from each other. We thus see from (8.15) that in the hard-sphere approximation of Oxtoby and Evans [1988] f 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. With fir) from (8.15) we can now use (8.8) to find the equation whose solution p*(r) represents the nucleus density profile. Since now f(r) is independent of the first derivatives of p, O(fp)/O Pi' = 0 for each i = x, y, z and (8.8) leads to the equation of Oxtoby and Evans [1988] pla[p*(r)] + f p*(r')Ua(Ir - r'l) dr' = Pold

(8.16)

in which/~h = O(fhp)/Op is the hard-sphere chemical potential. The analogy between this equation and eq. (4.4) is obvious, since in the scope of the hardsphere 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 soobtained 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

Nucleation: Basic Theory. with Applications

numerically for nuclei of spherical symmetry by employing a concrete/Jh(P) dependence and Yukawa or Lennard-Jones attractive potential Ua. 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 Pnew * 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/j, W* decreases faster than predicted by the classical equation (4.11) until vanishing at Ap = Aps 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 flee 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 fbecomes a function of r both explicitly and/or implicitly through/9 and its derivatives. The density-functional theory operates namely with f(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) = 1/v(r)

(8.17)

where v (m 3) is the volume occupied by a molecule at point r(x, y, z). Accordingly, the total number M of molecules in the system is given by M = ~v p(r) dr

(8.18)

where dr = dx dy dz 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} = ~vf(r)p(r) dr,

(8.19)

since f(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 f is independent of the derivative of p. Since in a system with non-uniform density the pressure is a tensor with components Pik(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 pef(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 &tr. As known from the theory of elasticity [Landau and Lifshitz 1965], the change of the volume from dr to dr + &tr is related to the change Seik(r) of the components eik(r) of the deformation tensor by the formula (i, k = x, y, z) d;dr = [ E i

t~eii(r)]

dr.

(8.20)

Accordingly, the elementary work Sw done by the system for the increase of the volume dr will be [Landau and Lifshitz 1965] 6w(r) = []E i

~,Pik(r)t~eik(r)] dr. k

(8.21)

On the other hand, due to energy conservation, the work done by the system leads to a decrease of the free energyf(r)p(r) dr in the volume dr. We shall therefore have d;w(r) = - p ( r ) d r ~ ( 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 ~ p dr) equals zero, i.e.

t~p(r) + p(r) 6dr = 0.

dr

(8.23)

Setting equal the fight-hand sides of eqs (8.2 l) and (8.22) and accounting for (8.20) and (8.23), we arrive at the formula

Of (r)/Op(r) = pef(r)/p2(r)

(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 Pef (r) = [52 E i

k

Pik(r) t~eik(r)]/[Ei

&ii(r)].

(8.25)

106

Nucleation: Basic Theory with Applications

Equation (8.24) is analogous to the known thermodynamic relation [Guggenheim 1957] Of/Op = p/p2 for systems with uniform density, pressure and Helmholtz free energy per molecule. Thus, the introduction of the effective pressure Pef 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 p e f ( r ) . For example, with the help of Pef eq. (8.21) takes the familiar form 6w = p e f t ~ d r , and the chemical potential p(r) at point r can be defined by p (r) = f(r) + pef(r)/p(r).

(8.26)

This expression follows from the thermodynamic formula [Guggenheim 1957] (cf. eq. (8.2)) lap dr = f p dr + Pef dr

for the Gibbs free energy/~19 dr of a uniformly dense phase of t9 dr molecules with Helmholtz free energy f p dr in a volume dr under pressure Pef. Also, the quasi-thermodynamic Gibbs-Duhem relation (at constant T) dpef(r) = p(r) dp(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 f is independent of the derivatives of p(r) (see Ono and Kondo [1960]). 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 Pef equals the tangential component PT 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 = P y y = P z z = p and P x y = Pxz = Pyx = Pyz = Pzx = Pzy = 0 [Ono and Kondo 1960]. Hence, Z Z Pikt~ik "- p ~., t~eii i k i

and from (8.25) it follows that Pef = PWhen the system is with r-dependent density p across a planar surface layer, Pxx = Pyy- PT(Z), Pzz -- PN = P and p~, = Pxz = P y x = P y z = 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 and PT are called, respectively, normal and tangential components of the pressure tensor). Also, &tA = dA(&~ + &yy) dr = dAdz,

&lr/&tA = dz

(8.28) (8.29)

Density-functional approach 107 where dA = dxdy is a differentially small area, and &tA is the small change of dA leading to the change &lr of the volume dr, Then, Per from (8.25) takes the form Pef = [PT(t~lexx +

l~eyy) + prezz]/(&xx + (~eyy -t- r

).

With the help of (8.20) and (8.28) this expression becomes Pef = [ ( P T -

p )( t~tA/dA ) + p( &tr/dr) ](dr/ &tr)

which, owing to (8.29), leads to Pef = PT(Z). The determination of Pef for a system with a spherical surface layer is analogous. In this case Prr - - pN(r), Poo = P~o~o= pT(r) and PrO = Pr~o = Pot = Polo = P t p r = P t p o = 0 in spherical coordinate system r,O,q) with origin at the centre of the sphere representing the surface layer [Ono and Kondo 1960]. In addition, t~dA = dA(~e00 + ~eq,~0) dr = dA dr,

t ~ d r / ~ = dr

(8.30) (8.31)

where dA = r 2 sin 0 d0 dip. With the aid of (8.20), (8.30) and (8.31) from (8.25) it follows again that Pef --PT(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 eliminatingflr) in (8.4) and expressing W{p} in terms of p(r) and p~f(r):

W[p} = ~v {[,u(r)- Pold]p(r)+ [P-pef(r)]} dr.

(8.32)

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 Pola and p to their local values p(r) and pef(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 p(r) and pef(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 pef(r) = pT(r) and dr = 4n-rz dr so that eq. (8.32) takes the form W{p} = 4zr

~0Rsys{[,u(r) - Pold]p(r) + [P - pT(r)]}r 2 dr

(8.33)

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 Rsy s (in practically all calculations it is possible to use Rsy s = 0% since p(r) = Pold and pT(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 thef(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 p*(r) = Pold

(8.34)

where p*(r) = [p(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 p(r) have the same value/.tol d 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 v [p - P e f * ( r ) ]

dr

(8.35)

where p e f * ( r ) = [pef(r)]p=p,(r) 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, pef*(r) = pT*(r) = [PT(r)]p=p*(r) and dr = 47rr2 dr so that this formula takes the form W* - 4to

~0Rsys[p - pT*(r)]r 2 dr

(8.36)

which follows also from (8.33) and which was used by McGraw and Laaksonen [1996] with Rsy s = co.

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 Pe~ of the components (respectively, through the value p~ of the tangential component) of the pressure tensor generated in the old phase by the appearance of the nucleus. The/~(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 p(r) dependence with statistical mechanical justification is the expression used by Oxtoby and Evans [ 1988], Zeng and Oxtoby [ 1991 ] and Hadjiagapiou [1994] in the 1.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 Pe~ (or P~r ) on r can be obtained by integrating the Gibbs-Duhem relation d Pe~ (r) = ,o*(r)d,Uol d

(8.37)

which is eq. (8.27) applied to the nucleus and in which, by virtue of (8.34), /1old stands for/~*(r). Since in principle for a uniformly dense phase pef(r) is r-independent, most generally, Pe~ (1") = p = constant for r sufficiently far from the nucleus and pe~ (r) = p* = constant when r is close enough to the centre of the nucleus provided that the nucleus has a uniform density P*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 p~ on r: in order for W* to be finite when Rsy s = 0% p~ must tend to p faster than r -3 for F ---~ oo.

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 p in the system to the spatially varying effective pressure Pef 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/~* 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 Pe~ on ~old which, along with the P(,Uold) one, makes W* from (8.35) a function of Poid and, hence, Aju. 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 Pold and employment of the Gibbs-Duhem relations (5.8) and (8.37) results in dW*/dp~

=-

fv

[p*(r)

- Pold]

(8.38)

dr.

According to (8.18), the integral of p*(r) in this equation represents the total number M 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 Po]dVof 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

dW*ldpold -- - 4 ~

f:

sys

[p*

(r)

- Pold] r2

(8.39)

dr

which follows also from (8.36) and in which the approximation R s y s = ,x, 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 p~*f(r) dependence in (8.35) we can determine the dependence of W* on/'/old (or Ap) 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*(Pold) = W*'+

[p*(r) - Pold] dr

dPola

(8.40)

]'/old

for arbitrarily shaped nuclei and W*(~old)

[p*(r)

"- W ' t - I - 4 7 r

- Pold] r

2dr }

d/2ol d

(8.41)

/./old

for spherical nuclei (W*' = W* (gold) and, as already noted, R s y s = ~ 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 /-told = Pold,s, as then W*' = 0 (see eq. (5.48)). In the absence of spinodal (then ~/old,s oo), an appropriate choice for ].Lold and W*' in the above formulae is JUold and W*' = 0 (see eq. (5.50)). Equations (8.35) and (8.40) thus offer two alternative ways of finding the W*(Pold) dependence which plays a key --'--

_-- o o

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 ( P e r ( 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 9* = q~*). Suppose that we want to perform the integration in (8.35) without knowing the concrete Pe~ (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 p* and p, . respectively. This means that per(r) in eq. (8.35) is a step function of r: 9 p. * Per ( r ) -" inside the nucleus and Per ( 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 ~ * = ~*) in which, mathematically, q~* compensates for the error introduced by the stepwise approximation of Per9 ( r ) . Physically, ~. is the nucleus total surface energy and it must obey the equation ~0" = (p* - p) V* +

[P

-

Per

(8.42)

(r)] dr

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 q~* 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 P e r ( r ) after mtegratlon. In the particular case of spherical nuclei ~* = 4~R'2o ", V* = (4zr./3)R.3, Pe~ (r) = p~ (r), dr = 4to .2 dr and (8.42) leads to the known formula (eq. (15.5) of Ono and Kondo [1960], eq. (1.26) of Rusanov [ 1967]) o

.

:~

,

f Rsys tr = (l/3)(p* - p ) R * +

o

,

* [P - PT (r)](rlR*) 2 d r

(8.43)

.10

for the specific surface energy (7 of arbitrarily positioned spherical dividing surface (Rsy s = oo for a system with sufficiently large volume). It is worth noting that eq. (8.42) defines cr also for a planar interface with a r e a Ai: then q~*. = AirY, p* = p and, if the z axis is perpendicular to the interface plane, P e r ( r ) = P T ( Z ) and dr = Aidz. 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

o" = j_~,[p - PT (z)] 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 and Hilliard 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 u(lr- r'l) between the molecules, using good enough approximations forf(r) and/~(r) and obtaining analytical results for the p*(r) and W*(A/~) 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 nucleation

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 nontrivial 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 D6ring [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 al. 1966; Kashchiev 1971, 1974, 1984a], we shall now formulate the m a s t e r 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 soformulated 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 onecomponent 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 [ 1950]; Sigsbee [ 1969]; Wilemski [ 1975]; Binder and Stauffer [ 1976]; Stauffer [ 1976]; Temkin and Shevelev [1984]; Greer et al. [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) (ii)

There exist clusters in the old phase which consist of different number n of molecules (or atoms) (n = 1, 2 . . . . ), 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 a r e 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 A/~) 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 nonuniformity 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 fnmZnwhich 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 n-sized cluster (as in Part 1, we keep treating the single molecules as clusters of size n = 1).

,,,.._ r ,,,..,,.._ =,,._

---,=

!

D

I i I .,, i ~,.,= I I I I i i

I

I I I I I I I I I I I I

I i I I

.,L ,qw

1

2

fmnZm

9.

.

n-2

i.q~--

fnmZn | |

a. w

n-1

9 w

n

,,

,,.._

~.

.k

,k

.._

w

w

v

v

n+ln+2

.

.

.

m

CLUSTER SIZE Fig. 9.1

Schematic presentation of the possible changes in the size o f a cluster o f n molecules. T h e c o n c e n t r a t i o n o f n-sized clusters diminishes because o f n --->m transitions (the arrows leaving size n) and increases thanks to m --~ n transitions ( t h e 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 fnm(t) 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 Z~(t) of n-sized clusters will be diminished per unit time by the quantity M(t)

~., fnm(t)Zn(t)

m=l

where M(t) 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 M(t) ~., fmn(t)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 nonaggregative processes occurring at certain rates which we shall denote by Kn(t) and Ln(t), respectively. On the other hand, the change of Z~(t) per unit time is expressed mathematically by the derivative dZn(t)/dt. The balance between the above quantifies thus leads to the sought master equation of first-order phase transitions in one-component systems [Kashchiev 1984a] (n - 1, 2 . . . . . M) M(t)

d--dZn (t) = m~,= l dt

[fmn(t)Zm(t) -- fnm(t)Zn(t)]

+ Kn(t) - Ln(t).

(9.1)

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 Zn(t). The solution Z,(t) of the master equation is unambiguous and physically acceptable when it satisfies the initial condition (n = 1, 2 . . . . . M) Zn(0) = Z.,0

(9.2)

where Zn,o is the (a priori known) cluster size distribution at the initial moment t = 0. Clearly, due to mass conservation, Zn(t) and M(t) are connected by the relation m(t)

]~ nZn(t) = m(t)/V

(9.3)

n=l

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 metastable) state into any other timeindependent (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) ~ O, Ln(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. I984; Zinke-Allmang et al. 1992] is an example of a process occurring in such a system: then usually K,,(t) = Ln(t) = 0 for n > 2, and K 1 and Ll are given by the rates of monomer impingement onto and desorption from the substrate, respectively. For a closed system, Kn(t) = L,(t) = 0 for any n so that M = constant. Equation (9.1) then becomes [Kashchiev 1971] (n = 1, 2 . . . . .

M) M

d Z , (t) = ]~ [fmn(t)Zm(t) - fnm(t)Zn(t)], dt m=l

(9.4)

eq. (9.2) remains the same and eq. (9.3) changes to M

~, nZn(t) = M / V (or M/As) = constant.

(9.5)

n=l

A great simplification of the master equation (9.4) for closed systems is achieved by assuming that the transition frequencies fnm a r e 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 Ap, since only then c a n fnm be constant with respect to t provided that the clusters gain and lose only monomers whose concentration Z1 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 fnm c a n 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 frequenciesf~m. 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 - 3 s - 1 or m - 2 s - l for HEN on a substrate) defined by [Kashchiev 1971, 1974, 1984a] (n = 1, 2 . . . . . jn(t) =

M(t) E

~

m=m'+l m=m"

M)

[fm'm(t)Zn (t) - fmm'(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, t-dependent netflux of clusters along the size axis. Figure 9.2 illustrates this perception: the arrows connect all sizes m' < n with an arbitrary size m > n, which means that the sum of all of them symbolizes the partial net flux gl

Jn,m(t) = ]~ [fm'm(t)Zm (t) -from (t)Zm(t)] m'=l

q

lm mi b mm

'4 - -

i

I I I , !

--~

,' fmnZm

I I

,&

.k

.k

1

2...m'...n-1

.k

.L

"',, fnmZn" .I,,.

! .k w

n n+l

m

C L U S T E R SIZE Fig. 9.2

Schematic presentation of the net flux j,,,,, through size n. This flux is the of the forward and backward fluxes (the arrows) due to m" --->m and m --->m" transitions with m" values from 1 to n.

sum

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 Jn,m over all sizes m > n yields the total flux jn(t) from eq. (9.6) which parallels the formula used by Katz et al. [ 1966] with time-independent transition frequencies f,m. With the help ofj~ 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 J,-1-Jn, after some algebra we find that (J0- 0)

Master equation 121 M j,,_ l(t) - j n ( t ) = Z

~n,,(t)Zm(t) - fnm(t)Zn(t)]

m=l

so that setting equal the left-hand sides of this equation and eq. (9.4) results in [Kashchiev 1974, 1984a] (n = 1, 2 . . . . . M) d Z,,(t) = Jn-1 (t) - jn(t)" d"-'t

(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 J~-l -J,,, this 'motion' is controlled by the frequencies f,,, 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 D6ring 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 [ 1942] 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 [Zettlemoyer 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(t))

cgt

Z(n, t) =

f

M(t)

~1

[f(m, n, t)Z(m, t) - f(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

f

M(t) nZ(n, t) dn = M ( t ) / V (or M(t)/As)

(9.9)

where Z(n, t) - Z,,(t), f(n, m, t) -fnm(t), K(n, t) - Kn(t) and L(n, t) - Ln(t). Accordingly, for a closed system the master equation (9.4) reads [Kashchiev 1971] (1 < n _<M) c9 Z(n, t) = ~1M [f(m, n, t)Z(m, t) - f(n, m, t)Z(n, t)] dm, Ot

(9.10)

122 Nucleation:Basic TheorywithApplications and in eq. (9.9) M/V (or M/As) is constant with respect to t. Again for a closed system the flux j(n, t) = in(t) from (9.6) becomes [Kashchiev 1971]

j ( n , t ) = ~,M{~ln [f(m', m,t)Z(m', t ) - f ( m ,

n

m', t)Z(m,t)] dm" } dm (9.11)

and the master equation (9.7) takes the standard form of continuity equation [Kashchiev 1974] (1 __ n _<M)

3t

Z(n, t) = - ~-~--j(n, t)

~n

(9.12)

upon using the truncated Taylor expansion

j(n-l,t)=j(n,t)+[--~nJ(n,t)][(n-1)-n] of j(n - 1, t) 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

<<

2fnm,mint

(9.13)

where fnm,min denotes the smallest among the transition frequencies fnm" This condition is based on the rigorous formula ofRee et al. [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 = o o . 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 firstorder 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 thermodynamically stable or metastable state to another such state when this process

Master equation

123

does not involve formation of a macroscopically 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) (2) (3)

early (nucleation) stage, advanced (coalescence) stage, 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 . . . . ) f,m(t) = 0

(9.14)

for in - 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 f,m(t) 0 only for In - ml = 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 , . . . )

in-1 ,

JR

,

gn+lZn+l

gnZn i

>

,

fn.lZn.1

fnZn

,,,

.k

n-1

'

v

.

lp'

n

'

.i,, lw'

n+l

CLUSTER SIZE Fig. 9.3

Scheme of the 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

[Zl] + [Z~] ~

[Z~ +~]

125

(9.15)

where [Z1] and [Zn] are the 'chemical' formulae of the monomers and the nsized clusters, respectively. Equation (9.15) is of great physical significance, for it is an expression of our concept concerning the m o l e c u l a r m e c h a n i s m 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. l) to (n = 1, 2 . . . . . M) d__ Zn(t) = fn l(t)Zn z(t) _ gn(t)Zn(t) _ f n ( t ) Z n ( t ) dt -

+ g~ + l(t)Zn + l(t) + K ~ ( t ) - Ln(t)

(9.16)

where fn(t) -f,,,n+l(t) and g~(t) - f n , ~ - l ( t ) are, respectively, the frequencies of monomer attachment to and detachment from an n-sized cluster (by definition, here and everywhere below f0 = 0, g l - - 0 and ZM+ 1 - 0 ) . Equation (9.16) is the master equation of nucleation in open systems. For closed systems Kn(t) = Ln(t) = 0 and it becomes (n = 1, 2 . . . . . M)

~

Zn(t) = f n - l ( t ) Z n _ l ( t ) - gn(t)Zn(t) - f n ( t ) Z n ( t ) + gn+l(t)Zn+l(t).

(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 frequenciesfn and gn, 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) d-d-Zn(t) = fn 1Zn dt .

l(t) - gnZn(t) - f n Z n ( t ) + gn + 1Zn + l(t) . . .

(9 18)

formulated first by Tunitskii [1941] with time-independent f, and g,. As already noted, this independence of the transition frequencies on time implies applicability of eq. (9.18) only to isothermal nucleation at constant supersaturation. 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 flux j,,(t) from (9.6) contains only two summands (see the white arrow in Fig. 9.3), i.e. (n = 1, 2 . . . . . M) j , ( t ) = f n ( t ) Z ~ ( t ) - g,+l(t)Zn+l(t).

(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 ] and Zeldovich

126 Nucleation: Basic Theory with Applications [ 1942] to transform eq. (9.18) into a partial differential equation of FokkerPlanck 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 gn, it has found a much wider use in the theory of nucleation. Denoting fn(t) by fin, t) and g,(t) by g(n, t), let us follow Tunitskii [1941] and approximate f,_l(t)Z~_l(t) and g,+l(t)Zn+l(t) by the truncated Taylor expansions about point n:

f (n - 1, t)Z(n - 1, t) = f (n, t)Z(n, t) + -~n [f(n, t)Z(n, t)] [(n - 1) - n]

+~

~ n 2 [f(

n, t)Z(n, t)]}[(n - 1) - n] 2,

g(n + 1, t)Z(n + 1, t) = g(n, t)Z(n, t) + -~n [ g ( n , t ) Z ( n , t ) ]

+-~

[(n+ 1 ) - n ]

}

-~Tn2 [g(n,t)Z(n, t)] [(n+ 1 ) - n ] 2.

Replacement of the first and the fourth summands in (9.16) by the right-hand sides of these expressions leads readily to (1 _
c)t

Z(n, t)=-O--~-([f(n, t) - g ( n , t)]Z(n, t) dn lv 2 On {[f(n, t)+ g(n, t)]Z(n, t)} + K(n, t ) - L(n, t). (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 frequenciesf(n,t) =fin) and g(n, t) = g(n) 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 f - 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

v(n, t)= f(n, t ) - g(n, t).

127

(9.21)

In real terms, v(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, t h e f + 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' (1/2)If(n, t) + g(n, t)] 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 f 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)

f~(t)C,(t) - g,+ l(t)Cn+ l(t) = 0

(9.22)

and employ this function to exclude g,(t) and g, + l(t) from eq. (9.16). The resulting equivalent master equation contains C~(t) instead of g~(t) and reads [Kashchiev 1969b] (n - 1, 2 . . . . . M)

d zn(t) = fn-l(t)Cn-l(t) [Zn l(t)/Cn_l(t) - Zn(t)/Cn(t)] dt - f~(t)C.(t)[Z.(t)/C.(t) - Z~+ j(t)/C~+l(t)] + K.(t) - L.(t). (9.23) Denoting Cn(t) by C(n, t), we can use twice the truncated Taylor expansion

Z(n + 1, t)/C(n + 1, t) = Z(n, t)/C(n, t)

to approximate the first and the second bracketed finite differences in (9.23) by the partial derivatives - O[Z(n - 1, t)lC(n - 1, t)llOn and - O[Z(n, t)lC(n, t)]lOn, respectively. Then, the so-obtained new finite difference can again be represented as a derivative with the aid of the analogous Taylor expansion

f ( n - 1, t ) C ( n - 1, t) ~

un

[ Z ( n - 1, t ) l C ( n - 1, t)]

= f(n, t)C(n, t)-~n [Z(n, t)/C(n, t)]

128

Nucleation: Basic Theory with Applications

We thus arrive at the master equation of nucleation (1 < n < M)

0_0Z(n, t) =

f(n, t)C(n, t) -~n [Z(n, t)lC(n, t)]

Ot

t

(9.25)

+ K(n, t) - 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, t) = c?t

f(n, t)C(n t) -~n [Z(n, t)lC(n, t)] . '

(9.26)

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 g(n, t) (the latter through C(n, t)) are allowed to change with time t. This ability of eq. (9.26) was used by Ptischl 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 timeindependentf(n, t) =fin) and C(n, t) = C(n), which corresponds to isothermal nucleation at constant supersaturation, eq. (9.26) takes the form of the known equation of Zeldovich [ 1942] (1 ___n _<M)

0 Z(n, t) =

-gi

f ( n ) C ( n ) -~n [Z(n, t)lC(n)]

9

(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) when jn(t) from (9.19) is represented as [Kashchiev 1969b]

j(n, t) = -f(n, t)C(n, t) -~n [Z(n, t)/C(n, t)].

(9.28)

This result forj(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 ) - f ( n , t) ~

Z(n, t)

(9.29)

6in

where (9.30)

Master equation 129 Comparing (9.20) and (9.25), we then note that j(n, 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 'motion' along the size axis. Indeed, the clusters of various sizes are not of equal number and the existing concentration gradient OZ/On causes cluster 'diffusion' with a 'diffusion coefficient'fin, t) 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, t-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' fin, t)/kT on the n axis and the 'force' Op/0n acting on the cluster because of the non-uniformity of the cluster 'chemical potential' p - kT In C(n, t) along the size axis. As already pointed out, in real terms v (n, t) is nothing else but the growth rate dn/dt of the n-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' (1/2)[f(n, t) + g(n, t)] and fin,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 g~ 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 fn, g,, and Cn. To emphasize that such a priori physical argumentation is completely unnecessary for the definition of C,(t) with the help of eq. (9.22), so far we did not address the question about the physical meaning of C,,(t) and will not do that until Chapters 10 and 12. Drawing attention to the fact that Cn(t) is free of whatever physical assumptions concerning the nucleation process is worthwhile because of the widespread incorrect opinion that the introduction of C,,(t) 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: C,,(t) is merely the solution of (9.22), given by [Kashchiev 1974, 1984a] (n = 2, 3 . . . . . M) -

Cn(t) = Cl(t)[fl(t)f2(t) . . .fn_l(t)/g2(t)g3(t) . . . g,,(t)],

(9.31)

and is mathematically well defined as long as its value Cl(t) at n = 1 and the transition frequencies fn(t) and gn(t) a r e 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 n, t dependence offn(t) 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) are 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, f~(t) is also known). As we shall see

13o Nucleation: Basic Theol. with Applications in Section 10.2 and Chapter 12, C,(t) has the physical significance of cluster concentration. The most notable with respect to C,(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 quantifies 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(1, t) = Zl(t)

Z(M, t) = 0.

(9.32) (9.33)

Equation (9.32) requires knowledge of the actual concentration Z1(t) 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 fnm of forward cluster-size transition, from size n to a larger size m, as proportional to the concentration Zm_ n(t) of the clusters of m - n molecules (m > n):

fnm(t) = ~On,m_n(t)Zm_,~(t). (9.34) Here the frequency factor cO.,m_. (m3/s or mZ/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 tOn,m_n 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 (.On,m_n = (Dm_n,n. 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 fmn(t) = 0,

(9.35)

but only for m - n > 1. Substitutingf~m andfm, from (9.34) and (9.35) in eq. (9.1) thus results in ( n = 1,2 . . . . . M)

--~Zn(t) = fn-l(t)Zn-l(t) - gn(t)Zn(t) - fn(t)Zn(t) + gn+ l(t)Zn+ 1(t) n-2 1 + ~ m~=2 r

M(t)-n ~, O)nm(t)Zm(t) n_m(t)Zn_m(t)Zm(t) - Zn(t) m=2

(9.36)

+ K n ( t ) - Ln(t). Here f,, given by fn(t) -- fn,.+ l(t) =COn,l(t)Zl(t),

(9.37)

and gn =f,,~- 1 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 M(t) ~,

m=n+2

O)n,m_n( t) Zm_n ( t ).

As before, by definition f0 - 0, g l - 0 and ZM+I = O. 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 in open systems. For closed systems K~(t) = L,(t) = 0 and with M = oo 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

132

Nucleation: 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 et al. 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 fourf,,,g~ terms in (9.36) can be represented as - Oj(n, t)/On with j(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) c) Z(n, t) =

f ( n , t) -~n Z(n, t) - v(n, t)Z(n, t)

3t

+1

to(m, n

-

m, t)Z(n - m, t)Z(m, t) dm

- Z(n, t) fl ~t~')-'' to(n, m, t)Z(m, t) dm + K(n, t)

-

L(n, t)

(9.38) in which to(n, m, t) = tonm(t), and v(n, t) - dn/dt is the monomer-mediated growth rate of the n-sized clusters. Equation (9.38) describes simultaneous nucleation (the 0/o~ term) and coalescence (the integral terms) in open systems and for that reason it 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, t) 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 g,, 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 [Overbeek 1952; Voloshchuk and Sedunov 1975] (n = 1, 2 . . . . . M)

M a s t e r equation

133

n-1

d Z, (t) = 1 ~ d--7 2 , =1

r

Z,_m(t)Zm(t)

' M( t ) - n

- Z,(t)

Z

r

)Zm(t ).

(9.39)

m=l

This set of ordinary differential equations was derived by von Smoluchowski [ 1916, 1917] with time-independent OOnmand M = ,,o 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]) (1 < n < M): O Z(n at

t) = 1

og(m, n - m, t)Z(n - m, t)Z(m, t ) d m

' M-n

- Z(n, t)

~1

oo(n, m, t)Z(m, t) dm.

(9.40)

With time-independent frequency oJ(n, m), M = ~o 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; M6tois 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 O s t w a l d r i p e n i n g 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 Ostwaldripening regime and, recalling that K , ( t ) = Ln(t) = 0, we obtain [Kashchiev 1969b] (n = 1, 2 . . . . . M) d--d-z,,(t) = f~ l(t)Zn l ( t ) - gn(t)Zn(t) -- fn(t)Zn(t) + g,, + l(t)Zn + 1(t).(9.41) dt -

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): r) Z(n, t) Ot

f(n

t) ~

]

Z(n, 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 Pfischl and Aubauer [1980] to study the overall kinetics of transition of thermally treated alloys to different thermodynamically stable states. It must be pointed out, however, that when eq. (9.26) or (9.42) is employed for analysing solely the stage of ageing, it can be simplified essentially by neglecting the 'diffusion' flux -fi)Z/tgn with respect to the 'drift' flux oZ. Hence, provided that If z/ l<< Iozl, eq. (9.42) becomes [Todes 1946, 1949a] ~9 c9 Z(n, t) - [v(n, t ) Z ( n t)] 8t -~n ' "

(9.43)

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, i986; Flynn and Wanke i974; 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 f,,m play the same central role as that of the nucleation work W(n) in the thermodynamics of nucleation. In principle, knowingf,,m 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,t) 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 frequencyfn of monomer attachment and relatively briefly the frequency g,, of monomer detachment. As we know from Section 9.2, the frequencies fnm 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 f,,, for instance, when the old phase is gaseous. Figure 10.1 illustrates this mechanism of monomer attachment to condensedphase clusters in the cases of H O N and HEN on a substrate. In the case of HON (Fig. 10.1 a), according to the kinetic theory of gases, the number Zn of collisions per unit time between Z, spherical n-sized clusters and Zl monomers can be represented as [Moelwyn-Hughes 1961] (n = 1, 2 . . . . ) Z, = 47r(R + Ro)2[(kT/2tc)(1/m, + 1/mo)]l/2ZnZl

(10.1)

Fig. 10.1

Direct-impingement mechanism of attachment of molecules (the circles) to (a) spherical cluster 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 = (3Oo/4~) 1/3 and m0 are the molecular radius and mass, respectively. As by definition fn = ~ z , / Z n , taking into account that the radius R and the mass m, of the n-sized cluster are given by R = Ro nil3 and m n = nmo, from (10.1) it follows that (n = 1, 2 . . . . )

f,(t) = ~ncvZ/3(kT/2tCmo)l/2[(n + 1)/n]]/2(nl/3 + 1)2Zl(t).

(10.2)

Here c - (36zc) 1/3, and ~, a number between 0 and 1, is the sticking coefficient which takes into account that some of the impinging monomers may not be attached to the cluster. When Z~ does not depend on time t, f, is constant with respect to time (provided that ~,, 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, ZI is nothing else but the equilibrium monomer concentration Cl 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 C1 from (7.5). As discussed by McDonald [1963], this assumption is based on the fact that ZI and C1 are very large numbers. They can thus coincide even though for a supersaturated system the actual and the equilibrium cluster size distributions Z~ and C(n) are increasingly different from each other with increasing n. Hence, identifying Z1 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 f, reads (n >> 1) fn(t) = )'nc 02/3 [p(t)/(21rmokT)1/2]n2/3.

(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 timeindependent attachment frequency f, represented as the product of the monomer sticking coefficient ~',, the Hertz-Knudsen impingement rate p/(27rmokT) ~/2 of monomers and the area 4~rR2 = c v 2/3 n 2/3 o f 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 = (361r)1/3 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. lb) we have

fn(t)

= )in[ (1 - COS

Ow)/211t2/3(Ow)](36~o~ ) l/3[p(t)/(2~mokT) 1/2] n 2/3, (10.4)

since the lateral area of the cap is 2zrR2(1 - c o s 0w) = (361ro2)l/3n2/3(1 - c o s 0w)/2 gt2/3(0w), gt(0w) being given by eq. (3.56). As required, eq. (10.4) passes into (10.3) at 0w = n: (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. 10.1b). 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 JrR2 sin 2 0w = (36rco~)l/3n 2/3 sin 2 0w/41F2/3(0w) o f the cap and (10.4) changes to (0w < ~2) f,(t) = )'n [sin 20w/411t2/3(Ow)](361ro2)l/31(t)n 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 f, 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 f, also in the case of 2D HEN of condensed-phase clusters on a substrate when the clusters attach only molecules striking their periphery (Fig. 10.1d). Considering clusters of monoiayer height, for the attachment area at the periphery of an n-sized 2D cluster we have dobn 1/2 so that the analogues of (10.4) and (10.5) read (n = 1, 2 . . . . ) fn(t) = ~nbdo[p(t)/(27rmokT) 1/2] n 1/2

(10.6)

fn(t) = ~'nbdol(t) n 1/2.

(10.7)

As in Section 3.2, here bnl/2 is the length of the cluster periphery, b is a shape factor (e.g. b = 2(It.a0) 1/2 for disks, b = 4a~/2 for squares, etc.) and it is assumed that the width of the attachment area is equal to the molecular diameter do. Equations (10.3)-(10.7) show that if the sticking coefficient 7, 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 o f f n on n calculated from eq. (10.3) for HON of spherical water droplets in vapours at T = 293 K under constant pressure p = 2pe (curve 2) and 4pe (curve 4). The parameter values used are those listed in Table 3.1. Figure 10.2 illustrates also the linear increase o f f , with p or I, which is equivalent to an exponential increase with Ap, since p - Pe exp ( A p / k T ) and I = I e exp ( A p / k T ) (see eqs (2.8) and (2.9)).

8x109

4

6x109 '7, t/) = 4x109 v

j2X109

0

,

1

,

.

.

I

,

100

..,

,

I

200

9

.,

,

9

I

i

300

'

'

.i

I

400

,

I

,

,

I

500

9

i .

..

,

600

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 c u r v e - detachment frequency g, under evaporation control according to eq. (10.72).

140

Nucleation: 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 ~,, on n. The problem of finding T,, as a function of n is an important problem p e r se, but is beyond the scope of our considerations. In principle, ~ 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, T,, 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 ),, [Bliznakow and Kirkova 1957; Bliznakow 1958; Hirth and Pound 1963]. Most generally, ~,, 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 ~,, as a product of the coefficients ~,,, and ~,,,, also numbers between 0 and 1, accounting separately for the effects of thermal accommodation and surface status, respectively: T,, = fi,,,Ts,,,.

(10.8)

The above considerations concern the frequency f, 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 1 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 f, is physically equivalent to the frequency g,, of monomer detachment from a condensed-phase cluster in vapours. For that reason, analogously to g, from (10.68), in the case of evaporation control f, in HON of a gas phase is given by eq. (10.3) with v 0 replaced by the volume k T / p , occupied by a molecule in the n-sized gaseous cluster and with p replaced by Pe,,: f~(t) =

~',c [pe,n/(27cmokT) l/z] [kT[pn(t)] 2/3 n 2/3.

(10.9)

Classically, the pressure p, inside the cluster and the cluster vapour pressure

Pe,n 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 f, 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.

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 f,, is given by f , ( t ) = "ynJd,n(t)4~R 2

(10.10)

where Jd,~ (m-2 s-l) is the incoming diffusion flux of molecules to the surface of the n-sized cluster of radius R and surface area 4~R 2 = c o~/3 n 2/3, c being equal to (36t01/3. The diffusion flux is calculated by the Fick formula Jd,n = D(cgZr[tgr)r=R

(10.11)

in which D (m 2 s-1) is the coefficient of monomer diffusion in the solution. Equation (10.11) shows that finding Jd,,, requires knowing the concentration Zr(t) (m- 3) of monomer solute as a function of time t 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]

(r 2~a Zr) "]

(10.12)

Zr(t) = 0 at r = R, Zr(t) = Z1 at r = oo.

(10.13)

o~Zr __ D c9 at - r 2 Or under initial and boundary conditions Z r ( t ) -- Z 1

at t = 0,

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 Z1 (m -3) in the bulk of the solution (i.e. at r = oo) 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 ) = ZI(1 - (R/r){ 1 - erf [ ( r - R)/(4Dt)I/2]})

(10.14)

where erf (x) = (2//r 1/2)

exp (- x '2) dx'

(10.15)

is the error function [Korn and Korn 1961 ]. From eqs (10.11) and (10.14) we thus find that Jd,n --

(DZIlR)[ 1 + (R2/~Dt)1/2].

(10.16)

This expression shows that after time t > R2]D the diffusion flux becomes time-independent and takes the stationary value

Jd,n = DZIlR

(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 gruEls 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 Jd,n may also be acceptable. Hence, introducing Jd, from (10.17) in (10.10) leads to

fn = ~'n4rCRDZI.

(10.18)

This relation can be generalized to cover the case of nucleation under conditions of variable concentration Zl(t) of the monomers in the bulk of the solution. If the variation of Z1 with time is sufficiently slow, in quasi-stationary

Transition frequencies 143 approximationjd.n is again given by (10.17), but with Zl = Zl(t). As to Z1, as already discussed, we must set it equal to the equilibrium monomer concentration C1 and calculate it from eq. (7.5) in which according to (7.12) Co = l/v0, and due to (2.14) and (3.44) W1 = A - k T l n (C/Ce) for sufficiently dilute solutions. Recalling that to a good approximation the solubility Ce and the molecular heat ;1. of dissolution are related by [Moelwyn-Hughes 1961 ] C e ~- C 0 e -21kT,

( 10.19)

we find that Z 1 -" C. This is an expected result, because in the case considered Z1 is just another notation for the concentration C of dissolved molecules. Thus, since for condensed-phase clusters R - (c/4rc)l/2vl/3n 1/3, in quasistationary approximation f,, from (10.18) can be represented as (n >> 1) f,,(t) = 7,,(4 ~C) 1/2 O~/3 DC(t)n 1/3

(10.20)

where c = (36zc) 1/3. In deriving eqs (10.18) and (10.20) we have treated implicitly the n-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 7/is sufficiently low, since the diffusion coefficient D,, of the cluster is determined by the equation [Einstein 1905]

D n - kT/6tcrlR.

(10.21 )

That is why a more accurate formula for f~ requires accounting for the diffusion motion both of the cluster and of the monomers whose diffusion coefficient is D = kT/6zcORo. 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 Dn + D and the effective radius R + R0. The latter change allows for the non-zero radius R o of the diffusing molecules. Hence, fn from (18) becomes

f,-

Tn4rC(R + Ro)(Dn + D)Z1.

(10.22)

With ~,, = 1, this is the von Smoluchowski formula for f,, when monomer attachment is controlled by volume diffusion (e.g. Overbeek [1952]). To express fn from (10.22) in terms of n for condensed-phase clusters we can use (10.21) in the form Dn = DRo/R. Recalling that R = Ro nil3, R 0 = (3v0/ 4/r) 1/3 and Z1 = C, we thus find (n = 1, 2 . . . . )

fn(t) = 9/'n(41rc)l/2vl/3DC(t)(n-1/3 + 1)(nl/3+ 1)

(10.23)

where c = (36n:) 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 off,, 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 = (367r) 1/3 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 nsized cluster. As these clusters compete to capture diffusing monomers, fn 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 fn increases with increasing cluster size n and concentration C of monomers in the bulk of the solution. If 7n = constant, the dependence offn 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 Ap, since due to (2.14) C = Ce exp ( A p / k T ) . The solid curves in Fig. 10.4 illustrate fn from (10.20) as a function of n at two fixed concentrations C = 5C e (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 ~, 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))i,n = 1 is a reasonable approximation. 3x107

i

2x107

' ,

"~ "

,,-'"

~" "

.,= ~

5

lx107

0

1

100

200

300

400

500

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 fn under volume-diffusion control according to eq. (10.20) at C/Ce = 5 and 10 (as indicated); dashed curve detachment frequency g, 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 directimpingement control, if the condensed-phase clusters are cap or disk shaped (Fig. 10.3b and c), fn 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 0w (e.g. 0w > rd4) eq. (10.20) can be used in the form

f,(t)

= 7'n[(1 - c o s

Ow)/lltl/3(Ow)](61r2vo)l/3DC(t)nl/3

(10.24)

which accounts for the fact that the lateral area of the cap is 2zrR2(1 - cos 0w) and that R = [3Vo/41rllt(Ow)]l/3n1/3. This equation is the analogue of (10.4). As required, at 0w = Jr (complete non-wetting) it passes into eq. (10.20). (3) Surface-diffusion control Surface diffusion can be the transport mechanism controlling fn 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 = ds2 Vs exp (-Esd/kT)

(10.25)

Fig. 11).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 nm) is the length of a molecular jump along the substrate surface, and Esd 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 TheorywithApplications incoming diffusion flux Jd,n ( m-1 s-l) of monomers to the cluster periphery. Indeed, analogously to (10.10), now f,, is given by f~(t) = Tja,~(t) 2zrR'

(10.26)

where R' is the radius of the cluster base (R" = (3o0/471:)1/3 [sin 0w/ V1/3(Ow)]nl/3 for cap-shaped clusters (Fig. 10.5a) and R' =- R = (ao/rC)l/2nl/2 for disk-shaped clusters of monomolecular height (Fig. 10.5b)). The incoming diffusion flux can be calculated from the Fick formula Jd,n = Ds(~

(10.27)

= R'

which is the 2D analogue of (10.11). Here Zr(t) (rn-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 ]

OZr -Ot -

Ds 3 r Or

( r t~zr )

+ I-~

~

Zr

(10.28)

"Co

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 Vdis the resorption time (i.e. the mean residence time of a molecule on the substrate), so that Zr/Vd is the local desorption flux of molecules. This flux is greater at higher substrate temperature, because v0 decreases with increasing T according to [Lewis and Anderson 1978] Vd= (l/Vs) exp (Edes/kT)

(10.29)

where Edes is the activation energy for resorption. Like (10.12), eq. (10.28) also has a stationary solution satisfying the initial and boundary conditions (10.13) with R replaced by R'. As in the volume-diffusion case, this solution is obtained from (10.28) upon setting OZrlOt = 0 and may be expected to hold for t > R'Z]Ds . This gives t > 1 ps with R' = 1 nm and a rather low surface-diffusion coefficient Ds = 1 pmZ/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 - Zl [ 1 - Ko(r/As)/Ko(R'/As)]

(10.30)

where K0 is the zeroth-order modified Bessel function of second kind. The quantity As, defined by the Einstein-type formula ~s = (DsVd)l/z,

(10.31)

is the mean diffusion distance travelled by an adsorbed molecule before desorption (typically As = 1 nm to 1 pm).

Transition frequencies 147 Employing Z,. from (10.30) for the calculation of the derivative in (10.27) leads to (e.g. Vetter [ 1967]; Sigsbee [ 1971 ])

Jd,n = (DsZI/As)[KI(R'/As)/Ko(R'/As)]

(10.32)

where K1 is the first-order modified Bessel function of second kind. In quasi-stationary approximation this formula for Ja,n can be used with timedependent concentration Zl(t) (m -2) of the adsorbed molecules far away from the cluster (i.e. at r = ,,o). In this approximation, from (10.26) we thus find that for both caps and disks

f,(t) = N2XDsZ~ (t)(R'/2s)[Kl (R'//q,s)/Ko(R'/2s)].

(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 K0 and K1 functions, we can simplify (10.33) to

L(t) = 7'~2ZrDsZl (t)/ln (2s/R')

(10.34)

for small clusters (R' << As) and to

f~(t) = ~'n2rCDsZ~(t)(R'/&s)

(10.35)

for large clusters (R' >> As). However, as shown by Kashchiev [1978], we can use the single approximate formula

2rc(R'/&s)[Kl(R'/2s)/Ko(R'/2s)] = 1.3 + 2zcR'/2s for clusters of any size as long as R'/As > 10-4. Since typically only clusters of sizes satisfying the condition 10-4 < R'/)~s < 1 are involved in the nucleation process, the above result means that for both caps and disks f~ from (10.33) can be approximated by the simple expression (n = 1, 2 . . . . )

f~(t) = ?'nc*DsZl(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 1OR' [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 fn [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 and Kashchiev [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 f~, just as in the case of volume-diffusion control, we must replace Ds 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 (Os, n is the surface-diffusion coefficient of a cluster of n molecules). As a result, since typically R' + R0 << As and Ds,n < Ds, in accordance with (10.34) and (10.36)f~ 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 fn also when the clusters on the substrate are mobile (in this casefn is given more accurately by eq. (10.95) at m - n = 1). It remains now to express Z1, the concentration of adsorbed monomers far away from the cluster, in terms of experimentally controllable parameters such as the impingement rate I 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, Z1 equals the equilibrium monomer concentration C1 determined by eq. (7.5) so that, using (3.86) and (3.89) with A1 = a0, we have Z 1 "- C 1 = C O

exp [(A/I -

~, + E 1 +

trsao)/kT].

(10.37)

This formula, in which Co = 1/ao in accordance with (7.8), shows that Z 1 is proportional to exp (Ap/kT), i.e. to p, I or C in the cases of Ap 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 C1) during nucleation. The increase of Z1 from zero at the moment t = 0 of putting the bare substrate in contact with the old phase to the time-independent concentration C1 corresponding to adsorption equilibrium at t = oo is governed by the differential equation (e.g. Lewis and Anderson [1978])

dZl(t)/dt = I - Zl(t)/'r d.

(10.38)

Transition frequencies

149

This equation quantifies the balance between the adsorbing (/) and the desorbing (Z]/va) 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 Z](0) = 0 reads: Zl(t) = Izd[1 - exp (- t/'t'd)].

(10.39)

This formula tells us that if HEN on the substrate occurs for t << Zd, Z1 increases linearly with time according to (e.g. Lewis and Anderson [ 1978]) Zl(t ) = It.

(10.40)

Conversely, for t >> zd, the concentration of adsorbed monomer is constant and given by (e.g. Lewis and Anderson [1978]) Z] = C1 = lVd = (lelVs) exp [(Zip + Edes)/kT].

(10.41)

This is the concentration of monomers on the substrate at adsorption equilibrium and it corresponds to that from (10.37) with Co from (7.8) and Ap from (2.9). This correspondence leads to the conclusion of general validity that, to a first approximation, the energy E1 + Crsa0 is merely the monomer desorption energy Edes. Indeed, since I e --- CoY s exp (- 2/kT) [Lewis and Anderson 1978], comparison of the r.h.s, of (10.37) with that of (10.41) yields E1 + Crsa0 = Edes 9

In many cases nucleation is observed experimentally over periods of time much longer than Zd. Combining eqs (10.31), (10.36) and (10.41), we find that in these cases f,, is given by the formula [Stowell 1974a; Lewis and Halpern 1976; Lewis and Anderson 1978] (n = 1, 2 . . . . ) fn = 7,c*DsZdl = YnC*/],21

(10.42)

which applies to clusters of arbitrary shape. Physically, this formula means that if ~n = 1, nearly all molecules impinging upon the diffusion zone around the cluster (the area of this zone for caps or disks is about 7rA,s2 when R' << As) are attached to the cluster independently of its size n. Equations (10.39)-(10.42) describe ZI andfn 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 p / ( 2 z m o k T ) 1/2. For example, in quasistationary approximation, eq. (10.42) then becomes (n = 1, 2 . . . . ) fn(t) = ~ c * D s C o K p p ( t )

(10.43)

Kp = Zd/Co(2zmokT) 1/2

(10.44)

where

is the Langmuir constant of adsorption [de Boer 1953]. Naturally, eq. (10.43) is readily obtainable from (10.36) with Z1 determined from the low-coverage form of the Langmuir adsorption isotherm (e.g. de Boer [1953]) ZI/C o = CI/C o = Kpp

(10.45)

150 Nucleation: Basic Theory with Applications

where C1/Co is the coverage of the substrate by monomers in adsorption equilibrium, and Co is given by (7.8). Similarly, when the substrate is immersed in a solution with concentration C of monomer solute, after time t >> ~rd,in quasi-stationary approximation f,, is determined by (10.46)

f~(t) = 7'nc*DsCoKcC(t)

where Kc is the constant in the low-coverage Langmuir adsorption isotherm [de Boer 1953] (10.47)

Z l / C 0 = C1/C 0 = g cC.

Covo = do, eq. (10.47) shows that when K~ = v0, the volume concentrations Cl/do and C of solute in the adsorbed monolayer and in the bulk of the

Since

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 f~ on n at two constant pressures p = 2pe (lines 2) and 4pe (lines 4). The calculation is done for hemispherical water droplets (for them R' = (3Vo121r)l/3n1/3) 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 As = 50 nm. 2x1012

tl

" iI

......

"7

O0 E

E~

l x l 012

k ~

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

, _ _ _ . . . . . . . . . . . . . . .

2_

" ' : : : : : -

....

..................................................

,~

o.,

n*

I

1

I

:::

.9. . . .

B,A,

'

1

I

100

!

I

I

I

I

200

I

I

I

l

'

--

300

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 fn under surfacediffusion 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 fe, nfrom the exact and the approximate formulae (10.33) and (10.43), respectively.

Transition frequencies

151

In the calculation Z1 in (10.33) is expressed with the help of (10.45) and it is taken into account that DsCoKp = A,s2/(2 ZrmokT)1/2. As seen from Fig. 10.6, the dependence off, on n is indeed rather weak and is practically negligible in the n < 100 size range which is of interest in the theory of nucleation. The linear increase offn 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 fn to exp (Ala/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 fn 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 = ~r/2), from (10.4) and (10.43) we find that the ratio rDI/SDbetween f,, by direct impingement and f,, by surface diffusion is given by rDI/S D =

(2rdc*)(R/As) 2.

(10.48)

Since 27c/c* -- 1, this formula shows that surface diffusion dominates over direct impingement as long as R < As [Sigsbee 1971, 1972]. This condition is usually fulfilled, for example, in HEN during condensation of vapours on a substrate (then very often As > 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 R 2 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 rvo/so betweenfn by volume diffusion andfn by surface diffusion we find

rvo/so = (2zc/c*)(RD/CoKcDs).

(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 < CoKcDs/D. As in most cases only clusters of R < 10 nm are of interest for nucleation, with Co = 1/ao = 1019 m -2 and D = 10-9 m 2 s-i, this condition will be satisfied for substrate/solution systems for which KcDs > 10 -36 m 5 s -1. When Kc and/or D s are sufficiently small (i.e. when the adsorption coverage and/or the monomer surface diffusivity are low), KcDs < 10 -36 m 5 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 nucleation of liquids or solids in condensed phases (e.g. melts, solutions, etc.). In attachment controlled by interface

152 Nucleation: Basic Theory with Applications

transfer, the molecule to be attached is in immediate contact with the condensedphase cluster (Fig. 10.7) and can join the cluster by making a random jump over a distance comparable with its diameter do. The probability for such a jump is characterized by the jump frequency [Frenkel 1955] Vj, n -"

VO,n exp (- Eit,n]kT)

(10.50)

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,n ( = 1013 s-l) is the vibration frequency of a molecule of the old phase at the surface of an n-sized cluster, and Eit,n > 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 Ac, n, in the volume d0Ac,n around the cluster there are ZldoAc,n molecules that can be attached to the cluster (in interfacetransfer control, by definition, the monomer concentration Z1 is uniform fight up to the cluster surface). The attachment frequency fn is then merely the product of vj,,,, ZldoAc,,, and the sticking coefficient )'~ so that we have (n = 1, 2 , . . . ) fn = ~ v0,, exp (- Eit,n/kT)Z1 doac,,.

(10.51 )

Equation (10.51) is the general formula for f,, in the case of interfacetransfer control. For each particular cluster shape it is possible to determine the contact area Ac,~ as a function of n and thus find the dependence offn on the cluster size n if ~, v0,~ and Eit,,, 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 ~,, 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 7~, but also of v0,~ and Eit,,. 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 v0,n and Eit,~, the common approach [Volmer 1939; Frenkel 1955; Zettlemoyer 1969] is to accept that, approximately, Vo,n = constant = Vd and Eit,n - constant = Ed. Then, as the monomer diffusion coefficient is given by [Frenkel 1955] D = do2 VOexp (-Ed/kT),

(10.52)

from (10.51) it follows that (n = 1, 2 . . . . ) (10.53)

f , = y,,(DZl/do)Ac, n.

This expression is useful for nucleation in solutions, since then D is often known independently and, as already discussed, Zl is related to the solute concentration C which is an experimentally controllable parameter. Alternatively, employing the Stokes-Einstein formula (cf. eq. (10.21)) D = kT/3 ru/0r/

(10.54)

allowsfn to be written down in terms of the viscosity 7/of the phase (solution, melt, etc.) around the cluster (n = 1, 2 . . . . ): fn = Yn( k T [ 3 ~ d211)Z1Ac,n 9

(10.55)

This equation is particularly convenient for calculating fn for nucleation of crystals in melts, as then 7/may be known independently, and Z1 -- 1/Vo. With this approximation for Z1, eq. (10.55) was used in a number of papers considering this case of nucleation [Toschev and Gutzow 1967a; Toschev 1973a; Rowlands and James 1979; Gutzow et al. 1985]. The possibility of introducing D or 7/as a parameter determining fn for nucleation of crystals in melts was utilized also, e.g. by Kelton et al. [ 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]) r/(T) = 7"/0exp [ E v / k ( T -

To)]

(10.56)

where 7/0 is a practically T-independent factor, Ev is the activation energy for viscous flow, and To 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 Z1 = 1/o0 leads to the qualitatively incorrect result that f~ from (10.51), (10.53) or (10.55) does not depend explicitly on A/a. Indeed, if we want to be consistent with the postulate that Zl equals the equilibrium monomer concentration Cl, according to (3.86), (3.89), (7.5) and (7.12), in the case of liON of condensed phases (Fig. 10.7a) the correct expression for Z1 reads Z 1 -- C 1 -

(1/Oo)

exp [(Ap - A)/kT]

(10.57)

154 Nucleation: Basic Theory with Applications

since then E1 = 0 and trs = 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 capshaped clusters with such a large wetting angle 0w that attachment of monomers to their periphery can be ignored). In the case of 2D HEN of condensedphase 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, Z1 in (10.51), (10.53) or (10.55) is equal to C1/do where C1, given by (10.37), is the equilibrium concentration of monomers on the substrate. Hence, g l = Cl/do =

(Co/do) exp [(A/a - A, + El + trsao)/kT]

(10.58)

where Co/do = l/aodo = l/vo. From the above two equations we see that the correction to the commonly used approximation Z1 = l/vo for nucleation of crystals in melts is the exponential factor which introduces infn from (10.51), (10.53) or (10.55) the same dependence on A/a as that o f f , 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/a from (2.23) and with typical values of Aselk = 1 to 5 we find that exp [(A/z - ~)/kT] = exp (- Ase/k) -- 0.4 to 0.007. In eq. (10.58), however, this factor is a function of T and depends on the energy E1 + Crsa0 which, as noted above, is approximately equal to the monomer desorption energy Eden. Comparison of the last two equations shows that when E1 + Crsa0 = 0, the concentrations (in m -3) 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 fn, because in them D and 7/ 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 77in (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 v0,n and Eit,n to one, Eit. This can be done by ignoring the n-dependence of Eit,n, i.e. by assuming that Eit,n = constant = Eit, and by employing the approximation v0,n = constant = kT/hp of Turnbull and Fisher [1949] which follows from the theory of reaction rates [Glasstone et al. 1941 ] (hp is the Planck constant). Then (10.51) transforms into the expression (n = 1, 2 . . . . )

fn = ~'n(kT/hp) exp (- Eit/kT)ZldoAc, n

(10.59)

which, with 7~ = 1, Z1 = 1/Vo and size-dependent Eit, is the formula for f , 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 0 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 forfn for nucleation of condensed

Transition frequencies

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 Z1 = 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 . . . . )

f~(t) = 7,,(c v~/3 /do)DC(t)n 2/3

(10.60)

for HON of spheres (c = (36z01/3), cubes (c = 6), etc. and that (n = 1, 2 . . . . )

fn(t) = ~,[(1 - cos Ow)]21lt2/3(Ow)](co2/3]do)DC(t)n 2/3

(10.61)

for 3D HEN of caps (c = (36zr)1/3). As already noted, the use of Z1 = 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 Z1 = C1/do where Cl is specified by eq. (10.47). Hence,

Zl = CoKcC/do= KcC/vo

(10.62)

in the concrete case of low-coverage Langmuir adsorption (this equation for Zl is a particular case of the general equation (10.58)). With Ac,n = dobn 1/2, from (10.53) it thus follows that (n = 1, 2 . . . . )

fn(t) = ~,,(b/vo)DKcC(t)n 1/2

(10.63)

for disks (b - 2(/r~a0)l/2), square prisms (b = 4 a 1/2 ), etc. of monolayer height. Whenever necessary, in eqs (10.60), (10.61) and (10.63) D can be replaced by the solution viscosity 7/ with the aid of the Stokes-Einstein formula (10.54). It is worth keeping in mind as well that in these equations D (or 77) 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 0w = ~r (complete nonwetting), 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 ZI, from (10.55) we find that (n = 1, 2 . . . . )

f , = ~,,( co 2/3/3 zcd~ )(kT/oorl) exp [(A/j -/].)/kT]n 2/3

(10.64)

for HON (c is the shape factor appearing in (10.60)) and that (n = 1, 2 . . . . ) f, - ~,,[(1 - cos Ow)/211t2/3(Ow)](c v 21313tcd 2)(kTIvorl) exp [(Ap - A)/kT]n 2/3

(10.65)

for 3D HEN of caps (c = (36zc)1/3). Similarly, using Z1 from (10.58) in eq. (10.55) yields (n = 1, 2 . . . . )

f,, = ~,,(b/3zcdo)(kT/vorl) exp [(A/J - ~ + E1 + Crsao)/kT]n 1/2 (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 f, for caps in HEN is smaller than f~ for spheres in HON. According to (10.65), f~ diminishes with the decrease of the wetting angle 0w. When 0w ---> ~r (complete non-wetting), as required, f, from (10.65) passes into f, from (10.64) with c = (36zc)1/3 (HON of spheres). We note as well that without the exponential factors in them, eqs (10.64)-(10.66) correspond to the expression forf~ used by Gutzow et al. [1985] in the scope of the Z1 = 1/Vo approximation. Equations (10.60), (10.61), (10.63)-(10.66) tell us that whenfn 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 ~ with n is negligible). As under diffusion control, in the case of nucleation in solutions f~ increases linearly with increasing solute concentration C, i.e. it is again proportional to exp(Ap/kT). The same explicit exponential variation off, with Ap is in force also for nucleation of crystals in melts. The dependence off, 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), Ap 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 7/rather than to Ap. This sensitivity is particularly high when nucleation occurs in a temperature range in which

12x109 ..,.,

8x109 cg

4x109t,,

. o

1

:,:...,..

100

,.,,,,,... 200 300

1, .... 400

, 500

.

.

.

.

600

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 c u r v e s - detachment frequency gn 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 Tg 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 = ( ~ 6 ) d 3) we find that for HON of spheres the ratio rVD/ITbetween f~ controlled by volume diffusion and by interface transfer is given by rVD/I w = 2 n -1/3.

(10.67)

As seen, rvD/iT < 1 for n > 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 Eit,~ and Ed.

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 C), 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 fn- For that reason, in obtaining gn it is convenient to use indirect methods based on a priori knowledge offn. 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 f n - 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 g~ and fn- 1 are equal, we can obtain gn simply by identifying it withfn_ 1 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 Pe,n (see Section 6.3). The cluster/

158

N u c l e a t i o n : B a s i c T h e o r y with A p p l i c a t i o n s

vapour coexistence has a dynamic character and is maintained thanks to the equality in the number of attachments, f n - 1, and detachments, g,, per unit time. In view of (10.3), this balance between g, and f n - 1 at p = Pe,n yields (n >> 1) g n = ~ - 1c 02/3 [ P e , n / ( 2 ~ m o k T ) l / 2 ] (

n -

1) 2/3

(10.68)

which is the formula used already in the pioneering works on nucleation [Farkas 1927; Kaischew and Stranski 1934a; Becker and D6ring 1935]. Since this formula is obtained with f,, 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 fn in HON of a gas phase from its own condensed phase (see eq. (10.9)) corresponds to g,, from (10.68) with u0 replaced by kT/p~. With appropriate modification, eq. (10.68) determines gn 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 p~ inside the cluster (this is so because g, for gaseous clusters corresponds to fn for condensed-phase clusters in vapours). For that reason, with Pe,n changed to p, and u0 replaced by the volume kT/pn_ l 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 gn in HON of a gas phase from its own condensed phase: g,, = ?'n - l c(kT/pn - 1 ! ~2/3t,., klle,n - l / P * ) [ P n / ( 2 7 c m o k T ) l / Z ] ( n

-

1)2/3.(10.69)

Classically, p, is determined by the Laplace-type equation (6.4), Pe,, is given by (6.24), p* is specified by (4.14), and the factor Pe,,,- l/P* is necessary to ensure the equivalence of eq. (10.69) to eq. (10.90) when in the latterfn from (10.9) and W(n) from (3.42) are used for the determination of gn 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 Pe,~ 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 g, = f~_ l(Ce,,), we find that (n >> 1) gn -

~'n - l ( 4 1 r c ) l/2 oo/3 D C e , n ( n -

1) 1/3

(10.70)

for volume-diffusion control and that (n = 2, 3 . . . . ) gn = 7n - 1(c v 2/3 Ido)DCe,n(n - 1)2/3

(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 Pe,n , Pn and Ce,n 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 gn 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, Pe,n and Ce,n 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 = (367r) 1/3 for spheres, 6 for cubes, etc.) gn

= fe, n -

1 exp (2c V 2/3 a/3kTnl/3).

(10.72)

Here fe,n- 1 is the value of the attachment frequency f n - 1 at the equilibrium pressure Pe or concentration Ce (i.e. at A/.t = 0): fe, n - 1 = ")/'n- 1c 02/3 [ P e / ( 2 7 t r r t o k T ) l / 2 ] (

n -

1)2/3

(10.73)

for HON in vapours, and fe, n- I = )'n- I(4~7"c)I/2ol/3DCe(

n -

1) 1/3

(10.74)

or

fe, n - 1 = Tn - I(C V~/3 /do)DCe(n - 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 fn, gn is insensitive to changes in the actual monomer pressure p or concentration C as long as 7'n and cr are also constant with respect to p or C. This means that unlike fn, gn does not depend on the supersaturation A/~ when this is varied isothermally. With increasing n, gn 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/~) 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 fOrpe,n and Ce,n, its applicability to the smallest clusters is questionable. For that reason, in the size range of,

160 Nucleation: 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 gn for liquid or solid clusters in condensed phases as (n = 2, 3 . . . . ) g, = ~'n- 1Vo,n exp ( - Edet, n[kT ) ZldoAc,n_ 1 9

(10.76)

This relation implies reversibility between the forward and backward 'reactions' of attachment and detachment: the same number ?;,,_ ~Z~doAc,,,_ l of molecules that can join the (n - 1)-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 v0,,, is sizeindependent and the same for both attachment and detachment: v0,n = kT/hp. Thus, it is the activation energy Edet, n for detachment from an n-sized cluster that makes g, from (10.76) essentially different from the corresponding attachment frequencyfn from (10.51 ). Clearly, in order to be able to determine g,, we need a knowledge of Edet, n. In the absence of such a knowledge it is convenient to relate Edet, n tO the activation energy Eit,n 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 i t , n - 1 = G a , n - G z ( n - 1) Edet, n = Ga, n - G 2 ( n ).

Here G2(n - 1) and G z ( n ) a r e the Gibbs free energies of the system in its initial and final states when the cluster in it is of size n - 1 and n, respectively, and Ga, n is the Gibbs free energy of the system in the activated state when the cluster size is somewhere between n - 1 and n. Since G z ( n ) is related to W(n) by eq. (3.4), it follows from the above two equations that Edet, n - - E i t , n _

1 + W(n-

1 ) - W(n).

In combination with ( 10.51 ), this formula allows representing gn from (10.76) as (n = 2, 3 . . . . ) g,, = fn - 1 exp { [W(n) - W(n - 1)]/kT}.

(10.77)

Equation (10.77) shows that, as in (10.72), thermodynamics enters again the description of the kinetic quantity gn- When concrete dependences of fn 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 gn/fn_ 1 ratio in various cases of nucleation and was used recently in numerical studies on the kinetics of HON of crystals in melts [Kelton et al. 1983; Kozisek 1988, 1990, 1991; Kelton 1991 ]. In this case of HON, with the aid of fn from (10.64) and the classical formula (3.39) for W(n), from (10.77) we find that

g~ = [}'n- 1(c v~/3/31rd~)(kT/voO) exp (- 2/kT)(n exp (2c v~/3 ty/3kTn 1/3)

1) 2/3]

(10.78)

since, as noted by McDonald [1963], n 2/3- ( n - 1) 2/3 -- 2/3n 1/3. We used this approximation to show that in the scope of the classical nucleation theory eq. (10.77) leads again to a dependence of g~ 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 fe,n- 1 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, A/~ in (10.64) is controlled by the melt pressure p, this pre-exponential factor is again the attachment frequencyfe,~_ 1 resulting from (10.64) at Ap = 0. This is so, since in this case A/I is given by [Volmer 1939] A/~ = AOe(p - Pe)

(10.79)

when compressibility is negligible (Pe is the equilibrium pressure, and AO e = O0,m - V0,c is the difference between the volumes V0,m and V0,c 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), A/~ from (2.8) and (2.14), Pe,n and Ce,~ from (6.17) and (6.28), the McDonald approximation given above and f, from (10.3), (10.20) and (10.60)" after some algebra eq. (10.77) leads to gn from (10.68), (10.70) and (10.71) in the respective cases. The dashed curves in Fig. 10.8 display the dependence of gn 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 A = TeAse and the parameter values listed in Table 6.2. As seen, despite that g~, in contrast to f~, does not depend explicitly on A/~ = Ase(Te - T), it changes significantly with T mainly as a result of the r/(T) dependence. Again, gn and f~ are equal at the nucleus size n* which is smaller at the lower T value (i.e. at the higher supersaturation A/~). 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 and Anderson 1978]. Let us consider 2D clusters of monolayer height on an atomically smooth substrate with adsorption sites of concentration Co given by eq. (7.8). For monomer attachment by the surface-diffusion mechanism f~ is determined by eq. (10.36) where Z1, the concentration of adsorbed monomers, is given

162

Nucleation: Basic Theol. 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 Co, 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 Co of molecules inside the monolayer cluster (the molecules are assumed to be close-packed). Dividing this concentration by Co 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 [- (Es,,,- Es.n- 1)/kT], because the cluster must change its 'surface' binding energy from Us, n t o Es, n _ 1 (Es,n 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 Halpern 1976; Lewis and Anderson 1978] (n = 2, 3 . . . . ) gn = 7'n- l c * D s C o exp [- (Es,,, - E s , n - 1 ) / k T ]

(10.80)

where 7,,- ~ 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 f,,, gn is also A/t-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 as = cr and cri = 0. In this case, with the help of (7.31) and the classical approximation (3.81) for E,,, from (10.80) it follows that gn = fe,n

-

1 exp (b r,/2kTn

1/2).

(10.81)

Here b = 2(~ra0)1/2 for disks, 4a~/2 for square prisms, etc. and, in conformity with (10.36) and (10.37), fe,n- 1 = ~ , - lC*DsCl,e

(10.82)

is the attachment frequency at equilibrium (i.e. at Ap = 0) when the concentration of monomers on the substrate is Cl,e (in general, Cl,e is given by Zx from (10.37) at A~ = 0, and in particular, by Clx = leVd, Cl,e = CoKop~ and Cl,e = CoKcCe as required by eqs (10.41), (10.45) and (10.47)). The exponential factor in (10.81 ) results from using the McDonald-type approximation n 1/2 - (n - 1) 1/2 = 1/2n 1/2. 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

Transitionfrequencies 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 (Co~C1) + E s , n - 1 - E s , n

(10.83)

by virtue of (7.31) and (10.37). When W(n) - W ( n - 1) from this formula and f n - 1 from (10.36) with Z1 = C1 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 gn 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 Pe,n, Ce,n, W(n) and Es, n 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 + 1)-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 . . . . ) fnC(n) = gn + 1C(n + 1).

(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 reactionrate theory. Regarded from this point of view, however, gn from (10.77) is not any more a purely kinetic quantity: it is determined by both kinetics (fn-l) and thermodynamics (W(n)). Equation (10.84) and, hence, eq. (10.77) are applicable only to nucleation

!64

Nucleation: Basic Theory with Applications

occurring under time-independent conditions. In practice, it is important to know g,, 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 Ap, T, etc. and defined as [Kashchiev 1969b] (n = 1, 2 . . . . )

C(n, t) = Cl(t) exp {- [W(n, t ) - Wl(t)]/kT(t) }

(~0.85)

where Cl(t ) -~ C(1, t) and Wl(t ) ~ W(1, t). This equation is a generalization of (7.6) and in it, according to (3.86),

W(n, t) = - n A p ( t ) + ~(n, t)

(10.86)

is the work to form an n-sized cluster at the supersaturation Ap existing at time t. The cluster effective excess energy 9 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, Ap, etc., it satisfies the condition for detailed balance in the form [Kashchiev 1969b] (n = 1, 2 . . . . )

fn(t)C(n, t) = gn + l(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 (n= 2,3 . . . . )

gn(t) = fn - l(t) exp { [W(n, t) - W(n - 1, t)]/kT(t) }.

(~o.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 Cn(t) in the master equation (9.23) of nucleation by considering fn(t) and gn(t) as independently known kinetic quantities. With respect to eq. (10.87) the situation is reversed: we presumed that we knew the function C(n, t) and used this function to define the unknown detachment frequency gn(t). The fact that eqs (9.22) and (10.87) coincide upon setting C,7(t) - C(n,t) means that Cn(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 f~, gn and (7,, 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 (fn, g~) and thermodynamics (Cn) 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

Ap(t)/kT = In [I(t)/Ie],

In [p(t)/Pe]

or

In [C(t)/Cr

Transition frequencies

165

and for fn we have eq. (10.33) and its respective approximation (10.42), (10.43) or (10.46). Thus, with the aid of the McDonald approximation n 2 / 3 (n - 1)2/3 _ 2/3nl/3, from (10.88) we find that

gn =fe,n- 1 exp [211tl/3(Ow)cU 213 G/3kTn 1/3]

(10.89)

where fe,n- 1, the frequency of monomer attachment at equilibrium (i.e. at Ap = 0), is given exactly by (10.33) at Z1 = Cl,e and approximately by (10.82) with Cl,e = le't'd, Cl,e = CoKppe or Clx = CoKcC~, respectively. We note that at gt = 1 (HON) eq. (10.89) takes the form of eq. (10.72) and that it predicts a constancy of gn with time despite the changing supersaturation in the system (this constancy is due to the general property of gn for condensed-phase clusters not to depend explicitly on Ap). 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 gt = 1/2, c* = 1.9 and As = 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 n*) 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) fn- l and W(n)- W(n - 1) must be replaced by f(n, t) and OW(n, t)/ On, respectively, and in eq. (10.80) )'n and dEs,n/dn should be used instead of ~n- 1 and Es,n - Es,n_ 1- The same replacement of the finite differences by the dW/dn and dEs,Jdn derivatives is necessary in eq. (10.83). For instance, eqs (10.88) and (10.89) take the form

g(n, t) = f(n, t) exp { [tgW(n, t)/On]lkT(t) } g(n) = fe(n) exp [2 g//3(0w)C U2/3 tr/3kTn 1/3],

(10.90) (10.91)

respectively, where fe(n) -fe,,. 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 ~t = 1) of condensed phases in vapours at constant T, for example, it is easily verified that eq. (10.91) follows from (10.90) withf(n, t) from (10.3), W(n, t) from (3.39) and Ap(t) = kT In [p(t)/pe]. Also, with f(n) from (10.9), W(n) from (3.42), Ap from (2.10) and p* 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 fnm (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 f~. For

166

Nucleation: Basic Theory with A p p l i c a t i o n s

completeness, however, we shall briefly consider fn,, in the two different cases of attachment controlled by mobility or growth coalescence of condensedphase clusters of size n and m - n which contact to make a new cluster 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 n-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 Z1 replaced by the mass mm- ,l, radius R m _ n and concentration Z m _ n of the (m - n)-sized clusters [Moelwyn-Hughes 1961]. For that reason, since m m _ n = mo(m - n) and Rm_n = R o ( m - n) 1/3, in the same way asf~ from ( l O . 2 ) , f n m c a n be written down as (m > n) f n m ( t ) = ]In,m _ n c O 213 (kTI2~mo)l/2[m/(m - n ) n ] 1/2

In 1/3 + ( m -

n)l/3]2Zm_n(t)

(10.92)

where ~,,,m- ,, is the sticking coefficient. As seen, in the m - n = 1 case (monomer attachment), (10.92) passes into (10.2) in which ~,l - ~,lAs 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, f n m is given by eq. (10.22) with D, R0 and Z1 replaced by D m _ n , Rm_ ,1 and Zm_ ~. Since the diffusion coefficient D m _ n and the radius R m _ 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) fnm(t) = ~l,m_n(4~c)l/201/3D[

[n 1/3 +

(m -

n - 1/3 + (m - n ) - 1 / 3 ]

n)l/3]Zm_n(t)

(10.93)

where c = (36z01/3. Again, in the m - n = 1 limitfnm from this equation turns into the monomer attachment frequency fn 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- 1/3 + (m - n ) - 1/3][nl/3 + ( m - n ) 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 casefnm 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 - n)-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 Z~(t) and diffusion coefficient Ds,, + Os,m_ n towards the periphery of an immobile cluster with radius R '+ Rm_~ 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 I and lrd 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 Z1 and R replaced by Zm_ ~ and R' + Rm-~, 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,n,m- n of (m -- n)-mers is time-dependent even asymptotically: for t >> (R' + Rm_~)2/ 4(Ds,n + Ds,m- n) it is given by [Carslaw and Jaeger 1959; Crank 1967] Jd,n,m- n(t) = 2(Ds,n + Os,m- n)Zm- n/( R" + Rm-n)

In [4(Ds,n + Ds,m _ n)t](g' + gin_ n )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 1/ln[4(Ds,n + Ds, m _ n)t/(R" + Rm_ n )2] _ E'

where g 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/in x function varies within the limits indicated (e.g. I/In x = 0.43 to 0.11 for x = 10 to 104). From the definition fnm(t) =- ~n,m-nJd,n,m-n(t)21t(g" + Rm_n), which is the analogue of (10.26), we thus find that in quasi-stationary approximation fnm is of the form (m > n) fnm(t) = ~n,m- nc*(Ds,n + Ds,m-n)Zm - n(t)

(10.95)

as long as t > (R" + gm_n)2/(Ds,n + Ds, m _ 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" + Rm_ n) ~--R'= 1 nm and (Ds,~ + Os,m_ n) = Ds,m- n = 100 nm2/s shows that the above formula for f n m 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* - 4zre' -- 1 to 5 is the respective capture number. An expression for f,m, 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 )'n,1 - ~ and Ds,1 - Ds, eq. (10.95) can be used for determination of the frequency f~ 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 that D s , , - 0).

168

Nucleation: Basic Theo~ with Applications

Equation (10.95) tells us that the dependence o f f , m on n at fixed m - n is governed by the size dependence of the diffusion coefficient Ds,,, (provided ?',,m-, is 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; Masson et al. 1971; Kern et al. 1971, 1979; M6tois 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]

Ds,n = D'n - a'

(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 D s , n 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) fnm(t)

= ~tn, m _ n c * D "

[n- o~ + ( m - n ) - a'] Zm_

n(t)"

(10.97)

(2) Growth coalescence An (m - n)-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 - n)-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 frequencyfnm w e assume that the clusters are randomly positioned and consider first HON. Two clusters of n and m - n molecules and respective radii R and R m _ n will coalesce when the distance between their centres becomes equal to R + R m _ n. As in the case of mobility coalescence, it is convenient to think again of an effective cluster of radius R + R m _ n, whose centre coincides with the centre of the n-sized cluster, and of Z m _ . mathematical points representing the centres of all the (m - n)-mers in unit volume of the system. These points move radially towards the centre of the effective cluster with velocity d(R + R m _ .)/dt which is the sum of the velocities dR~dr and d R . , _ n / d t of the n- and (m - n)-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 f.., is merely equal to the product of the number 4n'(R + g m _ ,)2[d(R + g m _ ,,)/dt]Zm_ ,, of points reaching the cluster surface per unit time and the sticking coefficient ~,,,,,,_ ,,, we find that for condensed-phase clusters (m > n) f n m ( t ) - ~,,,m-nVO[ n l / 3 + ( m - n ) l / 3 ] 2 [ V n ( t ) n + Um _ n ( t ) ( m -- n ) -

2/3]Zm_ ,(t)

2/3

(10.98)

Transition frequencies

169

where v,, = d n / d t is the cluster growth rate and it is taken into account that R = (41~[3vo)l/3n 1/3 and R m _ n = ( 4 ~ / 3 v o ) l / 3 ( m - n) 1/3. Clearly, eq. (10.98) is applicable only to clusters of supernucleus size, since On > 0 only when n > n* (the subnuclei tend to decay so that, deterministically, On < 0 for n < n*). However, the applicability of (10.98) can be extended over the subnucleus size range with the help of the approximation on = 0 for n < n*. This results infnm = 0 when both n < n* and m - n < n*, i.e. in ignoring growth-coalescence-controlled contacts between subnucleus clusters. We note also that as the cluster growth rate V n is largely determined by monomer attachment and detachment, in conformity with (9.21) we have On(t) = f n ( t ) - gn(t).

(10.99)

Hence, v~ = f , for large enough supernucleus clusters, because f~ >> gn for n >> n*. In the scope of these approximations, from (10.98) it follows, for instance, that (m > n) (10.100)

fnm(t) = ~'n,m_nVOfn(t)Zm_n(t )

for supernucleus clusters of size n >> n* (for them vn --fn) which attach subnucleus multimers of size m - n < n* (for them Vm-n -- 0). From (10.98) we also find that (m = 2n) (10.101)

fnm(t) = )in,m _ n8OOOn(t)Zm _ n(t)

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 frequenciesfn 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 fnm controlled by growth coalescence of immobile clusters. For example, for attachment of large enough supernucleus multimers ( m - n >> n*, Vm-n = f m - n ) to large enough supernucleus clusters (n > > n*, On----fn) in solid solutions, with the help of c = (36zr) 1/3 and v0 = (7r/6) d03, from (10.60) and (10.98) we find that (m > n) fnm(t) = ~'n,m - n(~n q"

)tm -

n)(/t'2/6) d 4 D C ( t ) [ nil3 + (m

-

n)ll3]2Zm

_

n(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 0w < n:/2 or disks of monolayer height in HEN of condensed phases on a substrate. To find the multimer attachment frequency fnm 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 21r(R' + R~,_~) where, in conformity with (3.13) and (3.52), R' = (3v0/4nr)l/3[sin O w / l l f l / 3 ( O w ) ] n l / 3 and R ~ , _ n = (3v0/4/r)l/3[sin Ow/lltl/3(Ow)](m - n) 1/3 for the caps and R' = R = (ao/Tr)l/2n 1/2 and R,~,_,, =

170

Nucleation: Basic Theory with Applications

Rm_ n = (ao]~)l/2(m - n) 1/2 for the disks. Consequently, since now the number

of points reaching the cluster periphery per unit time is determined as 27r(R' + R',_, )[d(R' + R,'~_,, )/dt]Zm_,,, the multimer attachment frequency is given by (m > n) fnm(t) = ~,,m_,,(7~/6)l/3[sin Ow/lltl/3(Ow)]Zv2/3[nl/3 + ( m -

n) 1/3]

X [Vn(t)n - 2/3 + V m - n(t)(m - n) - 2/3]Z m _ n(t)

(10.103) for caps with 0w < zr/2 in 3D HEN on a substrate and by (m > n) fnm(t) = )in,m _ nao[n 1/2 + (m - n)l/2][On(t)n- 1/2 + V m - n(t)(m - n ) - l 1 2 ] Z m _ n ( l )

(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, f n m -~ O. 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 = 2n) fnm(t) = ~'n,m -,,4aovn(t)Zm _ n(t)

(10.105)

which, with ~ , m - n -" 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) v~ --f,, 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 >> n*) fnm(t) = ~n,m_nC*aoDsZl(t)[n 1/2 + ( m + )'m- n(m - n ) - l 1 2 ] Z m _ n(t)

n)l/2][Tnn-1/2

(10.106)

provided that the monomer surface-diffusion length As is greater than the radii of the coalescing clusters. We recall that under conditions of adsorption equilibrium ZI 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 7,,- ~'m-,, = 1 and using the approximation [n ~/e + ( m n ) l / 2 ] [ n - 1/2 + (m - n ) - 1/2] = 4 (which parallels that of von Smoluchowski [ 1916, 1917]) may be helpful for analytical work with f n m f r o m ( 1 0 . 1 0 6 ) . When the product of the two bracketed factors in it is approximated by 4, eq. (10.106) follows directly from (10.105) with on replaced byfn from (10.36). Summarizing the above results, we see that in all cases considered, in accordance with (9.34), the multimer attachment frequencyf~m 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

Transitionfrequencies 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 ofeq. (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 ogn,m_n in the cases considered above.

10.4 Multimer detachment frequency Similar to gn, the multimer detachment frequency fmn (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)

wherefnm andfm, 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 fm.(t) = f~m(t) exp {[W(m, t) - W(n, t)]/kT(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 m > n, this equation is the general formula for the frequency fro, of detachment of a multimer of m - n molecules from an m-sized cluster. It shows that fm~ is readily obtainable once the multimer attachment frequency fnm and the work W for cluster formation are separately known. Using concrete expressions fOrfnm 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 [Kashchiev 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 f~ 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 f~. As noted in Section 10.1, it contains always the factor exp (Ala/kT). In all considered cases in which A/~ is not defined by eq. (2.20), with the help of A/~ from (2.8), (2.9), (2.14) and (10.79) used with timedependent 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 (n - 1, 2 . . . . )

f~(t) = fe,n(t) exp [Alu(t)/kT(t)]

(10.109)

where fe,~ is the monomer attachment frequency at equilibrium, i.e. at AV = 0. For instance, in the particular cases of nucleation considered in Section 10.2 fe,,, is given by eqs (10.73)-(10.75) and (10.82) with a concrete T(t) 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 A/~ controlled by T in accordance with eq. (2.20): in this case eqs (10.64)-(10.66) suggest a general presentation of fn in the form (n = 1, 2 . . . . )

f~(t)

= fo,n(t) exp

[Ap(t)/kT(t)].

(10.110)

This equation is similar to (10.109), as in itf0.n is also the value off,, at AV = 0. However, f0,,, 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 Te. Thus,f0,~ is an implicit function of A/~ mainly because of the temperature dependence of the parameters (e.g. 7/from (10.56)) taking part in its determination. Let us now consider gn. 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 gn holding only in the scope of the validity of (10.109). Namely, insertion of f,, from (10.109) in (10.88) and (10.90) and accounting for W(n, t) from (10.86) yields (n = 2, 3, .

.

~

Transition frequencies 173 g,,(t) =fe,,,-l(t) exp {[~(n, t) - ~ ( n - 1, t)]/kT(t)}

(10.111)

or, if n is considered as a continuous variable,

g(n, t) =fe(n, t) exp {[o3~(n, t)/tgn]/kT(t)}.

(10.112)

With concrete expressions for the monomer attachment frequency fe,n at equilibrium and the cluster effective excess energy q~ 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, Ap is not defined by eq. (2.20). These equations show that, in contrast to fn, g~ does not depend on the supersaturation in the system as long as Ap is changed isothermally and q~ is Ap-independent. This is not so, however, when Ap is varied by means of T as required by (2.20). Indeed, from (10.86), (10.88), (10.90) and (10.110) we then find that (n = 2, 3 . . . . )

g,,(t) = fo,,,- l(t) exp {[~(n, t) - ~(n - 1, t)]/kT(t)}

(lO.113)

when n takes only integer values and that

g(n, t) = f0(n, t) exp { [3t~n, t)/o3n]/kT(t) }

(lO.114)

when n varies continuously 0C0(n, t) -f0,n)-As seen from these equations, g~ is an implicit function of Ap through the temperature T. In the scope of the capillarity approximation, with 9 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 ~ 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 fnm of multimer attachment and detachment are eqs (9.34) and (10.108), respectively.

Chapter II

11 Nucleation rate

The central problem in the theory of nucleation is the determination of the nucleation rate J(t) (m -3 s-1 or m -2 s-1) which is the frequency of appearance at time t of supernuclei per unit volume or area of the system under consideration. Accordingly, the general definition of J reads

d~'(t) dt

J(t)=

(11.1)

where ~ (m -3 or m-2), given by M(t)

((t) =

Z

Z,(t),

(11.2)

n = n*( t ) + 1

is the concentration of supernuclei in the system. In the literature ( i s 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

~(t) =

f

M(t)

(11.3)

Z(n, t) dn.

9J n*( t )

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 supernucleus size, i.e. the solution Zn(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,,,(t)(t) -j[n*(t), t] through the nucleus size n* and of the actual concentration Z*(t) =- Z,,(t)(t) - Z[n*(t), t] of nuclei. Indeed, differentiating ((t) from (11.3) with respect to t, accounting that Z[M(t), t] = 0 (cf. eq. (9.33)) and substituting the result in eq. (11.1) leads to

J(t) =

f

M(t) o3Z(n, t) dn-Z*(t) Ot

dn*(t) dt "

(11.4)

d n*( t )

For a closed system or for an open system in which the supernuclei do not appear or vanish as a result of non-clustering processes, in eq. (9.1) or (9.8) we have Kn(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 0Z/ oat in the above integral by the size derivative - OriOn and carry out the integration. Taking also into account thatj[M(t), t] = 0, we can thus represent (11.4) as [Kahlweit 1970; Kashchiev 1974, 1984a]

Nucleation

J(t)=j*(t)-Z*(t)~

rate

dn*(t) dt

175

(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 j* 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 Ap: for instance, from eqs (4.32), (4.35) and (4.38) we see that, classically, d n * / d t ~: - d A p / d t for either 3D or 2D nucleation of condensed phases so that Ap = constant is a necessary condition for dn*/dt = 0. At variable Ap, due to the motion of n* along the size axis, J is thus greater or less than j* when Ap increases or decreases with time, respectively. In general, the flux j* is given by the equations j*(t) =

M( t )

n*( t )

]~

]~

(11.6t

[fm,,(t)Zm(t) -f,,m(t)Zn(t)]

n=n*(t)+ I m=l

j,(t) = f ' " ' I f

''*(')[ f ( m ,

] n, t) Z ( m , t) - f ( n , m, t ) Z ( n , t)] dm I dn

,In*( t ) [ , J l

,.J

(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 j* is of the form j * ( t ) = f * ( t ) Z * ( t ) - g,,.(,)+ 1Zn*(,)+ 1

(11.81 (11.9)

Here i f ( t ) - L . ( o ( t ) = f [ n * ( t ) , t] and C*(t) -- C,.(t)(t) - 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 j*, we can determine the nucleation rate without having any information about the concentration ~"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 ~"with the aid of this equation rather than of eq. (11.2) or (11.3). Indeed, integration of (11.1) under the initial condition ((0) = ~'0 results in ((t) - ~o +

J(t') dt"

(11.1 O)

where (0 ( m-3 or m -2) is the concentration of all supernuclei at the initial moment t = O.

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, n* 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 nonmorphological 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) (m -3 s-1 or m -2 s-1) of formation of clusters of size n > n', 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 J' 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

J'(t) = d (dt '(t) "

(11.11)

Here (" (m -3 or m -2) is the concentration of all detectable clusters in the system and is given by M(t)

]E Z,,(t)

('(t)=

(11.12)

n =n'+ 1

or, when n is treated as a continuous variable, by

~

M(t)

('(t) =

Z(n, t) dn.

(11.13)

7'

To express J' with the help of the fluxj'(t) -jn,(t) - j ( n ' , 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 soobtained d ( / d t in (11.11) and recalling that Z[M(t), t] - 0 andj[M(t), t] = 0 yields

J'(t) -j'(t).

(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 J' is always equal to the flux j' through the size n' provided the detectable clusters appear only as a result of clustering (i.e. if K,,(t) = L,(t) = 0 for n > n'). This is in contrast with the relation between the nucleation rate J and the flux j* (cf. eq. (11.5)) and is so because n' is time-independent by definition. As it should be, when n* does not change with time, in the particular case of n' = n* eqs (11.11)-(11.14) pass into eqs (11.1)-(11.3) and (11.5), respectively. As to the fluxj', it is given by eqs (9.6), (9.11), (9.19), (9.28) or (9.29) upon setting n = n'. For example, from (9.28) and (9.29) we have

j'(t) = - f ' ( t ) C ( n ' , t) -~n [Z(n, t)lC(n, t)] n-n" =v(n;t)Z(n;t)-f'(t)[~-~Z(n,t)]

(11.15) 11= n '

where f" -f,,,(t) -fin', t) is the frequency of monomer attachment to the n'sized cluster, and v(n', t) is the growth rate of this cluster. This formula gives j' 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

~'(t) = ~; +

J'(t') dt'

(11.16)

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 ~'~(m-3 or m -2) is the concentration of all detectable clusters at the initial moment t 0. When n* is time-independent and n" = n*, (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 J' when we knew the cluster size distribution function Z, 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, Zn(t) 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 Ap 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(n) is easily obtained from the condition j,,(t) = 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(n) 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 fluxj~(t) in (12.1) is given by eq. (9.6). However, since the concrete mechanism of cluster formation is immaterial for maintaining the equilibrium, j,, 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 j~(t) from (12.1) into the master equation (9.7) results in dZ,,(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(n). Setting Zn(t) = C(n) and introducing j,, from (9.19) into eq. (12.1) yields [Farkas 1927] (n = 1, 2 . . . . ) f n C ( n ) - g,, + lC(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(n) = C l ( f l f 2 . . . fn- l/gzg3 . . . gn)

(12.3)

where C~ - C(1) 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 A/.t < 0 and the system is in truly stable equilibrium, C(n) from (12.3) represents the really existing population of clusters in the system. When Ap > 0, however, as then the old phase is supersaturated and thus in metastable equilibrium, C(n) 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 f, and g,, are not functions of t, i.e. when T, Ap, etc. are kept constant. If, e.g. T and/or Ap are varied, the detachment and/or attachment frequencies change with time and C(n) 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 Ap. Indeed, since in this case the above derivation can be repeated, but with Z,,(t) set equal to C(n,t), 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 . . . . ) C(n, t) = Cl(t)[fl(t)fz(t) . . . f,_l(t)/g2(t)g3(t) . . . gn(t)].

(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 Cn(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 Cn(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) = C1 exp

{s (s,

In [ f ( m ) l g ( m ) ] dm

C(n, t) = Cl(t ) exp

}

(12.5)

}

In [ f ( m , t)lg(m, t)] dm .

(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 fn 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 fn and g,, (n = 2, 3 . . . . ): W(n) = Wl + kT

~ m =2

In (g,n/fm-l)

(12.7)

180

Nucleation: Basic Theo~ with Applications

at equilibrium and

W(n, t) = Wl(t) + kT(t) ~

In [gm(t)/fm-l(t)]

(12.8)

/7/"-2

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) = W1 + kT

W(n, t) = Wl(t) + kT(t)

In [g(m)/f(m)] dm

In [g(m, t)/f(m, t)] dm

(12.9)

(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 W~ - W(1) to form a cluster of size n = 1 is given by WI =-All + ~ - E 1 -

trsao

(12.11)

both for HON (with E1 = 0 and as - 0) and for HEN on a substrate (then E1 0 and as ~ 0). Accordingly, the quasi-equilibrium quantity Wl(t) is expressed as

Wl(t ) = - A p (t) + ~ - E l - Crsao

(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 ~ - E l - crsa0 is replaced by its counterpart ~1 resulting from (3.90) at n = 1. It is worth noting as well that E1 + Crsa0---Edes (see Section 10.1) and that W1 is connected with C1 and Co by eq. (7.5) and by the generalized equation

Wl(t) = kT(t) In [Co(t)/Cl(t)]

(12.13)

in the cases of equilibrium and quasi-equilibrium, respectively. This means that if the concentration Co of the sites on which clusters can form and the concentration C1 of the clusters of size n = 1 are known independently, the work W1 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 C1/Co 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 W1. 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

181

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 intruly stable equilibrium whenfn and gn 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 n*, 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 [Volmer 1939; Zinsmeister 1970; Toschev 1973a] f* = g*

(12.14)

or its generalized version [Kashchiev 1974, 1984a] f * ( t ) = g*(t)

(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 n* (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 tl*

W* = W1 + k T

]E In (gn/f~-1)

n =2

(12.16)

or //*

W* = W1 + k T

~l

In [g(n)/f(n)] 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 f,, and gn from these equations in (12.7) and recalling that in this case A/~ is determined by (2.8), we get (a = co 2/3)

182 Nucleation: Basic Theory with Applications

W ( n ) - W 1 = ~ [kT In (Pe/P) + 2acr/3ml/3] m=2 --- -(n - 1)Ap + (2act/3)

~1n m-1/3 dm

= (- nAp + acrn 2/3) - (- Ap + act). Thus, W(n) 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(n) can be obtained by using eq. (12.9) in combination withfln) and g(n) from (10.3) and (10.91). Also, if we employ fin) and g(n) from (10.3) and (10.91) in eq. (12.14) in order to calculate kinetically the nucleus size n*, we arrive again at the thermodynamic GibbsThomson equation (4.7). Similarly, as n replaces n - 1 when treated as a continuous variable, with fin) 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 f , 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 Z1 = C1, we obtain

C(n)

=

Cl(Cl[Co) n- 1 exp [(Es,2 - Es,1 + Es,3- Es,2 + . . . + Es,,,- Es,,,_ 1)/kT].

However, since E~;] = 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(n). The ability of the above kinetic formulae to reproduce known thermodynamic relations is a condicio 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 cr 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 =fe,n-1 exp ([JyQ/q,/kTnx)

(12.18)

where fe,,_ 1 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/~yQ > 0, rather than with only one,

Equilibrium

183

the product CO213 ~ (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 cr of the smallest clusters). Clearly, eq. (12.18) is a generalization of (10.72): it passes into it when x = 1/3 and ~yQ -" 2 cv~/3 a/3A.. As shown by f a n g and Qiu [ 1986], physically, x and ~yQ 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/~,yQ is analogous to the factor flSST--- 0.2 to 0.6 in the Stefan-Skapski-Turnbull formula cr - flSST/],102/3

(12.19)

which relates o"to the molecular heat & 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 cr and & is known to hold also for crystals in solutions [Nielsen and S6hnel 1971; Sangwal 1989; Mersmann 1990; Christoffersen et al. 1991 ]. With cr from (12.19) we thus find that, classically,/~yQ "- (2c/3)flSST where c = (36zr) 1/3 for spheres, c = 6 for cubes, etc. Considering n as a continuous variable (thenfe,,_ 1 in (12.18) is replaced byfe(n)), substitutingf,, and g, from (10.3) and (12.18) in (12.14) and accounting for (2.8) yields [fang and Qiu 1986] n* = (flyQ/]JA//)1/x

(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) = - n A p + []~yQ2/(1- x)]n l- x.

(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 * = ([3yQ~)l/X/( l / x - 1)A/~ l/x- 1.

(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 a n d ~yQ -- 2r u 213 0]3/~. 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 b e t w e e n ~yQ and/3SST we can conclude that with appropriately defined A/~ (see Chapters 2) and cr replaced by O'effrom (4.42), eqs (12.18), (12.20)-(12.22) are applicable also to onecomponent HON or 3D HEN of condensed phases in solutions and melts. Analysing experimental data, f a n g and Qiu [ 1986] found that x = 1/3 to 0.9 and ]~yQ -- 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 a n d / ~ y Q -- 0.33 for spherical water droplets at 293 K (this/~yQ value is calculated f r o m / ~ y Q - 2c v 2/3 a/32 with ~ = 7 • 10 -20 J and with v0 and cr 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, L(t) = constant = Js

(13.1)

for any n = 1, 2 . . . . . Accordingly, from the master equation (9.7) and from eq. (11.5) it then follows that dZfldt = 0 and that J(t) = Js,

(13.2)

since dn*/dt = 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 Js > 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 D6ring 1935]. We shall now determine the stationary cluster size distribution X n (m -3 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) = X,, in (9.18) and (9.19), recalling that d X J d t = 0 and using (13.1) leads to [Farkas 1927] (n = 1, 2 . . . . . M - 1) Js = f , Xn - g n + 1Xn + l"

(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 X1 in (13.3) we can write X1 = C1

(13.4)

where C1 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

XM = 0.

185

(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 Ss + g2X2 =fig1 I9s - - f n X . + g . + 1Xn + ~ = O,

(n = 2, 3 . . . . .

M-

2)

(13.6)

=0.

Js-fM_IXM_I

This is a set of M - 1 linear algebraic equations in the M - 1 unknowns Js, XM_ 1. It shows that Js and Xn can be time-independent and thus stationary quantities only when the transition frequenciesfn and gn are constant and when, in addition, the monomer concentration C1 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 C1 enters (13.6) not only explicitly, but also implicitly through the monomer attachment frequencies fl, f2 . . . . (see Section 10.1). The algebraic set (13.6) can be solved exactly by various procedures some of which are quite simple [Becket and D6ring 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, X~ can be represented as [Korn and Korn 1961] (n = 2, 3 . . . . . M - 1)

X2, X3 . . . . .

(13.7)

Xn - m n / m l .

Here A, and A ~ are the following determinants of order M - 1" lg 2

A ? !

m

0 -.-

0

1-f2 g3 "'"

0

0

o

1 0 -f3...

0

0

~

flC1

1 0

0 .... f,-1

0

1 0

0 ...

0

0

1 0

0...

0

0

~

o

g,, + 1 -fn+

o

o

o

~

" ' "

1 "'"

0

0

0

0

0

0

0

0

0

9

o

.

0

0

0

0

0

0

0

0

0

o

o

9

1 0

0 ...

0

0

.... fM-3

gM-2

0

1 0

0 ...

0

0

9""

0

--fM-2

gM-I

1 0

0-..

0

0

9""

0

0

--fM-

1

(13.8)

186

N u c l e a t i o n : Basic T h e o r y with A p p l i c a t i o n s

lg 2

0 .--

0

0

l-f2

g3 "'"

0

0

1 0 -f3"'"

0

0

0

0

0

~

0

0

0

o o o

0

0

0

o

o

0

0

0

"'"

0

0

0

- - f n + 1 "'"

0

0

0

.

.

o

~

~ 1 7 6

9

#

A 2----

1 0

0

.... fn-1

1 0

0

..-

0

1 0

0

..-

0

gn

~ 1 7 6

-f.

gn+l

0

1 0

0 ...

0

0

0

. . . . f M - 3 g M- 2

1 0

0 ...

0

0

0

"'"

0

1 0

0 ...

0

0

0

"'"

0

0

--fM-2 g M - I --fM- 1

0

(13.9) In order to evaluate the above determinants we first expand An by the minor offlC1 and then this minor by its minor of -f2- Continuing the latter kind of expansion until the minor of - f n - 1, we find that A~ can be written down as (n = 2, 3 . . . . . M - 1) (13.10)

A,, = (-1) n-1 C l f l f z - . . f n - l A ' + l Here the determinant A~ is given by 1

g,,

1 -fn 1

0

0

-..

0

0

0

gn+l

"'"

0

0

0

-f,-1

"'"

0

0

0

(13.11)

#

An= 1

0

0

....

1

0

0

...

1

0

0

...

fM- 3

g M- 2

0

0

-fM- 2

gM-1

0

0

-fM-l

for n = 2, 3 , . . . , M - 1 and, by definition, A]4 = 1. This determinant satisfies the recursion formula (n = 2, 3 . . . . . M - 1) t

An=(-1)

M-n

t fnfn+l '''fM-l--gnAn+l

(13.12)

which follows from its expansion by the minors of 1 and gn- We note also that A ~ 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 An and A~. 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):

Stationao, nucleation 187

A~, = (-1) M-'' f,,f,,+~ ...fM-i

[

~~' g,,g,,+,...gm]

1 + ,, _,, f,---~+~ (f,,~ .

(13.13)

Thus, using this expression for A,~ first in (13.10) and then at n = 2, we find that A, = (-1) M ( C l l f , , ) f l f 2 . . - f M - 1 for n = 2, 3 . . . . .

M-

1 + m-, f,,f,,+l

~f,,--~

(13.14)

]

(13.15)

1 and that

Ai-(-1)Mf2f3...fM

-

[

I 1 + , n~= 2 gf2f3 2g3."gm f,,,

.

Finally, substitution of A,, and A~ from these formulae into (13.7) and some rearrangement leads to (e.g. Zinsmeister [ 1970]) (n = 2, 3 . . . . . M - 1) flfz...ft,-I

x,,- c, -~;-~ .. g,, .

.

.

.

.

1 + ~., g z g 3 " " gm

,,--~ f~f3

g z g 3 " " gm

fm

"-" f~f~ ...fm

~

(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 Cn - C ( n ) from (12.3) (e.g. Andres [1969]; Abraham [1974a1) ( n = 1,2 . . . . . M - l ) : 1 x,, = c,, L '''-I

fmCm

-1

M-I

1

.~n fmCm"

(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 f,, and g,,. Alternatively, however, if C,, is unknown kinetically because of lack of information about the detachment frequency g,, 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

188 Nucleation: Basic Theory with Applications

(13.4) and (13.5) which are in fact (9.32) with Z1 = Cl and (9.33). Setting Z(n, t) - X(n) results in 3X(n)/Ot = 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)

[ ndmJ c' + c"

f(m)C(m)

'

The integration constants c' and c" are readily determined from the boundary conditions (13.4) and (13.5) as -1

c ' = 1, c " = -

9

f(m)C(m)

so that [Zeldovich 1942] (1 < n < M) M X(n) = C(n)

dm f (m)C(mi

M

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 frequenciesfn and gn 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 fn and gn or Cn (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(n), but without any loss of generality. To do that we note that, due to the exponentially sharp maximum of 1/C(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 f* at n = n*, and (iii) replace M by oo. The integration is then readily performed in terms of the error function defined by (10.15), the result being ~nu

dm 1 ~ ; e_//2( _ , , , ) 2 rcl/2{1-erf[fl(n-n*)]} f(m)C(m) = f'C* m dm = 2/3/*C* "

(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 < n <M):

Stationa~ nucleation 189 X(n) = C(n) 1 - e f t [fl(n - n * ) ] . 1- erf [fl(1 - n * ) ]

(13.20)

We note that, to a good accuracy, this equation can be approximated by [Nielsen 1964] (1 _< n _<M)

X(n) = 1 C(n){ 1 - erf [fl(n - n*)]}

(13.21)

fl(n* - 1) > 1.

(13.22)

provided that

This is so because when fl and n* satisfy (13.22), erf [fl(1 - n*)] = erf (- ,,,,) - - 1 with an error of less than 15%. The important point concerning eq. (13.2 l) is that it can be used for a reliable determination of X(n) in nearly all cases of interest, since under typical experimental conditions fl and n* obey (13.22). Indeed, with the help of fl from (7.39) and the approximation n* 1 -- n*, 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 [fl(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 (13.2 l) 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 fl which are quantities obtainable entirely by thermodynamic considerations (see Chapter 4 and Section 7.1). We could not expect that a priori, for X(n) 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(n) 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* 1 C* . X* = -~

(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

Nucleation: 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" 1,

(1 < n < n * )

X(n)/C(n) = O,

(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 p/p~ - 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 n*, C(n), fl and f,, from (4.10), (7.17), (7.39) and (10.3). The ~,,, v0, ty and Pe values used are those listed in Table 3.1. Under these conditions we have A#/kT = 1.5, n* - 53 and fl = 0.069. The sums in eq. (13.17) are computed with M replaced by the effective value M~f = 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(n) only at the left end nl = 40 of the nucleus region whose width A* = 26 is illustrated by the double arrow (these values of nl and A* are obtained 8x10 lo _

6x101~ r! v

E t-"

v

A* 4x101~

t-

X

2x101~

C*

"'"@"'"

. . . . l . . . . l . . . . l . . . . l . , , = J . . . . l . . . . l

10

Fig. 13.1

20

30

40

50

60

70

%1-.-.-~L

80

90

100

Stationary size distribution of water droplets during HON in vapours at T = 293 K and P/Pe =4.5: solid c u r v e - eq. (13.17); dashed c u r v e - 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 fight end n2 -- 66 of the nucleus region,

X(n) is already about 10 times less than C(n) and goes on diminishing for n > n2 in contrast to the steep rise of C(n). The variation of the X(n)/C(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(n) 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 < nl) 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 < n l, 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 = n] 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] 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(n) of all supernuclei outside the 'bottle' remains practically equal to zero. Hence, as an approximation, for X(n) it is possible to use the simple formula [Kashchiev 1969a] 1.2 91

1.0

_

.-~

. . . . . .

,

'

~ . . . . . . . . . . .

.

~'~. t-.

i

0.8

v

0 r

0.6

v

X

0.4 n1

/

0.2 ..... ,.

10

Fig. 13.2

,..,

20

....

30

n* \

n2

/

! ....

40

....

50

60

70

80

90

100

Size dependence o f the X(n)/C(n) ratio f o r H O N o f 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 l i n e - eq. (13.25); dotted l i n e - eq. (13.24).

192 Nucleation: Basic Theory with Applications

l,

X(n)/C(n) =

t

1/2

(1 < n < n l ) -- ( / ~ / Z 1/2

)(n - n*),

(nl

l 0,

< n < n2 )

(13.25)

(n 2 _< n < M)

which follows from (13.21) after expanding erf [/3(n - n*)] in a truncated Taylor series about n = n* and in which n l and n 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)/C(n) from (13.24) and (13.25), respectively.

13.2 Stationary rate of nucleation The stationary nucleation rate Js can be obtained from (13.6) without knowing the stationary cluster size distribution Xn. This possibility was used first by Farkas [1927] and then by Kaischew and Stranski [1934a] and Becker and D6ring [1935]. As already noted in Section 13.1, Js 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 D6ring [ 1935] who arrived at the exact formula for Js-According to the Cramer rule [Korn and Korn 1961 ], Js from (13.6) is given by (13.26)

Js = A j /A'2

where A ~ is specified by (13.9), and Aj is the following determinant of order M-l: fi Cl

g2

0

...

0

0

0

0

-f2

g3

"'"

0

0

0

0

0

-f3

"'"

0

0

0

Aj=

(13.27) 0

0

0

....

fM-3

gM-2

0

0

0

0

"'"

0

--fM-2

gM-I

0

0

0

"'"

0

0

--fM-I

Obviously, for Aj we have AJ= (-1) M Clflf2 9 9 9f M - 1.

(13.28)

From (13.15), (13.26) and (13.28) we thus find easily that the sought exact solution of (13.6) for Js reads

Stationa~. nucleation 193

Js =flC1

[

1+

g2g3 ... g,

,,-2 f2f3

f,,

1'

.

(13.29)

This is the known formula of Becker and Drring [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 fn and g, are known from model kinetic considerations. Owing to eq. (12.3), however, J~ from (13.29) can be written down also as (e.g. Andres [1969]; Abraham [1974a])

Js

=

1

L,=I f.C,,

],

(13.30)

"

This equation represents Js as a 'mixture' of kinetics (f,,) and thermodynamics ((7.) if in it the equilibrium cluster concentration is considered as given by the thermodynamic formula (7.4). Naturally, eq. (13.30) is entirely equivalent to (13.29) when Cn is determined kinetically from (12.3) by means of independently known f~ and gn. 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), Js is given by the formula Js = - f ' C *

IAd-~[X(n)/C(n)]t t~,, j

(13.31) /,/--- ?/*

which tells us that in order to find Js 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)/C(n) ratio with respect to n at n = n* and substitution of the result in (13.31) leads to the exact expression [Zeldovich 1942]

Is

M

Js =

dn 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 offn and gn or C n (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 = Mef >> n* (usually,

194

Nucleation: Basic Theory with Applications

it suffices to s e t M e f -- 2n* or e v e n M e f "-/'/2 -" r/* + zcl/2/2fl) [McDonald 1963; White 1969; Yang and Qiu 1986]. This reduction of the value of M to the effective value M e f is possible because of the sharp maximum of the 1/fnC, function at n = n*, which makes negligible the contribution of all summands with n outside the nucleus region of width A* = zcl/2/fl. 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~ = zf*C*

(13.33)

where C* is given by (7.44)-(7.49), and z = 2flhzl/2{ 1 - erf [/3(1 - n*)]}.

(13.34)

The numerical factor z in which 13 is specified by (7.38) is the so-called Zeldovich factor. When n* and/3 satisfy (13.22), as is usually the case, erf [fl(1 - n*)] = - 1 and the Zeldovich factor takes its known form [Zeldovich 1942; Zettlemoyer 1969] z = fl//r 1/2 = [( - d2W/dn2),, -- n./2rckT] 1/2

(13.35)

(actually, Zeldovich [1942] obtained this expression for z without zr1/2 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 fl 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 fl 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 (Ap > 0)

z

"-

(W*/31rkTn*2) 1/2 ---- 3Ap2/4(lrkTa 3O'ef 3 ) 1/2.

(13.36)

This particular form of the Zeldovich factor was found first by Becker and D6ring [ 1935]. In the case of 2D HEN of condensed phases on foreign (Act 0) or own ( A t r - 0) substrates, the classical formula for z from (13.35) is slightly different: with fl from (7.40) it follows that (e.g. Sigsbee [ 1969]) (Ap >__ aefAO" )

Z = ( W * [ 4 ~ k T n * 2 ) 1/2 = (All -aefAty)3/2](kT)l/2blc

(13.37)

As seen, z from (13.37) is insignificantly less than z from (13.36) for the

Stationary nucleation

195

same W * / k T 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 tile 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 / 2 p * ) 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 < Pe. The dotted curve in Fig. 13.3 illustrates the Ap dependence of z from (13.36) for HON of spherical water droplets in vapours at T = 293 K. The calculation is done with Ap from (2.8), a = (36zcv~) 1/3, O'ef - - O " and the v0 and cr values from Table 3.1 For comparison, the z(Ap) dependence (13.34) and the exact variation of z with A/.t 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(Ap) values are obtained from (13.30) upon dividing its r.h.s, by f ' C * and replacing M by M e f -" 2n*). The numbers at the points on the solid curve indicate the number n* 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(Ap) dependences is negligible in the supersaturation range of A p / k T < 4 which is typical for experiments. The good accuracy of the approximation (13.35) for 0.3 ,r

r

r

9

9

9

r162 9 r ," r 1 6 2 ."

0.2

0.1

,

t

,

,

I

I

I

I

2

I

, I

3

I

I

I

,,I

4

,a~ / 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 (13.33); 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 Theoo~ 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 Js = " 2~ f ' X * 9

(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 f*Xe*f corresponding to the bimolecular 'reaction' of aggregation of a nucleus and a monomer into a smallest, i.e. (n* + 1)-sized, supernucleus. 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 n2 = n* + A*/2 of the nucleus region without shrinking back to subnuclei. This is so, because, energetically, the supernuclei of size between n* and n2 are 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 f* 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 1/z -- 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 [Glasstone 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 f* 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])

Stationa~. nucleation

Js = A exp (- W*/kT)

197

(13.39)

which is applicable to any kind of nucleation and in which A (m -3 s-1 or m -2 s -1) is defined by

A = zf* Co.

(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 Co and is thus an essentially kinetic quantity accounting for the concrete kinetic (f*) and spatial (Co) peculiarities in each particular case of nucleation. Recalling that, typically, 0.01 < z < 1, f f = 1 to 1012 s-1 (see Section 10.1) and Co = 1015 to 1029 m -3 or 101~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 10 41 m -3 S-1 or A - 108 to 10 31 m -2 s -1. 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 t-independent Abt and T, we find that in all cases of nucleation of condensed phases

A = A" e ao/kT.

(13.41)

Here the factor A' ( m -3 S-1 o r m -2 s - l ) is given by

A'= z f ' C o

(13.42)

when Ap is not controlled by T according to (2.20) and by

A'= zf~Co

(13.43)

when T is the parameter used to change Ap in conformity with (2.20). In these expressions f * -fe,n* and f0* = f0,n* are the values off* at Ap = 0. While f * is physically the frequency with which the nucleus would attach monomers at equilibrium, f0* does not have this meaning, because it is determined at the actual temperature Trather than at the equilibrium temperature Te. Thus, f0* is a function of Ap 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 Ap if the supersaturation is varied without changing the temperature T. This is so, since A' from (13.42) is nearly Ap-independent. Indeed, f * is a function of Ap 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 f * with Ap is rather weak. In addition, eqs (13.36) and (13.37) show that, classically, the Zeldovich factor z is also only weakly changing with Ap. All this means that for practical purposes A' can

198

Nucleation: Basic Theory with Applications

be treated as a constant with respect to A/~, especially in view of the more strongly varying exponential factor in (13.41). In some cases A' is even strictly A/~-independent. This is so, e.g. when nucleation is 3D and f * is proportional t o n .2/3 ( c f . eqs (10.3)-(10.5), (10.60) and (10.61)). Then, according to (4.38) and (13.36), n .2/3 • A,u-2 and z ~ A/~e so that the product z f * and thus A' from (13.42) is not a function of A/~ 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 = Pe, (13.36) and (13.42) yields

A" = ~,~(c3ty/18TC2mo)l/2(pevo/kT)C 0

(13.44)

where ),* - ~,,., Co is given by (7.16) or (7.53), and c = (36zr) 1/3 for spheres, c = 6 for cubes, etc. We note that, owing to (7.15), (p~vo/kT) = (povo/kT) exp (-2/kT) ---exp (-2/kT), ~ being the molecular heat of evaporation or sublimation. We shall keep in mind also that with Co 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 3 is replaced by 36rd(2 + cOS0w) (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), 4gt(0w)/(1 - c o s 0w)2 - (2 + cos 0w)). 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 f0* is a strong function of Ap through T. For that reason, the overall A/~ dependence of A from (13.41 ) is actually governed by the dependence of A' on A~. 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'= ~'*[(c3crkT) In/9/t'3/2d2r/] Co e-z/kr

(13.45)

where ~ is the molecular heat of crystallization, c = (36zr) 1/3 for spheres, c = 6 for cubes, etc., and Co is given by (7.12). Similarly, using (10.65) instead of (10.64) shows that with c 3 = 36rd(2 + cos 0w) and Co 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 77 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 7/increases with lowering T, from (13.41) and (13.45) we conclude that A decreases strongly with increasing Ap provided 7* is Apindependent. 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 A/.t 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 nucleation

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/t-independent (provided 7* 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 Pe,n* is equal to the pressure p* inside the nucleus, from (4.16), (4.17), (13.36) and (13.40) we find that [Blander and Katz 1975; Blander 1979] A = 7"(c3o'/18~mo)1/2Co

(13.46)

where Co is specified by (7.12), and c - (36z01/3 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 c 3 = 36rd(2 + cos 0w) and Co 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 A/t 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 n* of water molecules in the nucleus droplet, crystal or bubble at the corresponding supersaturation (this 10 41 1040 I

""

o

o

bubble

o

10 39 10 38 10 37 10 36 !

v

E

<

droplet

10 35 ~ - - - ' - ' - - ' - ' - - - ~ 0 0 10 34 -

100

10

50

50

10 33 10 32 10 31 10

30

-

0

'

'

'

'

I

,

20,'~,

,

1

i

I

2

,

~

I

I

3

zXla/ kT 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 O'ef" - O ' ) . The A values for the water droplets and the ice crystals are computed from eq. (13.41) with A' from (13.44) and (13.45), respectively, and for the bubbles eq. (13.46) is used. The corresponding supersaturation Ap 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) = 4.47

• 10 -7

exp [1234.6/(T- 122.3)]

(13.47)

obtained by fitting the Vogel-Fulcher equation (10.56) to the r/(T) data of Skripov and Koverda [1984] (7"/is in Pa-s, and T is in K). The parameter values used are c 3 = 36zr and those listed in Tables 3.1, 3.2 and 6.2, Co in (13.44) is calculated from (7.53), and the o'(T) dependence of the ice nuclei is ignored. Figure 13.4 demonstrates clearly the difference in the course of A with Ap when the supersaturation is varied isothermally (lines 'droplet' and 'bubble') 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 Js splits up into two separate problems: (i) determining A from (13.40) in each particular case of nucleation by model kinetic considerations concerning f*, 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 f* 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 Ap. 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] (Ap > 0) Js = A exp (- B/Ap2).

(13.48)

Here the kinetic factor A is specified by (13.40) or (13.41), and the thermodynamic parameter B is defined as

B = 4c3V2oCr3f/27kT

(13.49)

where, according to (4.42), O'ef" - O" for HON and O'ef< O" 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 D6ring [ 1935]. Using their own general formula represented above by eq. (13.29), Becker and D6ring [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

201

classical formula for Js in the case of 3D nucleation of condensed phases (Ap >_ 0): Js = A' exp (Ap/kT) exp (-B/Ap2).

(13.50)

Here A' is given by (13.42) when Ap is not defined by (2.20) and is strictly or nearly constant with respect to Ap when this is varied isothermally (cf. eq. (13.44)). In this case, starting from zero at Ap = 0, Js from (13.50) is a monotonously increasing function of Ap. When Ap is controlled by changing T in accordance with (2.20), however, A' is given by (13.43) and decreases exponentially with increasing Ap. This makes Js 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 Ap is determined by (2.20), from (13.45) and (13.50) we find that (Ap > O) Js - (A"/r/) exp (Ap/kT) exp (- B/Ap 2)

(13.51)

where A" is of the form

A'" =

~}~[(c3GkT)l/2/91~3/2

d~]Co e- z/kr

(13.52)

with Co specified by (7.12). Without the factor exp (Ap/kT), eq. (13.51) parallels the Js(Ap) dependence of Volmer [1939]. We note also that in this equation the product A"exp (Ap/kT) with A" from (13.52) is virtually constant with respect to Ap provided 7* is constant, too. Indeed, using the approximation (2.23) for Ap, we have exp [(A~ - 2,)/kT] = 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; Zettlemoyer 1969] (Ap > aefAO") Js = A exp [-B/(Ap

(13.53)

- aefAO')]

for 2D nuclei on a foreign substrate (then Aa ~ 0) and that [Hirth and Pound 1963; Zettlemoyer 1969] (Ap _>0) Js = A exp (- B/Ap)

(13.54)

for 2D nuclei on their own substrate (then Aa = 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 = bex2/4kT.

(13.55)

In this formula b -- 2 ( ~ e f ) 1 / 2 for disks, b -" 4 r-ue for square prisms, etc., and aef = a 0 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 Js = A' exp (Ap/kT) exp [- B/(Ap Js = A' exp (Ap/kT) exp (- B/Ap)

-

aefAO')]

(13.56) (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, Ap 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 Ap-independent if Ap is varied isothermally, but is a strong function of Ap when this is changed by changing T in conformity with (2.20). That is why the character of the Ap-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 A/_t than is Js from (13.50) because of the lower power of Ap 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/.t < 0. Indeed, for such substrates Air< 0 and from (13.56) we see that Js > 0 already when Ap >-aeelAtrl. 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 1986]. 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 < Pe) Js = A exp [- B/(p* - p)2],

(13.58)

the kinetic and thermodynamic parameters A and B being given by (13.40) and the expression B = 4c 3o'ef3/27 kT

(13.59)

in which O'ef" - O" for HON and O'ef< O" for HEN. In eq. (13.58), the pressure p* inside the gaseous nucleus is a function ofp as required by (4.14) and can be approximated by the equilibrium pressure Pe when p e o o / k T << l. As tO A, it is defined by eq. (13.40) and is practically p-independent. For example, when f* 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 Js 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 Js 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 (-qr*lkT) exp (n*A~lkT)

(13.60)

Js = A (C~ICo)"* exp [(n*q~l - qr*)/kT]

(13.61)

where t/r* is the nucleus effective excess energy, and C1 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 q~* or n* q~l - qJ* 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 f* and Co. Atomistically, z = 1 is a good approximation for the Zeldovich factor [Walton 1962, 1969b], since the nucleus region is quite narrow (A* = 1) when n* ~ 1. Let us now exemplify the application of eqs (13.60) and (13.61) by considering atomistic nucleation of condensed phases. Then q~l is of the form (cf. (3.89)) q~l = & - E1 - asa0,

(13.62)

qJ* 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) C1 = Co exp [(A/t

-

~l)/kZ],

(13.63)

from (13.60) and (13.61) we get the equivalent expressions Js = A' exp [ ( E * - 2 n * + asA*)/kT] exp [(n* + 1)Ap/kT]

(13.64)

and Js = A" (CI/Co) n* + 1 exp ({2 + E* - (n* + 1) E1

+ as[A* - (n* + 1) ao]}/kT)

(13.65)

where A' is given by (13.42) or (13.43) when, respectively, Ap is not or is controlled by T 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, E1 = 0) and HEN (then as > 0, El ~ 0) on foreign (as ~ a) or own (as = a) substrate (we recall also that E1 + asao--- Edes and that for 2D HEN A* = n*ao if the nuclei are of monolayer height). These equations contain as particular cases the known atomistic formulae for Js in condensation of vapours on substrates [Walton 1962, 1969b] and in electrodeposition [Milchev et al. 1974; Milchev 1991]. Equation (13.64) tells us that when A/t is varied without changing T, atomistically, Js is a simple exponentially increasing function of zS/t. This is so, as then A' is A/t-independent, and n*, E* and A* are constant, too. Also, from eq. (13.65) we see that when A/t is varied isothermally, Js o~ 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 (n* + 1)-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 supernuclel must be proporuonal to C1n * + l , hence the proporuonality n*+l 9 of Js to C~ . 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 C1. 9

9

9

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 Js 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(Ap) 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] (S > 1): Js = A' S exp (- B'/ln 2 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" = 4c3v~cr3f/27 k 3 T 3

(13.67)

where O'ef crfor HON and Cref< o'for HEN (see eq. (4.42)). The supersaturation ratio S is defined as "

"

S = p/p~,

S = Hie

(13.68)

for nucleation in vapours (cf. eq. (7.19)) and as S = C/Ce,

S = a/a e,

S = H/II e

(13.69)

for nucleation in solutions. The kinetic factor A' is strictly or practically Sindependent 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 Co from (7.53) represents the self-consistent classical formula for Js, Js = 7* (c3~r/18~2mo)l/2(pe/kT) 2 Vo exp (c v~/3 cr/kT) S exp (- B'/ln 2 S),

given for spherical nuclei (c 3 = 36zr), 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 J~ [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)" Js = A' Sn* § 1 exp [(E*

~n* + r

(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 as > 0 for HEN. The kinetic factor A' is S-independent when S 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' = zT* co2/3[pe/(21r,mokT)l/2]n*2/3Co

(13.71)

where Co is specified by (7.16). We note that, with c = [(1 - cos 0w)/ 2~?/3(0w)](36n:)l/3 (cf. eqs (10.3) and (10.4)) and Co 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 < 2TeAse/Acp,e): Js = [A"/o(T)] exp (AseAT]kT) exp (- B'/TAT2).

(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' = 4c 3o00"ef/27 2 3 kAs 2 where O'ef--

O" for

HON and

O'ef < O" for

(13.73)

HEN. The kinetic factor A" is defined

as

A"= z fo OCo

(13.74)

and is T-dependent mainly through the factor f0"77 o~ exp (- ~l/kT) where q~l is specified by (13.62). For instance, in the case of HON under interfacetransfer control, eq. (13.74) passes into (13.52) which shows that then A"o~ exp (- 21kT) (according to(13.62), for HON q~l = ~,)- 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 f0* 77 ,,~ exp (- 2/kT) (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 (AseAT/kT) in (13.72) is practically T-independent: A" exp (AseAT/kT) o~ exp [(AseAT- ~,)/kT] = exp (- Ase/k) = 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 f ~ o o~ exp (- ~l/kT) = exp [(- ~ + E1 + asao)/kT] so that the product A" exp (AseAT[kT) in (13.72) may depend essentially on T:

206 Nucleation: Basic Theory with Applications A" exp (AseAT/kT) ": exp [(AseAT- ~, + E 1 + Crsao)/kT] o~ exp [(E 1 + trsao)/kT].

(13.76)

This means that in these cases of 3D HEN the T dependence not only of the melt viscosity r/, but also of this product may contribute to the dependence of Js from (13.72) on the undercooling AT. The magnitude of the contribution of this product is controlled by the energy El + Crsa0 -- Edes. (d) Atomistic HON or HEN (either 3D or 2D) in melts (0 <_ AT < 2T~AsJ Acp,e):

Js = [A"/rl(T)] exp [- (n* + 1)Ase/k] exp [(~ + E* + trsA*)/kT]. (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"o~ exp (- ~l/kT).

(13.78)

For example, in the case of HON (then ~1 = 2) if monomer attachment is controlled by interface transfer, according to (10.64) and (13.74) we have A" = z'y*(co2/3n*2/3/3 lrd~) (kT/oo) Co e-z/kT

(13.79)

with Co from (7.12). Thus, in this case the product A" exp (2/kT) in (13.77) can be treated as virtually T-independent. (e) Classical electrochemical 3D HEN [Volmer 1939; Vetter 1967] (Atp > 0): Js = A' exp (zieoAtp/kT) exp ( - B'/Atp2).

(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" = 4c 3O00"ef 2 3/27z~ e 2 k T

(13.81)

where O'ef< (5". The kinetic factor A' is defined by (13.42) with Co from (7.8) or (7.9). It depends negligibly on the overvoltage Atp when this is changed isothermally. (f) Atomistic electrochemical 3D or 2D HEN [Milchev et al. 1974] (Aq~>_.0): Js = A' exp [(E* - An* + crsA*)/kT] exp [(n* + 1)zieoAq~/kT].

(13.82)

This equation is obtained from (13.64) with the aid of (2.27) and in it ~ is the molecular heat of dissolution. The kinetic factor A" is given by (13.42) with z --- 1 and is again A~independent when T is kept constant while varying Atp. 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 > aoAcr/kT): Js = A' S exp [- B'/(ln S - aoAcr/kT)].

(13.83)

This expression follows from (13.56) with the help of (2.8), (2.9), (2.13),

Stationary nucleation

207

(2.14), (2.16), (13.68) and (13.69). In it Acr~ 0 for a foreign and Act= 0 for the own substrate. According to (13.55), the thermodynamic parameter B' is of the form

B" = b2h2/4kZT 2

(13.84)

where b = 2(~a0) 1/2 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 = le, (13.37) and (13.42) we find that in molecular beam condensation

A' = ?'* [c*kT(ln S - aoAcr/kT)3/2/btr ~2 le Co

(13.85)

where Co is given by (7.8) or (7.9). (b) Atomistic 2D HEN in vapours or solutions (In S > aoAcr/kT):

Js = A" S n* + 1 exp {[E* + (Crsa0 - &)n*]/kT}

(13.86)

or, equivalently,

Js = A'S (C1/Co) n* exp (Es*/kT).

(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*ao. 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* = Es,n. defined by (7.31). Naturally, eq. (13.86) follows also from (13.70) with A* = n*ao. 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 = Ie and (13.42) we find that

A'= z~'*c*22IeCo

(13.88)

where Co 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 ?'* = 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), &{ = Ds~rd, C~ =lva and S = I/Ie. (c) Classical 2D HEN of crystals in melts [Volmer 1939; Zettlemoyer 1969] (aoAG/As e <_ AT < 2TeAse[Acp,e):

Js = [A"/o(T)] exp

(AseAT[kT)

exp [-B'/T(AT-aoAty[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"

= b 2 x2/4kAse,

(13.90)

and the kinetic factor A" is given by (13.74) with Co from (7.8) or (7.9). As already discussed, this factor is T-dependent according to (13.78) with ~1 =

208 Nucleation: Basic Theory with Applications ~,- E1 - (rsa0, because now the monomers that join the monolayer nuclei are in contact with the substrate. For that reason, the product A" exp (AseAT/kT) 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" = ~'* [b(kT) l/2(AseAT - aoAtT) l/2/61rdovo] C O exp [ ( - ~ + E 1 + Crsao)/kT]

(13.91)

where O's ~ tr, Atr ~ 0 and as = tr, At7 = 0 for foreign and own substrate, respectively, and E 1 + O'sa0 --- Edes. (d) Atomistic 2D HEN of crystals in melts (aoAtr/Ase <- A T < 2TeAse/Acp,e):

Js = [A"/rl(T)] exp [ - ( n * + 1)Ase/k] exp [(~ + E* + t:rsaon*)/kT]. (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*ao. With as ~ tr and as = cr it is applicable to nuclei on foreign and own substrate, respectively. The kinetic factor A" is given by (13.74) with z = 1 and Co from (7.8) or (7.9). This factor depends on T according to (13.78) with ~l = A,- El - Crsa0 (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" = z'y*(bn*l/2kT/3~dovo)Co exp [(-/q, + E1 + Crsao)/kT] (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] (Aq~ >_.aoAcr/zieo): Js = A' exp (zieoA~/kT) exp [- B'/(Atp- aoAtY/zieo)].

(13.94)

This equation results from (13.56) upon using (2.27) and (13.55) and in it At~ 0 for a foreign and Ao = 0 for the own substrate. The thermodynamic parameter B' has the form

B' = b2h"2/4zieokT

(13.95)

where ~cis the specific edge energy of the monolayer nucleus. The kinetic factor A' is specified by (13.42) with Co from (7.8) or (7.9) and depends negligibly on Atp when this is varied isothermally. (f) Atomistic electrochemical 2D HEN (Aq9 >_ aoAtr/zieo): Js = A' exp {[E* + (O'sa0 -A)n*]/kT} exp [(n* + 1)zieoAtp/kT] (13.96) or, equivalently, Js = A' exp (zieoAq~/kT) (C]/Co) '~* exp ( E * / k T ) .

(13.97)

Stationa~. 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 Co from (7.8) or (7.9). This factor is Aquindependent when the overvoltage is changed at constant T. With ors ~ cr and as = a, eq. (13.96) applies to 2D HEN of monolayers on foreign and own substrate, respectively. This equation follows also from (13.82) with A* = aon* 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 C1 of adsorbed monomers and of the nucleus 'surface' binding energy Es* = Es,n, defined by (7.31). It shows how knowledge about the C1(Atp) dependence from adsorption theories can help in determining Js in electrodeposition of monolayers of condensed phases. 3 . 3 D 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 < Ap < Pe): Js = A exp (-B/Ap2).

(13.98)

This equation is an approximation to eq. (13.58) and its accuracy is good when PeVo/kT< < 1. The thermodynamic parameter B is defined by eq. (13.59). The kinetic factorA is given by eq. (13.40) with Co from (7.12) for HON and Co from (7.8)-(7.11) for HEN and is practically independent of the underpressure Ap - Pe - 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): Js = A exp [(-#* + p'V* + asA*)/kT] exp (- V*p/kT).

(13.100)

This formula follows from eq. (13.60) upon using @* from (4.50) and in it as = 0 for HON and as > 0 for HEN. The practically p-independent kinetic factor A is given by eq. (13.40) with z --- 1 and Co from (7.12) for HON and from (7.8)-(7.11) for HEN. The quantities p*, V*,A* and ~* in eq. (13.100) are unknown, but if they are independent ofp 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 S, AT and Ap. The curves labelled 'HON' and 'HEN' in Figs 13.5, 13.6 and 13.7 depict the classical Js(Ap) dependence for HON and HEN of water droplets, crystals and bubbles,

21o 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 7* = 1 and that the nuclei are spherical in HON and hemispherical in HEN. The foreign incompletely wetted substrate (the wetting angle 0w is equal to ~z/2) is considered free of active centres and having Co = 1019 adsorption sites per m E of its surface (see eq. (7.8)). The numbers at the points on the Js(Ap) 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 n* values are calculated from the Gibbs-Thomson equations (4.38) and (4.40) with c a = 367r, Ap from (2.8), (2.10) and (2.23) and with O'ef- " O " for HON and Gee = ( 1 / 2 ) 1 / 3 0 " for HEN (the latter equality follows from (4.42) upon accounting that according to (3.56) gt = 1/2 at 0w = zd2). As already noted many times, the resulting rather small values of n* imply that, being classical, the J~(Ap) curves in Figs. 13.5-13.7 may have only a qualitative character. The Js(S) 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), c 3 = 36n: and Co = 2.6 x 1027 m -3 are used for HON, the Co value following from (7.53). In the case of HEN, in (13.44) c 3 is replaced by 18zr which is the value of 36n:/(2 + cos 0w) at 0w = zd2. Figure 13.5 shows the sharp rise of Js with S (i.e. with Ap) both for HON and HEN, which is more 1025

.

10 20

~~

.

.

il-

HE

"

5X10 6

.

.

.

.

.

.

.

.

40

HON

1015

5O 0

1010

,7 .-

.

1

.

.

.

.

2

.

.

.

3

.

4 0

105

l o .5

o-,o

/ / oo

10-15 1

2

3

4

5

6

S

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 J~(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 m 3 per second. In the case of HEN this S value is smaller: already for S > 2.35 we have Js > 1 droplet per m 2 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 Apc 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 H O N Js = 10-18 m-3 s-l 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 m 3 of water vapour at the chosen T = 293 K and p ape 2.35Pe. The calculation of the Js(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 =

-

1030 1025

~ e'E

102~

'7,

1015

v

"

1010 105 1

0

/ boo 40

// 8o AT (K)

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 r/(T) dependence is taken into account by means ofeq. (13.47). In eq. (13.52) c 3 = 367r and Co from (7.12) are used in the case of HON. As for the droplets, in the case of HEN c 3 in (13.52) is replaced by 18Jr, and Co is given the value of 1019 m -2 mentioned above. The temperature dependence of O'efin (13.73) is ignored and this quantity is calculated from (4.42) with ~ = 1 and (1/2) 1/3 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 Tg -- 135 K [Skripov and Koverda 1984] is already less than 1 nucleus per m~per second and finally vanishes at T = 122.3 K, the temperature at which 7/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 Ap by lowering T leads to an impeded motion of the molecules in the old phase and thus to a lower frequency f* of their attachment to the nuclei. The lower f*, however, the smaller A from (13.40) and, hence, Js 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), c 3 is again set equal to 36zr and 18Jr for the spherical and the hemispherical bubbles in the cases of HON and HEN, respectively. Also, Co is determined from (7.12) for HON (as already mentioned, for HEN Co = 1019 m-2). The O'ef values used in (13.59) are calculated from (4.42) with ~ = 1 and (1/2) 1/3 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 m 3 and that one of the walls of this container is a foreign substrate free of active centres and characterized by wetting angle 0w = re/2. If the substrate area is 1 m 2, the stationary rate of nucleation in this system is directly read from Figs 13.5-13.7. We thus see that at lower A/.t values (S < 4.4, AT < 42 K and Ap < 0.92pe 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

,.~, 1020

..

_: ,x,o,I -

,!

1015 -

E

101o

--3

107t

o. . . .

HEN HON oi,

50~

j/ oo oo/

-,.o

/

/

//2~ ~

105

1

0

0.2

0.4

0.6

0.8

1.0

,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 O'ef< (7") and the resulting higher equilibrium concentration C* of nuclei on the substrate (see Fig. 7.2). However, for higher Ap 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, AT > 42 K and Ap > 0.92pe 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 Ap range, we can even register the HEN-to-HON transition occurring with increasing Ap. Experimental evidence for this transition was presented, e.g. by Butorin and Skripov [ 1972] 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 f* of monomer attachment to the nucleus and the concentration Co of sites on which the nuclei can form. In Chapter 4 and Sections 7.1 and 10.1 we have seen how W*, Co and f* 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*, Co and f* 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 Js as the shape factors c or b, the specific energies Gef, AG or ~r the binding energies E* or E*, the sticking coefficient ~ , the number Na 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/~ 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 ( o f 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 J~, i.e. then J(t) = Js. Using this result in eq. (11.10), we find easily that in this case ~' is a linearly increasing function of time t:

((t) = ~o + Jst.

(13.101)

Here (0 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 ((t) 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 [1995]; Rudek et al. [1996]; Fisk et al. [1998]). We shall now see how some of the Js(Ap) 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 Js(Ap) formula for interpretation of a given set of experimental Js(Ap) data is justified if we have a p r i o r i 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(Ap) dependence remain questionable. As an example, let us consider nucleation of condensed phases. When it is known for sure that certain Js(Ap) data correspond to HON, classically, eq. (13.48) can be employed for describing the observed dependence of Js on Ap. Alternatively, if A and B in (13.48) are unknown, the data are usually plotted in In Js-vS-(1/Ap 2) or (1/TAp 2) coordinates, because the classical theory suggests fitting by a straight line with equation In Js = In A - B(1/Ap 2) or

In Js = In A - B ' A s 2 e ( 1 / T A p 2)

(13.103)

when Ap is varied isothermally or by means of T according to (2.20) (B' is defined by (13.73)). From the best fit, the two free parameters A and B or B ' A s 2 are determined, and then B or B'As2e is used in (13.49) or (13.73) for calculation of the specific surface energy or. 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 ' A s 2 value may be O'efrather than cr 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(Ap) dependence in In Js-vS-[1/(Ap - aefAO')] or [ 1 / T ( A p - aefA(Y)] coordinates according to the equation In Js = In A - B[1/(Ap - aefAcr)] or In Js = In A - B ' A s e [ 1 / T ( A p - aefAO')]

(13.104)

when Ap is varied isothermally or via T in conformity with (2.20). Here A, B or B ' A s e and aefAO"are free parameters and the best-fit value of B or B ' A s e can be used in (13.55) or (13.90) for calculation of the specific edge energy to. It is worth keeping in mind that the interpretation of Js(Ap) data is unambiguous only for HEN on the own substrate, since the process can only

2!6

Nucleation: Basic Theory with Applications

be 2D. Moreover, Atr = 0 by definition so that, classically, the experimental Js(Ap) dependence can be linearized in In Js-vS-(1/Ap) or (1/TAp) coordinates" In Js = In A - B(1/Ap)

or In Js = In A - B'Ase(1/TAp).

(13.105)

This formula results from (13.54) or (13.104) and, like (13.103), contains two free parameters: A and B or B'Ase. It is important to note that, having found the thermodynamic parameters B, B'AS2e or B'As e 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* = 2 B k T / A p 3

or

n* = 2kB'As2e/AI.t 3

(13.106)

for HON or 3D HEN of condensed phases when Ap 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* = B k T / ( A p - aefAty)2 or

n*

= kBtAse/(A[l.l

-

aefAO') 2

(13.107)

for 2D HEN on foreign (Act, 0) or own (Act = 0) substrate in the respective cases of variation of Ap. 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 Js(Ap) data requires the usage of eq. (13.60) of the atomistic theory. Irrespective of the kind of the nucleation process, when n* is Apindependent, for nucleation of condensed phases this theory suggests a linearization of the Js(Ap) dependence in In Js-vs-Ap coordinates when Ap is varied isothermally. Indeed, from (13.64) there results In Js = [In A" + (E* - ~,n* + tYsA*)/kT] + [(n* + 1)/kT]Ap (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 Ap range studied (see Fig. 4.1b). The above method of using the classical nucleation theory for analysing experimental Js(A/~) data ignores the dependence of the kinetic factor A on A/~, 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 Ap 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 Js(Ap) dependence requires accounting of the fact that A may change appreciably with Ap, especially in a wider Ap range and/or when Ap 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(Ap) dependence can be achieved

Stationary nucleation

217

with a sufficient accuracy by using In (Js/S) and In A' instead of In Js and In A in eqs (13.103)-(13.105). Similarly, in these equations In Js and lnA have to be replaced by In (r/J s) and In A" (see eqs (13.72) and (13.89)) in a more accurate analysis of Js(A/~) data for nucleation in melts when Ap 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 Ap rather than the supersaturation A/~. In conformity with eq. (13.98), classically, linearization of experimental Js(Ap) data both for HON and for 3D HEN of bubbles in own liquid can be obtained in In Js-vS-(1/Ap 2) coordinates provided Ap is varied isothermally: In Js = In A - B(1/Ap2).

(13.109)

With the aid of the best-fit value of B it is possible to calculate O'efand n* by using, respectively, eq. (13.59) and the approximate relation n* = 2Bpe/Ap 3

(13.11 O)

which follows from (4.40) and (13.59) upon setting p* = Pe. When the condition pevo/kT << 1 is not satisfied, in (13.109) and (13.110) Pe 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(A/.t) data of Viisanen et al. [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 Js is a linear function of 1/(ln S) 2, i.e. of 1/A/~2. 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 B/(kT) 2 = 185.5 and 231.5 and A = 1.2 x 1032 and 7.5 x 1029 m -3 s-1 for water and n-butanol, respectively. Using these B/(kT) 2 values in (13.49), c 3 = 367r (spherical droplets), cref = cr (HON), v0 = 0.03 nm 3 for water and v0 = 0.145 nm 3 for n-butanol results in, respectively, cr = 82.5 and 28.8 mJ/m 2 for water and n-butanol. These trvalues compare with the known values of the specific surface energy of water at T= 259 K (tr= 77.6 mJ/m 2 [Viisanen et al. 1993]) and of n-butanol at T = 240 K (tr = 28.7 mJ/m 2 [Viisanen and Strey 1994]). The theoretically predicted A values, however, do not agree with the experimentally obtained ones. Indeed, assuming direct-impingement control with 7* = 1, with the help of Co from (7.53), m0 = 3 • 10-26 kg and Pe = 204 Pa for water at T = 259 K [Viisanen et al. 1993] and m0 = 1.2 x 10-25 kg and Pe = 4.52 Pa for n-butanol at T = 240 K [Viisanen and Strey 1994], from (13.44) it follows that A' = 1.8 • 1 0 33 and 6.1 x 1 0 30 m -3 s -1 for water and nbutanol, respectively. Using the mid-range S = 8 for water and S = 11 for nbutanol, we thus find that, theoretically, A = 1.4 x 10 34 and 6.7 x 1031 m -3 s -1 for water and n-butanol, respectively. These A values are about two orders of

218

Nucleation: Basic Theory with Applications

45

40 "7 O3 !

i

x~~ x~x

35

E

~" x

%

30

v

xx

N

t""

,,,,=-

\

25

20 0.12

,,

,

,

I

,

0.16

,

,

',,

t

I

\

0.20

-,,,

,

.~

I

0.24

,

,

.

,

I

0.28

,

9

0.32

1 /In2S Fig. 13.8 Dependence of the stationary nucleation rate on the supersaturation ratio in H O N of droplets in vapours: circles - data for water at T - 259 K [Viisanen et al. 1993]; triangles - data for n-butanol at T = 240 K [Viisanen and Strey 1994; Strey et al. 1994]; solid l i n e s - 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(Ap) data in In (Js/S)-vs-1/(ln S) 2 coordinates. The disagreement between theory and experiment is visualized in Fig. 13.8 in which the dashed lines display the Js(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 [i994] 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 / ( k T ) 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 J~(A~) data for nucleation of ice in water (the circles and the triangles [Butorin and Skripov 1972]) and for nucleation of gallium fl-phase crystals in gallium melt (the squares [Skripov et al. 1971]). In conformity with the second of eqs (13.103) and, hence, with the classical theory for HON or 3D HEN, In Js is a linear function of 1~TAT 2, since in the narrow AT range studied A = (A"/O) exp ( A s e A T / k T ) = constant

Stationary nucleation

32

.....

H

,

219

,

30 28

"7,

26

!

E "~ffl

24

E

22

20 18

0.55

0.60

0 . 6 5 0.70

II

3.0

3.2

3.4

3.6

3.8

1 / T(z~T) 2 (10 .6 K 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]; squares data for gallium [3-phase crystals in gallium melt [Skripov et al. 1971]; l i n e s - 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 O'ef 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 K 3 and In (A/m-3s -1) = 83.8 for ice and B' = 9.54 x 107 K 3 and In (A/m -3 s-1) = 87.2 for gallium. Using these B' values in eq. (13.73) with v0 = 0.0326 nm 3 and Ase = 2.63k for ice and with v0 = 0.0196 nm 3 and As e = 1.32k for gallium yields O'ee= 26.4 mJ/m 2 for ice in water and O'er= 40.8 mJ/m 2 for gallium 13phase crystals in own melt (these numbers differ slightly from 28.7 of Butorin and Skripov [ 1972] and 40.4 of Skripov et al. [ 1971 ]). Accordingly, A = 2.5 • 10 36 m -3 s-] for ice and A = 7.4 x 10 37 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 • 10 36 m -3 s-1 in the case of interface-transfer control (this value follows from eq. (13.51) with Ap, 77 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 Js(Ap) data of Milchev and Vassileva [ 1980] for electrochemical nucleation of silver crystals on a glassy carbon electrode in aqueous solution of AgNO 3 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

220

Nucleation: Basic Theory with Applications

26

a ' l , , , i , , , i , , , i , , , i

,

,

,

,

,

,

i",

, , ,

,

,

,

I,

I

24 "7

r.n 22

i

E '~ 20

~ Q 18

16

0

|

500 1000 1500

(v-2)

8

10

12

14

16

(A~0+0.04)"1 (V -1)

0.05

Aqi (V)

0.10

Fig. 13.10 Dependence of the stationary nucleation rate on the overvoltage in HEN on a foreign substrate: circles- data of Milchev and Vassileva [1980] for silver crystals on a glassy carbon electrode in aqueous solution of AgN03 at 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. (13.108)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/Atp2, i.e. on 1/Ap 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 Ap is again related to the overvoltage Atp via eq. (2.27). Figure 13.10b demonstrates that the experimental Js(AM) dependence (the circles) can be linearized in In Js-vS- 1/(Aq~- aefAtY[zieo) coordinates (cf. eq. (13.94)). The best-fit straight line corresponds to aefAtY[zieo = - 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 B/zieo = 1.04 V and A = 1.8 • 1014 m -2 s-1. With zi = 1 and a0 = 0.083 nm e, from (13.55) we thus get tr = 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 ~r 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/zieo = 1.04 V and aefAtY[zie 0 = - 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

Stationa~. nucleation

221

Vassileva [ 1980], graphically, the dependence of In Js on Atp is a broken straight line (Fig. 13.10c). The slopes of this line yield n* = 4.5 _+0.2 for Atp = 25 to 50 mV and n* = 1.2 _ 0.1 for Atp = 50 to 90 mV (these n* values are calculated from eq. (13.108) with Ap from (2.27) and zi = 1). The conclusion of Milchev and Vassileva [1980] is that the silver nucleus changes discontinuously 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 Js(Ap) 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 Js(Ap) 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 AgNO3 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 1/Ap~ 1/Atp. The slope and the intercept of the best-fit straight line drawn in Fig. 13.11 are B/zieo = 335 mV and In A = 49.4, the latter giving A = 2.8 • 1021 m -2 s-]. Since zi = 1, using this B/zieo value in eq. (13.55) results in tc = 27 pJ/m [Bostanov et al. 1983] under the assumption that the 2D nucleus is square 35

'

'

I

'

'

'

I

'

'

'

I

'

'

'

i

'

"

'

3,4

s"~

33

E

32

t""

31

9

[

]

==,...,.

30

29 44

,.,i.,,,.,, 46

48

i,.,i,,.,...,,..i.,, 50 52 54

1/ Fig. 13.11

56

58

60

(v

Dependence of the stationary nucleation rate on the overvoltage in 2D HEN on own substrate: c i r c l e s - data of Bostanov et al. [1983] for silver on the (100) face of a silver electrode in aqueous solution of AgN03 at T = 318 K; lines - best fit according to the first eq. (13.105).

222

Nucleation: Basic Theory with Applications

shaped (then b = 4a~/2 = 1.154 nm). Physically, both the tr and the A values are reasonable even though the nucleus size is rather small" setting Ap = zie0AtP and AG = 0 in the first of eqs (13.107) and using the above B/zieo 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 Js(Ap) data of Skripov [1972] for HON of bubbles of diethyl ether (the squares) and benzole (the circles) in own liquid at T = 421.15 and 498.95 K, respectively. In accordance with eq. (13.109) of the classical theory, In Js is a linear function of 1/Ap 2. The slopes and the intercepts of the best-fit straight lines drawn in Fig. 13.12 give B = 57.0 MPa 2 and A = 8.7 x 1024 m -3 s-1 for diethyl ether and B = 245 MPa 2 and A = 2.9 x 1038 m -3 s-1 for benzole. Using these B values in eq. (13.59) with c 3 = 36Jr leads to specific surface energies 0.el = 2.70 mJ/m 2 for diethyl ether and Cref= 4.65 rnJ/m 2 for benzole which are comparable with the independently known cr = 3.13 mJ/m 2 and tr = 4.68 mJ/m 2 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 = 1 0 39 m - 3 s -1 for HON, which follows from eq. (13.46) under the assumption for evaporation control and sticking coefficient 7" = 1. While the agreement between the theoretical and experimental values of A for benzole and the fact that for it 0.ef = O" are strongly in support of HON, the smaller experimental value of A and the obtained 0.ef -- 0 . 8 6 0 " for diethyl ether are an indication for HEN. Since Pe = 21

.

9

I

I

.

|

,

.

.

| '.

,

.

! ' ,

' |

,

|

.

I

.

.

,

,

',

'.

,

.

",

,

,

|

| "=I

,

20

"7or} o~ !

E

19 18 17

15 14

0.27

0.28

0.29

0.30

.

0.31

I I

0.65

0 . 7 0 0.75

.

.

.

0.80

1 /(,5,p) 2 (MPa "2) Fig. 13.12 Dependence of the stationary nucleation rate on the underpressure: circles - data f o r benzole bubbles in benzole at T = 498.95 K [Skripov 1972]; squares - data f o r 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 Ap 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 Js(Ap) 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 Ap we are faced with the question for the choice of a formula for J~ 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 Js as a function of A/~. 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(A/J) dependence is interpreted in coordinates which are not specific for any particular Js(A/J) 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 Js(A/J) data [Kashchiev 1982]. Suppose we have an experimental Js(Ap) dependence obtained by varying A/J 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 Js formula (13.39) rewritten as -W* = kT

In Js - k T In A.

(14.1)

Differentiating this equation with respect to A/J and invoking the nucleation theorem in the form of eq. (5.21) results in the expression An* = k T ( 1 - Po~d/Pnew)[d(ln Js)/dA/J - d(ln A)/dA/J]

(14.2)

which, owing to (5.18), can be represented equivalently as An* = kT[d(ln Js)/d/Jold - d(ln A)/d/Jold].

(14.3)

These formulae are applicable to all possible cases of one-component nucleation when/Jold and thus A/J is changed isothermally. They show that for the model-independent determination of An* it is necessary to plot the Js data in In Js-vs-A/J or In Js-vS-/Jold coordinates: then the slope at each point of the resulting line gives the corresponding An* value if the dependence of In A on Ap or Pold is neglected. To a first approximation, this is possible, because the kinetic factor A is a relatively weak function of A/J 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/~ at constant T, from eq. (13.41) which expresses the A(A/I) dependence in a very general form we find that kTd(ln A)/dA/I -1, since d(ln A')/dA/~ = 0. For nucleation of gaseous phases at isothermally varied Ap the contribution of the In A derivative is even less important: we can set d(ln A)/dA/.t = 0, as in this case A is practically Ap-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)[kTd(ln Js)/dAIJ - hA]

(14.4)

where n A =- kTd(ln A)/dA/J --- 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 - Pold/Pnew)/(1 - Pold/Pn*w)][kTd(ln Js)/dAIJ - hA].

(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 PnCw 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 Pn*ew = Pnew and eq. (14.5) simplifies to [Kashchiev 1982] n* = kTd(ln Js)/dAIJ - 1.

(14.6)

For nucleation of gaseo.us phases, again not too close to the spinodal, we have Pnew << Pold and Pnew << Pol0 SO that eq. (14.5) can be approximated by n* = kT( Pn*ew/Pnew)d(ln Js)/dA/J.

(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 Pn*w 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* = (kT/Pnew)d(ln Js)/dAIJ.

(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; Milchev et al. 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 et al. 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 A/j in terms of the corresponding experimentally controllable parameter, e.g. p, I, 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), from eq. (14.6) it follows that [Kashchiev 1982] n* = d(ln Js)/d(ln S) - 1 = d(ln Js)/d(ln p) - 1 = d(ln Js)/d(ln/) - 1 (14.9) for nucleation in vapours, n* = d(ln Js)/d(ln S) - 1 = d(ln Js)/d(ln C) - 1 = d(ln Js)/d(ln a) - 1 = d(ln Js)/d(ln H ) - 1 (14.10) for nucleation of condensed phases in solutions, n* = (kT/zieo)d(ln Js)/dAq~- 1 = -(kT/zieo)d(ln Js)/dtp - 1

(14.11)

for electrochemical nucleation of condensed phases and n* = (kT/Ave)d(ln Js)/d(p - P e ) - 1 = (kTIAve)d(ln J s ) l d p - 1 (14.12)

for nucleation of crystals in melts (or during polymorphic transformation) when A/J is varied by p at constant T. Similarly, under the condition pevo/kT << 1, treating the new gaseous phase as ideal gas (then Pnew = p/kT) and using (2.10) and (13.99), from eq. (14.8) we find that V* = kTd(ln Js)/dAp = - k T d ( l n Js)/dp

(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(A/J) data are obtained by changing A/J 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 A/J through T. Nevertheless, in the experimentally important case of changing A/J 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 A/.t and applying the nucleation theorem in the form of eq. (5.39), we find n* = k[(1 - Snew/Sold)/( 1

- Snew/Sold)][d(T

In Js)/dA/~ - d(T In A)/dA/~]. (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 Snewof a molecule in the EDS-defined nucleus. Unfortunately, since Snowis generally unknown, eq. (14.14) is practically usable only in its approximate form [Kashchiev 1982] n* = k d ( T In Js)/dA/l

-

nA

(14.15)

corresponding to the assumption Snew ---- Snew 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 o f n A = k d ( T In A)/dA/~ to n* can be significant. For instance, for nucleation of crystals in melts when Ap is changed according to (2.20), the analysis shows [Kashchiev 1982] that nA may depend considerably on Abt 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 " / r l ) e AIa/kT

(14.16)

in which despite that the product A" exp (Alu/kT) may be virtually constant, the melt viscosity 7/is a strong function of T (see eq. (10.56)), i.e. of Ap. The conclusion is, therefore, that due to the theoretical uncertainty in hA, 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(Aju) data obtained at non-isothermally varied A/~. The nucleation theorem has a valuable application also to multicomponent nucleation. With the aid of Js(Pold,i) 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 Poldi of component i in the old phase is vaned isothermally, the formula for A n i reads [Strey and Viisanen 1993; Viisanen et al. 1994; Oxtoby and Kashchiev 1994] An* = kTtg(ln Js)/tgl~old, i - hA, i

(14.17)

where n a , i - kTtg(ln A)/tgbtold, i 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 Pold,i 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-nonanebutanol 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 a universal method for determination of An*, n* and V* from experimental Js(A~t) 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 Js-vs-Ap or/~old coordinates. The slope of the resulting line then yields An* or n* with an accuracy of about 1 molecule. Thus, the Ap 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(Ap) 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 T = constant. The circles in Fig. 14.1a show the simulation data for In (Js/le) as a function of At~/kT (le is the impingement rate at equilibrium), and the curve represents the best-fit function In (Js/le) = - 1 3 . 4 3 2 6 2 / ( A M k T ) - 3.38439 + 1.93868(A/~/kT). The circles in Fig. 14. lb depict the n*(Ap) 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*(Ap) dependence for the EDSdefined nucleus with the classical Gibbs-Thomson dependence of n* on Ap for 2D square nucleus with monolayer height. Since in this case b = 4d0 and, to a first approximation, tr = el/2do [Kaischew and Stranski 1934a], from (4.35) we have n*

= (el/A,H) 2

(14.18)

where e~ is the energy of molecular interaction between nearest neighbours in the monolayer. The curve in Fig. 14.1 b represents this equation with e l / k T = 4, the value used inthe 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 Js(Ap) 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

-2 (a)

-4 m

u)

.....) V

m

r-

-6

-8 (b)

0 .1r

r-

0 0

1 1.6

2.0

2.4

0

2.8

0

3.2

/ kT 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]; l i n e - best-fit function used for calculating the derivative in eq. (14.6). (b) Dependence of the corresponding nucleus size on the supersaturation: circlesdata 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*(Ap) dependence obtained by Viisanen et al. [ 1993], Viisanen and Strey [1994] and Strey et al. [1994] with the help of eq. (14.9) from Js(Ap) 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 GibbsThomson equation (4.10) without using adjustable parameters, since in this

230

Nucleation: Basic Theory with Applications

60

50 IL \

(a)

40 r

O

9 u"--~...O...~_

3 0 -I

Q

O

20 10"

..'1 I

50

-~

i

I

I

_|

i

I

I

I

|

i

I

,

I

I

I

I

I

I...

,

I

|

I

I

I

I

I

I

9

4O 9

c:: 30

Q

20 10'

5

'

I

'

I

10

I

'

'

,I

15

9

20

,

25

S Fig. 14.2

Dependence o f the nucleus size on the supersaturation ratio in HON of (a) water and (b) n-butanol droplets in vapours at different temperatures (adapted from Viisanen et al. [1993]; Viisanen and Strey [19941; Strey et al. [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 v0 and cr are known. We thus have the first reliable experimental finding that this equation can be applicable to EDSdefined 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 fluxjn 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 fnm (i.e. the supersaturation, the temperature, etc.) and/or the non-aggregative fluxes K~ and L n are time-dependent. The point is, however, that even under conditions ensuring stationarity, the nucleation process cannot become stationary fight 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/~, as then fnm 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 X~ nucleation proceeds in non-stationary regime characterized by time-dependent both cluster concentration Z~(t) and nucleation rate J(t). This non-stationary cluster size distribution Z~(t) (m -3 or rn-2) is the solution of the master equation (9.4) or (9.10) (with t-independent Am because of the constancy of A~), which satisfies the initial condition (9.2). The respective non-stationary nucleation rate J(t) (m-3 s-1 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 fnm and n*. This equation follows from (11.5) upon taking into account that dn*/dt = 0 when A/~ 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 [KanneDannetschek and Stauffer 1981; Kelton et al. 1983; Volterra and Cooper

232

Nucleation: Basic Theory with Applications

1985; Trinkaus and Yoo 1987; Kozisek 1988, 1989, 1990, 1991; Shizgal and Barrett 1989; Shiet al. 1990; Shi and Seinfeld 1991 a, 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 Z~(t) of the clusters during isothermal non-stationary nucleation occurring at constant A/~ 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 t = 0 of supersaturating the system, in it there are only monomers whose concentration Z1 equals their equilibrium concentration C1 corresponding to the imposed supersaturation. The initial condition (9.2) then simplifies to

Z~(O) = C1;

Zn(O) = 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 Zl(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 (t _> 0)

Zl(t) = C1

(15.3)

ZM(t) = 0.

(15.4)

Thus, eq. (9.18) becomes dZ2/dt = f l U 1 -

Oe2 + g2)Z2(t) + g3Z3(t)

dZn/dt = f n - 1Zn- l(t) - OCn+ gn)Zn(t) at" gn + 1Zn + l(t),

(It = 3, 4 . . . . .

M-

2)

(15.5)

dZM_ 1/dt = f M - 2 Z M - 2 ( t ) - (fM-1 + g M - I)ZM - l(t) 9

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(t) in the form ( n = 2 , 3. . . . . M - l ) Zn(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.(t) from Xn. Substituting Z. from (15.6) in (15.5) and allowing for (13.6) results in dy2/dt = - (f2 + g2)Y2(t) + g3Y3(t) dyn/dt = L -

l Y n - 1 ( 0 -- (fn + gn)Yn(t)

+ gn + lYn + l(t),

(n = 3, 4 . . . . .

M - 2)

(15.7)

-- OCM- 1 + gM- 1)YM- l(t)

dy M _ 1/dt = f g - 2 Y M - 2 ( t )

where, according to (15.2), Yn satisfies the initial condition (n = 2, 3 . . . . . M-l) (15.8)

Yn(O) = -Xn.

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 Yni(t) of the form (i = 2, 3 . . . . . M - 1) (15.9)

Yni(t) = ani exp ( - )~it).

Here ani is a constant, and ~i > 0 is the ith eigenvalue, i.e. the ith root of the characteristic equation

+g2f2 - ~

-g3

-f2

+g3 -/]"

0

-f3

f3

9

0

-.

0

0

0

-g4

"'"

0

0

0

0

0

0

f4

+g4 -/~''" o

=0

,

0

0

0

fM-3 "''.t. g M _ 3 _ ~

0

0

0

....

0

0

0

...

fM- 3 0

--gM-2 fM-2

+ g M - 2 --/]' -fM- 2

0 --gM-1 fM-1 +gM-1 -~

(15.10) of the set (15.7). This equation, in which/~ is any of the ~i'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 and Korn 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 Le, ~3,

234

Nucleation:

Basic Theory with Applications

. . . . AM_ 1. 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 ani with the help of the recursion formulae (f2 + g2-/~i)a2i-

-fn-

lan-

g3a3i = 0 1,i + ( f n -!-

(n = 3, 4 . . . . .

gn - / ~ i ) a n i -

gn + lan + l,i-" 0,

M-2)

(15.11)

- f M - 2aM- 2,i + (fM- 1 + g M - 1 - ~ i ) a M - l,i = O.

These formulae result from substituting y~ from (15.9) in (15.7) and in them, without loss of generality, it is convenient to set (i = 2, 3 . . . . . M - 1) aM_ 1,i = 1.

(15.12)

We can now use the M - 2 linearly independent solutions Yni from ( 1 5 . 9 ) in order to represent the general solution y,(t) of (15.7) as a linear combination of them (n = 2, 3 . . . . . M - 1): M-1 yn(t)

-

E

Ciani

i=2

exp ( -

(15.13)

~it).

Thus, the last step is to find the M - 2 unknown constants ci. They are the solution of the linear algebraic set of M - 2 equations (n - 2, 3 , . . . , M - l) M-l ~,

(15.14)

Ci a n i = - X n

i=2

resulting from using the initial condition (15.8) in (15.13). In accordance with the Cramer rule [Korn and Korn 1961], ci is given by (i = 2, 3 . . . . . M-l) Ci -- d i / d " (15.15) where d" and d i a r e the following determinants of order M - 2: a22

a23

...

a2,i_ 1

a2i

...

a2,M_

a32

a33

"'"

a3,i-1

a3i

"'"

a3,M-1

1

(15.16)

dP ~

aM-l,2 aM-l,3

di =

"''aM-l,i-1

a22

a23

999 a2,i-

a32

a33

" 9 9 a 3,i - 1

aM-l,2

1

aM-l,i

"''aM_I,M-1

-X2

999 a2,M-

- X 3

9 9 9 a 3,M - 1

1

.

(15.17)

aM-I,3"''aM-I,i-I--XM-I'''aM-1,M-I

It remains now only to insert yn(t) 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

M-I

Zn(t) = Xn + ]~ (di/d') ani exp ( - ~lit) i=2

(15.18)

or, equivalently,

Zn(t) = X~

1 - [ ~., dia~i] -1 Y~ dia~i exp (-~,it) . i=2

i-2

(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, Zn(t) ~ Xn for t ~ ~ and that eq. (15.18) is similar to the Zn(t) formula of Shizgal and Barrett [.1989]. Mathematically, Zn(t) from (15.18)or (15.19) is the exact and complete solution ofeq. (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 nonstationary cluster concentration on the transition frequencies f,, and gn through the constants ~i, ani and di (as seen from (15.19), knowing d ' f r o m (15.16) is in fact unnecessary for the determination of Zn). 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 ~i of eq. (15.10) only for M - 2 _<4, i.e. for M _< 6. This means that for M > 6 we must resort to numerical methods for the determination of the ~ ' s as functions of fn and gn. 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 Mef = n2. Hence, eqs (15.18) and (15.19) are particularly useful for the analytical description of atomistic non-stationary nucleation characterized by n* _< 5 and z~* _< 2 so that for the fight end of the nucleus region we have n2 = n* + A*/2 _< 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 ;~i, ani and di are obtainable from (15.10)-(15.12) and (15.17) as exact and explicit functions offn and gn. 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* _ 2. Setting M = Mef = na = 4, from the characteristic equation (15.10) (which is now a quadratic one) we find that its two roots ~ and ~.3 are given by 2~ = (1/2){ (f2 + f3 + g2 + g 3 ) - [(f2 + f3 + g2 + g3) 2 -40e2f3 + g2f3 + g2g3)] 1/2}

(15.20)

&3 = (1/2){(f2 +f3 + g2 + g3) + [(f2 +f3 + g2 + g3) 2 - 4(f2f3 + g2f3 + gzg3)] 1/2 }.

(15.21)

Accordingly, from (15.12) we have a32=a33

= 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 a22 = (f3 + g 3 - A2)/f2, a23 = (f3 + g 3 - ~3)/f2.

(15.23)

Using these expressions for a22, a23, a32 and a33 in eq. (15.17) yields

d2 = -X2 + (f3 + g3 -/],3)X3[f2,

d3 = X 2 - (f3 + g3 - J~2)X3]f2 (15.24)

where, in line with (13.16), (15.20) and (15.21), X2 - fl ~3 + g3)Cl[~V2/],3,

X3 - flf2Cl[,~2~3

(15.25)

are the stationary concentrations of the dimers and the trimers, respectively. Substituting the ani'S and the di'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

(f3 4" S 3 - '~2)'~'3 (f3 + g3 - ~3)/],2 -23t] Ze(t) = X2 1 - (f3 + g3) (/'],3- ~2) e-a2' + (f3 + g3) ('~3-- '~'2) e (15.26) /]'3 -A2 t /]'2 -X3 t ] Z3(t) = X3 1 - 2 3 - ,'],"--'--~e + ,;1.3- Z2 e .

(15.27)

These equations, along with Zl(t) = Ci and Zn(t) = 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, C1 and the monomer attachment and detachment frequencies fl, f2, f3, g2 and g3. It is interesting to note that the relative time variations Z2(t)/X2 and Z3(t)/X3 of Z2 and Z3 do not depend on the monomerto-monomer attachment frequency f l - A l s o , we see that Zz(t) ,,~ t and Z3(t ) ,,~ t2 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,,(t)

~

t n-

1.

(15.28)

Curves 2 and 3 in Fig. 15.1 illustrate, respectively, the time dependence of Z2 and Z 3 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 = p/p~ = 50. The used transition frequencies are f2 = 3.08 ns -1, g2 = 5.33 ns -l and f3 = g3 = 4.02 ns -l. These values are calculated from eqs (10.3), (10.72) and (10.73) with c = (36zr) 1/3 and ~, v0, m0, Pe and o'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 f3 = 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*(t) = Z3(t) of nuclei in the water vapours. As seen, Z* first increases parabolically with t (in conformity with (15.28) Z* ~: t"*-1 = t2) and then slows down its rise

Non-stationary nucleation

237

1.0

0.8 x

0.6

X~ 0.4 NTM

0.2

0

0

0.2

0.4

0.6

0.8

1.0

t(ns) Fig. 15.1 Time dependence of the concentration of dimers and trimers in nonstationa~. 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 1/f3 = l/f* = 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, t) 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,O)=O,

(1 < n < M )

Z(1, t) = C1 Z(M, t) = 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 et 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 Z(n, t) in the form Z(n, t) = X(n) + C(n)y(n, t)

(~5.32)

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, t) is the unknown deviation of the Z(n, t)/C(n) ratio from the X(n)/C(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 _
Oy(n, t) 1 o3 [ Oy(n, t) ] Ot = C(n) On f(n)C(n) -~-n y(n, O) = -X(n)/C(n),

(1 < n _ M)

(15.33) (15.34)

y(1, t) = 0

(15.35)

y(M, t) = 0,

(15.36)

because by virtue of (13.18) we have d{f(n)C(n)d[X(n)/C(n)]/dn}/dn = O, X(1) = C1 and X(M) = 0. As seen, eq. (15.33) admits separation of the variables. Hence, analogously to (15.9), we shall have linearly independent particular solutions yi(n, t) of the form

yi(n, t) = ai(n) exp (- Ait)

(15.37)

where, however, i = 1, 2 . . . . . o% since now the number of these solutions is infinite. In (15.37) ~i > 0 and ai(n) are, respectively, the eigenvalues and the eigenfunctions of the equation of Sturm-Liouville [Korn and Korn 1961 ] ( i = 1, 2 , . . . , oo)

d { f(n)c(n) ~d [ai(n)] } + ~iC(n)ai(n) d---ff

= 0

(15.38)

which follows from (15.33) upon substituting yi(n, t) from (15.37) in it. According to (15.35)-(15.37), the boundary conditions which must be satisfied by the unknown ai(n) function in eq. (15.38) are (i = 1, 2, . . . . ,,o)

ai(1) = O, ai(M) = 0.

(15.39)

Having found Ai and ai(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(n, t) as (1 < n<M)

y(n,t) = ]~ ci ai(n) exp (- &it)

(15,40)

i=1

which parallels eq. (15.13). Here only the constants c~ are unknown so that the last step is to determine them from the equation (1 _< n <_.M)

C(n) ]~

r

ai(n) = -X(n)

(15.41)

i=1

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 (i, k = 1, 2 , . . . , ~; i ~e k)

Non-stationary nucleation 239

~

~4 C(n)ai(n)ak(n) 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)a k2 (n) dn so that (i = 1, 2 . . . . , oo)

ci = -

C(n) a2i (n) dn

X(n)ai(n) dn

(15.43)

where the subscript i stands instead of k. As we now know all of the quantities ~i, ai(n) and ci in (15.40), combining (15.32), (15.40) and (15.41), we finally obtain (1 _< n _<M)

Z(n, t) = X(n) + C(n) ~, ci ai(n) exp ( - ~it)

(15.44)

i=1

or, equivalently,

Z(n, t) = X(n)

{

1-

ciai(n)

1

Z ciai(n ) exp (-/],i t) .

(15.45)

i=1

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 t ~ oo. 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 ofeqs (15.18) and (15.19), the practical usage of (15.44) and (15.45) requires knowledge of A.i, ai(n) and ci as functions of the parameters controlling the monomer attachment frequency fin) and the equilibrium cluster concentration C(n). Unfortunately, in general none of &i, ai(n) and ci 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 ~i, ai(n) and ci 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 -- 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 n[ and n~ 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 (i = 1, 2 . . . . . oo) f * d 2 a i ( n ) / d n 2 + ~iai(n) = 0.

(15.46)

The boundary conditions to this equation, which replace (15.39), thus read ai( n;) = O,

ai( n~) = 0.

(15.47)

Here nl" = n* - A'/2,

n~ = n

+A'/2,

(15.48)

and A' is specified by [Kashchiev 1969a] A' = 41/3 = (4/zrl/Z)A*

(15.49)

where/3 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 fin) - f * 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(n, t) will evolve practically instantaneously from Z(n, 0) = 0 to Z(n, t) = X(n) for all subnuclei far enough from the nucleus region (i.e. for 1 < n < n~ ), and (ii) that Z(n, t) will be always vanishingly small for all supernuclei of size n > n~. Since Z(n, t) and ai(n) 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 . . . . . oo): /~i -" i2l~2f */A'2

ai(n) = sin [ i l r ( n - n~)/A'].

(15.50)

(15.51)

Introducing these eigenfunctions in (15.43), 'shifting' the limits of integration from 1 to n~ and from M to n~ and using the fact that within these limits C(n) -- C* and X(n) = C*[(1/2) - (~/zci/2)(n - n*)] (see eqs (7.37) and (13.25)), we find that ci is given approximately by the formula (i = 1, 2 . . . . . oo) ci =-(2/iJr)[sin 2 (iJr/2) + (4/zr 1/2) cos 2 (in;/2)].

(15.52)

Finally, substituting the above/1,i, ai(n) and ci 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)[ 1/2 - (~]~l/2)(n - n*)] e~

- (2/~)C(n) Z (l/i)[sin 2 (i~/2) i=1

+ (4/~ 1/2) cos 2 (ig/2)] sin [(itc~/4)(n- n* + 2//3)] • exp (-i2~2f12f*t/16).

(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. In - n*l < A*/4 = ~1/2/4fl) that the linear dependence (13.25) of the X(n)/C(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 ci is very sensitive to the approximate C(n) and X(n) dependences used for performing the integrations in eq. (15.43). In obtaining ci 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 n*. 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 = n* and recalling (13.23), mutatis mutandis, we get

Z*(t) - X* ~1 + (4/to) ~ [ ( - 1 ) i / ( 2 i - 1 ) ] exp [ - ( 2 i - 1)2t/4~r]~ (15.54) J i=l t where "r (s), called often the time lag of nucleation (see Section 15.3), is given by [Kashchiev 1969a]

= 4/Jr2/ 2f *.

15.55

In some cases it may be more convenient to use eq. (15.54) in its equivalent presentation [Carslaw and Jaeger 1959] oo

Z*(t) = 2X* ]E (- 1)i- l { 1 - erf [(2i - 1)IrT1/2/2t 1/2] }

(15.56)

i=l

where eft(x) 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[1~1/2x) exp (- x 2) for x ---) oo [Carslaw and Jaeger 1959], Z*(t) takes the simple form

Z*(t) = X*(16t/n3"c) 1/2 exp (- 7r2"c/4t).

(15.57)

This equation approximates Z*(t) from (15.56) with an error of less than 30% for t < 2r. Equation (15.54) shows that, in the scope of the approximations made in obtaining ~i, ai(n) and ci, the Z*(t)/X* ratio is a universal function of the dimensionless time t/z. This function is depicted in Fig. 15.2 by the solid

242

Nucleation: Basic Theory with Applications

curve. As seen, eq. (15.54) predicts correctly the sigmoidal shape of the Z*(t) dependence, expected for n* > 2 on the basis of eq. (15.28). Also, Z*

= 0 at t = 0 and Z* --~ X* as t ~ oo. For t > 10z the nucleus concentration is already practically equal to its stationary value X*, which means that, physically, 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*(t) from (15.54) is a reasonable approximation to the exact Z*(t) 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 t/z one where z = 0.158 ns is calculated from (15.55) with f* = 4.02 ns -1 and fl = zc]/2z = 0.799 (the z value used is obtained from the formula z = gEg3/(f2f3 + g2f3 § gEg3) 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*(t) 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 D e m o and Kozisek [1993] and by Abraham [1969], respectively. Curve 27 pertains to HON of Li20.2SiO2 crystals in their vitrified melt under conditions of n* - 27, and curve 71 is for HON of water droplets in vapours ...Z..ZZ_.;

1.0

............

s"s s t l s

0.8

~

s

i

s

9

i

II ,Ic

i

iI

ii

0.6

/

lII

~///t

,'3 ,,'71 /

X ,Ic

U

i

l II

0.4

I

ii

i

0.2

Il

,'

I It i

0

I

,'

~

// I# / ,/

/

i

.

iii I

/

/I

~1 I" _4'

,,'27

,'

i

,'

ii

i iI _ J ' " =

2

,"

i

.

I,

i

.

4

I

I

6

,

I,

I,,

I

I

8

|

-

"

|

10

tlx 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 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 curve 27-numerical data of Demo and Kozisek [1993] for HON of Li20.2Si02 crystals in vitrified melt under conditions of n* = 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*(t) dependence (curve 27) is described well by the expression

Z*(t) = X*[1 - erf {3/3n*(1 - n *-1/3) x [1 - exp (- 80t/77r2"c)] -1/2 exp (- 40t/77r2~')}] (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 ofeq. (13.3) (used atn = n*) and eq. (15.18) we find easily that forn* = 2, 3 . . . . . M - 1

M-1 J(t) = Js + ~, ( d i [ d ' ) ( f * a n * i - gn*+lan*+l,i) e x p ( - ~ i l)

i=2

(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 J(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) = Js 1 -

~., di(f*an, i - gn,+lan,+l,i)

i=2

M-1 ]~ d i ( f * a n , i - gn,+lan,+l,i) exp (-/],i t)

i=2

} (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) =flC1 - gzZ2(t) with Zz(t) 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 &i, ani, d~ and di which are quantities obtainable only approximately from (15.10), (15.11), (15.16) and (15.17) when M = Mef > 6. Let us consider again the case of M - Mef = n2 = 4 and n* = 3, for which we have explicit expressions for 2i, ani and di, 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,,+ 1,i = aMi = 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(t) from (15.27) into the

244

Nucleation: Basic Theory with Applications

relation J(t) =f3Z3(t) which follows from (11.8), (15.1) and (15.4). Recalling also that now Js = f3X3, we thus obtain J ( t ) = Js

[

1

/]'3 -

~3-

22 e

-Z2t

(1561)

J~2 e-Y~3t] + /13- A--------~

where Js, 2,2 and ~3 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 nonstationary nucleation rate in the exemplified case of M = M e f "- 4 when the trimers are nuclei (n* = 3). The dashed curve 3 in Fig. 15.3 illustrates this dependence in JIJs-vS-t/z coordinates. Since z, L2 and Z3 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, n* = 3. As noted in Section 15.1, the calculation of the values of z, L2 and ~3 by means of classical formulae for the monomer attachment and detachment frequencies implies that these values are questionable and that, thereby, the J(t) 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 ---> oo. It is worth noting that the J(t)/Js ratio and, hence, the position of 1.0

0.8 / t

O0

0.6

g lo

/(/~/" 0.4

tt

J

II

iii i I

,,"" 7

;i i I iI

i

s

/

0.2

/

r I

0

.",_.,

I...,

1

. . . . .

I"%

2

,

tlx

,

,

I

3

,

. . . .

I

4

. . . .

5

Fig. 15.3 Time dependence of the non-stationary rate of nucleation: solid curve- eq. (15.64); dashed curve 3 - eq. (15.61)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 2 7 - numerical data of respectively, Kelton et al. [1983] and Demo and Kozisek [1993] for HON of Li20.2Si02 cGstals 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 flThe 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 oo

J(t) = J s - f ' C *

p*

~, cia i exp (-~i t)

(15.62)

i=1 p*

where the stationary nucleation rate Js is given by (13.32), and ai - [dai(n)/ dn]n- ,,,. Since J = 0 at t = 0, this equation can be cast into the equivalent form

J(t) = Js 1 -

~, cia[*

Z cia[* exp (-~it) .

i-1

i=1

(15.63)

Equation (15.62) or (15.63) is the sought exact expression for the nonstationary 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 fn and gn + 1 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 [ 1989]. 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 ~i, a'* and ci. 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 ~,i, ai(n) and ci from eqs (15.50)-(15.52). From (15.48), (15.49) and (15.51) it follows that a[* ( i ~ / 4 ) cos (i~'/2) so that, due to (15.50) and (15.52), eq. (15.62) yields

J(t) = Js + (2~f *C*/~2112) ~ cos (i~:/2) exp (- i2~2~2f*tl16). i=l

Hence, invoking (13.33), (13.35) and (15.55) and rearranging the series, we get

J(t) = Js [1 + 2 ~ (- 1)i exp (-iztl~)].

(15.64)

i=1

This approximate formula for the non-stationary nucleation rate was derived by Kashchiev [1969a] with the help of Z(n,t) from (15.53). It shows that in the scope of the approximations used for the determination of)ci, ai(n) and ci, the J(t)lJs ratio is a universal function of the dimensionless time t/~', the nucleation time-lag 1: being given by (15.55). As seen from Fig. 15.3 in which the solid curve depicts the J(t) dependence (15.64), ~"sets up the time scale of the attainment of steady state" J = 0.95Js at t = 3.7"c. At short times

246

Nucleation: Basic Theory with Applications

(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]) oo

J(t) = Js(4rc'c/t) 1/2 ~, exp [- ( 2 i - 1)2zr2"t'/4t]

i=1

(15.65)

which shows that for t < 4~:, with an error of less than 1%,

J(t) = Js(4tc'r/t) 1/2 exp (- nr2"r/4t).

(15.66)

As noted above, according to Kashchiev [1969a], the time lag ~:in (15.64)(15.66) is defined by eq. (15.55). This makes J(t) from (15.64) numerically different from the J(t) 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 "r = 1/4zrf123"* (this expression corrects for the missing factor 4 in front of t in Collins's eq. (18)). Thus, Collins's 7: is about 5 times less than 7: from (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 ~:from (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 "r as 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*(t) dependence (15.54), is mathematically more justified when the nucleus size is large enough, e.g. when n* > 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 (- 80t/7~2"t')]-l/2 x exp {- [3fin*(1 - n * exp (- 80 t/7/17217)}.

1/3)]211

-

exp (

- 80t/7/rzr

-1

(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 et al. [1983]; Kashchiev [ 1984a]; Gutzow et al. [ i 985]; Kelton [ 1991 ]). Figure 15.4 exhibits the experimental J(t) data of K6ster [ 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 vin (15.64) as free parameters. In this way K6ster [ 1978] found that Js = 2.9 x 1012, 3.7 • 1013 and 3 x 1015 m -3 s-1 and 7:- 2.5 x 105, 6 x 104 and 2.5 x 103 s at T = 824, 831 and 920 K, respectively. 0.6

i

|

i

i

i

=

:

'

|

|

'

:

i

|

i

'

i

|

i

i

:

:

l

,

i

=

,

|

0.5 0.4 0.3 "3

0.2 0.1 amA

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

tlx 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 1978]; 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 djl/dt = -g2[Jl(t)-j2(t)] djn/dt = fn[Jn- ](t) --jn(t)] -- gn + l[Jn(t) --Jn + l(t)],

(n = 2, 3 . . . . .

M-

2)

(15.68)

djM_ 1/dt = f M - 1 [jM- 2(t) -JM-](t)] 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 f,, and gn, and (iii) eliminating the appearing dZn/dt derivatives with the help of (9.7). According to (9.19) and (15.2), the initial condition to (15.68) is

j~(O) = f~C1,

j,(O) = O.

(n = 2, 3 . . . . .

M-

1)

(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 n , ( t ) when one is not interested in the cluster size distribution Zn(t). 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 A1, 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, t) = j ( n , t) + [Oj(n, t)/On][(n - 1) - n] [C(n + 1)]-1 c)j(n + 1, O/On = [C(n)]-lc)j (n, O/On + (O{[C(n)]-lOj(n, t)/On}/On)[(n + 1) - n] about point n, we get (1 < n <__M)

c)t

= f (n)C(n)

C(n)

"

(15.70)

This basic equation for j(n, t) 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))

j(n, O ) = f l C 1 6 o ( n -

1)

(15.71)

where ~ is the Dirac delta-function [Kom and Korn 1961 ], and the boundary conditions ensuring stationarity for t ~ oo are

[3j(n, t)/c)n],,= 1 = O,

[c?j(n, t)/c)n],=M = 0.

(15.72)

The determination of the non-stationary nucleation rate J(t) -j(n*, t) is straightforward when the solution j(n, t) 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

15.3

249

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 Ap is set up by the time lag v of nucleation. In Chapter 17 we shall see that v 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 v appears as another basic quantity characterizing the kinetics of nucleation. According to Kashchiev [ 1969a], "t"is 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 "r (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 [1991 a] and Miloshev [ 1992] (see also Section 15.4). Using eqs (7.43) and (13.35), we can represent "t" from (15.55) in the following equivalent forms [Kashchiev 1969a] T = 4/~zef *

= 4A*2/~f *.

(15.73)

The last of these equalities allows a simple physical interpretation of v [Feder et al. 1966], because it is an Einstein-type formula (cf. eq. (10.31)) relating v to the width A* of the nucleus region and the attachment frequency f* of monomers to the nucleus. Namely, we can view 7: as the mean time needed for a cluster to 'cross' the nucleus region by 'walking' randomly along the size axis from n = n l to n = n2 with a size-independent 'diffusion coefficient'f* (see also Section 9.2). According to eqs (15.55) and (15.73), the nucleation time lag v is largely determined by f*, since the effect of the other transition frequencies (f,, and gn for n ~e n*) is amalgamated into the single parameter fl (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 z we may use the approximation (15.74)

= 10/f*

which results from (15.73) with typical z = 0.1. The conclusion is, therefore, that 7: is more or less a direct measure of the reciprocal o f f * and for that reason is very sensitive to the concrete mechanism of monomer attachment to the nucleus. This distinguishes v from the other basic nucleation parameter Js which is much more strongly dependent on, and thereby more informative about, n* or W* than about f * (see, e.g. eqs (14.9)-(14.12)). Let us now see what is the Ap dependence of v both in general and in some particular cases of nucleation. Using (10.109) and (10.110) with tindependent Ap and T, from (15.73) we find that for nucleation of condensed phases, in general, T is given by r = (4/z3Z2fe)

e -A~/kr

(15.75)

250 Nucleation: Basic Theory with Applications when Ap is not controlled by T according to (2.20) and by

"~= (4[TC3Z2f o ) e-/W/kr

(15.76)

when T is the parameter used to change Ap in conformity with (2.20). These expressions for 9 are valid regardless of the kind of the nucleation process (HON, HEN, 3D, 2D, etc.) and in them fe--fe,n* and fo -fo,n* are the values of f* at Ap = 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 z(Ap) 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 andfe* on Ap the nucleation time lag 1:decreases monotonously with the increase of the supersaturation A/.t provided this is varied isothermally. This, however, is not the case with vfrom eq. (15.76) which shows that when A~ is changed by means of T as required by (2.20), ~:first decreases with increasing Ap (i.e. lowering T), but then passes through a minimum and rises sharply as Ap goes on increasing by further lowering of T. This rise of 7: at lower temperatures is due to the decrease of fo with T. For instance, in crystal nucleation in melts fo ~ l/r/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 7:(Ap) behaviour in some particular cases of nucleation and for various mechanisms of monomer attachment to the nucleus by using concrete expressions for Ap, z and f* (or f~* and f0*) 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 --" 4( 2mok T) l /2/~/2z2 ~,*r

n,2/3S

(15.77)

where ct)i is a numerical shape factor. For HON coi = c with c = (36~z)1/3 for spheres, c = 6 for cubes, etc. For 3D HEN coi ~ c. For example, CDI = (36~Z)1/3(1 -COS 0w)/21p'2/3(0w)for caps on a substrate contacting the vapours (then S = P[Pe) and ct)i = (36z01/3 sin 20w]4~2/3(Ow) for caps with 0w <--~ 2 on a substrate during molecular beam condensation (then S = I/Ie). Equation (15.77) gives directly the z(S) dependence in the case of atomistic nucleation, as then z -- 1 and n* = 1, 2 . . . . is S-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)

"C= 16c(2mo)l/2cref/~/2~'*CDi(kT)l/2peS(ln S) 2

(15.78)

where tree = cr for HON and Cref< o" for HEN (see eq. (4.42)). Apart from the

Non-stationary nucleation

251

numerical factor, with CDI = r and O'ef= O" (i.e. in the case of HON) this z(S) 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 S. From eq. (15.77) we see that, atomistically, the "t(S) dependence is a broken straight line when plotted in "t-vs-(1/S) 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 n* 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 S values, "t is greater for substances with lower vapour pressure Pe and/or higher specific surface energy or. Another point to emerge from eq. (15.78) is that the time lag "tHENin HEN is related to the time lag ZHONin HON (then CDI - c and O'ef= O') by the formula "tHEN -- (C/CDI) ~('/"tnON

(15.79)

where ~, a number between 0 and 1, is the activity factor appearing in eq. (4.42). In particular, hu = ~l/3(0w) for cap-shaped nuclei, ~(0w) being given by (3.56). Hence, for such nuclei we have THE N --

[2gt(0w)/(1 -

cos

0w)]"tHO N

(15.80)

when the substrate is in contact with the vapours and (0w - 7r/2) 'tHEN---- [4gt(0w)/sine 0w]"tnON

(15.81)

during molecular beam condensation. Equation (15.80) was obtained by Toschev and Gutzow [1967a] who concluded that THEN < "tHON when the nucleus/substrate wetting angle 0w is small enough. The relation between "tHEN and "tnON 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 T = 293 K. The calculation is done with col = c, O'ef = O" and the parameter values given in Table 3.1. The numbers at the circles on the curve indicate the number n* 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 "t is due to the relatively high frequency f* of monomer attachment to the nucleus droplet (see Fig. 10.2). (b) 3D or 2D HEN under surface-diffusion control In this case, when f* is given by (10.42) or (10.43), from (2.8), (2.9), (13.68) and (15.73) it follows that t = 4/~3zg~'*c*~2sleS

(15.82)

where/re = pe/(2ZcmokT)1/2. With z --- 1, this equation represents the atomistic t(S) 9 dependence in the cases of both 3D and 2D HEN. Classically, however, z is 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 100 SD 80

60 v

t.t..=

70 40

50

20

.,,,,.,i..,...._

1

2

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 DI) and eq. (15.83) for surface-diffusion control (curve SD). The numbers at the symbols indicate the nucleus size at the corresponding supersaturation ratio. 3 / 9 z r 2 7 , c , ( k T ) 3 As2ieS(ln S)4 "t"= 64C3Vo20"ef

(15.83)

for 3D HEN and into (In S > aefAtT/kT) "t"= 4b2tc2/zt'37*c*(kT)2 As2IeS(ln S - aefAty/kT) 3

(15.84)

for 2D HEN on a foreign ( A c t , 0) or own (Act = 0) substrate. Equations (15.82)-(15.84) reveal that the time lag "r decreases again monotonously with increasing S and that it is shorter for longer mean distance As of monomer surface diffusion. Curve SD in Fig. 15.5 depicts the z(S) dependence (15.83) for HEN of hemispherical water droplets on a foreign substrate held at T - 293 K in vapours and characterized by a relatively small 3 A,s l0 nm. The calculation is done with c = (367r) 1/3, c* _ . 1.9, Cref = (1/2)o 3 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 77is less than several nanoseconds for S > 2.5, which reflects the fact that f * 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 1: 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

Non-stationary nucleation

253

nucleus is strongly impeded, z could 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 (15.85)

.r = 2]~7/2Z2~+,.~1/2 I "VD ,,1/3 v0 D C e n , l / 3 S

where Cvo is a numerical shape factor. For HON cVD = c with c = (36z01/3 for spheres, c = 6 for cubes, etc., and for 3D HEN of caps with 0w > zr/4, approximately, CVD= (36Z01/3(1 -COS 0w)2/4I/t2/3(0w). Equation (15.85) gives the "r(S) dependence for atomistic nucleation, since then z -- 1 and n* is Sindependent. We see that due to abrupt changing of the nucleus size n*, as in the case of nucleation in vapours, this dependence may appear as a broken straight line in z-vs-(1/S) coordinates. Classically, however, n* and z depend on the supersaturation ratio S according to (4.38) and (13.36) when nucleation is 3D so that from (15.85) we obtain (S > 1) "t: = 16C2VOa2ef[3 lrs/27*cvD1/2(kT)2DCeS(ln

S) 3

(15.86)

where O'ef- - O " for HON and O'ef --- ID'l/3(0w)O"for 3D HEN of caps (see eq. (4.42)). Qualitatively, the z(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 "c(S) 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 ' e f - " O " and cVD - c = (36701/3. As seen from Fig. 15.6, z 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 off* in (15.73). Recalling again (2.14) and (13.69), we thus obtain "r = 4do/zr3z 2 ~ C l T O213 DCen*2/3S.

(15.87)

In HON, for the numerical shape factor CIT we have CIT = c with c = (36701/3 for spheres, c = 6 for cubes, etc. In HEN, if the nuclei are cap shaped, CIT = (367r)1/3(1-cos 0w)/2gt2/3(0w). With z = 1 and S-independent n*, eq. (15.87) is the atomistic z(S) dependence which, graphically, is again a straight (possibly broken) line in z-vs-(1/S) coordinates. Classically, using n* and z from (4.38) and (13.36) transforms (15.87) into (S > 1) "r = 16cdoaef/lr2y*ClTkTDCeS (ln S) 2

where O'ef= O" for HON and O'ef< O" for HEN.

(15.88)

254 Nucleation: Basic Theory with Applications 0

....

VD

o0 v

I

1

I

'

|_

5

_

i

.

.

.

I

10

I

~

i

9

.

I

,

15

I

..... I

'

20

S Fig. 15.6

Time lag of nucleation as a function of the supersaturation 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 interfacetransfer control (curve IT). The numbers at the circles indicate the nucleus size at the corresponding supersaturation ratio.

Comparison of eqs (15.86) and (15.88) shows that the "r(S) dependence is nearly the same under volume-diffusion and interface-transfer control. In the latter case, however, ~ is less sensitive to the effective specific surface energy O'ef of the nucleus/solution interface. Also, from eq. (15.88) it follows that when monomer attachment is controlled by interface transfer, the connection between ~rHEN for 3D HEN of caps and ~:r~oy for HON of spheres is again given by eq. (15.80). The "c(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 f * 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

"c = 4Vo/~3z2)/'* bDKcCen*l/2s

(15.89)

where b = 2(/ra0) 1/2 for disks, b = 4a~o/2 for square prisms, etc. With z ~- 1 and S-independent n*, this equation is in fact the 1:(S) dependence of atomistically small 2D nuclei of monolayer height (vvs 1/S is again a straight line with possible breaks). To obtain the classical dependence of 7: on S we

Non-stationary nucleation 255 set a e f = a0 in eqs (4.32) and (13.37) for n* and z and combine them with eq. (15.89), the result being (In S >_aoAa/kT)

,t: = 8VolC/~ y* kTDKcCeS(ln S - aoAcr/kT) 2

(15.90)

where Aa ~ 0 for foreign and Aa = 0 for own substrate. As seen, this "r(S) dependence is analogous to those given by eqs (15.86) and (15.88). In fact, for 2D nuclei on own substrate v from (15.90) shortens with increasing S in precisely the same way as does 7rfrom (15.88) although eq. (15.88) applies to 3D nuclei. With re= doa, KcCe = 0.001 (according to (10.47), this corresponds to 0.1% adsorption coverage of the substrate at C = Ce), A a = 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), vcould 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 r/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

"C= [12dgo~/3o( T)[~2 z2T*ClTkTn*2/3] e (z-6u)/kT

(15.91)

w h e r e CIT is the shape factor appearing in (15.87),/1, = TeAse, and A/,t 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 7/ depends on T (see eq. (10.56)) and thereby on A/~, eq. (15.91) shows that the r/(T) function has a strong effect on the dependence of lr on A/~. Again, with z -- 1 and A/~-independent n* eq. (15.91) is the atomistic formula for the time lag lr 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/~ > 0):

"c= [48C d~voaefrl(T)/~7*CiTAi ~2] e (z-A")/kT.

(15.92)

If the undercooling A T - T e - T is related to A/~ by (2.23), this equation leads to the following explicit dependence of "t"on T (0 < AT < 2TeAse/Acp,e): "t"= 48c dgvo

exp

(Ase/k)GefFl(T)/g'2~'*ClTZ~Se2AT 2.

(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 O'ef "- O', tIT = r 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 - Te (then AT = 0) "t is infinitely long and that with lowering T it first shortens due to the AT term. However, as 7/is higher at lower T, after passing through a minimum "t begins to increase with further undercooling. Thus, in contrast to the time lag for nucleation in vapours and solutions at isothermally varied Ap, "t is not a monotonous function of Ap when this is controlled by T in accordance with (2.20). Equation (15.93) tells us that ~: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 et al. [1977, 1981, 1997]; K6ster [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 Tg) at which the melt viscosity is sufficiently high. The curve in Fig. 15.7 depicts the RAT) dependence (15.93) for HON of spherical ice crystals in water at atmospheric pressure. The calculation is clone with qv = C, O'ef = O" (assumed T-independent), r/(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 103 102 10

-

1o

1 10 -1 10 .2 ~-~ 1 0_3 10 .4

-

20

10-5 10 .6

ion

10 .7 10-6

I

0

'

'

looo .

20

,

. _ I

40

.

9

.

I

,

60

.

.

I

80

,

.

,

I

100

,

i

i

I

120

140

AT (K) 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 the dotted line indicates the undercooling at the glass-transition temperature.

Non-stationary nucleation

257

is 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 = [12dovoTl(T)/~z27*bkTn *1/2] exp [(~,- E 1 - O'sa0 - Ai.t)/kT]. (15.94) We note again that with z -- 1 and A/~-independent n* this is in fact the atomistic formula for z"as a function of A/~. Classically, n* and z change with A/~ according to eqs (4.32) and (13.37) so that eq. (15.94) becomes (Ap _ a0Ao') r = [24d0v 0 h-r/(T)/zrzy*(A/.t - a0Acr) 2] exp [(~ - E l - tYsa0 - Ap)/kT] (15.95) where Act ~ 0 for foreign and Act - 0 for own substrate. Hence, in terms of the undercooling AT, using (2.23) we find (aoAcr/Ase <- A T < 2TeAse/Acp,e) z = [24d0u0 exp (Ase/k)K77(T)/Ir2~I*Ase 2 (AT-aoAo/Ase) 2] exp [ - ( E l

+ Crsao)/kT].

(15.96)

Equations (15.94)-(15.96) are analogous to (15.91)-(15.93), which means that all of the above conclusions concerning the z'(AFt) dependence for HON or 3D HEN are valid for 2D HEN, too. The new knowledge about ~r from (15.94)-(15.96) is the presence of the exponential factor which contains the energy term El + tYsa0 and thus takes into account that the molecules joining the nucleus periphery are adsorbed on the substrate. Physically, El + Crsa0 corresponds to the desorption energy Edes of a molecule (see Section 10.1). 4. 3D nucleation of gaseous phases under evaporation control In this case the time lag zis also given by eq. (15.73) so that using (10.9) at n = n* and recalling that the nucleus vapour pressure Pe,,,* is equal to the pressure p* inside the nucleus yields "t"= 4(2m0)l/2[It'5/2z2~,'*cEv(kT)l/6p, 1/3n,2/3"

(15.97)

Here cEV is a numerical shape factor: for HON cEv = c with c = (36zc) 1/3 for spheres, c = 6 for cubes, etc., and for 3D HEN of caps cEV = (36Z01/3(1 - cos 0w)/21/t2/3(0w). 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) .r = 16c(2mo)l/2p,tTef/lr3/2~,cEv(kT)l/Z(p,

_ p)2

(15.98)

where O'ef - " (5" for HON and O'ef < (5" for 3D HEN. Since p* depends on p according to (4.14) in which A p - Pe - P, when PeUo << kT, in (15.97) and (15.98) we can use the approximations p* = Pe and p* - p = Ap.

258

Nucleation: Basic Theory with Applications

Equation (15.98) shows that the time lag ~:is infinitely long at Ap = 0 (i.e. at saturation) and that it shortens monotonously if the underpressure Ap is increased isothermally. This behaviour of 7: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 1tHEN and THONfor 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 l:(Ap) dependence (15.98) for HON of steam bubbles in water at T = 583 K. The calculation is done with cEV = c, O'ef= (3", p* = p~ 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 o f f * is the reason for the negligibly short time lag 7:in bubble nucleation controlled by evaporation. It must be pointed out, however, that for other mechanisms of mass transport lrmay have much higher values. For instance, under viscousflow control, analogously to (15.91)-(15.96), iris proportional to the viscosity /7 of the liquid [Carlson and Levine 1975] and can be as long as, e.g. an hour or a day. 300

200 oo ct~

100

~ 5 0 0 0

,

0

,

,

.I

0.2

,

,

,

300 i

I

0.4

I

0.6

150 ,

.

.,

I

0.8

100 J

=

,

1.0

zXp/ Pe Fig. 15.8 Time lag of nucleation as a function of the underpressure in HON of steam bubbles in water at T = 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 supemuclei 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. 15.3). The practically complete saturation of J at its stationary value Js occurs after a period of the order of z. The time lag ~ris thus the parameter characterizing the duration of the non-stationary regime in isothermal nucleation at constant A#. It is clear, however, that this duration can be equivalently characterized by other time parameters such as, e.g., the time tl/2 at which J = (1/2)Js. This is so because these parameters are unambiguously related to 7:. For instance,

s

r,,, ,

|

i,,,

i, '1,1"

.

"|'_

j . '. . ~ - - - - - ' ~

r'--

l

"

(a)

1;

--j

tl/2

e

i

A - ~ _J

0

1 I

9

I /,"

IAs I/

0

t Time dependence of (a) the non-stationary rate of nucleation and (b) the concentration of supernuclei. The arrows indicate the nucleation time lag z, the time ts/2 at which J = (1/2)J s and the delay time 0 of nucleation, the latter being defined by the equality of the hatched areas A and B or by the intercept of the t --> oo asymptote (the dashed line) of the ((t) function. F i g . 15.9

260

Nucleation: Basic Theory with Applications

with J(t) from (15.64) we calculate that tl/2 = 1.38~. The time tl/2 is indicated in Fig. 15.9a where it is seen also that there exists a characteristic time 0, called often delay time of nucleation, which makes equal the area A between the J(t) curve and the time axis in the 0 < t < 0 range and the area B between the J(t) curve and the Js line from t = 0 to t = . A mathematical expression of the equality of these areas is the following definition of the delay time 0 [Andres and Boudart 1965] g~o 0 = J 0 [1 - J(t)/Js] dr.

(15.99)

Physically, at no pre-existing supernuclei in the system 0 determines a thought process of non-stationary nucleation occurring at zero rate from t = 0 to t = 0 and at stationary rate Js for t > 0 (see Fig. 15.9a). In other words, this thought non-stationary nucleation is a delayed stationary nucleation, the time of the delay being 0. We note that because of (11.10), when ~'0 = 0 (no pre-existing supernuclei), 0 from (15.99) can be defined equivalently by

~(t) = Js(t - O) at t ~ oo,

(15.100)

i.e. 0 is merely the time intercept of the long-time asymptote of the ((t) function (Fig. 15.9b). Equation (15.100) tells us that in the t ---) oo limit the concentration Js(t- O) of supernuclei 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 0 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 (n* = 2, 3 . . . . . M - 1): M-1

M-1

0 = [ ~, d i ( f * a n , i - gn* + lan * + 1, i)]-1 i=2

E i=2

(di/~i)(f*an,i-

gn* + lan * + 1, i)

M-I

= -(1/d'Js)

]~ (di[~i)(f"*an*i- gn* + lan*+l, i)"

i=2

(15.101)

The same result is obtained also with the help of J(t) from (Al.12), but then 0 is expressed in terms of the polynomial PM- n*-I. The problem with the application of eq. (15.101) is that &i, ani and d i a r e obtainable only approximately from (15.10), (15.11) and (15.17) when M = M e f > 6. Thus, in general, the dependence of Ai, ani and di and, thereby, of 0 on the attachment and detachment frequencies fn and gn cannot be revealed easily. To get a feeling about the relation between 0 and these frequencies we can consider the particular case of M = M e f = 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 ~,i, ani and d i o r 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-stationa~. nucleation 261 0 = (f2 + f3 + g2 + g3)/(f2f3 + g2f3 § g2g3)

(15.102)

where f3 - f * . We observe that 0 does not depend on the monomer-tomonomer attachment frequency fl. Also, with the f2, f3, g2, g3 and T values given in Section 15.1 we calculate 0 = 0.298 ns and 0/z = 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 oo

oo

oo

p

p

O : []~., ciai*] -1 Z (ci/~i)ai* = ( f * C * / J s ) ~, (Ci/,~i)ai i=1

i=1

i=1

t~

(15.103)

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 0. Instead of doing that, however, we can obtain such a formula directly from eq. (15.99) with the aid of available approximate J(t) dependences. We shall now make use of J(t) from (15.64) in order to find an approximation to 0 from eq. (15.103). Performing the integration in (15.99) with J(t) from (15.64), taking into account that [Gradshtein and Ryzhik 1962] oo

~, (-1)i/i 2 = -ZC2/12

(15.104)

i=1

and recalling eqs (15.55) and (15.73) leads to the formula [Kashchiev 1969a] 0 = (n2/6)T = 2/3f12f * = 2/3n'z2f*.

(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 0 and the time lag 9is only in a numerical factor: 0 is merely 7r2/6 = 1.6 times greater than ~. This implies that all results and conclusions in Section 15.3 concerning v are directly applicable to 0 from (15.105) after allowing for the presence of the factor Jrz/6. In particular, eq. (15.105) says that 0 is in fact determined only by the frequency f* of monomer attachment to the nucleus - the effect off, and g,, for n ~ n* is minor and is taken into account by the virtually constant factor fl 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 "r from experimental data for the delay time 0 [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 0 from (15.99) does not allow the exact dependence of 0 on the transition frequencies f~ 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 J(t) function. This 'indirect' method of finding 0 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 0 on f, 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*: "*Z(n t) dn = t~t

'

f(n)C(n)

[Z(n t)/C(n)]

m

dn. (15 106)

'

"

Using eqs (11.9), (13.18), (13.31), (13.32) and (15.1), for the integral on the fight of (15.106) we can write f(n)C(n)

[Z(n, t)/C(n)]

dn

m

= - J ( t ) - f ( m ) C ( m ) O [ Z ( m , t)/C(m)]/c)m =-J(t) + Js[d(X/C)/dm]-~ [c)(Z/C)/c)m]

so that (15.106) becomes -O(Z/C)/Om + [J(t)/Js][d(X/C)/dm]

We now integrate this equality from m - 1 to m - M and get -

~ m (ZIC) dm + [J(t)lJs]

~

(XIC) dm

Employing the boundary conditions (13.4), (13.5), (15.30) and (15.31), we find (Z/C) dm = [Z(m, t)lC(m)]m- M - [Z(m, t)lC(m)]m- 1 = - 1,

I

M " d~ (X/C) d m = [X(m)/C(m)]m =M

-

[X(m)/C(m)]rn = 1 = -1 ,

Non-stationarynucleation 263 ~IMI~~*Zdn]I~-~(X/C)] dm

=

I[~ n* ] } Z(n, t) dn [X(m)lC(m)] m m"'M

- ~1M [X(m)/C(m)][-Z(m, t)] dm ~1n*Z(n, t) dn +

~1M[Z(n, t) X(n)/C(n)]

dn.

Hence, eq. (15.107a) transforms into

1 - J(t)/Js = (1/Js) 0at {~1n*Z(n,

t) dn -

fl M [Z(n, t) X(n)/r(n)] dn} 9 (15.107b)

This form of 1 -J(t)/Js is ready for integration in accordance with (15.99). Thus, finally, owing to the initial condition (15.29) and the fact that Z(n, oo) = X(n), we arrive at the sought exact formula for the delay time 0 [Wu 1992a; Shneidman and Weinberg 1992a]" n*

0= (l/Js)

X(n) dn -

sl

}

[X 2 (n)lC(n)] dn .

(15.108)

Here C(n) is given kinetically or thermodynamically by eq. (12.5) or (7.4), respectively, and X(n) and Js are specified by eqs (13.18) and (13.32). Hence, depending on the chosen description of C(n), eq. (15.108) represents 0 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 0 in the case when n is treated as a discrete variable. This is indeed s o - in this case there holds [Hile 1969; Wu 1992a, 1992c; Shneidman and Weinberg 1992b] n*

0= (1/Js) [ ]~ X , n=2

M-1

]E (X2/C,)].

n=2

(15.109/

When here Cn, Xn andJs are given by (12.3), (13.16) and (13.29), eq. (15.109) represents 0 as an explicit function of the transition frequencies f~ and g~.

264

Nucleation: Basic Theory with Applications

For instance, in the particular case of M = Mef = 4 and n* = 3, from the above-mentioned equations we have C2 = Ctfl/gz, C3 = C2f2]g3, X2 = Clfl(f3 + g3)/0c2f3 + g2f3 + g2g3), X3 = X2f2](f3 + g3) and Js = f3X3 so that eq. (15.109) passes into eq. (15.102) obtained from (15.99) by direct integration of the corresponding exact J(t) 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 0as a 'mixture' of kinetics and thermodynamics. Inspection of eqs (15.108) and (15.109) shows also that, as already mentioned in respect to 0 from (15.102), the delay time is independent of the monomerto-monomer attachment frequency f~. 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 0 [Wu 1992c; Shneidman and Weinberg 1992a, 1992b]. Following Wu [ 1992c], let us employ the approximations (7.37) and (13.21) for C(n) and X(n) in order to perform the integrations in (15.108). In lieu of (15.108) we get

2 ~l sO/C * = I~

exp (X2)[1 - erf (x)] dx (1 -n*)

-- (1/2)

f

fl(M-n*)

exp ( x 2 ) [ 1 - e l f (x)] 2 dx

,Jfl(l-n*)

.- I~( n*- 1) exp (X 2) err (x) [1 - erf (x)] dx _

(1/2)

ffl(M-n*)

exp (X2)[1 - erf (X)] 2 dx.

(15.110)

J/~(n*-l)

Since this result is valid under the condition (13.22), i.e. for large enough nucleus size n*, using the approximations erf (x) = (2/tcl/2)x for x < 1 and 1erf (x) = (1/rcl/2x) exp (-x 2) for x > 1 [Carslaw and Jaeger 1959] and recalling that exp (x2) --- 1 + x 2 for x < 1, we find that 3( n*- 1)

f0

exp (X 2) erf (x) [1 - erf (x)] dx

= (2/zr u2)

(1 + x 2) x(1 - 2re-rex) dx + (1Dr u2)

f fln*(l/x) dx ,tl

= [ ( 4 5 ~ 1/2- 64)/30tr] + ( 1 / ~ 1/2) In (fin*),

f

fl(M-n*)

f/~l,~

exp (x2)[1 - erf (X)] 2 dx = (1Dr)

a/3(n*- 1)

(1/x 2) exp (--X 2) dx .*

= (l/zrfln*) exp (- flZn*2) - (1/to In) [1 - erf (fin*)] = O.

Non-stationar), 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] 0 = [ln (fin*) + 0.3]/2flzf * = [ln (zrl/2zn*) + 0.3]/2zczZf * = {~[ln (fin*) + 0.3]/8}~'.

(15.111)

This approximate formula for the delay time 0 is valid for fin* > 1 and shows again that 0 is proportional to the time lag 7r (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 n*, 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*/ 3kT) 1/2 and fin* = (W*/4kT) 1/2 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 Ap we have fin* o~ 1/Ap for 3D and fin* o~ 1/(Ap aefAa) 1/2 for 2D nucleation. Since under typical experimental conditions 10 < W * / k T < 60, fin* usually has values from 1.8 to 4.5. Hence, the value of the 0/z" ratio from (15.111) is typically between 1.1 and 2.2 and is comparable with the value ~ / 6 = 1.64 predicted by eq. (15.105). This result is important, because it is an evidence for the reliability of the time lag z defined by (15.55) or (15.73). Figure 15.10 displays O/z as 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 (15.111) is reasonable, especially in view of the many uncertainties in the theoretical determination of the monomer attachment frequency f*. The dotted line in Fig. 15.10 visualizes the prediction 0/~:= n3/ 96 = 0.32 following from the first equality in (15.105) with the time lag 7: = 1/4rcfl2f * 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 0/'r value calculated from eq. (15.102) in the n* 3 case considered in Sections 15.1 and 15.2. The down triangle and the squares, circles, up triangles and stars also represent exact 0/7: values: they were obtained numerically by Shizgal and Barrett [ 1989], Abraham [ 1969], Kelton et al. [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 (15.11 l) is important for more elaborate considerations and in this respect it must be noted that higher order approximate formulae for the 0/z ratio are also available [Shneidman and Weinberg 1991, 1992a, 1992b]. Figure 15.11 displays the experimental 0(AT) data of Gutzow et al. [ 1968] (the circles) and of Koverda et al. [ 1974] (the squares), obtained in the case

266 Nucleation: Basic Theory with Applications

&X

......

~o

- _ -~o

. . . . . . . . . . . . . . . . . . . . . . .

II

U

.~ .....................................................................................

1

2

3

4

5

6

7

13n* Fig. 15.10 Dependence of the O/'t ratio on the thermodynamic parameter fin*: 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 of Shizgal and Barrett [19891; squares-numerical data of Abraham [19691; 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, NaPO 3 crystals in NaPO3 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 ((t) dependences by using eq. (15.100), they have to be regarded as representing with some uncertainty the actual delay time 0 of the nucleation process. The dashed and the solid curves in the figure depict the 0(AT) dependences calculated, respectively, from eqs (15.105) and (15.111) with the help of 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 0/'r 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 2 • 10-6 for the ice and the NaPO3 crystal nuclei, respectively (the latter 7* value is comparable with the value of 10-6 reported by Gutzow and Toschev [1968]). If it is assumed that 7* = 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

106

. . . , . .

':/ 'i '

.

, :~

2X10-3 J ]

::

:It

~

10'

M t

103

, ....

//

2X10-6/'

:

9

.

i

,,~ ,'/

80

1 O0

120

.-

,,g

:l "/'

~0 ~

. . . . . . .

P

i

,

,,:

267

,"

t

9

I

: glass

/~

transition

"

: :: -

lass

~I transition

140

250

300

350

400

AT (K) Fig. 15.11 Delay time of nucleation as a function of undercooling: squares-data for ice crystals in vitrified water [Koverda et al. 1974; Skripov 1977]; circles- data for NAP03 crystals in NAP03 glass [Gutzow et al. 1968]; solid curves- eq. (15.111); dashed curves - eq. (15.105). The numbers at the curves give the ~* values used in the calculation, and the dotted lines indicate the undercoolings at the glass-transition temperatures of water and NAP03.

Table 15.1 Values of various quantities used for calculation of the delay time 0 in 3D nucleation of crystals in NAP03 glass. 1o

do

Te

(rim3)

(nm)a

(K)

0.066

0.50

898

Ase[k

3.0

O'ef (mJ/m2)

/70 (mPa.s)

Ev/k

Td

(K)

(K)

100

12

1782

493

acalculated from do = (6v0/zr)1/3 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(A T) and Js (AT) dependences to experimental data for crystal nucleation in, respectively, Li20.2SIO2 glass and water.

15.5 Concentration of supernuclei In Section 13.4 we have seen that the time dependence of the concentration 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 ~'(t) dependence contains one more piece of information - it allows determination also of the delay time 0 (see Fig. 15.9b) and, thereby, of the time lag 7:. 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 ~'(t) dependence can be found easily by using J(t) from (15.59), (15.60), (15.62), (15.63) or (A1.12) 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 ~(t) function is of the form (n* = 2, 3 , . . . , M - 1) M-1

r

= J~t + Z ( d i / ~ i d ' ) ( f * a n * i

- g,,,+lan,+l,i)[1

- exp (- 2it)].

i=2

(15.112) In conformity with eq. (15.100), this exact formula shows that in the limit of t --~ oo the concentration of supernuclei increases linearly with time (see Fig. 15.9b). The slope of this asymptotic linear ~'(t) 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 0 defined by eq. (15.101). We can therefore say that no delay in the ~'(t) 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 = Mef = 4 and n* = 3 eq. (15.112) takes the simple form 23 -;t2,_ 22 -~3t ] ~(t) = Js t - 0 + ~,2 (/t,3 - )1,2) e 23 (/1,3 - 22) e

(15 113) .

which follows from employing J(t) from (15.61) in (11.10). Here ~ , ~,3 and 0 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 ~(t) 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 ~'0- 0. Equation (15.112) is exact, but its usage is hampered by the mathematical difficulties associated with the determination of )ti, ani and d i when M = Mef > 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 ~'0 = 0 yields [Kashchiev 1969a]

~(t) = J s { t - ( ~ / 6 ) r - 2~: ]E [(- 1)ili 2] exp (- fit~T)}. i=1

(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 --~ oo (i.e. when t >> T), as required, ((t) tends asymptotically to ((t) from (15.100) with 0 given by (15.105). Equation (15.114) can be represented in the equivalent form oo

((t) = 2Js't" ]~ {(4n't/'0 u2 exp [ - ( 2 i -

1)2nr2"c/4t]

i=1

- (2i - 1)n:2{ 1 - erf [(2i - 1)rc'cl/2/2tl/2]} }

(15.115)

which follows from using J(t) from (15.65) in (11.10). Equation (15.115) is convenient for analysing ~'(t) in the t ~ 0 limit. For instance, retaining only the first summand the infinite sum in (15.115), we find that for t < 0.251:, with an error of less than 15%, ~(t) = 8Js'C(tllr'r 3/2 exp (- navl4t).

(15.116)

From eq. (15.114) we see that the (/JsTrratio is a universal function of the dimensionless time t/T. Since the ((t) dependence is accessible to a direct experimental determination, this means that various ((t) data plotted in (~'/ JsV)-vs-(t/v) coordinates should produce a master curve. Figure 15.12 illustrates the validity of this prediction: we observe a good agreement between the theoretical ((t) dependence (15.114) (the curve) and available experimental ~'(t) data for nucleation of crystals in Li20.2SiO2 glass at T = 727 K (the 10

6 ~.P

4

w

0

2

4

6

8

10

12

tlx Fig. 15.12

Time dependence of the concentration of supernuclei in non-stationary nucleation: circles- data of Kalinina et al. [1980]for nucleation of crystals in 2Na20.CaO.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 2Na20.CaO-3SiO2 glass at T = 753 K (the circles [Kalinina et al. 1980]). Non-stationary ~'(t) 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 et al. 1977, 1980, 1997; Fokin et al. 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 ~" and of the Ap dependences of 77and 0. The fact that eq. (15.111) provides a higher-order approximation to the 0(Ap) dependence suggests replacing the time lag 77in eqs (15.54), (15.64) and (15.114) by the delay time 0 and using these equations in the form o~

Z*(t) = X* { 1 + (4/zr) E [(- 1)~/(2i - 1)] exp [- (2i - 1)2n2t/240] } i=1

(15.117) oo

J(t) = Js [1 + 2 ]~ (- 1)i exp (- i2zc2t/60)]

(15.118)

i=1 oo

((t) = J s { t - O - (12Dr2)0 E [( - 1)i/i2] exp (- i2tc2t/60)}. (15.119) i=l

When 0 is considered as related to z by the simplest approximation (15.105), these equations are merely an equivalent representation of (15.54), (15.64) and (15.114). If, however, 0in (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 Co under concrete experimental conditions. Moreover, the attachment frequency f* of

Non-stationary nucleation

271

monomers to the nucleus and the concentration Co of nucleation sites are coupled into a single parameter, the product f'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 f* or Co is available experimentally, we can split the product f ' C 0 and, thereby, reduce the theoretical uncertainty in A. Such an information about f* is contained in the nucleation time lag Tfrom eq. (15.55) or (15.73) and, accordingly, in the delay time 0 from eqs (15.105) and (15.111). As pointed out by Kashchiev [ 1972a, 1972b], this means that experimental data for the Ap dependence of the product JsZ or JsO 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 vfrom (15.55) and accounting for (7.44) and (13.35), we find that Js T = ( 4 ] l t 5 / 2 f l ) C * =

(4/~z)C* = (4/~z)Coexp (- W*/kT).

(15.120)

This relation between the product JsT and the equilibrium concentration C* of nuclei (or the nucleation work W*) was derived by Kashchiev [1972a, 1972b], and in another context a similar Js~r formula was used by Zeldovich [1942]. Since the factor 4/zr3z is practically Ap-independent and is a number usually between 0.5 and 5, eq. (15.120) tells us that the experimental determination of the product JsviS 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 ~ values of K6ster [1978], given in Section 15.2, from eq. (15.120) with assumed 4/~z = 1 we find that C* = 7 • 1017, 2 • 1018 and 8 • 1018 crystal nuclei per m 3 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 JsO product than with the JsT one. This is so because Js and 0 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),

JsO = ~C* = K exp (- W*/kT).

(15.121)

Here the pre-exponential factor K (m -3 or m -2) is defined by

K = ~Co,

(15.122)

and the numerical factor ~ is given by = 2/3zl/2fl = 2/3 zz

(15.123)

to a first approximation and by = [In (fin*) + 0.3]/2rcl/2fl = [In (zcl/2zn*) + 0.3]/2zrz

(15.124)

272

Nucleation: Basic Theory with Applications

to a higher-order approximation. Similar to the bracketed factor in (15.120), is practically AM-independent and is a number typically between 0.5 and 5. Hence, as in the Jslr case, the calculation of the product JsO from available ((t) 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 JsO (or JsT) 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 f*, is virtually AM-independent and roughly equal to the concentration Co of nucleation sites. For instance, for classical HON or 3D HEN of condensed phases we have (cf. eq. (13.48))

(A~ _>0)

JsO = K exp (-B/A/~ 2)

(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) Js 0 = K exp (- B"/ln 2 S),

(15.126)

B' being given by (13.67). Accordingly, when the process takes place in melts, similar to (13.72), there holds (0 < AT < 2TeAse[Acp,e) JsO = K exp (- B'/TAT 2)

(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 (AM > aerate) JsO = K exp [- B/(Ala - aefAty)]

(15.128)

where Atr~ 0 or Act= 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)) JsO = K exp (- tIl*/kT) exp (n*Ala/kT)

(15.129)

where the nucleus effective excess energy ~* is specified by (4.49). Obviously, with properly defined supersaturation A/.t and K (i.e. Co, 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(Js0) on AM when the supersaturation is varied isothermally and the stepwise changes in the value of n* occur in the AM range studied (see Sections 4.4 and 13.5). Finally, we note the classical Js0 formula for HON or 3D HEN of gaseous phases, which is the analogue of (13.58) (0 < p < Pe):

Non-stationary nucleation 273

JsO = K exp [- B/(p* -p)2].

(15.130)

Here p*, B and K are given by (4.14), (13.59) and (15.122), and p* - p = Ap is a good approximation when pevo/kT << 1. The above expressions show that when analysing an experimentally obtained Ap dependence of JsO we can apply directly the methods used with respect to Js(Ap) data (see Section 13.5). The great advantage now is that the preexponential factor K is completely insensitive to the particular kinetics of monomer attachment to the nucleus. Figure 15.13 depicts the dependence of JsO on AT, obtained by Koverda et al. [ 1974] (see also Skripov [ 1977]) for ice nucleation in amorphous water. The 0(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 ln(JsO)-vs-(1/TAT 2) coordinates. The best-fit values of B' and K are B' = 19.23 x 106 K 3 and K = 9.2 x 1029 m -3. Since K = Co, and for HON in water Co = 1/Vo = 3.1 x 1028 m -3, vis-a-vis the uncertainty in the experimental determination of Js and 0, this value of K is reasonable and indicative for HON. Also, with c 3 = 36m v0 = 0.0326 nm 3 and Ase]k = 2.63, in accordance with (13.73), the above B' value leads to (Tee= 27.0 mJ/m 2 which compares with (Tee ---- 26.4 mJ/m 2 found in Section 13.5 with the help of classical analysis of the Js(Ap) 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

o?, E

59.0

"3 c-

=....,

58.5

58.0 0.48

. . . . .

'

0.50

'

'

'

' 0.52

. . . .

'

0.54

'

'

'

0.I

i6

1 / T(AT) 2 (10 .6 K 3) Fig.

15.13 Dependence of the JsO 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 Js(Ap) data. The uncertainty in this determination is associated with the presence of the kinetic factor A in eq. (13.39) and is negligible when the supersaturation Ap is varied isothermally, but may be considerable when the temperature is the parameter used to change Ap. We shall now see that this uncertainty can be reduced significantly if in addition to Js(Ap) data we dispose with corresponding independently obtained 0(Ap) data so that we know also the product Js0 as a function of ACt.The application of the nucleation theorem to the JsO product is highly advantageous, for it eliminates the necessity to seek a concrete formula for JsO (see Section 15.7) which would provide the most plausible interpretation of the available dependence of JsO on Ap. 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 Ap dependence of JsO. To that end we rewrite eq. (15.121) in the form -

W*

= kTln

(JsO) -

kTln

K

(16.1)

and consider first the case when the JsO data are obtained by varying Ap isothermally. Differentiating both sides of this expression with respect to Ap, taking into account that d(ln K)/dAp "= 0 (see eqs (15.122)-(15.124)) and employing the nucleation theorem in the form of eq. (5.21), we get

An* = kT(1 - Pola/Pnew)d[ln (JsO)]/dAp

(16.2)

or, in an equivalent presentation resulting from using (5.18), An* = kTd[ln (JsO)]/dpola.

(16.3)

These formulae are applicable to all possible cases of one-component nucleation when Pola and thus Ap are changed isothermally. We see from them that for the model-independent determination of the excess number An* of molecules in the nucleus the JsO data must be plotted in In (Js0)-vsAp or Pola coordinates. Then the slope at each point of the resulting line gives An* at the corresponding value of Ap or Pold. 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 the nucleation theorem

275

from JsO data if in them Js is replaced by JsO and nA or the unity is omitted. For instance, for the determination of the number n* of molecules in the onecomponent EDS-defined nucleus we have the universal formula

n* = kT[(1 - Pold/Pnew)/(1 - Pold/Pnew*)]d[ln (JsO)]/dAlu

(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 (JsO)]/dA,u

(16.5)

for condensed-phase nuclei and by (cf. (14.8))

V* = (kT/Pnew)d[ln (JsO)]/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 (JsO)]/d(ln S)

(16.7)

for nucleation of condensed phases in vapours and solutions,

n* = (kT/zieo)d[ln (JsO)]/dAtp

(16.8)

for electrochemical nucleation,

n* = (kT/Ave)d[ln (JsO)]/d(p - Pe)

(16.9)

for nucleation of crystals in melts (or during polymorphic transformations) when Ap is varied by means of p at constant T, and V* = kTd[ln (JsO)]/dAp

(16.1 O)

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 JsO 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 fight-hand sides of these equations contain only the experimentally obtainable derivative of In (Js0) with respect to A/.t or/1old. Let us now consider the case when the JsO data are obtained at different temperatures. When A/~ 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[(1

-

Snew/Sold)[(1 -

- d(T In K)/dA~/}.

Snew/Sold)]{

d[T In (JsO)]/dAlu (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 JsO 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 Snow of a molecule in the EDS-defined nucleus. To a first approximation, however, it can be assumed that Snow = S~ew.Then eq. (16.11) simplifies to (cf. eq. (14.15))

n* = kd[T In (JsO)]/dAp + nx

(16.12)

where nK = - kd(T In K)/dAla, and K is expressed in the units of JsO. Compared with (14.15), eq. (16.12) is much more convenient for a modelindependent determination of n*, since nK can be estimated with a much greater confidence than hA. Indeed, recalling that K is virtually Ap-independent and close to Co (see eq. (15.122)), we find that nK is well approximated by

nK =-k(dT/dAla) In Co

(16.13)

where Co cannot exceed 1029m- 3 for HON or HEN in the bulk of the old phase and 1019 m -2 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 dAp/dT = - A s ( T ) - - Ase so that, approximately,

n~: = (k/Ase) In Co.

(16.14)

If JsO is measured in m - 3 or m - 2, this formula gives nK = 25 and nK = 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 Co = 3.1 • 1028 m -3) 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 < 2TeAsffAcp,e, AI.t is given by (2.23) and eq. (16.12) can be used in the simpler form

n* = (k/Ase)d[T In (JsO)]/dAT + nK =-(k/Ase)d[T In (JsO)]/dT + nK

(16.15)

where nK is given by (16.14), and JsO and Co must be expressed in the same units. The above considerations show that even when Ap 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 Ap dependence of the product JsO in [T In (Js0)]-vsAp (or AT or T) coordinates. When JsO 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 nx from (16.14) with the rather low Ase/k = 1 (typically, Asffk = 1 to 5) and with the maximum possible Co = 1029 m -3 or C O = 1019 m -2. The circles in Fig. 16.1a represent the experimental T In (Js0) 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

(a)

o v

0'3

0.98

i

E ~e-

0.96

!---

(b)

15

..!r .r

==,

10 ,=

=

=

1 102

i

i

i

i

,

104

,

,

I

,

106

,

,

I

,

.

108

,

I

,

110

,

,

I

,

112

,

.

114

AT (K) Fig. 16.1 (a) Dependence of the T In (JsO) product on the undercooling: circles - data for nucleation of ice crystals in vitrified water [Koverda et al. 1974; Skripov 1977]; l i n e - 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, l i n e - G i b b s - Thomson eq. (4.38).

of T In (JsO) on AT is fitted by the function T In (Js0) = 1 0 4 ( - 49.72278/AT + 2.3042 - 0.00805AT) (the curve in Fig. 16.1 a) allowing the evaluation of the d[T In (JsO)]/dAT derivative. Considering nK (i.e. Co) as a free parameter in eq. (16.15), with the aid of this derivative and Ase/k = 2.63, we can use the limiting value of Co = 3.1 • 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. 16.1 b. 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 Ap = AseAT, c = (36z) 1/3 (spherical or cap-shaped nuclei) and O'ef = 26.4 mJ/m 2 (this is the aef 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 Ap 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 theoryindependent way. What can be done in such a way is in fact the determination of the dependence of n* on Ap, 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 dAp/dT = - As(T) (see eq. (16.13)). The so-determined n*(Ap) 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 Co.

Chapter 17

Nucleation at variable supersaturation

When the supersaturation Ap in the system varies with time t, the transition frequencies f~m in the master equation (9.1) are time-dependent. For that reason, the cluster concentration Z,, and, hence, the flux j,, from (9.6) are also functions of t. Moreover, such a function is the nucleus size n*, 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 Ap. While the latter is only temporary and terminates after a period of the order of the time lag znecessary 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 Zn into the stationary one Xn(t) - Xn[Ap(t), T(t)] which corresponds to the momentary value of A/J. It is clear, therefore, that when Ap varies sufficiently slowly, Zn will have time to adjust itself to Xn(t), i.e. then Zn(t) --- Xn(t). Accordingly, nucleation will then occur at quasi-stationary rate Jqs(t) given approximately by Jqs(t) = Js(t)

(17.1)

where Js(t) - Js[A~t(t), T(t)] is the stationary nucleation rate corresponding to the value of Ap 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 Ap 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, t) of the clusters is then the solution of the master equation (9.26) in which the monomer attachment frequency f(n, t) and the quasi-equilibrium cluster concentration C(n, t) are also known functions of t (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, Z(n, t) must obey the initial condition (15.29) and the boundary conditions (15.31) and Z(1, t ) = Cl(t),

(17.2)

the latter being the analogue of (15.30). Thus, introducing the new unknown function

Y(n, t) = Z(n, t)/C(n, t),

(17.3)

we can represent eqs (9.26), (15.29), (15.31) and (17.2) in the following form [Kashchiev 1970] (1 _< n _<M):

c)Y(n, t) tgln [C(n, t)] + Y(n,t) Ot tgt - C(n, t) ~On f ( n , t) C(n, t) c)Y(n,o3~t) Y(n, O) = O, Y(1, t) = 1,

]

Y(M, t ) = O.

(17.4) (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 nl(t) < n < n2(t ), the r.h.s, of (17.4) is approximately equal tof*(t)O2y(n, t)/3n 2, because for these n values f(n, t) = f[n*(t), t] - f * ( t ) and C(n, t) = C[n*(t), 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 A/j varies with t sufficiently slowly, the boundary conditions can be 'shifted' to the ends nl(t) and n2(t) of the nucleus region (we note, however, that now n l 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 Co of nucleation sites is t-independent, we can replace (17.4) and (17.5) by the equations (nl(t) < n < n2(t))

c)Y(n, t) = f * ( t ) c)2 Y(n ' t) + I d [ W * ( t ) ] } Y(n t) Ot c)n 2 -d-[ k T ( t ) ' Y(n, O) = O,

Y[nl(t), t] = 1,

Y[nz(t), t] = 0.

(17.6) (17.7)

Nucleation at variable supersaturation

281

Here W*(t) - W[n*(t), t] is the nucleation work corresponding to the values of Ap and T at time t and can be calculated from (10.86) at n = n*(t). The nucleus size n* is also a known function of t because of the presumption that we know the temporal variation of Ap 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 OYqs/Ot = 0, since it depends only implicitly on t through Ap, T and their time derivatives. It is clear that the function Yqs will be physically acceptable after elapsing of sufficiently long time tqs 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)/C(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 (nl(t) <- n <_ n2(t))

dZYqs(n)/dn 2 +

2yqs(n) = 0

(17.8)

Yqs(nl) = 1, Yqs(n2)= 0.

(17.9)

Here the dimensionless quantity

2 is defined by

2(t) = [1/f*(t)]d[W*(t)/kT(t)]/dt

(17.10)

and may be positive or negative depending on whether Ap diminishes or increases with time. It depends on t implicitly through Ap(t) and T(t) 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 > tqs. Evidently, tqs 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 "t"calculated from (15.55) or (15.73) at suitably chosen Ap and T values. Before proceeding further let us obtain an equivalent, but simpler expression for 2 applicable to all cases of nucleation of condensed phases when Ap varies with time (e.g. through the pressure p(t) in the system), but T remains constant. Invoking the nucleation theorem in the form of eq. (5.29) and using the relation dW*/dt = (dW*/dAp)(dAp/dt), from (17.10) we find that for EDS-defined nuclei [Kashchiev 1970] 2(t) = -n*(t)Ap'(t)/kTf*(t)

(17.11)

where Ap'(t) - dAp/dt. This formula tells us that[ 2 [ is small enough and, hence, quasi-stationary nucleation is possible only when Ap varies not too rapidly (then [Ap'[ = 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 nindependent 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 - nl)] - [cos ( A*)/sin ( A*)] sin [ ( n - nl)] (17.12) where A*(t) - nE(t) - nl(t) is the variable width of the nucleus region. According to (7.38) and (7.43), A* is given by A*(t) =

~l/2/fl(t)

=

{ 21rkT(t)/[

-

o~2W(n,

t)/OnE]n = n,(t)} l/2 (17.13)

and is an implicit function of t through Ap and T. It must be pointed out that from (17.12) is physically meaningful not only for 2 > 0, but also when q2s< 0. Indeed, from mathematics [Korn and Korn 1961 ], in the latter case we have = i[ I, cos (il I x) = cosh (1 [ x) and sin (i1 [ x) = i sinh ( [ I 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(n) in the nucleus region is given by Zqs(n) = C(n,t)Yqs(n). This relation can be used for determination of the quasi-stationary concentration Zq*(t) = Zqs[n*(t)] = C*(t)Yqs[n*(t)] of the nuclei in the system. Setting n = n*(t) and accounting that nl(t) = n*(t) - A*(t)/2, from (17.12) we get Zq*s(t) = C*(t)/2 cos [ ( t)A*(t)/2].

(17.14)

As already noted above, the quasi-stationary approximation is physically realistic only for sufficiently small I [ 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 -x2/2 and (1 -x2/2) -1 - 1 + x2/2 which are valid for x < 1 [Korn and Korn 1961 ], from (17.14) we find that Zqs(t ) = (l/2)C*(t)[1 +

2(t)A*2(t)[8]

(17.15)

provided

[d[W*(t)/kT(t)]Jdt[< 4f*(t)/A*2(t) = 16/~'r(t).

(17.16)

Here 2 is specified by eq. (17.10), and v(t) - "r[Ap(t), T(t)] is the time lag corresponding to the momentary values of Ap and T. It is defined by (15.55) or (15.73) and is an implicit function of t through Ap 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 Ap with time t, (17.16) simplifies to

[ zxp'(t) I < 4kTf*(t)/n*(t)A*2(t) = 16kT/~n*(t)'c(t),

(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 Ap' 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 z = 0 (then = 0), according to (17.15), the quasi-stationary

Nucleation at variable supersaturation

283

concentration Zqs (t) of nuclei is exactly equal to their stationary concentration X*(t) = (1/2)C*(t) corresponding to the momentary values of A/~ 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)/C(n, t) = Zqs(n)/C(n, t) = Yqs(n) in eq. (11.9), we find that the quasi-stationary flux jq* (t) through the 'moving' point n*(t) on the size axis is given by Jqs(t) =f*(t)C*(t)(

t)/2 sin [ ( t)A*(t)/2].

(17.18)

Since the value of l l is restricted by the condition (17.16) (i.e. by l I A* < 2), we can simplify (17.18) with the aid of the approximations sin (x) -- x - x 3 / 6 and (1 - x e / 6 ) - 1 _ 1 + xe/6 valid for x < 1 [Korn and Korn 1961]. Hence, approximately, Jqs = [f*(t)C*(t)/A*(t)][1 +

e(t)A*e(t)/24].

(17.19)

Setting Z* and j* in eq. (11.5) equal to Zq* and jq* from (17.15) and (17.19), we finally obtain the following expression for the quasi-stationary rate of nucleation: Jqs(t) = Js(t) {1 - A*(t)n*'(t)/2f*(t)

+ [ e(t)A*2(t)/24][ 1 - 3A*(t)n*'(t)/2f*(t)] }.

(17.20)

Here n*'= dn*/dt, 2 is specified by eq. (17.10), and Js(t), defined in analogy with (13.33) and (13.39), by Js(t) = z(t)f*(t)C*(t) = A(t) exp [- W*(t)/kT(t)],

(17.21)

is the stationary nucleation rate corresponding to the momentary values of A/~ and T (the Zeldovich factor z(t) = 1/A*(t) and the kinetic factor A(t) = z(t)f*(t)Co, cf. eqs (13.35) and (13.40), depend implicitly on t through A~t and, possibly, T). Equation (17.20) describes the time variation of Jqs when i 2(t) I A*2(t) < 4, i.e. when A/~ 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 dn*' = (dn*/dAp)A/l', from eq. (17.20) we see that eq. (17.1) is valid when the rate of change of the supersaturation has absolute value I A~' I satisfying not only (17.16) (respectively (17.17)), but also the inequality I A~'(t) i < 2f*(t)/A*(t)ldn*(t)/dAla(t)l

= 8A*(t)/n3v(t)ldn*(t)/dAla(t) I.

(17.22)

284

Nucleation: Basic Theory with Applications

This inequality ensures that A*(t)ln*'(t)[/2f*(t) < 1 and in it "r(t) is again the time lag from (15.55) or (15.73) corresponding to the momentary values of Ap 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*/dAp 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

I At~'(t) I < 47rkT(t)f*(t)/A*3(t) = 16kT(t)/rc2A*(t)'c(t)

(17.23)

both for 3D and 2D nucleation of condensed phases. Let us now consider the case when Ap 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* _
n*(t)A*2(t)Ap'(t)/24kTf*(t)] ~n*(t)T(t)Ap'(t)/96kT]

(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 Jqs(t) 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 "r and the absolute value [Ap'I of the rate at which Ap 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 Ap (this is the "r = 0 limit), Jqs(t) would be strictly equal to Js(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 Jqs(t) function will be different depending on whether [ Ap' I 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 Jqs(t) = Js(t){1 + [A, 2 (t)/24f*(t)]d[W , -Js(t){1 +

(t)/kT(t)]/dt} [n3z(t)/96]d[W (t)/kT(t)]/dt}

(17.25)

which is the analogue of (17.24) and follows from (17.20) after neglecting the n*' terms and accounting for (17.10). Hence, in this case eq. (17.1) is

Nucleation at variable supersaturation

285

again a good approximation for Jqs(t) as long as T and A/~ 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(t) dependence becomes Jqs(t) = Js(t)[ 1 -

A*(t)n*'(t)/2f*(t)]

= Js(t)[ 1 - ~'r

(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*/ dA/~)A/~', 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

A*3(t)Ap'(t)/4zckT(t)f*(t)] = Js(t)[ 1 + zc2A*(t)'t(t)Alu'(t)/16kT(t)].

Jqs(t) = Js(t)[ 1 +

(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 I Ap' I is already so high that it obeys the inequality I A/~'(t) I >

4tckT(t)f*(t)/A*3(t) = 16kT(t)/tc2A*(t)~(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/~. Hence, despite the fulfilment of the condition (17.16) for quasi-stationarity, eq. (17.1) cannot be used as a good approximation to the Jqs(t) 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 Jqs 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 T, the inequality (17.22) (or (17.23)) must also be satisfied if we want eq. (17.1) to be valid. With appropriately defined Ap 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

I dS(t)/dtl< 16S(t)/~n*(t)v(t),

(17.29)

286 Nucleation: Basic Theory with Applications

eq. (17.1) with Js(t) from (17.21) is a reliable approximation for the Jqs(t) dependence. We thus see that the greater the supersaturation ratio S and/or the smaller the nucleus size n* and the time lag z, 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/lOOz(t)

(17.30)

I dC(t)/dtl<

(17.31)

Ce/lOO'c(t)

for nucleation of condensed phases in vapours or solutions, respectively. Considering HON as an example, with the high enough values of z = 100 ns and z = 10 ps for these two cases (see Figs 15.5 and 15.6) and with, e.g. Pe = 1 kPa and Ce = 1 0 23 m -3 (see Tables 3.1 and 6.1), for the r.h.s, of (17.30) and (17.31) we find 100 MPa/s and 1 0 26 m -3 s -1, 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 Tunitskii [ 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 A/.t from (2.27) in (17.17), we find that this condition is of the form

I dAtp(t)/dtl< 16kT/~zieon*(t)'t:(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* = 50 and z= 10 ps so that, e.g. for bivalent ions (zi = 2) in solutions at room temperature we get

I dAtp(t)/dt]< 10 V/s.

(17.33)

Experiments on electrochemical nucleation in galvanostatic regime (then the overvoltage Atp varies with time) [Schottky 1962; Klapka 1971; Bostanov et al. 1972] evidence that the rate of change of Aq~ can have absolute values comparable or surpassing that in (17.33). Hence, verifying the fulfilment of (17.32) before the usage of the Jqs(t) 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

I dp(t)/dtl< 2f*(t)/A*(t)ldn*(t)/dp(t)l = 8A*(t)/~t)ldn*(t)/dp(t)[

(17.34)

Nucleation at variable supersaturation

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 EDSdefined nuclei in the form of eq. (5.28). With/'/old -" /-/e -t- U0( p - - P e ) for the chemical potential of the liquid (cf. eq. (2.6)), P*w = p*/kT (ideal-gas approximation) and Pold "- l/U0, after neglecting the unity with respect to the Pold/P*w ratio we find from (5.28) that dW*/dp = n*kT/p*. Thus, with the help of the relation dW*/dt = (dW*/dp)(dp/dt), (17.16) becomes

[ dp(t)/dt [ < 4p*(t)f*(t)/A*E(t)n*(t) = 16p*(t)/rcan*(t)'c(t) (17.35) where the pressure p* =-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

I dp(t)/dt I < Pe/1000 "t'(t)

(17.36)

which parallels (17.30) and results from approximating p* by Pe and setting n* = 500. For evaporation-controlled detachment of molecules from the nucleus bubble the value of 1:is similar to that of the time lag for nucleation of droplets in vapours (see Fig. 15.8). Hence, with exemplary Pe = 1 MPa and the high enough 7: = 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 can 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 I dp/dtl values for other mechanisms of mass transport, because then the time lag "t"can be quite long (e.g. under viscous-flow control "cis proportional to the viscosity 77 [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* = Pe. Using (4.40) to calculate the dn*/dp 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 (16p*/zc3n*'c)(ZCpen*/p*A*). This quantity is greater than that on the fight of (17.35), because always Pe-> 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 Ap 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*/kT)/dAla](dAl~/dt) and dA/l = - AsdT, from (17.16) we get

i dT(t)/dtl< 8kTE(t)f*(t)/n*(t)A*E(t)lAla(t) - 2T(t)As(t)l = 32kT ,2(t)/nan*(t)'c(t)lAl~(t)- 2T(t)As(t)[

(17.37)

where As(t) - As[T(t)], and A/.t 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 Ap in the denominator is replaced by A/~ - a e f A G (the latter change disappears for 2D HEN on own substrate, as then Atr= 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/~ from (2.23) and the approximation As = Ase we can give (17.37) the simpler and sufficiently accurate form

] dT(t)/dt ] < 8kT2(t)f*(t)/ASen*(t)A*2(t) ] 3T(t)- Tel = 32kT2(t)/rc3Asen*(t)z(t)13T(t)-

Te I"

(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/.t from (2.23), we rewrite (17.23) as

[ dT(t)/dt[ < 47rkT(t)f*(t)/AseA*3(t) = 16kT(t)/It2AseA*(t)~(t). (17.39) Inspection of (17.38) and (17.39) reveals that, as always A* < n*, the former is stronger than the latter if 2T < zrl3T- Tel. This means that when 0 < T < 0.3Te or 0.4Te < T < Te, the condition for the validity of the approximation (17.1) for the Jqs(t) dependence during nucleation of crystals in melts is the inequality (17.38). Only when T is in the relatively narrow range between 0.3Te and 0.4Te 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 = Te and the high enough n* = 50, from (17.38) we find that when T is varied between 0.4Te and Te, crystal nucleation in melts will be quasi-stationary and, in addition, eq. (17.1) will be valid provided

] dT(t)/dt i < kTe/lOOAseV(t).

(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 timedependent temperature below Te. As seen from (17.40), the higher the melting temperature Te and/or the smaller the melting entropy Ase and the time lag 9 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 to the Jqs(t) dependence for T in the range of 0.4Te < T < Te. If, however, in this range the I dT/dt ] value is close to or somewhat higher than the value of the r.h.s, of (17.40), Jqs(t) may deviate noticeably from Js(t) 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 Li20.2SiO2 melt at oscillating temperature. In their study Te = 1313 K, Ase/k = 5, 9= 25 s (at the chosen average temperature of 850 K), and the maximum value of the rate of temperature variation is [ dT/dt] = tOoscTosc

Nucleation at variable supersaturation

289

= 0.125 to 3 K/s (tOoscand Tosc are, respectively, the oscillation frequency and amplitude). Comparing these ldT/dtlvalues 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 roost and/or Tosc values. This is in line with the finding of Kozisek and Demo [ 1993a] that the deviation of Jqs(t) from Js(t) increases linearly with tOoscand Tosc 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 ~r < 1 ps (see Fig. 15.7), with Te = 273 K and Ase/k = 2.6 we find from (17.40) that the process will be quasi-stationary and eq. (17.1) will be valid when I dT(t)/dtl< 106 K/s. The experimental data of Butorin and Skripov [1972] (the triangles in Fig. 13.9) evidence that, indeed, constant cooling rates i dT/ dt I 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 7r> 1 s (see Fig. 15.7), according to (17.40), I dT/dtlshould 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 z in limiting the regime of quasi-stationary nucleation of crystals in melts.

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Part 3 ,,,

,

,

,

,

,

,

,,

Factors affecting nucleation

,

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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 W for cluster formation and, thereby, the nucleation work W*. In the case of 3D HEN this effect is quantified by the activity factor ~ i n the effective specific surface energy O'efdefined by eq. (4.42). As ~ 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 s e e d . 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 et al. 1965; Krastanov 1970]. Following Fletcher [ 1958], we consider a moon-shaped condensed-phase cluster of radius R on a spherical seed of radius Rs (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 R s.

294

Nucleation: Basic Theory with Applications

(18.1)

Os,0 = 4zrO's R 2

~(R) = 47rtr~ R 2 - 2trOts R2[1 0(R) = 27rtrR2[ 1 +

(18.2)

cos 0w)]

u(x)(1 - x

+ 2troti R~ [ 1 -

cos 0w)]

u(x)(x -

u(x)(x -

cos 0w)].

(18.3)

Here o', crs and tri 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 (18.4)

x = Rs/R u(x)

= (1 - 2x cos 0w + x2)-1/2.

(18.5)

Since we are considering a condensed-phase cluster, according to (3.15), the excess energy Gex 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 9 (R) = 2n:o-R2 {[ 1 + + x 2 cos

u(x)(1 - x

Ow[u(x)(x -

cos 0w)]

cos 0w) - 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/.t(rd3o0)R3{2 + 3u(x)(1 - x cos 0w) - u3(x)(1 - x cos 0w)3 -

x312 -

3u(x)(x

+ 2~rcrR2{[ 1 + + x 2 cos

-

cos 0w) +

u(x)(1 - x

Ow[u(x)(x -

u3(x)(x

-

cos 0w)3]}

cos 0w)]

cos 0w) - 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 ( ~ 3 ) R 3 and the expression in the curly brackets behind R 3. We note that eq. (18.7) takes the simple form W(R) = - At~(47r/3Vo)(R 3 -

R3s ) +

4~ztr(R 2 - Rs3)

(18.8)

in the limiting case of 0w = 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 GibbsThomson 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

Fig. 18.2 Spherical nucleus of radius R* on (a), (b), (c) seeds with different radii, and (d), (e), (f) seeds with different wetting angle. W * = I 6/t'V00"ef/3A~ 2 3 2.

(18.9)

Here the effective specific surface energy O'ef(J/m 2) is defined by (4.42), and the activity factor ~ is given by W3= (1/2){ 1 + u*3(1 - x * cos 0w) 3 + 3x .2 cos Ow[u*(x*-cos 0 w ) - 1] + x'312 - 3u*(x* - cos 0w) + u*a(x * - cos 0w)3] }

(18.10)

where x* = Rs/R*

(18.11)

u* = (1 - 2x* cos 0w + x*2) -1/2.

(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 tref is entirely determined by the wetting angle 0w, now O'ef is a function not only of 0w, but also of the supersaturation A/~. This is so, because the activity factor W from (18.10) depends on the nucleus radius R* which varies with A/~ as required by eq. (4.9). The implication is that, classically, the familiar dependence of W* on 1/A/~2 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 0w is fixed, from (18.10) we find that ~ - 1 at Rs = 0 and that W = g//3(0w) at Rs = ~,, ~ 0w) 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 Rs is fixed, but the wetting angle 0w has the limiting values 0w = 0

296

Nucleation: Basic Theory with Applications

(complete wetting) or 0w = 7r (complete non-wetting), from eq. (18.10) it follows that = (1 - 3x .2

+

2x .3) 1/3

(18.13)

in the former case [Krastanov 1947/48] (this is 3D HEN of shells, Fig. 18.2d) or that W = 1 in the latter case (this is HON of spheres, Fig. 18.20. Equation (18.10) reveals also that, geometrically, unlike in HEN of a cap or a lens (see Section 4.3), the activity factor W in HEN of a moon-shaped nucleus is not equal to the ratio between the volume of the nucleus and the volume (4/3)7rR .3 of the corresponding homogeneously formed spherical nucleus. As noted by Fletcher [1958], this distinction between ~ for caps or lenses and W for moons is a result of the lack of proportionality of the moon volume t o R 3 and of the moon surface area to R 2. 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 Wand, hence, the greater the nucleation activity of the considered seed. This means that diminishing the seed radius Rs from infinity (fiat 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/~ in eq. (18.7), setting in this expression R = R*, x = x* and u = u* and allowing for (4.9), we get n * = (8zro~o'3/3A~t3) {2 + 3u*(1 - x * cos 0w) -

-

U*3(1 -- X* COS 0w)3 -

x'312

-

3u*(x* - cos 0w)

+ u*3(x * - cos 0w)3] }.

(18.14)

This is the Gibbs-Thomson equation for the moon-shaped nucleus. It shows that, due to the A/~ dependence of x* and u*, the number n* of molecules in such nucleus is a rather complicated function of the supersaturation A/~. As it should be, in the limits of Rs = 0 or 0w = 7r eq. (18.14) passes into eq. (4.10) for HON of spheres, and in the Rs ~ oo 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 Js(m -2 s-l). Thus, classically, using eqs (4.42), (13.40) and (18.9) in (13.39), we find that [Fletcher 1958], Js =

zf*Co

exp [- ~3(A/~, Rs,

Ow)B/AI~2]

(18.15)

where the thermodynamic parameter B is given by eq. (13.49) with c = (36z01/3 and tref = ty. The Zeldovich factor z changes relatively slightly with the seed radius Rs (through n* and W*) and may be treated as virtually Rsindependent. This is valid also for the frequency f * 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 "ccannot 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 T - 293 K and supersaturation ratio S =-P/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 A/.t = 0.92kT, the nucleus droplet is of radius R* - 1.2 nm, and B - 89(kT) 2. The kinetic factor A - z f ' C o is expressed as A ' S (cf. eqs (13.40)-(13.42)), and A' is assumed to have the exemplary value of 1026 m - 2 s -1 corresponding to Co

10 30 30 ~

1025 10 20

60 ~

1015 10 lo 't/)

105

90 ~

10 -5 10-10 10-15 10 -20 0.1

1

10

100

R s / R* Fig. 18.3 Dependence of the stationary nucleation rate on the seed radius according to eq. (18.15)for H E N at T - 293 K and P/Pe = 2.5 o f 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 typical z = 0.1 and f * = 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 Rs = R*. When the seed is sufficiently smaller than the nucleus, Js is virtually independent of both Rs 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 A/~, 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(A/~) 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 Ap = k T In S as required by eq. (2.8). The seeds are assumed to have the same radius Rs -- 2 nm, but different wetting angle (as indicated). Figure 18.4 shows that, due to the dependence of the activity factor ~ from (18.10) on the supersaturation, the stationary nucleation rate is a more complicated function of A/~ 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 A/~ dependence of the stationary nucleation rate in the presence of seeds cannot be linearized in the classical In Js-vS-(1/A/~)2 coordinates for 3D nucleation if the A/~ range studied corresponds to nuclei of size comparable with that of the seeds. In this situation a reliable interpretation of the Js(A/~) data is possible only in the scope of the first application of the nucleation theorem (Chapter 14). Indeed, employing eq. (14.6) allows a theoryindependent determination of the A/~ 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*(A/l) data by eq. (18.14) would be an evidence for 3D HEN of moonshaped 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 fiat parts of the substrate surface. At a given A/~, 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

1025 30 ~

1020 60 ~

90 ~

1015 I

!

E ~

010

180 ~

10 5

1

2

3

4

5

6

S 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 fiat 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 Navascurs and Tarazona [1981 ], Navascurs 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 et 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 0w in the range from 0w = 0 (complete wetting) to 0w = (complete non-wetting) (Fig. 19.1). The total surface energies ~0s,0and ~s of the substrate before and after the cluster formation are again given by eqs (3.51) and (3.54), respectively. Compared with ~ from (3.53), however, now the total surface energy q~of the cluster contains an extra term equal to the total line energy tc'2~R sin 0w of the cluster/substrate contact line whose length is 2zrR sin 0w: ~(R) = o'2 tcR2(1 - cos 0w) + o'i/t'R2 sin 2 0w + tc'27rR sin 0w. (19.1) Here tO' (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 tO' term in (19.1) is analogous to the tr term in ~ for 2D HEN of disks (see eq. (3.68)), there exists a basic distinction between to' and to. Namely, whereas the disk specific edge energy tr is always positive, the cap specific line energy ~ can be either positive, zero or negative. This distinction between ~r and tc is a manifestation of their different physical meaning. Since tc is 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, tO' 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 et al. [1980] demonstrated experimentally that ~ can change sign in dependence of the molecular interactions in the contact-line region (concerning the sign of ~' see, e.g. Exerowa et al. [ 1994]). Since we are considering condensed-phase clusters, according to (3.15), the cluster excess energy Gex is given by ~ 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]" 9 (R) = 2zro-R2(1 - c o s 0w) + ~r(cri- Crs)R2 sin 2 0w + 2~r

sin 0w. (19.2)

In principle, R and 0w 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 9 from (19.2) at a fixed cluster volume Vn from (3.52). Hence, assuming hereafter that R and 0w 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 d ~ = 0 and dV,, = 0. Doing that yields the formula [Shcherbakov and Ryazantsev 1964; Gretz 1966a] COS 0 w = (O"s - - O ' i ) / O ' - -

tC'lRcr sin 0w

(19.3)

which can be represented equivalently as cos 0w = (O's - o'i)/o'- (41r/3vo)l/3lC'lffl/3(Ow)/nl/3a sin 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 (gt(0w) is 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 ~ = 0. Understandably, it passes into (3.55) also in the limit of R or n ~ oo. For clusters of smaller size, however, the effect of the line energy is considerable if ]r'] is high enough: for such small clusters 0w 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 0v and rewrite (3.55) in the form

(19.5)

COS Oy ----(0"s -- O~)/0".

It is clear that, physically, 0y is the angle of wetting of the substrate by a macroscopically large new phase. The 0w(n) dependence (19.4) for water droplets in vapours at T = 293 K on a substrate with macroscopic wetting angle 0v = 80 ~ is depicted in Fig. 19.2. The calculation is done with v0 and o'from Table 3.1 and with assumed

180 160 140 120 100 "tD

v

0

80

-20 60 40

20

1

50

100

150

200

250

300

n Fig. 19.2 Size dependence o f the cluster wetting angle: lines - 20, 0 and 20 - eq. (19.4) f o r 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

h-" = - 20, 0 and 20 pJ/m (as indicated in the figure). We see that in the U < 0 case 0w is an unambiguous function of n and that the smaller the capshaped 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 to' > 0, the smallest clusters cannot be with the corresponding equilibrium shape either. Indeed, Fig. 19.2 shows that the 0w has no value for n < 11. Besides, in the tO' > 0 case a given cluster of large enough size can have two wetting angles. As we shall see below, the smallerangle 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 ~ ( 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 tYi - O's by the line energy and recalling that the cluster volume is given by (3.52) leads to

W(R) = gt[ 0w(R)][ - (4rcAp/3vo)R 3 + 4rco'R 2] + ~zh'R sin [ 0w(R)] (19.6) where gt(0w) is defined by (3.56). In terms of n, this expression can be given the equivalent form

W(n) = - n A p + (36~z02) 1/3G~l/3[Ow(n)] n 2/3 + (3/r200/4)1/3/r ' sin [0w(n)]gt-1/3[Ow(n)]nl/3.

(19.7)

We note that these two equations turn into (3.59) and (3.60) in the particular case of to' = 0 when 0w is n-independent and equal to 0v. Figure 19.3 exhibits the W(n) dependence (19.7) for condensation of water vapours at T = 293 K on a substrate again with 0v = 80 ~ The calculation is done with the parameter values used for Fig. 19.2 and with A/.t determined from (2.8) at supersaturation ratio S = pipe = 2.2. Curves - 20, 0 and 20 correspond to to' = - 20, 0 and 20 pJ/m, respectively. As seen, in the ~ < 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(n) 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 0w value which we read in Fig. 19.2 at n = 10). In the opposite case of ~ > 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 greaterangle 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 0w(n) curve 20 in Fig. 19.2. 160 140 120 100

..----- W*

80

20

40 20 0 -20

1

50

1O0

150

200

250

300

350

400

Fig. 19.3 Dependence of the work f o r cluster formation on the cluster size: c u r v e s 20, 0 and 20 - eq. (19.7)for cap-shaped water droplets with specific line energy - 20, 0 and 20 p J/m, respectively. The droplets are on a substrate in vapours at T = 293 K and p i p e -- 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 to' 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 0w 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 0* - 0w(R*) 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 capshaped nucleus and the nucleus wetting angle 0* are given by n* = IF(0*) 32/rv~o'3/3 A/23 cos 0* = cos 0 y - K'Ap/200 O'2 sin 0".

(19.8) (19.9)

Regretfully, these two equations express only implicitly the dependence of n* and 0* on the supersaturation Ap. Yet, as sin 0* > 0 in the physically relevant range of 0* = 0 to/9* = ~r, eq. (19.9) tells us that 0* > 0v or 0* < 0y

Line energy

305

for ~r > 0 or 1r < 0, respectively. Accordingly, the Gibbs-Thomson equation (19.8) shows that when tO' > 0 or to' < 0, the nucleus size n* is, respectively, larger or smaller than that at to' = 0. This is visualized in Fig. 19.3 and follows from the monotonous rise of gtwith the wetting angle (see Fig. 3.6). Also, eq. (19.9) reveals that the effect of the line energy on the nucleus wetting angle 0* (and, thereby, on n *) is increasingly important at higher supersaturations and for nuclei with higher] tO' ] and/or lower tr. Moreover, sufficiently high values of the parameter] tO' 16/1/2o0o2 make the r.h.s, of eq. (19.9) either less t h a n 1 (when tO' > 0) or greater than 1 (when to' < 0). This equation then has no solution with respect to 0", 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 0*(A/~) dependences (19.8) and (19.9) again for nucleation of water droplets in vapours at T = 293 K on a substrate with 0v = 80 ~ The calculation is done with the parameter values already used for Figs 19.2 and 19.3. As seen, when ~:' ~ 0 (curves - 2 0 and 20), some 20 ~ departure of 0* from 0v already changes considerably the nucleus size n* with respect to that in the ~ = 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 (9* satisfying eqs (19.8) and (19.9) at tO' ~ 0 when Ap is sufficiently low. In the to' < 0 case, whereas the greater n* and 0* 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 U > 0 the nucleus which is effective at the given supersaturation is characterized by the smaller n* and 0* values. Figure 19.4 shows also that when to' ~ 0, it is not possible to speak about nucleus (eqs (19.8) and (19.9) have no real solutions with respect to n* and (9") when A/~ is higher than the supersaturation corresponding to the 'noses' of curves - 2 0 and 20. This means that in the range of these high A/J values nucleation of caps is not involved in the condensation process on the substrate. Now, knowing that R* and/9" 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, 3 /3 A,U2 W* = 16 n:Uo2tref where

(19.10)

O'ef "- t/tO" in accordance with (4.42), and the activity factor 7~ is

defined by ~3 = gt(0*)+ (3~/8v0cr2)A/~ sin 0 " = gt(0*) + (3/4)(cos 0 v - cos 0") sin 2 0".

(19.11)

In a different appearance, eq. (19.10) was obtained by Gretz [1966a]. At h-" = 0 (then 0* = 0y) it passes into eq. (4.25) for W* for cap-shaped nuclei

306

Nucleation: Basic Theory with Applications

50 (a)

40 30 "It

t'-

20 20

10

-20 (b)

150 t~ "0

v

-It

20

120 90 60

-20

30 1

5

10

15

S Fig. 19.4

Dependence of (a) the nucleus size, and (b) the nucleus wetting angle on the supersaturation ratio f o r 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 is 80 ~.

with zero line energy, but when ~ > 0 or tO' < 0, it predicts nucleation work higher or lower, respectively, than that in the ~' = 0 case. When to' > 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~ in a rather complicated way, and not just proportionally to A/~- 2 as required by (4.25). This is due to the fact that now O'ef is a function of Ap through the activity factor which is controlled not only by the Young wetting angle 0v as in the ~ = 0 case (then gt(0*) = ~r(0v) and ~ = Iprl/3(0y)), but also by the A/t-dependent

Line energy 307

wetting angle 0* of the nucleus. Thus, with regard to their Ap dependence, O'ee in (19.10) and ~ from (19.11) are congenial with O'ef and ~ for moonshaped nuclei (cf. eqs (18.9) and (18.10)). It should be noted that a more accurate calculation of W* in the x" < 0 case may require accounting that in this case, if large enough, the stable clusters which correspond to the minimum of W(n) (curve - 2 0 in Fig. 19.3) could act as seeds for the nuclei [Chakarov et al. 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 U~ 0, we are now in a position to determine the corresponding stationary nucleation rate Js ( m-2 s-l) 9Thus, classically, substituting W* from (19.10) in the general formula (13.39) and accounting for (4.42) and (13.40) yields [Gretz 1966b] .Is = zf*Co exp [- ~3(Ap,

Oy)B/A].l 2]

(19.12)

where the thermodynamic parameter B is given by (13.49) with c = (36zr) 1/3 and Cref= or, and the activity factor ~ i s specified by (19.11). Owing to the relatively weak dependence of the Zeldovich factor z and the frequency f* of monomer attachment to the nucleus on W* and/or n*, both z and f* can be treated as practically unaffected by the specific line energy U. We note that, in view of eq. (15.73), this implies also that h" has virtually no effect on the nucleation time lag ~. Depending on the mechanism of monomer attachment, however, some effect of to' on f* and, thereby, on ~ may be expected when the value of r ' is such that the nucleus wetting angle 0* is close to its limits 0* = 0 or 0* = zc corresponding to complete wetting or non-wetting, respectively. Shown in Fig. 19.5 is the Js(Ap) dependence for condensation of water vapours at T = 293 K on a substrate which, macroscopically, is characterized by 0u = 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 10 26 m -2 s -1 corresponding to Co = 1019 m -2 ( n o active centres on the substrate) and to typical z = 0.1 and f * = 108 s-1. The supersaturation Ap is determined from (2.8) in dependence of the supersaturation ratio S = P/Pe. The variation of the activity factor ~ with Ap is taken into account with the help of eq. (19.11) in which 0* is considered as a function of Ap as required by eq. (19.9). The parameter values used are given in Table 3.1 (according to (13.49) they result in B = 89(kT)2), and the specific line energy U of the cap-shaped nuclei is assumed again to b e - 20, 0 and 20 pJ/m (as indicated in the figure). We see from Fig. 19.5 that while to' < 0 leads to higher nucleation rate in comparison with that in the U = 0 case, tO' > 0 acts in the opposite direction, the effect being considerable under the chosen conditions. The line energy affects also the character of the Js(Ap) function: due to the Ap dependence of the activity factor ~ from (19.11) at to' ~e 0, Js(Ap) from (19.12) is not practically linear in the classical In Js-vS( 1 / A p ) 2 coordinates when[ to' [ is sufficiently high. This means that a nonlinearity in such coordinates of the plot of experimental Js(Ap) data for 3D HEN on a substrate might be caused by non-vanishing line energy. Whether

308

Nucleation: Basic Theory with Applications

10 20

-20

1015

0

"7 !

v

E

1010 20 10 5

,

1

,

, I ,

II,

2

,

,

,

3

4

5

S Fig. 19.5 Dependence o f the stationary nucleation rate on the supersaturation ratio f o r H E N o f cap-shaped water droplets on a substrate in vapours at T = 293 K: curves - 20, 0 and 2 0 - 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 n* from the slope of the experimental Js(Ap) curve in In Js-vs-Ap coordinates. The so-obtained experimental n*(Ap) 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 Js as a function of Ap -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.

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 W(n) to form a strained n-sized cluster in an initially unstrained matrix must contain as a summand the total energy tibstr(n ) of the strain fields generated in the cluster and the matrix as a result of the cluster formation [Fisher et al. 1948]. Hence, in lieu of eq. (3.86) we shall have

W(n) =-nA/~ + ~ ( n ) + t~str(n ).

(20.1)

This formula is of most general validity and with appropriate expressions for the supersaturation AH, the cluster effective excess energy 9 and the total strain energy t~st r 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 EDSdefined condensed-phase clusters in a solid matrix such as vitrified melt, one-component crystal or solid solution. In this case Ap is determined approximately by eqs (2.14), (2.23) and (2.25), and ~(n) is obtainable from (3.15) and (3.87) at ~0s= Cs,0 = 0 so that the actual problem is the calculation of tibstr(n ). Let us see how t~str(n ) 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 mv = (Vm - Vo)/Vo

(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 [19691). respectively. The density of the energy of elastic strain in the cluster, •str, c, and in the matrix, Estr, m, 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 Estr, c and Estr, m a r e given by (e.g. Christian [1975]; Ulbricht et al. [1988]) Estr,c = 2(1 - 2v~) m2v E2Y,mEyx/3[2(1 - 2Vc)Ey,m (20.3)

+ (1 + Vm)Ey,c]2, (0 -< r _
(R < r < oo).

(20.4)

Here Ey, c and Vc are, respectively, Young's modulus and Poisson's ratio for the cluster, and Ey, m and Vm are the same quantities, but for the matrix. Both Ey, c and Vc are treated as independent of the cluster radius R. Equations (20.3) and (20.4) show that the dilatational 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, t~st r is readily obtained by integration of e~tr,c and Estr, m in accordance with ~ s t r ---- 4zr

e~tr,cr E dr + 4 zr

estr, m r

2 dr

(20.5)

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] t2bstr(n) = ~strn

(20.6)

where ~str (Jr), given by Cstr - m2vvoEv, cEy.m/3[2(1 - 2Vc)Ev,m + (1 + Vm)Eu

(20.7)

is the strain energy per molecule of the cluster. This energy increases quadratically with the volume misfit my 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 ~str is independent of the cluster size n (since the possible dependence of Eu and Vc on n is ignored) so that t~st r is proportional to n, i.e. to the volume of the cluster. It turns out that the size independence of ~str is preserved also for non-spherical clusters as long as their shape is f i x e d - then t~str 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 (Ey, c = Ey.m and Vc = Vm) is ~str independent of both shape and size [Nabarro 1940a, b; Christian 1975]. The point to remember is that the independence of ~str 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 tibstr(n) 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 tibstr(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 ~0(n) when the cluster is defined by the EDS. Hence, using in eq. (20.1) the classical q~(n) dependence (3.20) and tibstr(n) from (20.6) yields [Fisher et al. 1948; Christian 1975; Russell 1980] W(n) = - ( A p -

~str)n + (36/17)1/3 o~/3 tyn 213.

(20.8)

This is the classical formula for the work to form a spherical condensedphase 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 ~str -> 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 Ap - ~str which is lower than the actual supersaturation A/~. As a consequence, even if the old phase is supersaturated, the occurrence of the process is possible only when A~t > ~)strThis 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* = 327r0~ O3/3(Ap - ~str) 3,

(20.9)

W* = 16zro02O3/3(A/~ - ~str) 2.

(20. l 0)

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 ~str = 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 C/~str(n) from (20.6) is treated as A/~-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 > ~str) Js = zFC0 exp [- n/(Al~

-

~str) 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~) 1/3 (spherical nuclei) and O'ef "- O' (HON). As seen, the strain energy retards nucleation by effectively lowering the supersaturation and can even arrest the process when the condition A/I < ~str is fulfilled. The effect is largely due to the exponential factor in (20.11), because the Zeldovich factor z and the frequency f* of monomer attachment to the nucleus are relatively weak functions of ~str through n* and W* (see eqs (13.36) and (13.40)-(13.43)). It must be noted, however, that ~ 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 f* is influenced significantly by the presence of strain, the nucleation time lag ~rwill 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 Cvo = c = (36z01/3. Using the first equality in (13.36) and eqs (20.9) and (20.10) to express z and n *1/3 in (15.85) through ~str and accounting for eq. (2.14), we obtain (kT In S > ~str)

= 32OocrZ/~'*(kT)ZDCeS(ln S

-

~str]kT) 3

(20.12)

where S = C/Ce is the supersaturation ratio. This equation corresponds to eq. (15.86) for HON of spheres: it turns into (15.86) at ~str = 0. Clearly, when the sticking coefficient 7'* and/or the diffusion coefficient D of the solute molecules are affected strongly by the strain field around the nucleus, the above dependence of "r on ~str 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 ~str -" 0, 1.2 •

Strain energy

313

10 -21 and 4.8 x 10 -21J. 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 Ev, c = Ey, m = 100 GPa, Vc = Vm = 0.3 and v0 = 0.03 nm 3. 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 o" = 100 mJ/m 2 and T = 800 K in eq. (13.49). The kinetic factor A - z f * C o is represented as A = A ' S (see eq. (13.41)), and A' is assumed to be Ap-independent and equal to 3 x 1031 m -3 s-1. According to eq. (13.42), this A' value corresponds to z = 0.1, C 0 - 3 x 1028 m -3 and f * = 10 4 S-1 (this rather small value of fe* 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

10 20

1015

0.05

"T, 03 ! V

E

1010

O0

10 5

1

2

3

S Fig. 20.2 Dependence o f the stationary nucleation rate on the supersaturation ratio: curves O, 0.05 and 0.1 - eq. (20.11)for H O N o f condensed phase under dilatational strain in solid matrix at volume misfit mv = O, +. 0.05 and +_.0.1, espectively.

314

Nucleation: Basic Theory with Applications

J~(A/z) dependence: this dependence cannot be linearized in the classical In Js-vS-(1/Ap) 2 coordinates discussed in Section 13.5. This means that nonlinearity of the plot of experimental Js(A/~) 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/~ isothermally, plotting them in In Js-vs-Ap coordinates allows a reliable experimental determination of the n*(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 tJbstr(n ) in eq. (20.1) as long as this energy is Ap-independent. Indeed, the general form (5.4) of the nucleation theorem remains unchanged, because there is no contribution from the partial Ap derivative of tJbstr(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 et al. 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]; Chizmadzhev 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. W*) of the nucleation process, but also the kinetics of monomer attachment to the nucleus (i.e. f*). In this chapter we shall consider the impact of the electric field on the stationary nucleation rate Js and on the nucleation time lag 7: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/m 2) of, respectively, the electric field and the electric displacement are related by D = eoeE, where e is the E-independent relative permittivity or dielectric constant, and eo = 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 W ( n ) 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

G O _ (1/2) f E(r) 9D(r) dr d

(21.1)

316

Nucleation: Basic Theory with Applications

where G o 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 ~ 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, G 0, 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) =-nAp + ~(n)

+_

~el(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 t~

~el(n)

(1/2) J [E2(r ) 9D2(r ) - E l ( r ) 9Dl(r)] dr

(21.3)

is the change in electrostatic energy due to the replacement of the old electric field E 1 by the new field E 2 which appears in the system as a result of the cluster formation. Unfortunately, usually it is hard to calculate ~el from (21.3), because the integration is over all space and it is necessary to know E 2 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 19821 i"

~el(n) = -(1/2) Jv~ E1 (r) 9[D2(r) - e0em(r)E2(r)] dr

(21.4)

where e 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 ~el(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, D 2 = e0ecE2 and treating e m and the dielectric constant e c of the cluster as r-independent, we can rewrite (21.4) as

~el(n) = (e0/2)(em- ec)

fv,

E l ( r ) " E2(r) dr.

(2~.5)

As seen, when the dielectric constant e c of the cluster (i.e. of the new phase) is greater than that of the medium, ~eliS 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 e c = e m, and the process is inhibited by the field when ec<e

m.

Electric field

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 et al. 1983; Kuni 1984a; Rusanov and Kuni 1984; Shchekin et al. 1984; Warshavsky and Shchekin 1999a, b]. The ion is regarded as a sphere of radius R i 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 in eq. (21.2) we can use the classical expression

Fig. 21.1 Spherical cluster of n molecules (a) on a positively charged ion, and (b) in an externally applied uniform electric field (in both cases e c > em).

which follows from (18.6) at 0 w = 0 and R s = R i and which is valid for EDSdefined clusters. We note also that n and R are related by n = (4rd3v0)(R 3 - Ri3).

(21.7)

The electric field about the ion before and after the cluster formation is spherically symmetrical so that for the dot product E 1 9E 2 we have E 1 9E 2 = Er, lEr,2 , Er, 1 and Er, 2 being the radial components of E 1 and E 2 in the space occupied by the cluster. According to electrostatics, Er, 1 = Q/4zCeoemr2,

(R i _< r _< R)

(21.8)

Er, 2 = Q/4~eoec r2,

(R i _< 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 q~el" Substituting E r 1 and Er, 2 from (21.8) and (21.9) in this equation, accounting that dr = 4 ~ r 2dr and integrating over r from R i to R yields

318 Nucleation: Basic Theol. with Applications

@el(R) = ( l / e c - 1/em)(Q2/8tCeo)(l/R i - 1/R).

(21.10)

Thus, from eqs (21.2) (taken with +q)el), (21.6), (21.7) and (21.10) we find that the work to form a spherical EDS-defined condensed-phase cluster on an ion is given by (R > R i )

W(R) = - A p ( 4 z r / 3 V o ) ( R 3 -

R 3) + 4n'o'(R 2 - Ri2)

+ (1/e c - l/em)(QZ/8xeo)(1/R i - 1/R).

(21.11)

Owing to (21.7), in terms of n this formula takes the form

W(n) = - n A p + 47rcrR 2 {[1 + (3Vo/4zcR3)n] 2/3 - 1} + (1/e c - l/em)(QZ/8tceoRi){1 - 1/[1 + (3Vo/4tcR3)n]l/3}.

(21.12)

At e 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 cr is not affected by the ion charge, W depends quadratically on Q and is, therefore, insensitive to the sign of Q provided the positive and negative ions have the same radius R i (for the effect of the sign of Q see, e.g. Russell [1969]; Castleman, Jr. [1979]; Rusanov [1979]; Rusanov and Kuni [1984]; Shchekin et al. [1984]). The ion stimulates the cluster formation when e c > e m, as then the Q term in (21.11) or (21.12) is negative. In the opposite case of e c < e m this term is positive and the cluster formation is inhibited. Naturally, the electric field about the ion has no effect on W when e c = e 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 a r e 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 R s = R i. Figure 21.2 displays the W(n) dependence (21.12) in the practically more interesting case of e c > e m. The calculation is done for water droplets (e c = 80) in vapours (e c = 1) at T = 293 K and supersaturation ratio S = P/Pe = 2.5. The supersaturation A/j is evaluated according to eq. (2.8), the Pe, v0 and o" values used are listed in Table 3.1, and it is assumed that the ion is with charge Q = + e 0 and radius R i = 0.15 nm (e 0 = 1.6 x 10- 1 9 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 e c > e m is congenial with HEN of cap-shaped clusters with negative line energy (see Chapter 19 and curve - 2 0 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 n* = 203 water molecules, the ion 'jacket' is constituted of n m i n = 14 such molecules. In fact, it is the 'jacketed' rather than the 'naked' ion which acts as a charged completely wetted seed in the e c > e 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

-50

-60 -70

I-.

-80

W* / kT

-90 .

.

.

.

.

-100

n* -110 -120

1

1

....

' .... 50

' .... 1 O0

' .... 150

' .... 200

i .... 250

, .... 300

, ...... 350 400

Fig. 21.2

Dependence of the work for cluster formation on the cluster size at e c > e m according to eq. (21.12)for water droplets on unit-charge ions in vapours at T = 293 K and P/Pe= 2.5. The double arrow indicates the height of the nucleation barrier.

difference between the values Wmax and Wmin of the maximum and minimum of W ( R ) or W(n), i.e. we have W * = Wmax - Wmin.

(21.13)

In the case of e c < e m, however, just like in HON, W f r o m (21.11) or (21.12) displays only a maximum at the nucleus size n* so that then W* is given by W* = Wmax 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 e c with respect to the cluster size n (or radius R) and using the result in (4.1) leads to (e.g. Volmer [1939]; Hirth and Pound [1963]) R* = 2v0cr/Ap + (1/e c - 1/em)Q2vo/32tC2eoAl u R .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* = (32tcv02cr3/3 A/13)[1 + ( 1 / e c - 1/em) • Q 2 / 4 8 X e o v o c r ( n * + 4zcR~/3Vo)] 3 - 4zcR3/3Vo

(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 A/.t and are the Gibbs-Thomson equations for spherical condensedphase nuclei on completely wetted ions or electrically charged seeds. At Q = 0 or e c = e m they turn into eqs (4.9) and (18.14), respectively, the latter taken

320

Nucleation: Basic Theory with Applications

at 0 w = 0 and R s = R i. The curves in Fig. 21.3 represent the n*(Ap) dependence (21.15) for water droplets in vapours at T = 293 K and supersaturation Ap = k T In S. The parameter values used are those for Fig. 21.2. Curve ' H O N ' refers to the case of Q = 0 and R i = 0 corresponding to HON, and curve 'ion' is for nuclei on ions. This curve thus illustrates eq. (21.15) in the e c > e m case. As seen from Fig. 21.3, at a given sufficiently low value of A/,t > 0 there are two n* values satisfying the Gibbs-Thomson equation (21.15) at Q ~ 0 and e c > e m (the same is true for R* from eq. (21.14)). While the greater n* value gives the number of molecules constituting the nucleus, the smaller one represents the number nmi n of molecules of the stable cluster, i.e. of the 'jacketed' ion. As already noted, this means that in the e c > e m case nucleation on ions is analogous to HEN of cap-shaped clusters with negative line energy (cf. curves ' - 2 0 ' 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 Rmi n and the size nmi n of the stable cluster which exists in the case of e c > e m can be determined from eqs (21.14) and (21.15) upon replacing R* and n* by Rmi n and nmin: Rmin = 2Vocr/A p + ( ] / e c - 1/em)a2vo/32zC2eoAp R3in

(21.16)

nmi n = (32zcv 2 o'3/3Ap3)[ 1 + (1/e c - 1/e m) X

Q2/48~eoVoO(nmin + 4zcR3/3Vo)] 3 - 4zR.3,/3Vo .

400

300

t,.-

200

100

1

2

3

4

5

(

S Fig. 21.3

Dependence of the nucleus size on the supersaturation ratio: curve ' i o n ' eq. (21.15) at ec > era for nucleus droplets of water on unit-charge ions in vapours at T = 293 K; curve ' H O N ' - the corresponding Gibbs-Thomson eq. (4.10)for homogeneously formed nucleus droplets.

Electric field

321

In the e c < e m case, however, the course of the n*(A/~) or R*(Ap) 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/~ just as it does in HON (see curve ' H O N ' 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 e c > e m case, as required by eq. (21.13), we have to find Wmax and Wmi n by setting R = R* and R = Rmi n in (21.11) and using R* and Rmi n from (21.14) and (21.16). After some algebra we obtain (Q ~ 0, e c > e m) W* = (47r/3)o'(R . 2 -

R2in)+ (1/e c -

1/em)(Q2/6ZCeo)(1/Rmin- l/R*). (21.17)

In a similar way, in the e c < e m case, from eqs (21.11) and (21.14) we get

W* = 47rR3A~/3Vo + (47r/3)o'(R . 2 - 3Ri2) + (1/e c - 1/em)(Q2/6ZCeo)(3/4Ri- l/R*),

(21.18)

since now W* = W(R*). Equations (21.17) and (21.18) represent the Akl dependence of W* only implicitly through R* and Rmi n f r o m ( 2 1 . 1 4 ) 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 ec and e m. When e c > e m, the field stimulates the nucleation process - then the Q term in (21.17) is negative. The opposite is true when e c < e 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 ~u from (18.13). This has to be so because the ions are treated as completely wetted seeds with radius R s = R i. Naturally, at Q = 0 and R i = 0, W* from (21.18) becomes identical with W* from (4.11) for H O N of spheres. In the particular case of e c >> e 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 til~(n) _ ~el(n) in (21.2) plays the role of a Ap-independent t/J(n) in (3.86). When e c < e m, 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 e c > e m case, however, W* from (21.17) obeys the nucleation theorem in the form of eq. (5.21) with An*/(1 - ,Oold/Pnew ) = n* - nmin. This is so because in this case W* is given by (21.13), and for Wmax and Wmi n eq. (5.29) applies in the form dWmax/dA~ = - n* and dWmin/dA/l = - nmi n. Using W* from (21.17) and (21.18), we can now determine the stationary rate Js ( m-3 s-l) of HEN of condensed phases on ions with concentration C i (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 C O = C i so that with the help of (13.39) and (13.40) we find that

322

Nucleation: Basic Theory with Applications

Js = z f * C i exp {- ( 4 g c r / 3 k T ) [ R * 2 ( A p ) -

R2in(At./)]

- (lIE c - l / e m ) ( Q 2 / 6 ~ e o k T ) [ 1 / g m i n ( A ~ )

- 1/g*(A/.t)] }

(21.19)

in the case of e c > e m and that Js = z f * C i exp { - ( 4 g R 3 / 3 v o k T ) A p

- (4rccr/3kT)[R*2(Ap) - 3R 2]

- (1/e c - 1/em)(Q2/6ZceokT)[3/4Ri - 1 / R * ( A p ) ] } when e c < e m. At e m = 1 eq. (21.19) is the formula of Russell [1969]. The Zeldovich factor z and the frequency f* of monomer attachment to the nucleus change only weakly with n* and W* and can be regarded as practically independent of both Q and R i. According to eq. (15.73), this means that the nucleation time lag 7r 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(Ap) 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 Js(Ap) 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 Rmi n from (21.14) and (21.16), and the kinetic factor A , z f * C i is expressed in the form o f A ' S (cf. eqs (13.40)-(13.42)). The factor A - Zfe ~ C i is assumed to have the exemplary value of 1 0 1 6 m - 3 s -1 corresponding to C i = 109 m -3 and typical z = 0.1 and f * = 108 s-1. The used values of Q, R i, e c, e m, v 0, cr and Pe 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(Ap) dependence for HON under the same conditions is displayed by curve 'HON' 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 Js 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 C i is much lower than the concentration C O 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 Js(Ap) 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 e c > e m, using eq. (14.4) with An*/(1 - Pold/Pnew) = n* - nmi n and n A = 1 allows a model-independent determination of the Ap dependence of the difference n* - n m i n 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

10 20

10 ~5

ion /

"7 I v

E

/ HON

1010

U')

10 5

1

2

3

4

5

S Fig. 21.4 Dependence o f the stationary nucleation rate on the supersaturation ratio: curve ' i o n ' - eq. (21.19) at e c > e m f o r H E N o f water droplets on unit-charge ions in vapours at T = 293 K; curve ' H O N ' - t h e corresponding eq. ( 1 3 . 6 6 ) f o r homogeneously f o r m e d water droplets.

the Gibbs-Thomson equation (21.15). In the opposite case of e c < e 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 i 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 Nucleation 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 Oel which enters (21.2) with a plus sign. The electric field E 2 inside a spherical body in uniform external field E is also uniform and given by (e.g. Landau and Lifshitz [ 1982]) E2 = [3em/(e c + 28m)]E.

(21.20)

Substituting E 2 from (21.20) in (21.5), setting E 1 = E and accounting that E 2 = E 2 readily yields ~el(n) = [3E0em(em - ec)/2(Ec +

2Em)]E2Vn

(21.21)

where E (V/m) is the strength of the externally applied electrostatic field. This expression for q~elwas obtained by Isard [ 1977] (see also Cheng [1984]; Warshavsky and Shchekin [1999a, b]). It corrects the similar formula for q~el [Kashchiev 1972a, 1972b], in which the factor 3 is absent and e m - e c enters as e c - e m because of inaccurate choice of the limits of integration in the general eq. (21.3). As seen from (21.21), q~el < 0 when e c > e 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 Vc of a parallel-plate capacitor held at constant potentials q~l and ~02 of its plates. Electrostatics tells us that for such a capacitor, if C (F) is its capacitance, E . D = C(q~2 - q~l)Z/Vc inside and E = 0 outside, provided edge effects are negligible. Hence, since E 1 = E 2 = 0 outside the capacitor, the integration in eq. (21.3) is in fact only over the volume of the capacitor. Taking into account that q~l and q~2 are the same before and after the cluster formation, neglecting the possible change of Vc and setting E 1 9D 1 = Cl(q~2 - qh)2/Vc and E 2 9D 2 = C2(~02 - qh)2/Vc leads to ~ e l = (1/2)(C2 - C1)(q~2 - q~l )2"

(21.22)

Now, as the capacitance C 2 in the presence of a small enough spherical cluster in the capacitor is related to the capacitance C 1 of the cluster-free capacitor by [Landau and Lifshitz 1982] C2 = C 1 _

3 e 0 e m ( 8 m - ec)Vn/(8 c + 2em)d 2,

(21.23)

for q~el from (21.22) we get ~el(n) =-[380em(e m -ec)/2(e c +

2em)]E2Vn

(21.24)

Electric field 325

where E = (r - q91)/dc is the strength of the uniform electric field between the plates of the capacitor, and d c is the distance between them. This equation shows that ~el > 0 when e c > em, which means that the electrostatic energy is increased by the cluster formation. This is in contrast with q~el for a system with fixed charges (cf. eq. (21.21)). The absolute value of q~el from (21.24) is, however, precisely equal to that of ~el from (21.21). We can now substitute in eq. (21.2) either q~el from (21.21) or q~el 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 q~el from (21.21) and (21.24), respectively, we obtain W(n) = - n A p + qg(n) + [3e0em(em - ec)/2(e 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 << d c, since the usage of (21.23) is justified only under this condition [Landau and Lifshitz 1982]). We see also that, like the strain energy ~)str (see Chapter 20), the change q~el in electrostatic energy is a volume-type contribution to W. The proportionality of q~el to V n, however, is a consequence of the assumptions for fixed cluster shape and n-independent cluster dielectric constant ec. 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 ec > e m (then the E term in (21.25) is negative). There is no effect at ec = em, and the process is inhibited for e 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), q~(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 q~(n) is merely equal to the cluster total surface energy 0(n) so that accounting for eqs (3.13) and (3.20) transforms (21.25) into [Isard et al. 1978] W(n) = - ( A p + ceE2)n + (361r)l/3v~/3 crn 2/3.

(21.26)

Here the parameter c e (F 9m 2) is defined by c e = 3e0em(ec -em)VO/2(ec + 2e m)

(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 e from (21.27). Equation (21.26) shows that the change ceE2 in the electrostatic energy per molecule is additive to Ap. Since ceis n-independent, this means that the

326

Nucleation: Basic Theol. with Applications

formation of spherical condensed-phase clusters in uniform electric field occurs at an effective supersaturation Ap + cEE2 which is a function of the field strength E. Depending on the sign of c e, i.e. of the difference e c - e m, this supersaturation can be smaller (when c e < 0, i.e. e c < e m) or greater (for c e > 0, i.e. e c > em) than the actual supersaturation A~. The magnitude of ceE2 even at strong fields of E = 1 to 100 MV/m is relatively small. For example, for HON of water droplets (e c = 80, v 0 = 0.03 nm 3) in vapours (e m = 1) at E - 1 MV/m, with the help of (21.27) we find that CeE2 = 10 - 7 k T at T = 293 K. Hence, in this case the field does not contribute to A/~. If, however, both e m and E are higher, the contribution may be of significance: for metal clusters (e c = oo) formed, e.g. in a medium of e m = 1000, with v 0 = 0.03 nm 3 it follows that CeE2 = 0.01kT 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 Ap and E. Using (21.26) in conjunction with eqs (4.1) and (4.2) yields [Isard et al. 1978] n* = 327ru~cr3/3(Ap + ceE2)3

(21.28)

W* = 167to02o'3/3(Ap + 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 e from (21.27). At E = 0 they pass into (4.10) and (4.11) and show that for c e > 0, i.e. when e c > e m, HON is stimulated by the applied electric field, since then both n* and W* are smaller than at zero field provided A/~ 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 Ap-independence of the sum q~(n) + q~el(n) in (21.2) (this sum is the analogue of q~(n) 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-l) of HON of condensed phases. With the help of eqs (13.39), (13.40) and (21.29) we get (Ap > - c~E 2) Js = z f * C o exp [-B/(Ala + ceE2)2]

(21.30)

where the thermodynamic parameter B is specified by (13.49) with c = (36z01/3 (spherical nuclei) and O'ef = (3" (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/.t is appreciable. The Zeldovich factor z and the frequency f * of monomer attachment to the nucleus can be treated as Eindependent, 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 f * , 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 f * may take over the thermodynamic exponential factor in governing the Js(AP) dependence. Also, the possible dependence of the specific surface energy cr (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 f * is influenced only indirectly by E through n*, the nucleation time lag 7rwill 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~z)1/3. Expressing z and n .1/3 in (15.85) as functions of E 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 ( k T In S > - ceE2) "f = 32UoG2/Ir2T*(kT)2DCeS(ln S + ceE2/kT) 3

(21.31)

where S = C / C e 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 T*, the nucleus specific surface energy cr and/ or the solute diffusion coefficient D are affected considerably by the applied electric field, 7: will depend differently and more strongly on E than as predicted by eq. (21.31). Figure 21.5 displays the Js(AP) dependence (21.30) for HON of condensed metal phase (e c = ,,~) in a dielectric solid solution with e m = 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 Ap = k T In S (cf. eq. (2.14)), the thermodynamic parameter B = 3.73 x 10-42 j2 is calculated from (13.49) with c = (36z01/3 (spheres), u 0 = 0.03 nm 3 and o'= 10 mJ/m 2, c E is evaluated from (21.27), and the kinetic factor A - zffC 0 is represented as A = A ' S (see eq. (13.41)) with Ap-independent A ' = 3 x 1031 m -3 s-1. According to (13.42), this A" value corresponds to z = 0.1, C O = 3 x 1028 m -3 and f * = 104 s-1 (this rather small value of f * 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 e c > e m, i.e. c e > 0, the field stimulates the nucleation process. In this case the field can even make a saturated old phase (then Ap = 0) nucleate at a certain, though quite low rate. We note that a reliable experimental determination of the nucleus size n* from Js(AP) 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 n*(A/.t) 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 Js(E) dependence (21.30) at S = 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

10 20

1015

15

/0

1.06

1.08

"7

03

!

v

E

10 lo

03

105

1

1.02

1.04

1.10

S Fig. 21.5 Dependence of the stationary nucleation rate on the supersaturation ratio: curves 0 and 15 - eq. (21.30) at ec > em f o r H O N 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 E can stimulate the nucleation process (or inhibit it if e c < em). 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 al. [ 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

1020

1015 _ =,.,

=.,

"2, ~

!

V

E

10 lo

_

.,=

,.,,.

.=

10 5 _

1 0.1

I

I

I

I

,llll

'

I

I,,'

1

~,~1

10

'

'

I

t

t , ~

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 o f e c > em: curves 1.06 and 1 . 0 7 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 Js 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 e c < e m, which is even in qualitative disagreement with theory. This suggests that the influence of the electric field on other parameters (e.g. cr and f*), 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 carder 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~) and n*(A/.t) 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 Js 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 cartier 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 Ap, since A/.t is the major parameter controlling the various characteristics of the nucleation process - nucleus size n*, 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 T 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/-told and ~new are the species 1 chemical potentials in the gas mixture and in the condensed phase, respectively. Hence, in order to determine Ap as a function of p and P we must know separately the Pold(P, P) and Pnew(P, P) dependences. According to the thermodynamics of binary gas mixtures [Guggenheim 1957], Wold can be written down as ~/-/old(P, P) = ]-/e +

kT In (P/Pe) + bll(P + P -Pe)

- (bll - 2b12 + b22)P2/(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), bll (m 3) and b22 (m 3) are the second virial coefficients of the condensing vapours and the cartier gas, respectively, and b12 (m 3) 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 bll = 0). The p, P dependence of Pnew 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, Y-/new(P, P) = ]-/e +

vo(P + p -Pe).

(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]

Ap(p, P) = kT In (P/Pe) + (bll - Vo)(P + P -Pe) - (bll - 2b~z + b22)P2/(P + p).

(22.3)

This formula shows that when both the condensing vapours and the carrier gas behave as ideal gases (then bll = b22 = 0) and when there is no interaction between the molecules of type 1 and 2 in the gas mixture (then b12 = 0), the presence of the carrier gas (P > 0) results in a decrease of the supersaturation with respect to its value Ap(p, 0) at P = 0. Indeed, then eq. (22.3) reduces to

Ap(p, P) = Ap(p, O ) - voP

(22.4)

where Ap(p, 0) is given by the r.h.s, of eq. (2.7). The same decrease of Ap(p, P) with P is in force when only the condensing vapours behave as ideal gas (then bll = 0), but the gas mixture as a whole is ideal (by definition, for such a mixture bll - 2b12 + b22 = 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 (piPe) + vefP. Here Vef (m 3) is an effective molecular volume, defined by

(22.5)

332 Nucleation: Basic Theory with Applications Vef =

2b12-

(22.6)

b22 - Oo,

which depends on T and takes account of the nature of the carrier gas through b12 and b22. 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 Ap 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 Ap as a function of p and P. As seen from eq. (22.5), the sign of Vef controls the character of the change of A/~ in the presence of carrier gas: v~f > 0 or v~f < 0 leads, respectively, to higher or lower A/I with respect to A/~(p, 0) = kT In (P/Pe). However, since usually I Vef [ < 1 nm 3, only relatively high carrier-gas pressures (e.g. tens or hundreds of the atmospheric pressure) can cause such a departure of A/a from A~ (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 = Pe which are saturated in the absence of carrier gas (then P = 0 and Ap - 0) become either super- or undersaturated in the presence of such a gas, for then P > 0 and A/.t ~ 0. Therefore, the condensing vapours should have equilibrium pressure Pe,P ~ Pe if they have to be in the state of saturation after mixing with carrier gas. The dependence ofpe,p on P is readily obtained from the condition for saturation, A / . / ( p e , p , P) = 0, with the aid of Ap from (22.3) [Kashchiev 1996]"

kT In (p~,p/pe) + -

(bll

(bll

- 2b12 +

-Oo)(P

+

Pe,P- Pe)

bE2)p2/( P + Pe,P) =

0.

(22.7)

This equation expresses only implicitly Pe,P as a function of P for any P >_0. In the practically important case of P >> Pe it can be solved with respect to Pe,p to yield explicitly the dependence of this quantity on P in the approximate form

Pe,P = Pe exp ( - vefP/kT)

(22.8)

which follows also upon using (22.5) in the saturation condition A~ (Pe,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 p~,p 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 Pe,P = Pe at P - 0. When there are no interactions between the gas molecules (bll - b12 = b22 = 0) or when only the condensing gas behaves as ideal (bll = 0), but the gas mixture itself is ideal ( b l l - 2b12 + b22 = 0), owing to eq. (22.6) Oef = - 0 0 and both eqs (22.7) and (22.8) take the form of the known dependence ofpe,p on P [Glasstone 1956]. It must be noted that Pe,p is a relatively weak function of P because of the rather small absolute value of Oef. For example, with Vef -" -- O 0 = -- 0 . 0 3 nm 3

Carrier-gas pressure 333 for water vapours at T = 293 K in ideal mixture with a cartier gas having the atmospheric pressure P = 0.1 MPa eq. (22.8) predicts an increase ofp~,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" pe,p[pe - 1.08. It is instructive to note as well that combining (22.5) and (22.8) allows representing the supersaturation in the form of A/~ from (2.8): AIu(p, P) = kT In (P[Pe,P)"

(22.9)

This expression is an acceptable approximation even when P < p, for it gives correctly A/.t in the P = 0 limit (provided, of course, the o0 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 carder 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 number n* of molecules in the one-component EDS-defined nucleus of species 1, the nucleation work W* and the stationary nucleation rate Js are given by [Kashchiev 1996] (kT In S > - vefP) n* = 8c 3v02O-ef3/27(kTln S + Uef p)3

(22.10)

S + oefe) 2

(22.11)

W * = 4 C 3 O l O ' e3f / 2 7 ( k T l n

Js = zFC0 exp [- B'/(ln S + vefe/kT)2].

(22.12)

Here S PIPe 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 O'ef and the thermodynamic parameter B" are specified by eqs (4.42) and (13.67) (tree = cr 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 trwith P is small and in what follows we shall treat O'ef and B' 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 p,P-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 - z f * C o can be represented in the usual form A = A'S where A" - Z fe * Co is virtually constant with regard to both p 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 carder 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 P-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 A/~ 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 carder gas, f * depends on P only through n* and, according to eq. (15.73), the nucleation time lag 27is a relatively weak function of the cartiergas pressure. For instance, under conditions of direct-impingement or surfacediffusion control the classical dependence of 7: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 + VefP/kT and in this way we find easily that (kT In S > - oefP) r = 16c(2mo)l/2tyef/lr3/2~*Coi(kT)l/2peS(ln S + vefP/kT) 2

(22.13)

for HON (then Coi = c and O'ef= O') or 3D HEN controlled by direct impingement and that (kT In S >_.- VefP) 27-" 64C3U20"ef3 [9~2 ~*c*(kT)a~2Ie S (In S + VefP/kT) 4

(22.14)

for 3D HEN under surface-diffusion control. These formulae show that the sign of the effective molecular volume Uef governs the effect of the carriergas pressure P on z:. while Uef > 0 leads to shortening of 27with increasing P, when Uef < 0, 27 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 Uef is decisive for the effect of P also on the stationary nucleation rate. Indeed, depending on it, Js can either increase (when vef > 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 carder gas can stimulate the nucleation process at a given temperature and inhibit it at another temperature if at this temperature/3ef has the opposite sign. The more general conclusion is that in a T range in which the Oef(T)P/ kT ratio is considerable with respect to In S, an experimentally obtained Js(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 (l/In 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* = (kT/vef)d(ln Js)/dP

(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/~ = uefdP (see eq. (22.5)) and using the approximations P n*e w = P n e w a n d n A = 0 . A s 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 b22 = 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 supersaturation ratio S, for then n* is greater. Figure 22.1 displays the dependence of Js on S for HON of water droplets at T = 293 K in a carrier gas for which Uef "- 0.1 or - 0.1 nm 3 (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 O'ef = O', C = (36zr) 1/3 and v0 and cr from Table 3.1). The kinetic factor A - z f * C o is represented as A = A'S, and A' is given the value of 5.5 • 1034 m -3 s-l used for curve 'HON' in Fig. 13.5. For comparison, the Js(S) 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 J~ in the P = 0 limit, especially when used for HON. Indeed, as shown by Barrett et al. [ 1993], at low cartier-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 Js 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 Oef 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 Uef "- 0.1 and - 0.1 nm 3, 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 Uef i s , 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

10 20

-0.1 1015

S"

O0

03 i

v

E

1010

"3

10 5

1

2

3

4

5

6

7

8

S Fig. 22.1

Dependence o f the stationary nucleation rate on the supersaturation ratio: curves - 0 . 1 , 0 and 0.1 - eq. ( 2 2 . 1 2 ) f o r H O N o f water droplets at T = 293 K in a carrier gas which is under pressure P = 10 M P a and f o r which vef = - 0.1, 0 and 0.1 nm 3, 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 the experimental findings of Wilemski et al. [ 1992], Wagner et al. [ 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 supersaturations, 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

337

10 20

1015 0.1 '7,

3.5

03 03

'F: 10 l~

-0.1

v

0.1

105

-0.1

1 0.01

0.1

1

10

P (MPa) Fig. 22.2 Dependence o f the stationary nucleation rate on the carrier-gas pressure: curves - 0.1 and 0.1 - eq. (22.12) f o r H O N o f water droplets at S = 3 and 3.5 (as indicated) and T = 293 K in a carrier gas f o r which Vef = - 0.1 and 0.1 nm 3, respectively.

theoretically ruled out. Indeed, if o" and/or f*, i.e. the thermodynamic and/or the kinetic parameters B' and A = zFC0 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

Ap(a, Po) =//old(a, P0) -//new(a, P0) = kT In (a/ae)

(23.1)

where a is the solute activity, and a e is the equilibrium activity at pressure Po. 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 Vs of a solute molecule in the solution and the volume v0 of such a molecule in the nucleating one-component condensed phase allows using the known thermodynamic formulae [Lewis and Randall 1923; Guggenheim 1957]

//old( a, P) =//old( a, P0) +

vs(P- PO)

Pnew(a, P) = Pnew(a, P0) + Vo(P- Po)"

(23.2) (23.3)

With the help of eqs (2.1), (23.1)-(23.3) we thus find that [Kashchiev and van Rosmalen 1995]

Ap(a, P) = kT In (a/ae) + (Vs - Oo)(P- Po).

(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 v0. In the v0 < Vs case Aft increases with increasing P, but when v0 > Vs, higher pressures lead to lower supersaturations. Examples of these two cases are benzene solutions of naphthalene for which V s - v0 = 0.022 nm 3 [Lewis and Randall 1923] and aqueous solutions of CaCO3 for which Vs - v0 = - 0 . 0 7 5 nm 3 [Stumm and Morgan 1981]. At room

Solution pressure

339

temperature, if P0 is equal to the atmospheric pressure, the rather small absolute value of Us- v0 (usually ]us - u0l < 0.1 nm 3) makes the effect of the solution pressure P on Ap 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 Alu(a, P) from (23.4) can be represented in the form of Ap(a, Po) from (23.1), i.e. as [Kashchiev and van Rosmalen 1995]

(23.5)

Ap(a, P) = kT In (a/ae,e),

if the equilibrium activity ae, P of the solute at the new pressure P of the solution is defined by ae,P = a e exp [- (Us - Uo)(P - Po)/kT].

(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, a e and ae, e 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 7r 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 S > - (Us - Vo)(P- Po)/

kT) n* = 8 c 3 u 2 0"ef 3/27[kT In S + (Us - Vo)(P W*

= 4 c 3 u0Cref/27 2 3

P0)] 3

(23.7)

[kT In S + (Us - Uo)(P- P0)] 2

(23.8)

Js = zf*Co exp {- B'/[ln S + (Us- Uo)(P- Po)/kT] 2 }

(23.9)

where the supersaturation ratio S at the pressure P0 is defined by (13.69). The effective specific surface energy O'efand the thermodynamic parameter B' are given by (4.42) and (13.67) (tree = o'for HON) and can be treated as P-independent when the possible or(P) dependence [Rusanov 1967, 1978] is neglected. As in eq. (13.66), the kinetic factor A = zf*Co can be represented in the form A'S where A" - z f * Co 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 f* and f * 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 Ap 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 Vo > 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 Vs - v0 = -0.1 and 0.1 nm 3, respectively. The calculation is done with T = 293 K and with P/Po = 1 (curve 1) and P/Po = 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 c 3 = 367r, cref = cr and the v0 and cr values in Table 6.1), zf*Co is expressed as A'S, and A' is given the exemplary value of 3 x 1035 rn-3 s-1 which corresponds to z = 0.1, f * = 108 s-1 and Co = 3 • 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 Vs- v0 > 0 or Vs- 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 nm 3 (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 Js in less supersaturated solutions is also clearly demonstrated. Concerning the Js(S) and Js(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 S, the formula

n* = [kT/(vs- v0)]d(ln Js)/dP

(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 dAp= (v~ v0) dP (see eq. (23.4)) and, approximately, Pnew * = Pnew and n a = 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

10 ~5 100

"7 oo 03! v

E

1010

t~

0.1 /

/

/ - 0.1

10 5

1

1

10

20

30

40

50

S Fig. 23.1

Dependence o f the stationary nucleation rate on the supersaturation ratio: curves 1 and 100 - eq. ( 2 3 . 9 ) f o r H O N at T = 293 K o f condensed phase in a liquid solution f o r which Vs - Vo = - 0.1 and 0.1 nm 3 (as indicated) and which is under pressure P = Po and l OOPo, respectively, Po 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 S and P. It remains now to see what is the effect of P on the nucleation time lag 7:. The conclusion based on the general formula (15.73) is that ~: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 f*, we can determine the S, P dependence of "rby employing eqs (13.36), (23.4), (23.7) and (23.8) to express z and n* in (15.85) and (15.87) as functions of S and

342

Nucleation: Basic Theory with Applications

1020

1015

0.1 30

-0.1

C" 03

'E 101~ t~

0.1 20

105

1 0.01

-0.1

0.1

1

10

100

P (MPa) Fig. 23.2 Dependence o f the stationary nucleation rate on the solution pressure: curves - 0 . 1 and 0.1 - eq. (23.9) f o r H O N o f condensed phase at S = 20 and 30 (as indicated) and T = 293 K in a solution f o r which v s - Oo = -0.1 and 0.1 nm 3, respectively.

P. We can, however, avoid the tedious algebra, if we replace In S by In S + (Vs - Vo)(P- Po)/kT directly in eqs (15.86) and (15.88). We thus find that (ln S _>- (Vs - Vo)(P- Po)/kT)

= 16C2Ootref2/3 I~5/2~'*cvD1/2(kT) 2 DCeS[ln S + (Os - Vo)(P

- Po)/kT] 3

(23.11) for H O N (then cVD = c and O'ef = O') or 3D H E N under volume-diffusion control and that (ln S >__- (Vs - Vo)(P - Po)/kT)

"c = 16cdocref/~ClTkTDCeS[ln S + (Vs - Oo)(P- Po)/kT] 2

(23.12)

Solution pressure 343

for HON (then C and O'ef (3") 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 v0 > Vs) nucleation time lag. The above general results for the effect of P on Ap, n*, W*, Js and T find 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]) r

--

"-

P = P 0 - 2Crsol cos Ow,soi/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, CrsoI (J/m 2) is the surface tension of the solution/fluid interface, and 0w,sol is the wetting angle. It characterizes the wetting of the pore walls by the solution: 0w,sol= 0 and 0w,sol= nr correspond to complete wetting and non-wetting, respectively, 0w,sol between 0 and Jr/2 results in a concave meniscus, and when 0w,~ol is between ~r/2 and Jr, 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 nonwetting (then cos 0w,sol = -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 0w,sol> 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,sol< 0, P increases above P0 with diminishing r. The case of bulk solution, the pressure of which equals the pressure P0 of the fluid contacting the pore or the droplet, is described in the limit of r ~ o,,. We can, therefore, refer to P0 as the pressure of the bulk solution and use the r -+ oo 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 A/~, ae, P, n*, W*, Js and ~rare 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] (In S > - ro/r) Js = z f * C o exp [- B'/(ln S + r0/r)2].

(23.14)

The kinetic factor A - z f * C o can again be represented in the form A ' S with practically S, r-independent A' - z f * Co, and the characteristic 'radius' r0 is either positive, zero or negative, since it is given by

344

Nucleation: Basic Theory with Applications

ro = 2 ( v 0 - Vs)O'sol cos

(23.15)

Ow,sollkT.

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 ~ oo 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

1023 10 22 1021

S t,D

m

10 20

'EE 1019

bulk

'

1018 1017 1016

1015 0.01

0.1

1

r (l m) Fig. 23.3 Dependence o f 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 ' b u l k ' - eq. (23.14) for HON under the same conditions in the corresponding bulk solution.

Solution pressure

345

by cos 0w,sol> 0 (pore with solution with concave meniscus) and v0 > Vs, and the r0 < 0 one by cos 0w,sol = - 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 r 0 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 S = 50 and r 0 = 3.6 nm for the pores and r 0 = - 3.6 nm for the droplets. These r 0 values are illustrative for aqueous C a C O 3 solutions in contact with air, since they are calculated from eq. (23.15) with T = 293 K, Vs - v0 = - 0.1 nm 3, Crso1 = 73 mJ/m 2 and cos 0w,sol = 1 for the pores and cos 0w,so~ = - 1 for the droplets. In the calculation, the kinetic factor A - zf* Co 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 < Vs 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 A/~ > 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/~ the old phase is held long enough at a fixed under- or supersaturation Ap0 < Ap. If Ap0 ---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, Ap0) corresponding to Ap0 (see Section 7.1). When Ap0 > 0, however, the old phase is presupersaturated and this concentration is described by the respective stationary cluster size distribution X(n, Ap0) considered in Section 13.1. In this case Ap0 must be sufficiently low, as only then can virtually no nucleation take place before putting the old phase at the working supersaturation Ap. 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)

y,(O)

= Zn, 0 -

Xn,

(24.1)

M-1

]E cia,i = Zn,o- X,.

(24.2)

i=2

Hence, the sought non-stationary cluster size distribution Z,,(t) coincides with Zn(t) from (15.18), i.e. we can write again (n = 2, 3 . . . . . M - 1) M-1 Zn(t) = X n +

]~ ( d i / d ' ) ani i=2

exp ( - ~i t)

(24.3)

where ~i, ani and d' are specified by eqs (15.10)-(15.12) and (15.16), and X, is the stationary cluster size distribution (13.16). The only parameter affected by the non-zero initial cluster size distribution Z~.0 is di: now it is given by the equation a22

a23

"'"

a2,i-i

Z2,0

X2

""

a2,M-i

a32

a33

"'"

a3,i- 1

Z3,0 - X3

"'"

a3,m- 1

aM-l,2

aM-l,3

-

d i -... aM_l,i_

1 ZM-I,O-

XM_ 1 ... aM_I,M_ i

(24.4) which is a generalization of (15.17) and, as required, passes into it in the limit of Z,,,0 = 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 fn and gn and on the concentration Zn,0 of pre-existing clusters only when M = M e f __. 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, O) = Zo(n)

(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)

y(n, 0) = [Zo(n) - X(n)]/C(n)

(24.6)

348 Nucleation: Basic Theory with Applications oo

C(n) Y~ ci ai(n) = Zo(n) - X(n).

(24.7)

i=1

Consequently, the sought non-stationary cluster size distribution Z(n, t) is of the form of Z(n, t) from (15.44) (1 _< n <_.M): oo

Z(n, t) = X(n) + C(n) Y~ ci ai(n) exp (- ~it).

(24.8)

i=1

Here ~i and ai(n) are again the eigenvalues and the eigenfunctions of the Sturm-Liouville equation (15.38) with boundary conditions (15.39), and C(n) 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 ci in (24.8) is defined by the equation ci =

[Sa

C(n) a2i (n) dn

]lsl

[Z0(n) - X(n)] ai(n) dn

(24.9)

which turns into eq. (15.43) at no pre-existing clusters (then Zo(n) = 0). We note also that eq. (24.8) is analogous to the Z(n, t) formula of Shizgal and Barrett [19891. Although Z(n, 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 ~i, ai(n) and ci. To express Z(n, t) from (24.8) in a physically instructive form we must, therefore, resort to approximate results for these quantifies. For example, we can substitute ;ti and ai(n) from (15.50) and (15.51) into eq. (24.8) in order to approximate Z(n, 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 ni' and from M to n~, with the help of ai(n) from (15.51), the approximation C(n) = C* (see eq. (7.37)) and the truncated Taylor expansion Zo(n) - X(n) = (Zo* - X*) + (Z;* - X'*)(n - n*)

(24.10)

we find from (24.9) that ci is given approximately by ci = (2/izr) [(Z~/X* - 1) sin 2 (ira'2)

+ (4/zr 1/2) (Z(3*/X'* - 1 ) cos 2 (i~2)]

(24.11)

where Z~ - Zo(n*) and Z~* - (dZ0/dn), = ,,. To a good accuracy, from (13.25) for X'* =- (dXldn)n = ,, we have X' * = -(fl/ zrl/2)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

Z(n, t) = C(n)[ 1/2

([3]~1/2)(n

-

349

- n*)]

+ (2/zc)C(n) ]E ( I / i ) [ ( Z ~ / X * - 1) sin 2 (izr/2) i=1

+ (4]~l/2)(Z~*]X

"* -

1) COS 2

(DZ:/2)]

• sin [(ixfl/4)(n - n* + 2/fl)] exp (- izt/4"c).

(24.13)

Like eq. (15.53) into which it passes when Zo(n) = 0 (no pre-existing clusters), this expression is applicable only for In - n* [< 7rl/Z/4fl, i.e. only for clusters of near-nucleus size. As it is most accurate just for the nuclei, upon setting in it n = n*, accounting for (13.23) and rearranging, we find that the time dependence of the non-stationary concentration Z*(t) -- Z(n*, t) of nuclei is given by oo

Z*(t) = X* { 1 + (4/to) (1 - Z~/X*) Z, [(- 1)//(2i - 1)] i=1

exp [- ( 2 i - 1)2t/4~]}.

(24.14)

Equivalently, this equation can be represented as (cf. eq. (15.56)) oo

Z*(t) = Z0* + 2 ( X * - Z ~ )

]E (- 1)i-1{1- erf [ ( 2 i - 1)x'cl/z/2tl/2]} (24.15) i=1

so that, analogously to (15.57), for t < 1:

Z*(t) = Z o + (X* - Z~)(16t/zt3"c) 1/2 exp (- n'zv/4t).

(24.16)

Equations (24.14)-(24.16) show that while initially (at t = 0) Z* = Z~, in the t ~ oo 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 Z~ = 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 Z ~ / 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 Zn,o -- 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 Ap at t = 0 the system is held long enough at a fixed Ap0 _<Ap. As already noted above, in this case we shall have [Kashchiev 1969c; Kashchiev and Kaischew 1969]

Zo(n) = C(n, Alto, To) = Co exp [- W(n, Ap0, To)/kTo]

(24.17)

when Ap0 _<0 (then the system is previously undersaturated or saturated) and

Zo(n) = X(n, Apo, To)

(24.18)

when Ap0 > 0 (presupersaturated system). Here To 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~0 > 0, eq. (24.17) is a good approximation for Zo(n) provided n < n~ [Kashchiev 1969c] ( n~ - n*(Ap0,T0) is the nucleus size at A/~0 and To). 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, Ap0,T0) differs less than 50% from the corresponding equilibrium cluster size distribution C(n, Ap0, To). For that reason, hereafter we shall use eq. (24.17) regardless of the sign of Ap0. 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*(Ap/kr-

A/~0/kT0)

+ qJ*(Ala, T)/kT-q~*(Alao, To)/kTo].

(24.19)

This expression takes a very simple form for EDS-defined clusters of condensed phases in all cases when To = T so that the change of A/~0 to Ap is isothermal. Indeed, according to (5.20), for such clusters in either HON or HEN on weakly adsorbing substrates the nucleus effective excess energy t/r* does not depend explicitly on Ap and (24.19) reduces to

Z~/X*

= exp (-

n'Z)

(24.20)

where X > 0, defined by Z = (A/~ -

Apo)/kT,

(24.21)

is the experimentally controllable dimensionless difference between the working and the pre-existing supersaturation (when negative, Ap0 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 Z = In

(P/Po),

Z = In

(I/Io)

(24.22)

X = In (H/H0)

(24.23)

for nucleation in vapours and to X = In

(C/Co),

Z = In

(a/ao),

for nucleation in solutions, Po, I0, Co, a0 and H0 being the pre-existing pressure, impingement rate, solute concentration, solute activity and solute activity product, respectively. Equation (24.20) is free of assumptions concerning ~ * and is therefore applicable to any kind of nucleation (classical, atomistic, 3D, 2D, etc.) when To = T so that Z is varied isothermally. However, with redefined Z it can be used also when To ~ T, but only in those concrete cases of nucleation in which, on the basis of model representation, ~*(Apo, T0) can be treated as practically equal to ~*(Ap, 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 ~*(Ap, T) = CO'ef(T ) v 2/3 ( T ) n .2/3 = (3/2)n*Alu and ~ * ( A p 0 , T 0 ) = CtTef(To) v 2/3 (To)n .2/3 so that under the assumption ~*(Ap0,T0) = qr*(Ap, T) eq. (24.19) leads again to (24.20), but with Z given by

Pre-existing clusters

Z = ( 3 T / T o - 1 ) A p / 2 k T - Apo/kTo.

351

(24.24)

Another example is classical 2D HEN of condensed phases. Then, under the same assumption concerning ~*, for Z we have Z = (2T/To - 1)Ala/kT - Alao/kTo + (aefAt~/k)(1/T - l/T0),

(24.25)

since in conformity with ~(n) from Table 3.3 and eqs (4.32) and (4.33) 9 *(Ap, 7) = n*aefAa + b~cn*1/2 = n*(2Ap -aefAcr) in the cases of both foreign (Aa ~ 0) and own ( A t y - 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) Z from (24.24) becomes (To > T) Z = (Ase/2k)(Te/To- Te/T + 3 - 3T/To).

(24.26)

This formula was used by Kelton et al. [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 (To

_>T) Z = (Ase/k)(Te/To- Te/T + 2 - 2T/To) + (aefA(y/k)(1/T- 1/To)

(24.27)

for classical 2D HEN of condensed phases on foreign (Act ~ 0) or own (Atr - 0) substrates. Equation (24.20) shows that the magnitude of the Z ~ / X * ratio, i.e. the effect of the pre-existing clusters, is controlled by one single quantity, Z, which can be varied experimentally by means of AM0 and/or To in accordance with eqs (24.21)-(24.27) in the respective cases of nucleation at fixed working supersaturation Ap and temperature T. As seen, Z ~ / X * - 1 at Z = 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 Z o / X * -~ O. Actually, however, already Z > 5/n* suffices to practically prevent the pre-existing clusters from playing a perceptible role in the process. For example, when the T and Ap values are such that n* = 50, if Z > 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 e 5In* 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 P0 < p/e 5In* = 0.90p). 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 (Z -> 0) oo

Z*(t) = X*{ 1 + (4Dr)(1 - e -n-x) ~ [(- 1)i/(2i - 1)] exp [- (2i - 1)2t/41r] } i=l

(24.28) with n* Z = 0, 1 and 3 (as indicated). This formula follows from (24.14) and (24.20), and the n* Z values used correspond to Z ~ / Z * = 1, 0.37 and 0.05, respectively. The solid curve represents the Z*(t) dependence (15.54) (or

352

Nucleation: Basic Theory with Applications

1.0

0.8 .It

X

0.6

"1{

N

,//7

0.4 -

0.2 - ~ .

0

,

I

2

,

,

.

I

.

.

4

,

I

,

9

6

.

I

8

,

.

.

10

tlx Fig. 24.1 Time dependence of the non-stationary concentration of nuclei: solid line eq. (15.54) for no pre-existing clusters; dashed lines O, 1 and 3 - eq. (24.28) for preexisting clusters at n*Z = O, 1 and 3, respectively.

(24.28) at n* Z = ~) describing non-stationary nucleation at no pre-existing clusters. We see that, as already pointed out, only sufficiently small n* Z values (n*X < 5) result in concentration Z~ of pre-existing nuclei, which is high enough ( Z~ > 0.0 IX*) 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 ~ is replaced by the delay time 0 in the way discussed in Section 15.6. For instance, eq. (24.14) becomes (cf. eq. (15.117)) oo

Z*(t) = X*{ 1 + (4/7r) (1 - Z ~ / X * ) ]~[(- 1 ) i / ( 2 i - 1)] i=1

exp [- (2i - 1)21t2t/240]}

(24.29)

with Z ~ / X * specified by (24.19) and (24.20) in the cases considered above. When 0 is regarded as expressed by (15.105), eq. (24.29) is an equivalent representation of (24.14) or (24.28). If, however, 0 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 Ap0 is sufficiently less than A/I, eqs (24.19) and (24.20) are better approximations for the Z ~ / 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 Ap0 -<0 (previously undersaturated or saturated system), the expression Z ~ / X * = 2e -n*z 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 -'*z instead of e -n*Z when Ap0 is not too close to Ap.

24.2 Non-stationary rate of nucleation Following the procedure in Section 15.2, we can now determine the nonstationary 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 = n*) and (24.3) we find that J(t) is again given by eq. (15.59), i.e. by (n* = 2, 3 . . . . . M - 1) M-1

J(t) = Js + E (di]d')(J~an,i - gn* + lan * + i=2

1,i) exp

(- ~it),

(24.30)

but with d i 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 *

Y c i a ; * exp (- ~i t)

(24.31)

i=1

where now ci is determined by (24.9). Equations (24.30) and (24.31) in which Js 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/-dependent quantities in them. For example, using 2i, ai(n) and ci from (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) = Js[1 + 2(1 - Z~*/X*) ]~ (- 1) i exp (- i2t/v)]

(24.32)

i=1

or, equivalently, as (cf. eq. (15.65))

J(t) = Js {Z~*/X'* + (1 - Z~*/X'*)(4tcv/t) 1/2 e~

E exp [ - ( 2 i - 1)2nr2v/4t]}.

(24.33)

i=1

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 < 7rthe sum in (24.33) converges rapidly so that, initially,

J(t) = Js[Z~*/X'* + (1 - Z~*/X'*)(4z'c/t) 1/2 exp (- ~2T/4t)]. (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~*/X'* of

354

Nucleation: Basic Theory with Applications

the derivatives (at n = 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/~ so that even initially the nucleation rate is not zero. According to (24.34), the initial nucleation rate J(0) is given by

J(O) = (Z~*/X'*) Js.

(24.35)

Naturally, the stationary nucleation rate Js is not affected by the pre-existing clusters" from (24.32) we see that J -o Js for t ---) oo. As it has to be, in the case of no such clusters (then Z~* = 0) eqs (24.32)-(24.34) turn into eqs (15.64)-(15.66), and J(0) = 0. Equations (24.30)-(24.35) are valid for whatever pre-existing cluster size distribution Zo(n) and for any kind of nucleation (classical, atomistic, HON, HEN, etc.). We shall now consider the case of Zo(n) given by (24.17) or (24.18), assuming again that eq. (24.17) can be used regardless of the sign of A/~o (this assumption implies that A/~o is sufficiently less than A/~). Thus, with the help of (4.6), (7.44), (24.12) and (24.17) we find that (Apo < A~)

Z~* /X'* = (Irl/2/flkTo)[dW(n, A/~o, To)/dn]n =n, • exp [- n * ( A p / k T - Al~o/kTo) + t/r*(A/.t, T ) / k T - q~*(A/~0, To)/kTo].

(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(n, A/~0, To)/dn]n- ,,. = - A/~0 + A/~, and the exponential factor is given by the r.h.s, of eq. (24.20). Hence, eq. (24.36) takes the form (Z > 0)

Z(~*/X'* = (zcl/2ZI fl) exp (-n* Z)

(24.37)

where Z is specified by (24.21) in general and by (24.22) and (24.23) in particular. Substitution of this expression for Z~*/X'* into eq. (24.32) yields (Z > 0) [Kashchiev 1969c] oo

J(t) = Js { 1 + 211 - (tcl/2Z/fl)e - ~*z] E (- 1)i exp (- i2t/r) }.

(24.38)

i=1

Accordingly, from (24.35), for the initial nucleation rate J(0) we get (Z > 0)

J(O)

=

(lrl/2z/fl)J s e -,~*x.

(24.39)

The important point with eqs (24.37)-(24.39) is that in them Z cannot assume its limiting value of zero [Kashchiev 1969c]. Actually, in them Z is restricted by the condition Z > 1/n*, since only then does Z~*/X'* from (24.37) increase with decreasing Z. This restrictive condition for Z, like its stronger version [Kashchiev 1969c], reflects the fact that eq. (24.36) is an acceptable approximation for the Z(~*/X'* ratio only when A/~0is sufficiently less than Ap, e.g. when the system is previously undersaturated or saturated

Pre-existing clusters

355

(then Ap0 < 0). Experimentally, however, of greatest interest are the smallest Z values" from Section 24.1 we know that the smaller Z, the greater the role of the pre-existing clusters. To avoid the inconvenience with respect to the above restriction for Z we can trade accuracy for generality by neglecting the pre-exponential factor in (24.37) and representing this equation as (Z > 0) Z~*/X'* = exp ( - n ' Z ) .

(24.40)

Hence, eqs (24.38) and (24.39) can be used in the less accurate, but more general form (Z > 0) oo

J(t) = Js [ 1 + 2(1 - e - n'x) ]~ (_ 1)i exp (- i2t/v)]

(24.41)

i=1

J(0) = Js e - ,,*z.

(24.42)

Equations (24.40)-(24.42) have another important advantage over eqs (24.37)-(24.39). Like (24.20) and (24.28), with Z defined by (24.24)-(24.27), eqs (24.40)-(24.42) are applicable also when To ~ T provided the temperature dependence of the cluster effective excess energy ~ is negligible so that t/r*(Ap0, To) = t/)*(Ap, T). These equations show again that the effect of the pre-existing clusters is strongest at Z = 0 (then Zo(n) = X(n, A p , T) and, expectedly, J(t) = J(0) = Js) and that this effect is of practically no significance when Z > 5/n* (then J(0) < 0.01Js, 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 1.0 0.8 i

//

z z~

ii I

0.6

II

i/

sI

I/

st

0.4

i/

0.2 - : ' ~ r

. . . .

1

I

2

. . . .

I

3

. . . .

|

4

. . . .

5

Fig. 24.2 Time dependence of the non-stationary rate of nucleation: solid line eq. (15.64)for no pre-existing clusters; dashed lines O, 1 and 3 - eq. (24.41)for pre-existing clusters at n* Z = O, 1 and 3, respectively.

356 Nucleation: Basic Theory with Applications

n*Z = 0, 1 and 3 (as indicated). The solid curve represents the limiting J(t) dependence (15.64), i.e. (24.41) at Z = o o . 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 v with the delay time 0. Analogously to (15.118), we can therefore write oo

J(t) = Js [1 + 2(1 - Z ~ * / X ' * ) E (-1) i exp (-i2rc2t/60)]

(24.43)

i=1

where Z~*/X'* is given by (24.36), (24.37) or (24.40) in the cases noted above. This formula is equivalent to (24.32) when 0 is expressed through ~: by means of eq. (15.105), but can be used also with 0 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 ~'(t) of supernuclei in the system. Since we already know the non-stationary nucleation rate J(t), for the determination of the ((t) 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) M-1

((t) = ~0 + Jst + ~, (di/~,id')(f*an,i - g,* + la,, + 1,i)[1 - exp (- Zit)] i=2

(24.44) when n assumes only integer values and oo

((t) = ~0 + Js t - f ' C *

~, (cia;*]/],i)[1 - e x p

i=1

(-

~it)]

(24.45)

when n is treated as a continuous variable. These equations represent the exact ~'(t) dependence for nucleation at non-zero concentration of pre-existing clusters whose effect is taken into account by the quantities d i, c i and ~'0 from (24.4), (24.9) and (24.46) or (24.47). When no such clusters are present in the system (then Zo(n) = 0 and (0 -- 0) eq. (24.44) coincides with (15.112). We note also that eq. (24.45) is similar to the ~'(t) formula of Shizgal and Barrett [1989]. As seen from (24.44) and (24.45), while at t ~ oo the concentration of supemuclei increases again linearly with time (cf. eq. (15.112)), initially, it does not vanish" ((0) = ~0 ~ 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

~0 =

~2 Z,,,0 lz=?/*+l

(24.46)

Pre-existing clusters

357

or by

~o =

Zo(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 ~'(t) formulae. One such formula can be obtained either from (24.45) with the help of ~i, ai(n) and ci from (15.50), (15.51) and (24.11) or by integrating J(t) from (24.32) as required by (11.10). The result is ( ( t ) = (0 + Js { t - (/1:2/6)(1 - Z;*/X'*) "r oo

- 2 (1 - Z6*/X'*) v ~, [(- 1) i/i 2] exp (- i2t/'c)}

(24.48)

i=1

where ~'0 is given by (24.47). Equivalently, we can write (cf. eq. (15.115))

((t) = (o + (Z~*/X'*) Jst + 2(1 - Z~*/X'*) J~ oo

• ]~

[(4/t't/'t') 1/2 exp [- (2i -

i=1

1)2/r2"t'/4t]

1)wcl/Z]2t 1/2] } ]

(24.49)

3/2 exp (- lr 27:/4t).

(24.50)

- (2i - 1 ) ~ { 1 - erf [(2i so that for t < < 1: we have

( ( t ) = (o + (Z~*/X'*) Jst + 8(1 - Z~*/X'*) Js "r

As it has to be, in the case of no pre-existing clusters (then Z~*= 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 AM 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 Z~*/X'* 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 (Z > 0)

~(t) = ~o + Js{ t - (zt2/6)(1 - e - n*Z)'t" oo

- 2(1 - e-n'z)1: E [(-

1)i/i2]

exp (- i2t/~) }

(24.51)

i=1

where Z is defined by eqs (24.21)-(24.27) in the respective cases of nucleation. Similarly, employing Z~*/X'* from (24.37) in (24.48) leads also to ((t) from (24.51), but with the quantity rcl/2Z/fl as a multiplier of e -'*z [Kashchiev 1969c]. Though more accurate than (24.51), this (Tr1/2Z/fl)-containing formula for ~'(t) has the disadvantage of being applicable only in the range of Z > 1/ n* and when To = T.

358 Nucleation." Basic Theory with Applications

To reveal fully the effect of the pre-existing clusters on the ~'(t) dependence in the case when Zo(n) is specified by eq. (24.17) or (24.18), it remains now only to determine the initial concentration of supernuclei, ~'o, 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

~o =

C(n, A/t o, To) dn

(24.52)

when A/t0 < 0 and

~o --

X(n, A/to, To)dn ---

C(n, Apo, To) dn

(24.53)

when Ap > 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 Apo is not too close to the higher working supersaturation Ap, since only then is the size n~) of the nucleus at Ap0 sufficiently larger than the nucleus size n* at Ap. 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 ~'0from (24.52) and (24.53) we can invoke the formula [Zeldovich and Myshkis 1972]

Ix

"2f(x) dx = f2(x~)/I (df/dx)x=xl I

(24.54)

1

where f > 0 is an exponentially decaying function ofx between Xl and x2, and x2 is sufficiently greater than xl or, possibly, x2 = oo. Applying (24.54) to both (24.52) and (24.53) leads to the same approximate expression, namely, ~0 = C2(n*, Ap0, To)/l[dC(n, Ap0, To)/dn]n=n,[, so that with the aid of (4.6), (7.44) and (24.17) we obtain (Ap0 < Ap) ~0 = { kTo[[dW(n,

A/t0, To)/dn]n=n, }C*

x exp [- n * ( A p / k T - A/to/kTo) + q~*(A/t, T ) / k T - ~*(A/t 0, To)/kTo] (24.55) where C* is the equilibrium concentration of nuclei at the working supersaturation Ap. This general approximate formula for ~'0 is valid regardless of the sign of A/t0 as long as Ap0 is sufficiently less than A/t. 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 q)*(Ap0, To) = ~*(A/t, T) and [dW(n, A/t0, To)/dn]n=n* = A/t - A/t0 (see Sections 24.1 and 24.2), eq. (24.55) takes the form (Z > l/n*)

Pre-existing clusters

~o = (1/Z) C*e-n*z

359

(24.56)

where Z is given by (24.21) in general and by (24.22) and (24.23) in particular. When To ~ T, again for EDS-defined condensed-phase clusters, eq. (24.55) leads to (A~o < A~, Z > l/n*) ~o = [kTo/(A/~ - Al~o)]C*e -~*z

(24.57)

in all cases of classical HON, 3D HEN and 2D HEN in which the approximation qr*(A/~0, To) -- qr*(A/l, T) is acceptable. Here Z is defined in general by (24.24) or (24.25) so that in the concrete case of A/~0 and A/~ related to To and T by (2.23) the above formula becomes (To > T, Z > l/n*)

~o = [kTo/Ase(To- T)]C*e -n*z

(24.58)

where Z is expressed in terms of To 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/~ (then n~' ~ n*), eqs (24.56)-(24.58) cannot be used in the important Z ~ 0 limit. To avoid this inconvenience we can unite them into the less accurate, but general formula (Z > 0)

~o = C*e-n*z

(24.59)

in which Z 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 Z is diminished down to Z = 0. In the concrete case of classical HON or 3D HEN either at TO= T or TO~ 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 ~'0 with a reasonable accuracy in the range of Z values of practical interest (e.g. between 0 and 10/n*). For instance, under conditions characterized by Z = 0.079 (calculated from eq. (24.26)) and n* = 23, Kelton et al. [1983] found numerically that ~o = 0.17C* (see their Figs 2 and 11) for HON of crystals in vitrified Li20.2SiO2 melt. This finding for ~'0 agrees well with ~'0 = 0.16C* following from (24.59) with the above values of n* and Z- The accuracy of eq. (24.59), however, cannot always be so good. Indeed, as shown in Appendix A3, the more accurate calculation of ~0 in the case of A/~0 > 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/J0 = A/~ and To = T (then Z = 0), while in accordance with (A3.9) ~'o = (A*/4)C*, from (24.59) it follows that ~'0 = C*. This means that in the Z ~ 0 limit eq. (24.59) tends to underestimate ~'o, since for the width A* of the nucleus region at the working supersaturation A/~ 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 Ap and temperature T. This is so because, experimentally, ~'0 is accessible to a direct measurement, and Z can be varied by means of A/~o and/or To at fixed A/~ and T. According to eq. (24.59), (o(Z) data plotted in In (o-VS-Z 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 (0 and the bracketed factors in them. Repeating the experiment at a number of fixed Ap and T values, we can thus obtain experimental n*(A~) and C*(Ap) 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 oo

~(t) = ~o + Js{t- 0p - (12/zr2)0p i~ [(- 1)i/i2] exp ( - i2z~t/60)}. (24.60) Here the quantity 0p(S) is defined by 0p = (1 - Z~*/X'*) 0

(24.61)

with 0 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 and 0p = 0). With 0 specified by (15.105), eq. (24.60) is identical with (24.48), but when 0 is determined from the higher-order approximation (15.111), this equation may provide a better description of the ~'(t) dependence. It is seen also that in the t ~ ~ limit ~'(t) ~'0 from (24.60) becomes asymptotically a linear function of time: -

~'(t) - ~'0 = J~ ( t - 0p).

(24.62)

Comparison of this equation with (15.100) reveals that 0p is merely the delay time of nucleation at pre-existing clusters in the system. We shall examine 0p in the next section, but already now we see from eq. (24.61) that the closer the size distribution Zo(n) of these clusters to the stationary cluster size distribution X(n) corresponding to A/I and T, the shorter 0p. In particular, 0p vanishes when Zo(n) = 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 Z~*/X'* from (24.40). The dashed curves correspond to n* Z = 0, 1 and 3 (as indicated), and the solid curve represents eq. (15.119) (or eq. (24.60) at n*Z = ~) describing the ~(t) 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 nonstationarity: when their size distribution coincides with the stationary one (then Z = 0), the ~'(t) dependence is linear right from the onset of the process. In Fig. 24.4, the ~'(t) dependences resulting from eq. (24.60) with 0p from (24.61) and Z~*/X'* from (24.40) at n*Z = 4.4 (the dashed curve) and

Pre-existing clusters

O0

O I

V

s

0

s,

0

1

2

3

t/o Fig. 24.3

Time dependence of the concentration of supernuclei: solid l i n e - eq. (15.119) for no pre-existing clusters; dashed lines O, 1 and 3 - eq. (24.60)for pre-existing clusters at n* Z = O, 1 and 3, respectively. 1.5

,

i

'

.... '

I

'

'

'

'

I

"

'

'

'

I

'

'

'

'

I

'

'/' .I'~..~.

//-/ //." /" ,o" /~ 9

1.0 '-3 o ~.P

j~ o /O/o~ f" o" /~ 9 f~ .o f" ~ j" o, jOfoO

I

v

0.5

0.5

1.0

1.5

.

2.0

2.5

tie Fig. 24.4 Time dependence of the concentration of supernuclei: dotted and dashdotted c u r v e s - numerical data of Kelton et al. [1983] for isothermal HON of Li20 . 2Si02 crystals in vitrified melt at, respectively, n* Z = (no pr e-existing crystals) and n*Z = 4.4; solid and dashed curves - eq. (24.60) also at n*Z = and n*Z = 4.4, respectively.

361

362 Nucleation: Basic Theory with Applications

n*Z = oo (the solid curve) are compared with the numerical ~'(t) dependences obtained by Kelton et al. [1983] for HON of crystals in Li20 92SIO2 glass also at n*Z = 4.4 (the dot-dashed curve) and n*Z = oo (the dotted curve). Due to the somewhat higher value of 0p predicted by eqs (24.40) and (24.61), eq. (24.60) describes less accurately the numerical data for ~'(t) at pre-existing clusters in the glass.

24.4 Delay time of nucleation The delay time 0p of nucleation at pre-existing clusters is given approximately by eq. (24.61) which is in force for arbitrary size distribution Zo(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 0p when Zo(n) is specified by eq. (24.17) or (24.18). Then, in general, the Z~*/X'* 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 To = T, from (24.61) and (24.37) it follows that (Z > I/n*) [Kashchiev 1969c] Op = [1 --

(lr,l/2~[l~')e-n*X ] 0

(24.63)

where Z is specified by (24.21)-(24.23). Similarly, from (24.61) and (24.40) we obtain (Z > 0) Op = (1 - e -n-z) O.

(24.64)

This formula is less accurate than (24.63) when Z >> l/n*, but has the advantage to hold true in the whole range of Z values down to Z = 0 and to be applicable also at To ~ T. That is why, for Z 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 0p = 0 when Z = 0 (i.e. at To = T and A/~0 = Ap), the reason being that the pre-existing cluster size distribution coincides with the stationary one at the working temperature T and supersaturation A/~. In the other extreme, i.e. at Z = oo (no pre-existing clusters), 0p = 0 according to both (24.63) and (24.64). We note that these equations are liable to a direct experimental verification, since 0 and 0p are directly obtainable from experimental ~'(t) curves (they are the time intercepts of the asymptotic ~'(t) dependences (15.100) and (24.62)), and Z can be varied experimentally by means of A~0 and/or To at fixed Ap and T. As predicted by eq. (24.64), plotting experimental 0p(Z) data in -In (1 - Op/O)-vs-Z coordinates should yield a straight line whose slope is directly equal to the size n* of the EDSdefined nucleus at T and A/~. A similar linearization of the 0p(Z) data is possible also when eq. (24.63) is used for their analysis, since in it/3 is a constant with respect to To and Ap0. With the aid of a family of 0p(Z) data obtained at different fixed T, A/~ values we can thus determine experimentally the n*(Ap) dependence and use it for verification of the respective theoretical

Pre-existing clusters

363

Gibbs-Thomson equation (see Chapter 4). Experiments aimed at obtaining 0p(Z) dependences in various cases of nucleation have yet to be carded out. The effect of pre-existing clusters on the delay time is illustrated in Fig. 24.5 which depicts the 0p(Z) dependence for HON of crystals in vitrified Li20 92SIO2 melt at T = 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 = 4.8k (these are the Te and Ase values used in the study of Kelton et al. [1983]) the values of Z on the Z axis in Fig. 24.5 correspond to previous undercoolings at different temperatures To in the range of T < To < 0.7T~, i.e. to a presupersaturated system. The circle in the figure represents the 0p/0 value of 0.87 following from the numerical data of Kelton et al. [1983] for 0 and for 0p at Z = 0.19 (these authors found that 0 = 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 To = 800 K). As seen, eq. (24.64) predicts with an error of some 13% the magnitude of the 0p/0 ratio at the Z 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 T is lower (then n* is smaller, because the working supersaturation Ap is higher). From this figure we can conclude as well that, since Z > 0.3 when To > Te and T = 660, 750 or 840 K, if the Li20 92SIO2 glass is held previously at saturation or

1.0

__..___.____._

08t t:D

0.6

0.4

0.2

0

0

0.1

0.2

0.3

Fig. 24.5 Dependence o f the Op/O ratio on the thermodynamic parameter Z: circle numerical finding o f Kelton et al. [1983] for H O N o f L i 2 0 . 2 S i 0 2 crystals in vitrified melt at T = 750 K; curves 660, 750 and 840 - eq. (24.64)for the same case o f H O N at T = 660, 750 and 840 K, respectively.

364 Nucleation." Basic Theory with Applications

undersaturation (i.e. at To > Te), 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 ((t). 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 Z~*/X'* ratio. Fortunately, we can obtain also exact formulae for 0p, 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 ---> o,, limit for determination of the asymptotic ~'(t) dependence and to compare this dependence with that from (24.62). Doing that yields (n* = 2, 3 . . . . . M-l) M-I

Op =-(1/d'Js)

]~

i=2

(di/Zi)(j~an,i

- gn* + l a n *

+ 1,i)

(24.65)

when n assumes only integer values and oo

(24.66)

Op = ( f * C * / J s ) ~, (ci]~i)a;* i=1

when n is considered as a continuous variable. In these exact formulae for 0p solely the quantities d i and ci are affected by the initial cluster size distribution Zo(n). As these quantities are given by eqs (24.4) and (24.9), 0p from (24.65) or (24.66) becomes equal to 0 from (15.101) or (15.103) at Zo(n) = 0 (no pre-existing clusters). Also, since di = 0 and ci = 0 at Zo(n) = X(n), the above expressions tell us that 0p = 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 0p 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 0p 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] Op = (l/Js)

-

I1

[X(n) - Zo (n)l dn

[X(n) - Zo(n)][X(n)lC(n)] dn

}

(24.67)

Pre-existing clusters

365

where C(n),X(n) andJs 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] n :'!':

Or, = (1/Js) [ Z (X,,n=2

M-1

Zn,o) -

Z (X n --

gn,o)(Xn/Cn)

].

(24.68)

n=2

Here C,, X~ and Js are given by (12.3), (13.16) and (13.29) as functions of f, and gn. That is why the above two exact formulae unveil the explicit dependence of 0p on the monomer attachment and detachment frequencies and the concentration of pre-existing clusters. As it has to be, at Zo(n) = 0 and Zn,o = 0 eqs (24.67) and (24.68) turn into (15.108) and (15.109), and at Zo(n) = X(n) and Z,,0 = Xn they yield 0p = 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 0p of the kind of the formula for 0. For example, performing the integrations in (24.67) with Zo(n) from (24.17) or (24.18), we could find 0p 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 supernuclei 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 supernuclei formed in the system and the duration of the nucleation process. Indeed, during the process the active centres are progressively exhausted by the supernuclei 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 al. [ 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 supernuclei 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 supernuclei on the centres at time t we have N(t) = ~(t)V

(25.1)

for HEN in the volume V of the old phase and N(t) = ff(t)As

(25.2)

for HEN on a substrate with surface area As. The total concentration ~'(t) of supernuclei 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 Co 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 Js = Ja,sNa/V

(25.3)

for HEN in the volume of the old phase and Js = Ja, sNa/As

(25.4)

Active centres

367

for HEN on a substrate. Here the quantity Ja, s (s-l), given by Ja,s = zf* 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 dN(t) dt = Ja,s[ Na - N(t)],

(25.6)

because at time t not all Na active centres, but only Na - N of them are unoccupied by the N already formed supernuclei and, hence, able to generate new supernuclei. Equation (25.6) is a linear ordinary differential equation of first order with known solution [Korn and Korn 1961 ]. At no pre-existing supernuclei 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]

N(t) = Na [1 - exp (- Ja, st)]-

(25.7)

This formula shows that in stationary HEN on active centres the number of supernuclei in the system increases linearly with time only initially when the exhaustion of the centres is insignificant. Indeed, for t << 1/Ja,s we have

N(t) = Ja,sNat

(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 Js. The effect of the active centres is manifested at longer times (t > 1/Ja,s) when N slows down its rise with t and finally (t ~ ,,~) assumes its limiting value N a. 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 supernuclei formed in the system is merely equal to the number Na of active centres: Nm = Na.

(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) = Ja(t)Na/V

(25.10)

for HEN in the volume of the old phase and as

J(t) = Ja(t)Na/As

(25.11)

for HEN on a substrate, Ja (s-l) 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

Ja(t) = Ja,s [ 1 + 2 ]~ (-- 1)i

exp

(--

i2t/'C)]

(25.12)

i=1

where Ja,s is specified by (25.5), and the time lag Iris given by (15.73). Using eqs (25.1), (25.2), (25.10) and (25.11) in (11.1) we thus arrive again at eq. (25.6) in which, however, Ja,s is replaced by Ja(t):

dN(t) = Ja(t) dt

[Na - N(t)].

(25.13)

The solution of this equation again under the initial condition N(0) = 0 is of the form [Avrami 1939, 1940; Gutzow and Toschev 1971 ] N(t)

= Na

{

1 - exp

-

Ja (t') dt'

]} .

(25.14)

This expression is a generalization of eq. (25.7) and passes into it at Ja(t) = Ja,s = constant (stationary nucleation). We see also that N---) Na at t ---) oo, i.e. again Nm = Na. Employing Ja(t) 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: oo

N(t) = Na{ 1 - exp [ - Ja,s{t- (tr2/6)'r - 2"r ]~ [(- 1)i/i2] exp (- i2t/'O}]}. i=1

(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

N(t) = Na{ 1 - exp [- Nex(t)/Na]}

(25.16)

where

Nex (t)

= Na

Io

Ja(t dt'.

')

(25.17)

Sticking to the terminology of Avrami [1939, 1940, 1941 ], we may call ]Vex

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 Ja(t) = constant = Ja, s). Equation (25.16) tells us that the effect of the active centres on the N(t) dependence is controlled by Nex: while initially Nex < Na so that N(t) --- Nex(t ), at later times/Vex > > Na and N(t) --- N 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 T = 654 K and impingement rate I = 9 • 1015 m -2 s-l. As seen, this dependence is fitted well by eq. (25.7) (the

Active centres

4x1015

'

' ......'

=

=

I

'

'

'

I

,

,

,

I

'

'

I

=

,

'

I

'

'

'

,

I

,

,

,

I

,

,

3x1015 Oq i v

E

2x1015

Z

lx1015

,

200

400

=

I

600

=

,

I

I

800

I

I

I

I

I

l

1000

I

1200

t(s) Fig. 25.1

Time dependence o f the concentration o f supernuclei on active centres: circles - data f o r H E N o f Au crystals on (lO0)MgO substrate at T = 654 K and impingement rate I = 9 x 1015 m -2 s -1 [Robins and Rhodin 1964]; curve - best fit according to eq. (25.7)for stationary nucleation.

'

'

I

'

'

'

i

,

,

,

I

'

'

'

I

,

,

,

I

'

'

'

I

,,

,

,

1.0

0.8

Z Z

0.6

0.4

0.2

0

i

0

, 9

,

20

40

60

80

100

t (h) Fig. 25.2

Time dependence o f the concentration o f supernuclei on active centres: circles - data of Fokin et al. [1997] f o r nucleation o f l.t-cordierite crystals on a polished surface of cordierite glass at T = 1123 K; curve - eq. ( 2 5 . 1 5 ) f o r nonstationary nucleation.

369

370

Nucleation: Basic Theory with Applications

curve) with Na/A s = 3.6 • 1015 m -e 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-Zs -1 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 o f / l cordierite crystals on a polished surface of cordierite glass at T = 1123 K, and the curve is drawn according to eq. (25.15) with 7:= 7.88 • 104 S and J a , s = 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-JohnsonMehl-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 (26.1)

a - Vc/V

of volume crystallized till time t (Vis the initial volume of the melt). Obviously, for a closed system, the direct determination of the Vc(t) function requires finding the crystallite size distribution Zn(t) or Z(n, t) (m -3) as a solution of the master equation (9.4) or (9.10) and using it in the expression M

Vc(t) =

voV

Z

nZn(t ) = ooV

n Z ( n , t) dn.

n=n*+l

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 J(t) (m -3 s-1) which then only expand irreversibly in radial direction with growth rate G(t) (m/s) (Fig. 26.1).

Fig. 26.1 Overall crystallization by the polynuclear mechanism corresponding to appearance and growth of statistically many supernuclei (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 Vn(t', t) of any individual n-sized crystallite depends only on the earlier moment t' < t of its nucleation provided it is additionally assumed that the crystallites are isomorphic during growth. Hence, for the crystalline volume dVc formed between t' and t' + d t ' we can write dVc = Vn(t', t)J(t')V dt"

(26.2)

where, according to (11.1), J(t')V dt' is the number of crystallites formed between t' and t' +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 t 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

dVc = Vn(t', t ) J ( t ' ) ( V - Vc) dt'.

375

(26.3)

Thus, integration of this equation under the initial condition Vc(0) = 0 with allowing for eq. (26.1) leads to a(t) = Vc(t)/V = 1 - e x p [-Vex(t)/V]

(26.4)

where the so-called extended volume Vex [Avrami 1939, 1940, 1941 ] is given by Vex(t) = V

(26.5)

J ( t ' ) V n ( t ; t) dt'.

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 V by 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 V~(t) = V

(26.6)

J ( t ' ) V n ( t ; t) dt'

with an error of less than 10% (this result follows from using the truncated Taylor expansion exp (-Vex/V) = 1 - V~xlV). Naturally, eq. (26.6) is directly obtainable by integration of eq. (26.2). Since the crystaUites 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 t" of its formation. The individual crystallite volume Vn will then be given by Vn(t', t) = cgR d = Cg

(26.7)

G(t") dt"

where d = 1, 2, 3 is the dimensionality of growth, cg (m 3- d) is a shape factor (e.g. Cg = 4 z l 3 for spherical crystallites; see also Table 26.1), t - t" is the period of growth of the crystallite, and G(t) - dRIdt is the crystallite growth rate. Combining eqs (26.4), (26.5) and (26.7) yields the KJMA formula ct(t) = 1 - exp

- Cg

J(t')

t-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 Shapefactor CgfOr 1D growth of needles with constant cross-sectional area A o, for 2D growth of disks or square prisms with constant thickness Ho and for 3D growth of spheres or cubes. Growth

Shape

Cg

1D 2D 2D 3D 3D

needle disk square prism sphere cube

2A0 lrH0 4H0 4~3 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 - t', in the Kolmogorov expression the respective limits are t' and t. As discussed by Belenkii [1980], this difference of (26.8) from the rigorous Kolmogorov expression for a(t) does not matter when G is tindependent, but leads to a certain inaccuracy in the determination of the time dependence of ct 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 [Korn and Korn 1961 ], this means that the average time tav for crystallizing the melt, i.e. the lifetime of the melt in undercooled state, is given by [Belenkii 1980] tav = f : tdot(t) = ~ : exp [- Vex (t)/V] dt

(26.9)

where Vex is specified by (26.5) with Vn 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 dr'. Therefore, using eq. (25.1) in the differential form dN(t') = [V- Vc(t')] d((t'), recalling (11.1) and (26.4) and integrating under the condition N(0) = 0 yields [Kolmogorov 1937] N(t) = V

J(t')[1 - a ( t ' ) ] dt' = V

J ( t ' ) exp [-V~x(t')lV] dt"

(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 Nm 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 = oo [Kolmogorov 1937]:

Overall crystallization

377

too

Nm = V J 0 J(t) exp [- Vex(t)/V] 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/Nmused by Volmer [1939]. With Nm from (26.11) it thus follows that

Vav =

[So

J(t) exp [-Vex(t)lV] dt

}

.

(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 Na (see Chapter 25). Mathematically, therefore, IN is characterized by a J(t) dependence of the form

J(t) = (Nm/V)t~>D(t)

(26.13)

where t~D is the Dirac delta-function. Employing this expression for J(t) in (26.5) and recalling that for the product of t~o(t') with an arbitrary function y(t', t) there holds [Korn and Korn 1961]

~o t~D(t')y(t', t) d t ' = y(O, t), in conformity with (26.7) we get

Vex(t) = Nm Vn(O, t) = cgNm

G(t") dt"

.

(26.14)

This result is obvious: as in IN all Nm 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(t) 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) = VGVctv-1

(26.15)

implying crystallite growth according to the power law R(t) = (Get) 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 [1964]; Lyubov [1969, 1975]; Christian [1975]; Doremus [1985]; Srhnel and Garside [1992]; Mullin [1993]), and Gc(m 1Ivs-1) is obtained by kinetic considerations analogous to those used in Sections 10.1 and 10.2 for finding the monomer attachment and detachment frequencies fn 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 Vex(t ) =

cgNmGVat vcl

(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 [-

(t]lg) vd]

(26.18)

tar = F(1 + l/vd)O

(26.19)

1~ ----(g[cgNm)l/Vd( l/Gc),

(26.20)

where O (s), defined by

is the time constant of the process, and

F(X) = ~0 yx-le-Y dy

(26.21)

is the gamma-function [Korn and Korn 1961]. For 3D growth (d = 3) of spheres (Cg = 4zr/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 t~ 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 J(t) from (26.13) in (26.11) and (26.12) returns the identities Nm = Nm and Vav V/Nm regardless of the Vex(t) dependence, i.e. for whatever crystallite growth law. =

Overall crystallization

#f

lo

L

e~

~

0.61.L L

0.2 LL

II

:

,','

,,'

,' ,'

,'

.."

1

,'

,' ,'

: : :

// L // 0

i

,, ,','

'

II II II //II

L

0

,"

:

1 : ~ , //~ '

!.L

0.4 L L L

,"

379

2

,'

,'

,'

3

4

5

6

tie Fig. 26.2

Time dependence of the fraction of crystallized volume in overall crystallization by the polynuclear mechanism: curve I N - instantaneous nucleation according to eq. (26.18) with vd = 3; curve P N - 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 "c/O = 1 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) = Js, the stationary nucleation rate Js (m -3 s-1) 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 Vex(t) = [Cg/(1 + vd)] GVcdJs tl+vd.

(26.22)

Accordingly, using this result in (26.4), (26.9)-(26.12) yields (e.g. Belenkii

[1980])

vd]

(26.23)

tav = r[(2 + vd)/(1 + vd)] #

(26.24)

or(t) = 1 - exp [- ( t / O ) 1 +

f

tlz9

exp ( - x 1 + vd) dx

(26.25)

Nm = F[(2 + vd)/(1 + vd)]VJsO o: V(Js/Gc) vd/(~ + va)

(26.26)

Vav = l/F[(2 + vd)/(1 + vd)]Jsz9 o: (Gc/Js)Vd/( ~ + vcO

(26.27)

N(t) = VJsz9

,10

380 Nucleation: Basic Theory with Applications

where the time constant t~ is given by t~ = [(1 + vd)/cgGcVdJs] 1/(1§ vd).

(26.28)

When the crystallites are spherical with radii growing linearly with time (then Cg = 4zr/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 0 and tar 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 tav, N m 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 tav and Nm from independent measurements, Js is readily obtainable with the aid of the formula

Js = Nm/Vtav

(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 timeindependent 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 nonstationary 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, J(t) from (15.64) and G(t) from (26.15) yields [Gutzow and Kashchiev 1970, 1971] a(t) = 1 - exp [- (t/va)4y(t/~)].

(26.30)

Here the time-dependent factor Y, a number between 0 and 1, is given by

Y(x) = 1 - 27r2/3x + 7zca/30x2 - 31zr6/630x 3 + 127zr8/25200x 4 oo

+ (48/x 4) ]~ [(- 1)i/i8] exp (i=1

i2x)

(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 vfrom (15.55) and (15.73) on the a(t) dependence. As seen, when "r = 0, Y = 1 and eq. (26.30) passes into eq. (26.23) (with vd = 3) which corresponds to stationary PN. If 9~ 0, however, Y = 0 for t << 9 and the whole process

Overall crystallization

381

of overall crystallization is delayed, the delay being largely determined by "t', 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 a(t) dependence (26.30) at v/va = 1 and 5, respectively. Compared with the solid curve PN which corresponds to stationary nucleation (then v = 0), the dashed curves preserve nearly the same sigmoidal shape, but are significantly shifted towards longer times, the shift being about 0.57:. The short-time asymptotics of the a(t) 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(t[19)a(t]~) 15/2 exp (- ZC2V/4t).

(26.32)

This formula is useful for describing the beginning of overall crystallization, since it gives cr from (26.30) with an error of less than 10% for t < 0.2T provided v/t~ < 103. In the case of non-stationary PN the N(t) dependence and the quantities tav, N m and Vav are obtainable by employing ct(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 t h a t tav 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 qr (see Fig. 26.2), tar c a n be approximated by the formula tav = bpT + F[(2 + vd)/(1 + vd)]O

(26.33)

which passes into (26.24) at ~r= 0, i.e. when nucleation is stationary. Although the numerical factor bp in this formula should depend on the ~/0 ratio, this dependence is relatively weak [Kashchiev 1989a] and in many cases of practical interest the approximation bp = 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 7:is replaced by 60/zr 2, and the delay time 0 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 ct(t) dependence can be represented by the unified KJMA formula ct(t) = 1 - exp [-

(t/Lg) q]

(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 0 is governed by the growth constant Gc and either by the maximum number Nm of crystallites in IN or by the stationary nucleation rate Js in PN (see eqs (26.20) and (26.28), respectively). Usually Js 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 Applications

the nucleation process, especially in PN. When PN is non-stationary, the a(t) dependence is more complicated than that given above (see eq. (26.30)) and, in addition to Js, the time lag 9 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 [1975]) 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 0 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 = 4 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. glassforming and polymer) melts have shown that the a(t) dependence can be affected considerably by the nucleation non-stationarity [Westman and KrishnaMurthy 1962; Scott and Ramachandrarao 1977; Budurov et al. 1986; Spassov and Budurov 1988; Dobreva et al. 1996].

0 I v

r

,...,.

c--

-2

n

-3

-4 -5 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]; s q u a r e s - 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 N m > 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 supernucleus in 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 Nm 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 < [cgGVcd/(1 +

vd) J~sd]1/~l+vd).

(26.35)

Fig. 26.4 Overall crystallization by the mononuclear mechanism corresponding to appearance and growth of a single supernucleus (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 supernucleus 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') = gmeltt~D(t- t'), since 60 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 Rmelt 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 d shape (then V = cgRmelt ) and using eq. (11.10) at (0 = 0, we find that a(t) = 1 - e x p [- V

J(t') dt'] = 1 - e x p [- V((t)].

(26.36)

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 a(t) from (26.36) is identical with the probability Pl(t) for appearance of at least one supernucleus until time t. The experimentally accessible Pl(t) dependence is, therefore, given by the formula

Pl(t) = 1 - e x p [ - g

J(t') dt'] = 1 - e x p [- V((t)]

(26.37)

which, as shown by probabilistic considerations [Bigg 1953; Skripov et al. 1970; Gutzow and Toschev 1970; Toschev et al. 1972; Toschev 1973a; Skripov 1977; Skripov and Koverda 1984], 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) = Pl(t) = 1 - exp (- JsVt)

(26.38)

tav = t 1 = l[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 tl = f ~ tdPl(t) ~ U

for appearance of at least one supernucleus in it. It is worth noting as well that the above formula for tl is readily obtained from eq. (13.102) by setting = 1/V (one supernucleus in the volume of the system) at t = tl. Also, comparison of eq. (26.38) with the general KJMA formula (26.34) shows that the a(t) dependence in crystallization by the mononuclear mechanism is describable by this formula with q = 1 and ~ = 1/JsV. This q value corresponds to d = 0, and the expression for O follows formally from eq. (26.28) upon setting d = 0 and Cg = V. In the case of non-stationary nucleation the analysis is more complicated. Employing ~'(t) from (15.114) in (26.36) and (26.37) leads to the relation [Gutzow and Toschev 1970; Toschev et al. 1972]

or(t) = Pz(t) = 1 - exp [- JsV{ t - (/r2/6)z" oo

- 2'r Z [(- 1)i[i2] exp (- i2t['c) }]

(26.40)

i=1

which passes into (26.38) at z = O. When z ~ O, initially, a and P1 remain vanishingly small because of the delayed appearance of the first supernucleus.

Overall crystallization

385

For that reason, in contrast to a and P1 from (26.38) which initially are linear with time, a and P1 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 JsVZ = 1 and 5 (as indicated) in the latter. Experimentally, therefore, an S-shaped ct(t) or Pl(t) 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 P1 from (26.40) it is convenient to use ((t) from (15.116) in (26.36) and (26.37) and approximate a a n d P1 by V~'. Doing that results in the formula .0

................... s

/ /

//

i

0.8

/ /

i

/

lI ,

i

!

s/I

i is

0.6

I

tI

/

iI i

5

/

/

t

!

/

0.4 #

!

i

t

i

0.2

t t

i i

I

i t

i t

t

i /

t t

9

/

/ ,",

0

~-"

,

I

,

,

,

~

5

I

10

15

JsV t 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 JsV'C = 1 and 5, respectively. F i g . 26.5

tx(t) = Pl(t) = 8JsVr(tlTr't:)3/2 exp (-/t'2"c/4"t ")

(26.41)

which, compared with (26.40), gives a and P1 with an error of less than 15% for t < 0.25 z provided z < 104/JsV. The determination of tav or tl requires performing the integration in eq. (26.9) with a or Pa from (26.40). As the analytical solution of this problem is not available, the explicit dependence of tav or tl on Js and 7: remains unknown. Similar to eq. (26.33), however, the relation [Gutzow and Toschev 1970] tav = tl - bm't" +

1/JsV

(26.42)

may be an acceptable approximation in many practical situations. The numerical

386

Nucleation: Basic Theory with Applications

factor b m 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 JsVV 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 ~r is replaced by 6 0 / ~ with the nucleation delay time 0 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 et 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 ]; Stoyanova et al. [ 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]; Tumbull and Cormia [1961]; Clausse [1985]; Kashchiev et al. [1994, 1998]). Experimentally, P1 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 t). The circles in Fig. 26.6 represent the Pl(t) data of Skripov [ 1977] for repeated (91 times) crystallization at T - 383.4 K of a tin droplet with radius R = 35 pm. The experimentally found value of tl is 35 s which, in view of (26.39), leads to J s V = 0.028 s-1 or Js = 1.6 • 10 lj m -3 s-j at the temperature studied. The good description of the experimental Pl(t) dependence by the theoretical curve drawn according to eq. (26.38) with the above J s V value is an evidence for crystallization by the mononuclear mechanism under conditions of stationary nucleation. 1.0

'

'

'

'

I

'

'

-

'

"

I

L

'

'

'

'

|

i

'

,'

,i~

,

,

i

0.8 0.6 fl_ 0.4 0.2

,

0

t

t

I

[

50

,

,

i

i

I

100

t

i

i

,

,

.i

150

.

,

,

,

200

(s)

Fig. 26.6 Time dependence of the probability for appearance of at least one supernucleus: circles - data for repeated clajstallization at T = 383.4 K of a tin droplet [Skripov 1977]; 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 thermodynamically 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 o f 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 thermodynamically 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. S6hnel 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 [Turnbull 1981; Rasmussen 1982b], quasi-crystals [Shechtman et al. 1984; Kelton 1991], ice [Takahashi 1982], fatty acids [Sato 1988, 1993], amino acids [Kitamura et al. 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 twostage 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 Js and Gs in a metastable phase (the light grey circles) whose rates of nucleation and growth in the old phase are Jm and G m, 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 am of metastable phase crystallized till time t we can write am(t) = Vm(t)/V = 1 - exp [- (t/Om) m]

(26.43)

where m and Om 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 Vmof 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 Om by the kinetic exponent s and the time constant t~s of formation of the stable phase in the

388 Nucleation: 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 t' and t' + dt', at a later time t >_ 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 t' as initial moment of the crystallization process in the volume dVm, analogously to (26.43) we shall have dVs = (1 - e x p {- [ ( t - t')/t~s]S}) 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 t" = 0 to t' = t and its 1.h.s. from Vs = Vs(0) to V~ - Vs(t), accounting that Vs(0) = Vm(0) - 0 and dividing by V thus leads to [Kashchiev and Sato 1998] as(t) = am(t) -

exp {- [ ( t - t')/O~]'}[dam(t')ldt'] dt"

(26.45)

where a~(t) - Vs(t)lV, and v~n(t) is given by (26.43). Equation (26.45) describes the time dependence of the volume fraction as 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 as 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 d a m i d t 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), Om = (4[Cg, m G 3c,mJsm) TM,

(26.46)

Os = (4/Cg,sG3c,sJs,s) TM,

(26.47)

the subscripts m and s indicating that the shape factors Cg,mand Cg,s, the timeindependent rates Gc,m and Gc,s of growth and the stationary rates Js,m and Js,s 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/Os ratio are known, it is desirable to have this dependence in explicit analytical form. As shown by Kashchiev and Sato [1998], for t[l~ m > b', approximately, as(t) = am(t)(1 - exp {- (t~m/t~s)S[(t/Om) - b']S}) where b' - ( m -

(26.48)

1 / m ) 1In is a number, and am(t) is specified by (26.43). For

0 < t/Om < b', as(t) = 0 can be a sufficiently accurate approximation.

Curve oo in Fig. 26.8 displays the time dependence of am from eq. (26.43) at m = 4. The as(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 (Om/ Os)4 = 0.1, 1, 10 and oo (as indicated). We observe that the time at which as starts departing appreciably from zero is controlled by the value of the ratio 1.0 I II

0.8

0.6

0.4 !

0.2 t

0

1

I

s

2

3

t/0 m Fig. 26.8 Time dependence of the volume fraction o f the stable crystalline phase in two-stage overall crystallization: solid c u r v e s - eq. (26.45) at (Om/Os) 4 = 0.1, 1, 10 and oo (as indicated) and m = s = 4; dashed curves - the corresponding approximate eq. (26.48).

390

Nucleation: Basic Theory. with Applications

Om/Os between the time constants Om and t9s for formation of the metastable and stable phases. Namely, small Om/Os values (i.e. large Os 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 oo and of the 'beginning' of curve 0.1 in Fig. 26.8). Conversely, for large Om/Os values (i.e. fast stable-phase formation and, hence, small time constant Os) as(t) is close to cr~n(t) and in the Om/Os = oo limit we have as(t) = ~ ( t ) . That is why curve oo in Fig. 26.8 represents both the as(t) function (26.45) at Om/Os = oo and the ~n(t) 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/Os. The as(t) curves in Fig. 26.8 reveal that the main parameter to control the appearance of the stable crystalline phase is the Om/Os 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] Os/Om > 1.

(26.49)

In the concrete case of t9m and Os specified by (26.46) and (26.47) this inequality becomes

(Cg,mG3,mJs,m/Cg,sG3,sJs,s) TM > 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 Gc,s sufficiently smaller than Js,m 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 Cg,s and Cg,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 s m o o t h - 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 Ap, 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 tav 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 where do is the molecular diameter.

= do]tav

(27.1)

392 Nucleation: 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 tav. Noting that the net flux of molecules to a given site is fs - gs, for tar we shall, therefore, have tav = l/(fs- gs) wheref~ (s -1) and gs (s-l) 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 fs = a o f /ac,,,

(27.2)

gs = aog/Ac,,,.

(27.3)

Heref(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, n is the area of the crystal surface in contact with the old phase, and Ac,,,/ a0 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 tav, fs and gs in eq. (27.1) leads to the formula Gc = d0(fs - gs) = d0fs(1 - g/f).

(27.4)

To proceed further we recall the general relation (10.90) between f and g, in which now 3W/3n = - Ap, because for a macroscopically large crystal the effective excess energy qJ is negligible with respect to the A/~ term in (10.86). Hence, eq. (27.4) becomes Gc = d0fs(1 - e -A~/kT)

(27.5)

which is a general presentation of the formula given by Hertz [1882] and Knudsen [1909] for growth from vapours and by Wilson [ 1900] and Frenkel

C~. stal growth

393

[1932] for growth from the melt. Equations (27.2) and (27.5) show that at a given supersaturation A~t, Gc is time-independent when f is proportional to Ac,, (then fs 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 7n has a constant value which we shall denote by ~s in order to indicate that it refers to sticking to a growth site. It should be noted also that with ~s = 1, Gc from (27.5) is the maximum rate at which a crystal face can grow at a given supersaturation. To find Gc as an explicit function of A/t, we must take into account the difference in the fs(A~) dependence when A/.t 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

,4. f e,s--aAla/kT{,, 1 -- e -A~/kT)

--- - w

(27.6)

where fe,s = aofe/Ac,, is the value of fs at Ap = 0, i.e. at phase equilibrium. In the concrete case of growth from vapours Ap is given by eq. (2.8) or (2.9). Then, for monomer attachment controlled by direct impingement, from eqs (10.3) (used with p = P e , ~'n = ~s, c02/3n2/3 -- A c , n ) , (13.68) and (27.6) it follows that [Hertz 1882; Knudsen 1909]

Gc = 7~vole(S - 1)

(27.7)

where Ie = pe/(2ZcmokT) 1/2, and v0 = aodo 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 = (Ts/do)OgcCe(S- 1).

(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 = o0, such a difference does not exist (see Section 10.1) and (27.8) simplifies to

Gc = ~saoDCe(S - 1).

(27.9)

This formula could be derived with the help o f f 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 Ap or S. 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/~ controlled by T in accordance with (2.20), for f in (27.2) we can employ eq. (10.55) in combination with (10.58) at O's = or, because the crystal face is the own substrate. Setting ?'~ = ~ and d g -a0, from (27.2) and (27.5) we thus find that for crystal growth from the melt under interface-transfer control Gc = (7'skT/3~ZZZorl) exp [(- A, + El + trao)/kT]eA~/kT(1 - e-~/kT).

(27.10)

Here the energy term E1 + tra0 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 E1 + tYa0= 0. Then (27.10) takes the following simpler form corresponding to f from (10.64): Gc = ( TskT/3 ~aoO)e ~A~'-;t)/kr(1 - e-A~/kT).

(27.11)

As seen from eqs (27.10) and (27.11), in growth from the melt Gc is a function of A/~ not only directly, but also implicitly via T and the melt viscosity 77. Without the factor e ~Au-x)/~r, eq. (27.11) is essentially the formula of Wilson [1900] and Frenkel [1932]. In the simplest case of Ap given by (2.23), eq. (27.11) becomes Gc = (yskT/3rtaorl) exp (- Ase/k)[1 - e x p (- AseAT/kT)].

(27.12)

This Gc(T) dependence parallels that of Gilmer [1993] and differs from the Wilson-Frenkel Gc(T) dependence by the T-independent numerical factor exp (-Ase[k) 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/~ < 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 supersaturations (A/~ < 0.2kT) Gc depends linearly on Ap according to Gc = KgA/~

(27.13)

where Kg is a kinetic factor characteristic for continuous growth. This factor can be considered as Ap-independent in growth from melts, because in the range of small supersaturations r/(T) and exp [(El + crao)/kT] are practically T-independent, and (A/~ - &)/kT ~ - - Ase/k. The linear Gc(Ap) dependence (27.13) is a known experimental criterion for continuous growth. Curve CG in Fig. 27.2 displays the Gc(Ap) function (27.6) and visualizes the linearity between Gc and Ap for small enough Ap values. The linear dependence of Gc on Ap in continuous growth at low supersaturations is exemplified in Fig. 27.3 in which the squares and circles represent experimental Gc(Ap) data [Georgieva and Nenow 1967; Nenow and Georgieva 1968] for growth from vapours of concave faces of diphenyl crystals at T = 332 and 339.7 K, respectively. Being curved, the crystal faces

Crystal growth

0.6[

' ' 7 0.0006 . . . .

0.4 .O

i ,," ;" N, PN -"

0

-% CO

i

0.0002

j

-----7-

/

MN

0 0004

395

02

....... "

0

0.2

0

0

0.5

1.0

1.5

A~t / kT Fig. 27.2

Supersaturation dependence of the rate of c~. stal 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 M N - eqs (27.40) and (27.41) for polynuclear and mononuclear growth, respectively.

0.08 0.07 0.06 0.05

E

=L 0.04 0

I

0.03

9

q

0.02 0.01

0

0.05

0.10

0.15

0.20

0.25

1 / R (t.tm1) 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 1/R. 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.

Fig. 27.4 Cross-sectionof a molecularly smooth crystal face which advances (a) by mononuclear monolayer growth, (b) by 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

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 = doA fJs.

(27.14)

This general formula implies time-independent nucleation. In it Js (m-2 s-l) 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 Ap is varied isothermally, eq. (27.14) can be used in the form

Gc = doCoAf z fe*e ~u/kr e -w*/kr

(27.15)

which is obtained with account of eqs (13.39)-(13.42). Here W*(A/.t) is the nucleation work for 2D HEN on own substrate (see Sections 4.3 and 4.4), Co (m -2) is given by (7.8) or (7.9), and the pre-exponential factor doCoAfzf* is practically Ap-independent. With f * replaced by f0* (cf. eqs (13.42) and (13.43)), eq. (27.15) is applicable also to crystal growth from the melt when Ap 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 Af < [c'go2v[(1 + 2v)

j2v]l/(l+2v)

(27.16)

provided the cluster radius R grows according to the power law (26.16) represented as

R(t) = (Vst)v.

(27.17)

Here Cg is a numerical shape factor ( Cg = zr for circles, Cg = 4 for squares, etc.), and Vs (m 1/v s-1) is the growth constant of the clusters. For linear growth law (v = 1) Vs (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~ and Vs, respectively. Hence, from (27.1) it follows that Gc = {(1 + 2v) 1/(1+ 2V)F[(2 + 2v)/(1 + 2v)]} -l do(cg'o2VJs)l/(l+2v) (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 = 0.78do(cgV 2 Js) 1/3

(27.19)

which was derived by Todes [ 1949b], Nielsen [1964] and Hillig [1966] with a slightly different numerical factor. 3. Polynuclear polylayer 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 p

Gc = IVvdo(cgO2s v Js) 1/(l+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 gtv is close to unity and its value depends on the approximation used for the determination of G~. For example, again in the case of clusters growing linearly with time (v = 1), the most reliable value of gt1 is thought to be 0.97 [Gilmer 1980; Obretenov et al. 1986]. Then eq. (27.20) leads to the expression Gc = 0.97do(cgV 2 Js) 1/3

(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 G~ on Vs. 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 righthand sides of (27.14) and (27.20). Doing that yields Gc = doAfJs{ 1 + [Af/~vCg,1/~1+2v)](Js/Vs)2V/~l+2v)}_l

(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 = 1 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(Ap) and vs(Ap) functions in order to find out the Gc(Ap) dependence itself. For the Js(Ap) function we can use the results in Sections 13.2 and 13.3 for 2D HEN on own substrate (then Ao'= 0 and O's = o'). As to the vs(Ap) 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), v s (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 = d0fs(1 - e -Au/kr)

(27.23)

where now fs (s- ~) is the R-independent frequency of monomer attachment to a growth site at the monolayer step bordering the cluster. This means that fs can again be obtained from (27.2), but with Ac, n = d o b n 1/2 where b n 1/2 is the length of the cluster periphery (b = 2(~a0) 1/2 for circles, b = 4a~/2 for squares, etc.). Hence, in order for fs to be independent of the cluster size, f in (27.2) has to be proportional to n 1/2. This is so when monomer attachment is controlled, e.g. by direct impingement or by surface diffusion (but only for R > A,s) in growth from vapours and by interface transfer in growth from solutions or melts. Let us find Vs 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 dg = a0 and = 7s, we get the expression Vs = dofe,se Au/kr (1 - e - ~ / k r ) = 7svole(S - 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 > As, combining (2.9), (10.31), (10.35), (10.41), (13.68), (27.2) and (27.23) and taking into account that Ac, n = 2~rdoR yields Us -" -we,s'-'q^f aAplkT(, 1 - e - ~ / k T ) = 27saoAsl~(S - 1)

(27.25)

where fe,s = 27sd0Asle is the doubled value offs from (27.2) at Ap = 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, Vs from (27.25) is greater than Vs from (27.24) when As > do. 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 vs(A/~) dependence retains the above simple form also for growth

400

Nucleation: Basic Theory with Applications

from solutions under interface-transfer control. From eqs (2.14), (10.63), (13.69), (27.2) and (27.23) it the.n follows that Os = dofe,se~/kr(1 - e -~/kr) = ('Ys/do)DKcCe(S - 1)

(27.26)

where fe, s = (t's/ao)DKcCe is again the value offs from (27.2) at A/~ = 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 ~s 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, f i n (27.2) is given by eq. (10.66) with trs = or. The v~(A/~) function is now complicated because of the relation (2.20) between A/~ and T and of the dependence offs on the melt viscosity 0. Using eqs (10.66), (27.2), (27.23) and the approximation v0 = aodo results in (cf. eq. (27.10)) Vs = ()'skT/3rCaorl) exp [(-/l + E1 + trao)/kT]e~/kr(1 - e-~/kr).

(27.27)

When the A/~(T) dependence has the simplest form (2.23), this expression becomes Vs = ()'skT/3r~aorl) exp (-Ase/k)

exp [(g I + Crao)/kT][1 - e x p (-AseAT/kT)].

(27.28)

These two equations show that Vs, like Gc from (27.10)-(27.12), has a maximum with respect to Ap or T. This is in contrast to Vs from eqs (27.24)-(27.26), which increases monotonously with A~uor S because of the isothermal variation of the supersaturation. We are now in position to determine the A/~ 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, Gc = doCoAf z f * eA~akr e -8/A~ = ~'*c*doCoAfzA, 2 leS exp ( - B ' / l n S)

(27.29)

for mononuclear growth, where Co, z, B and B' are given by eqs (7.8) or (7.9), (13.37) (at Atr = 0), (13.55) and (13.84). For polynuclear growth, however, the Gc(Ap) dependence remains unknown, since Vs 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 >> )~s for the validity of (27.25). Actually, due to the complicated dependence o f f 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/ls, 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 ),* = ~s, we get Gc = d o C o A f z fe* eA~akr e -B/Au = ~(b/ao)CoAfDKcCezn*l/2s

exp (-BTln S)

(27.30)

for mononuclear growth and Gc = kldofe,se

At~/kT(1 - e-Al~/kT)2/3e-B/3Alu

= (k 1~s/do)DKcCeS]/3(S - 1)2/3 exp (-B'/3 In S)

(27.31)

for polynuclear growth where kl = Igl(CgaoCozfe * ]fe,s) 1/3 =

~1 (Cg"bdoCozn*l/2) 1/3

(27.32)

Here V1 = 0.97, and Co, z, n*, B and B' are given by eqs (7.8) or (7.9), (13.37) (at Act = 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 kl is nearly independent of A/~ or S. Also, it is close to unity for crystal faces which are free of nucleation-active centres, as then Co = 1/ao.

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 7* = ~s we obtain Gc = ~'*bCoAfzn* 1/2(kT/3 Zcvorl)

exp [(- ~ + E1 + crao)/kT]eA~/kre - B/~

(27.33)

for mononuclear growth and Gc = (kl ?'skT/37raorl) exp [(- ~ + E 1 + aao)/kT] e~/kr(1 - e-Z~/kT)2/3 e - B/3z~

(27.34)

for polynuclear growth. Here C 0, z, n* and B are again given by eqs (7.8) or (7.9), (13.37) (at Act = 0), (4.35) and (13.55), and the factor kl is specified by (27.32) and is practically A/~-independent. As seen from these expressions, Gc depends on A/~ both directly and implicitly through T and 77. When A/~ is related to T in the simplest way according to eq. (2.23), eqs (27.33) and (27.34) lead to Gc = ?'*bCoAfzn*l/2(kT/3JrVorl) exp (-Ase/k)

exp [(El + Gao)/kT] exp (-B'/TAT)

(27.35)

402

Nucleation: Basic Theory with Applications

for mononuclear growth and to

Gc = (klT'skT/3~aorl) exp (- Ase/k) exp [(El + trao)/kT] x [1 - exp (- AseAT/kT)] 2/3 e x p (- 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 T is due mainly to the increase of the melt viscosity 7/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(Ap) dependence for polynuclear growth of crystals as follows:

Gc = kvdofe,seAp/kT (1 - e-Ap/kT)2vl(1 + 2V)e-W*(hp)/(1 + 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 itfe,s (s -1) is the value at Ap = 0 of a properly defined molecular attachment frequency per growth site at the cluster periphery, and the numerical factor kv is virtually A/a-independent and given by

kv = Iltv(CgaoCoz f*/fe,s) 1/(l+2v).

(27.38)

Here gtv -- 1 may depend slightly on v, and Co and z are specified by (7.8) or (7.9) and by (13.37) with Atr= 0. Since z f*/f~,s --- 1, when the crystal face is free of nucleation-active centres (then Co = l/a0), kv = 1. It is important to note that at v = 0 eq. (27.37) describes mononuclear growth (cf. eq. (27.15)) if k0 is defined as

ko = AfCoz fe* /fe,s.

(27.39)

This expression follows from (27.38) at v = 0 by setting formally gt0 = 1 and Cga o = A f . 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 = kvdofe,seAl~/kT(1 - e-Ap/kT)2v/(1 + 2V)e-B/(1 + 2v)Au,

(27.40)

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~d~f v ~ e,se~/kre -mau

(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 k0: e.g. for a crystal face which is free of active centres, while kv-- 1, for k0 we have k0 = AfCo (because z f*/fe,s -- 1) so that k0 > 107 when the face is, say, of a r e a Af > 1 pm 2.

Crystal growth

403

Equations (27.40) and (27.41) reveal that Gc is a monotonously increasing function of Ap 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(Ap) dependence for nucleation-mediated growth basically different from that for continuous growth (see eq. (27.6)). This difference disappears only for high enough supersaturations (A~ > B/(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 et al. [ 1989], we can now sum the reciprocals of the fight-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)

Gc = kodofe,seA~/kre- ~/~ {1 + (ko/kv)[(1 - e-~/kr)eS/~]-ev/(l+Zv)} -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 B/Ap with W*/kT. Since ko/kv >> 1, as required, while for Ap --~ 0 eq. (27.42) passes into eq. (27.41) for mononuclear growth, with increasing Ap 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 Ap is varied according to (2.20) provided in them r ,,A~/kr is replaced by the attachment frequency fs determined from eqs (10.66) and (27.2). Thus, Gc in this case is proportional to l/r/and has a maximum with respect to Ap or T. The conclusion about the transition from mononuclear to polynuclear and then to continuous growth retains validity, since with increasing A/.t (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 Ap > B~ (1 + 2v). Also, it is worth noting that, as in the case of continuous growth, the above results are equally applicable when Ap < 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(Ap) 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 Je,s

"-~

404

Nucleation: Basic Theory with Applications

v = 1, kl = 1, k0 = 107 and B = 8 k T (this choice of B corresponds to tr = 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. [ 1974] 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 [ G c / S I / 3 ( S - 1)2/3]-vs-1/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. l , l ,

l l , | U W

0 I.",,-,-I

-1

CN

"7

CO v

-2

co ,,on.

CJ)

-3

o "--'

-4

,,,...

-5 -6

9 1

2

3

4

5

6

7

9 8

9

1/InS Fig. 27.5 Crystal growth rate as a function of the supersaturation ratio in nucleation-mediated growth: circles - data for the (110) face of paraffin C36H74 crystals in solution of petroleum ether at temperatures between 290 and 295 K [Simon et al. 1974]; line - best fit according to eq. (27.40) with v = 1 (Gc is in pro~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 nucleationmediated 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 do and let them spread with time-independent velocity Vs (m/s). Then each step will travel the distance ds to the neighbouring step in front of it within the same time ds/vs. 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 tar = ds/vs which in combination with (27.1) leads to Gc = dovs/ds.

(27.43)

This relationship shows that Gc is determined by two parameters" the interstep distance ds 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

Nucleation: Basic Theory with Applications

ds = go'R*

(27.44)

where ~ = 19 is a good approximation for the proportionality factor gt' [Cabrera and Levine 1956; Budevski et al. 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 Ap. According to (4.34) and (27.44), ds decreases with increasing A~. This means that for Vs 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 Az 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 As on both sides of the step so that Az = 2~s. The condition for step propagation in isolation is, therefore, ds > Az. Due to the decrease of ds with increasing A~t, 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 ds/Az ~ oo 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 As is smaller than the step radius of curvature). Also, Az = 2A,s and, as shown by Burton et al. [1951 ], Yz(ds, Az) = tanh(ds/2As). 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 - dofe,s(do/2/q,s)e~/kT(1 -e-~/kT)(Al-t/A, u') tanh(A/~'/A/~) = 7svole(S- 1)(ln S/ln S') tanh(ln S'/ln S)

(27.45)

where the characteristic supersaturation A/~' and supersaturation ratio S' are defined by A~t'= zrgt'd02td8~,s--- 7.5 d2td~s In S ' = Ap'/kT.

(27.46) (27.47)

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 -do. We can, therefore, use the Yz factor of Burton et al. [1951] in the form

Crystal growth

407

Yz(ds, Az) = tanh(ds/do) 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 = dofe,se~/kr(1 - e-a~/kr)(Ala/Aia') tanh(Ap'/Ap ) = (Ys/do)DKcCe(S- 1)(In S~ In S') tanh(ln S'/ln S)

(27.48)

where A/~'= (Trgt74)d0tr- 15d0r In S ' = AI~'/kT.

(27.49) (27.50)

Naturally, eq. (27.49) follows directly from (27.46) upon setting 2&s = do. 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 --- do and Y~(ds, Az) = tanh(ds/do). Combining eqs (4.34), (27.27), (27.43) and (27.44), we find that Gc = (7skT/3zCaorl) exp [(- ~ + E1 + aao)/kT] e~u/kT(1 -- e-Z~/kT)(Al.l/Ala') tanh (A/I'/A/I)

(27.51)

where Ap' is again given by (27.49). This formula shows that Gc is a complicated function of Ap 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 E1 + aao -- 0 and in the particular case of AMrelated to T via (2.23), eq. (27.51) results in the following Gc(T) dependence: Gc = (YskT/3xaorl) exp (- Ase/k) [1 - e x p (-AseAT/kT)](AT/AT') tanh(AT'/AT).

(27.52)

Here, in conformity with (27.49), the characteristic undercooling AT' = bp'/ As~ is given by A T ' = tcgt'dor/4A se --- 1 5 d o l c / A s e.

(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 x - 1 = x and tanh(1/x) --- 1 for x --->0 [Korn and Korn 1961 ], when simultaneously Ap < 0.2kT and Ap < 0.5Ap', we have Gc = (Kg/AI.t')A!a2

(27.54)

408 Nucleation: Basic Theory with Applications

where Kgis 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 Ap > 2Ap', tanh (Ap'lAp) = Ap'lAp [Kom and Kom 1961 ] and the Gc(Ap) 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 Ap', i.e. on the nucleation parameter ~r the specific edge energy of the 2D clusters of monolayer thickness. This is of experimental interest, since analysing Gc(Ap) data in accordance with (27.54) allows a determination of ~r In this respect it could be noted that the rather high values of Ap' 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 do = 0.3 nm, to- 10 pJ/m and ASe = 3k, from (27.50) and (27.53) we calculate S ' = 5 x 104 at T = 300 K and A T ' = 1100 K. Such high supersaturation ratios S' and undercoolings AT' are not common in experiments. In contrast, the presence of ~ in Ap' from (27.46) leads to considerably smaller S' values, which means that the departure of the Gc(Ap) dependence from the parabolic law (27.54) is more easily observable in growth controlled by surface diffusion. For example, with the above values of do, tr and T, from (27.46) and (27.47) it follows that S' < 1.7 when ~,s > 10d0. The Gc(Ap) dependence for spiral growth is illustrated in Fig. 27.2 by curve SG which is drawn according to eq. (27.48) with Ap'/kT = 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(Arp) data of Bostanov et al. [ 1969] for, respectively, (100) and (111) faces of Ag in the case of electrocrystallization in an aqueous solution of AgNO 3 at T = 318 K. In this case Ap is related to the overvoltage Atp through eq. (2.27), and Gc is obtainable with the help of the measured current density ig (A/m 2) which results from the growth of the face: Gc = Voig/zieo where zi = 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 Kgz2e2/Al, t" = 6.74 and 15.2 mm/sV 2 for the (100) and (111) face, respectively. As Ap' is related to ~r the good agreement

Crystal growth 70

. . . .

I ' ' ' " 1 ' ' ' ' 1 ' ' ' ' 1 ' 1 ' ' ' 1

. . . .

409

I " ' ] ' '

60 50

E

40

r v

o

(.9

30 20 10 |

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Am (mY) Fig. 27.7 Crystal growth rate as a function of the overvoltage in spiral growth: squares and circles- data of Bostanov et al. [1969]for, respectively, the (100) and (111)face of an Ag single crystal electrode in aqueous solution of AgN03 at T = 318 K; curves- best fit according to eq. (27.54).

between theory and experiment allows calculating the specific edge energies tq00 and ~qll of the monolayer steps of Ag on the (100) and the (111) face of Ag: tq00 = 25 pJ/m and ~r = 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(Ap) dependences are applicable also when A~ < 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 nucleationmediated crystal growth on the supersaturation Ap is practically entirely governed by the Ap 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)Ap - 2 v k T In (1 - e -au/kr) - (1 + 2 v ) k T In

(kvdofe,s).

Differentiating this equation with respect to Ap, employing the nucleation theorem in the form (5.29) and treating the product kvdofe,s as constant with respect to Ap, we get n* = (1 + 2v)kT d(ln Gc)/dAp - (1 + 2v) - 2v/(e a u n t - 1).

(28.1)

This general formula shows that the Ap 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(Ap) data in In Gc-vSAp 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 A,u/kT > In (1 + 2 v), 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)ld~p - 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)/dAp -- 3 - 2/(e au/kr- 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 Ap 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) - 2 v / ( S -

1)

(28.4)

for growth from vapours or solutions,

(a) =,.

~ 'Q ~

0

o

~ v

-2

c-

-4 I

,

.

.

,

I

.

,

,

,

I

,

,

,

i

,

9

,

,

..

,

,

I

(b)

8 7 6 -Ir

r-

5

3

@ _

m

~176 i

l

1.5

n

I

"

n

I

2.0

n

I

9

e-

n

2.5

3.0

3.5

/ kT 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. (28.1). (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 data from Fig. 14.1b; l i n e - Gibbs-Thomson eq. (14.18).

412

Nucleation: Basic Theory with Applications

n* = (1 + 2 v)(kT/zieo)d(ln ig)/dAtp - (1 + 2 v)

(28.5)

- 2v/[exp (zieoAq)/kT) - 1] for electrocrystallization and n* = (1 + 2 v ) ( k T / A v e ) d ( l n G c ) / d p - (1 + 2v) - 2v/{ exp

[AVe(,/9

- 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 T. In eq. (28.5) Gr is substituted by the experimentally measurable current density ig (A/m2), since these two quantities are related by Gc = Voig/zieo. The circles in Fig. 28.1a represent the Gc(Ap) 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 l / k T = 4 where el 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/dofe,s) = - l O . 3 0 0 8 1 / ( A p / k T ) + 1.72739 + 1.00258(Ap/kT) used for the calculation of d(ln Gc)/dAp. 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. lb. The curve in Fig. 28. lb depicts the n*(Ap) dependence following at e l / k T = 4 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*(Ap) 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 t 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. As a result, under otherwise equal conditions, t i may not have the same value when measured by different experimental techniques. This means that the theoretical interpretation of ti data must always be done with a ti formula which corresponds to the particular experimental method used for the measurement of t i. So far, two approaches have been most widely used for the theoretical determination of ti. The first one is based on the assumption that the appearance of the first supernucleus is the event that brings the system out of its metastable equilibrium [Volmer 1939; Nielsen 1964, 1967; Hirth and Pound 1963; Kubota 1983; S/Shnel and Mullin 1988; Kashchiev et al. 1991; S/Shnel 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 ti formula obtained is adequate for analysis of experimental t i data only when they are obtained by experimental techniques which allow counting the number of supernuclei or detecting the presence of a single supernucleus 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; Stihnel and Mullin 1979, 1988; Kashchiev 1989a; Kashchiev et al. 1991; S/Shnel 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 t i formula is legitimate when the ti 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 ti 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]; S6hnel and Garside [ 1992]; Jakubczyk and Sangwal [ 1994]; Sangwal and Polak [ 1997]). Let us now see how t i i s related to the nucleation rate when either the mononuclear or the polynuclear mechanism is operative. We begin with the former. One way to define t i in this case is to identify it with the moment of formation of one supernucleus in the system [Volmer 1939]. Mathematically, this definition reads V~(ti) = 1 or as~(ti)= 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 t i 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; S6hnel and Mullin 1988] ti = 1/JsV

(29.2)

where Js is the stationary nucleation rate specified by (13.39). Here and in what follows we present t i 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 Ap is varied isothermally ti can be represented by the general expression ti

=

(I/z f * C o V )

e (w*-Au)/k~

(29.3)

in which the concentration Co of nucleation sites is given by (7.10)-(7.12) or (7.16). Due to the relatively weak dependence of z and f * on Ap, the preexponential 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 ti is difficult, because ~'(t) is a complicated function of time (cf. eq. (15.114)). Nonetheless, for practical purposes it may suffice to use the approximation t i = bi,m't" + l[JsV

(29.4)

where the first summand accounts for the time delay in the ~'(t) function (Fig. 15.9b), and bi, m = 1 can be considered as independent of 7r, J~ and V. In the ~r= 0 limit (stationary nucleation) eq. (29.4) passes into (29.2), but when T >> 1/JsV, ti is entirely determined by the nucleation time lag v. Physically, the induction time can also be identified with the lifetime tav of the supersaturated old phase or, equivalently, with the mean time t~ for appearance of at least one supernucleus 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 i values, but

Induction time 415 the difference is small, because the numerical factors b m and bi,m in (26.42) and (29.4) are practically equal. Let us now determine t i according to the polynuclear mechanism. Clearly, ti can again be identified with the lifetime tav 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 ti = F[(2 + vd)/(1 + vd)][(1

+

vd)/cgGVdJs] 1/(l+vd)

(29.5)

for stationary nucleation and t i = bp't" + F[(2 + vd)/(1 + vd)][(1

+

vd)/cgGVcdJs] 1/(l+vd)

(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 t i = F(1 + 1]vd)[V[cgNm]l/vd(1/Gc).

(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 t i 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 i as the time needed for the formation of the volume Vd or, equivalently, of the detectable fraction ad = Vd/V of V. Mathematically, this definition of t i reads tZ(ti) = a d

(29.8)

where the fraction a of transformed volume is given by the KJMA formula (26.8), and aa << 1. This algebraic equation for t i is easy to solve again for progressive nucleation in stationary regime and particle growth obeying eq. (26.16). Then cr is given by (26.23) and with account of (26.28) and usage of In (1 - ad) - - - a a [Korn and Korn 1961], from (29.8) it follows that [Kashchiev et al. 1991 ] t i = [(1 + vd)O~d/Cg GVcdJs] 1/(1 + vd).

(29.9)

In the particular case of spherical particles (d = 3, Cg = 47r/3) with radii growing linearly with time (v = 1) this equation passes into that obtained by Chepelevetskii [ 1939]. We observe also that, not surprisingly, t i from (29.5) and (29.9) differ from each other solely by a numerical factor controlled by a d (i.e. by the resolution limit of the experimental technique). When nucleation is progressive, but non-stationary, ct(t) is a complicated function (see eq. (26.30)) and t i can be determined only approximately. It turns out that in the case of 1 + vd = 4, similar to (29.4), the nucleation time lag z appears as a summand in the r.h.s, of (29.9) [Kashchiev 1989a]. Generalizing this finding, we can thus represent ti in the form t i = bi,pr + [(1 + vd)ad/CgGVcdJs] 1/(1 + vcO

(29.10)

416 Nucleation: Basic Theory with Applications where for practical purposes the factor bi, p c a n be treated as a number close to unity. For example, in the 1 + vd = 4 case numerical analysis [Kashchiev 1989a] shows t h a t bi, p = 0 . 4 to 0 . 5 , because it depends weakly on Gc, Js and T according to bi, p = (4ad/CgG3JsT4) 1/16. 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 z = 0 (stationary nucleation), it passes into eq. (29.9). As to the usage of the definition (29.8) for determination of ti 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 f r o m (26.18) in (29.8) so that, upon accounting for eq. (26.20) and the approximation In (1 - ad) = ad, we get [van der Leeden et al. 1991]

-

ti - ( ad V/CgNm) l/vd(1/Gc).

(29.11)

Understandably, t i from (29.7) and (29.11) differ from each other only by an Crd-dependent numerical factor. We note also that in the vd = 3 case eq. (29.11) leads to formulae similar to those given by S6hnel 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 Nm/V 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 o n t i is via the maximum concentration Nm/V of supernuclei or the stationary nucleation rate J~ 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 t i 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 t i which is valid for whatever number of supernucleus particles formed in the old phase [Kashchiev et al. 1991]: t i - 1/JsV+ [(1 + v d ) ad]CgG:dJs] 1/(l+vd).

(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]

g < [cgGVcd/(1

+

vd) adJVsd] l/(l+vd)

(29.13)

eq. (29.12) passes into eq. (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(Ap), Js(Ap) and "r(Ap) 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 Ap is varied isothermally, from (13.39), (13.41), (13.42) and (29.9) it follows that, most generally, ti =

[(1

+

vd)ad/CgZfe*CoGVd] 1/(1 + vd) e(W*-Ap)/(1 + vd)kT

(29.14)

provided nucleation is stationary. With vd = 3 this equation leads to formulae similar to those given by S6hnel and Mullin [1988]. Inspection of eqs (29.3) and (29.14) shows that the latter turns into the former at d = 0 if then Ctoand Cg a r e 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 ti(Ap) dependence when the growth constant Gc ( m 1Iv s-1) is the following function of the supersaturation: Gc

z 1Ivde,s'-" r p~u/krC,,-1 - e-A~/kr),

--"*0

(29.15)

the factorfe,s (s -1) being the value at Ap = 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 fe,s = 2doDCe [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 ti(A/.t) dependence is of the form [Kashchiev et al. 1991 ] t i = Kvde-Ap/kT(1 -- e - A p / k r ) - vd/(1 + vd) exp [B/(1 + vd)Ap2].

(29.16)

Here the nucleation parameter B is specified by (13.49), and the practically Ap-independent factor Kvd (s) is defined by Kvd = [(1 + vd) ad/Cg Z fe * Cod~ fe,s vd ] ll(l+vd)

(29.17)

where Co (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 Zfe* --fe,s and d 3 -- v0 eqs (29.16) and (29.17) lead to t i = K3e-Au/kT(1 -- e-AU/kT) -3/4 exp ( B / 4 A p 2)

(29.19)

K3 = (4ad/Cgfooo)l/4(1/f~,s).

With

Cg =

4, ad = 10-8 to 10-4 and fe,s = 104

(29.18)

to

1010 S-1 , from (7.12) and

418

Nucleation: Basic Theory with Applications

(29.19) it follows that K3 = 1 ps to 10 ps for HON. In the case of HEN Co is given by (7.10) or (7.11) and K3 is expected to have higher values. As already noted above, at d = 0 eq. (29.16) describes the ti(A/l) 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]) ti =

Ko e-Ap/kr exp (B/A/~2)

(29.20)

the nucleation parameter B being given by (13.49). As follows from eq. (29.17) at d = 0, ad = 1 and Cg = V, the practically A/~-independent factor K0 is specified by

Ko = 1~zf * Co V

(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 f * = 104 to 101~ s-1, from (7.12) and (29.21) we estimate KoV = 10 -39 to 10 -33 m3s for HON so that then Ko = 10-30 to 10-24 s, e.g. for V = 1 mm 3. Like/(3, in the case of HEN Ko is expected to have higher values, because then Co is determined by (7.10) or (7.11). Comparing the above values of K0 and K 3, w e see that K0 is many orders of magnitude less than K 3. This means that experimentally obtained relatively high values of Kvd 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 Js from (13.50) and Gc from (29.15) in eq. (29.12), we can find also the unified ti(A H) 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 fight-hand sides of (29.16) and (29.20), the result being [Kashchiev et al. 1991] (vd > O) ti = KO e -zxlu/kT exp

(B/Ap2){ 1 + (Kvd/Ko)[(1 - e -Au/kr)

exp (B/AIu2)] -vd/(l + va)}.

(29.22)

This unified formula gives t i in the case of progressive nucleation in stationary regime provided the growth constant Gc depends on A/~ 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 A/~ ~ 0 limit (29.22) passes into (29.20) for the mononuclear mechanism. With increasing A/~, however, the second summand in (29.22) increases and since Kvd/Ko >> 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 ti(A/.t ) dependence (29.22)at K0 = 10-25 s, Kvd/Ko = 109-~ d = 3, v = 1 and B = 300(kT) 2 (according to (13.49), this B value corresponds to O'ef~ - 80 mJ/m 2 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 i is a monotonously decreasing

Induction time

419

100 10 PN 1 i

80

60 v

03 0 2.1

o.-

40-

2.3

2.5

20Ii!

0

i

0

,

i

i

I

,

i

i

i

1

I

2

,

,

3

,ap, / 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 Ap 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 Ap range in which they are valid. This implies that the correct choice of a ti(Ap) formula is of crucial importance for the reliable quantitative interpretation of ti(A/.t) data: full correspondence must exist between the definition of t i, which is behind the chosen formula, and the experimental technique employed for obtaining the ti(Aju) data. To further emphasize this point let us consider one more definition of t i and the resulting ti(Ap) dependence. Suppose we can monitor optically the supersaturated old phase by measuring the temporal evolution of the intensity It 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 It,0. At a later time t particles are already present in the old phase and It is smaller than It,0 in accordance with the Bouguer-Lambert law It(t) = It,0 exp

[-ke(t)Ls]

(29.23)

where Ls (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 >> ZI, ke is given by [Rayleigh 1899] k e = 2~zaeR6~

(29.24)

420

Nucleation: Basic Theory with Applications

provided multiple scattering is negligible. Here the factor a e (m -a) depends on the light wavelength A,l and the refractive index m r of the particles relative to the surrounding medium: ae = 6 4 / l : 4 ( m 2 - 1) 2 / 3 (mr2 + 2)2/], 4.

(29.25)

When R = 21, ke is a complicated function of R and ~l, which is given by the theory of Mie [1908]. In the diffraction limit (then R >> 21), ke is merely equal to the product of ~"and the doubled area 2~R 2 of the particle geometrical cross-section, i.e. (e.g. Born and Wolf [ 1968]) ke = 2/t'R2~.

(29.26)

As shown by Todes and Chekunov [1957], the expression ke = 2/r,aeR6~/(1 + aeR4)

(29.27)

is a useful interpolation formula valid for all values of the R/A,1 ratio. We see that for R << &l (Rayleigh limit) and R >> 2,1 (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 ((t') and that at a later time t _> t" all have the same radius R ( t - t'). Therefore, the change of ke in the period between t'and t" + dt" can be described by eq. (29.27) in the differential form dke = 2~Z~aeR6(t- t')d((t')/[ 1 + aeR4(t- t')].

(29.28)

Invoking eq. (11.1), integrating over all t" from 0 to t and allowing for the initial condition ke - 0 at t - 0 yields

ke(t) = 2~ae

J(t'){R6(t- t')l[1 + aeR4(t- t')]}dt"

(29.29)

where, according to (26.7), the particle radius R and growth rate G are related by

R ( t - t') =

~i-t'

G(t") dt".

(29.30)

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: It(t) = It,0 exp (- 2~aeLs

J(t'){R6(t- t')l[1 + aeR4(t- t')]} dt').

(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 It(t) = It,0 exp {- 2r~aeLsNmR6(t)/[1 + aeRa(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 It(t) = I,,o exp [- 2TcaeLsN m Gc6v t6v/(1 + aeGc4 v t4v) V].

(29.33)

This expression simplifies to It(t) = It,0 exp [ - 2/r,aeLs(Nm/V) G6Vt 6v]

(29.34)

when the condition

t < 1/a~e/4vGc

(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 ae(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 Nm/V is sufficiently high to allow the departure of It from It,0 to be detectable at times satisfying (29.35). When this detection is possible only at later times

t > 1/ael/nVGc,

(29.36)

the particles are already so large that the unity in eq. (29.33) is negligible and the It(t) dependence takes the form It(t) = It,0 exp [- 21rLs(Nm/V ) G2cVt 2v]

(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 R ( t - t') = G~c(t - t') v so that we get It(t) = It,0 exp [- (21r.aeLsJs/ae (l + 2v)/4VGc) Yv(aeG4Vtav)]

(29.38)

where

Yv(x) = (1/4v)

[y(1 + 2v)/4v]( 1 + y)] dy.

(29.39)

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

Y1/2(x) = ( l / 2 ) [ x - In (1 + x)],

(29.40)

in this case eq. (29.38) leads to It(t) = It,0 exp {-(lrLsJs/aeGc)[aeG2c t2 - In (1 + aeG2ct2)]}.

(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 It(t) = It,0 exp [- (td2)aeLsGc3Jst4],

(29.42)

since In (1 + x) = x - x2/2 for x ~ 0 [Korn and Korn 1961 ]. We cannot be satisfied with finding It 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(1 +2v)/4Vdy = [1/(1 + 6v)]x (1 +6v)/4v

Yf,(x) = (1/4v)

for x ~ 0 and of the function

y(1-2v)/4Vdy= [1/(1 + 2v)]x (1 +2v)/4v

Yv'(x) = (1/4v)

for x ---) =,. Hence, one way to approximate Yv(x) is to use the relation 1/Yv = 1/Y~ + 1/Y~' which leads to

Yv(x )

... x ( 1 +

6v)lav[[1 +

6v + (1 + 2v)x].

(29.43)

As seen, this interpolation formula describes correctly Yv in the x ~ 0 and x---~ oo 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) 10 2

i

,

.

= w . , .|

, |

|

= = | ==|

i

|

,

........

i

|

i = =|1

10

10-1

10-2 10 1

,

1

.,,,I

10

10 2

Fig. 29.2 Dependence of Yv on x: solid curves 1/2 and 1 - eq. (29.39) at v = 1/2 and 1, respectively; dotted curves--the corresponding approximate eq. (29.43).

Induction time 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 Yl(x) calculated numerically from (29.39) at v = 1). The dotted curves show the approximate Yv(x) dependence (29.43) at the same v values. Substituting Yv from (29.43) in (29.38), we thus arrive at the desired formula for It(t) in the case of progressive stationary nucleation of particles which grow according to the power law (26.16) with arbitrary v > 0: It(t) = It,0 exp { - 2 ~ a e LsG6VJstl+6v/[1 + 6v + (1 + 2v)aeG4Vt4v]}. (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 It(t) = It,0 exp {- [2to/(1 + 6v)]aeLs G 6vJs tl§ 6v}.

(29.45)

We see that at v = 1/2 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 It(t) 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 It(t) dependence is of the form It(t) = It,0 exp {- [2n:/(1 + 2v)]Ls Gc2VJst'l + 2v}.

(29.46)

Having obtained analytical expressions for the It(t) dependence, we are in position to define the induction time ti that can be determined by means of the corresponding optical technique. Usually, ti is identified with the moment at which the relative decrement fld << 1 of It with respect to It,0 is large enough to be reliably detected. Mathematically, therefore, this definition reads

lit,0- lt(ti)]/It,O = ~d

(29.47)

and in conjunction with (29.34) and (29.45) leads to t i = [(1 + 6v)~d/2r~aeLs Gc6Vjs]1/(1 + 6v)

(29.48)

for progressive nucleation and to

ti = (~dV/21r~aeLsNm]l/6v(1/Gc)

(29.49)

for instantaneous nucleation, because In (1 - fld) --- -fld. Equations (29.48) and (29.49) are valid provided t i 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 t i = [(1 + 2v)flj2tCLs GZVJs]l/(1 + 2v)

(29.50)

424

Nucleation: Basic T h e o r y with Applications

for progressive nucleation and that

ti

=

(~dV/2ZCLsNm]~/2v(1/Gc)

(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 i 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 t i o n G c, 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 i already on Gc and Js. Thus, although according to both the volumetric and the optical definitions t i is qualitatively the same function of Gc and Nm or Js, quantitatively, inadequate usage of some of the formulae for ti for a fit to t i 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 Gc from eq. (29.15) in order to find the explicit ti(A~u ) dependence when A/~ 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 d 3 --- v0 we get

ti

=

Kve-A~/kT(1

_ e-Al~/kT)-6v/(1 + 6v)

exp [B/(1 + 6v)Ap 2]

(29.52)

[B/(1 + 2v)A/.t 2]

(29.53)

in the Rayleigh limit and

ti

=

K v e - Au/~T(1

- e-Z~u/kT) -2vi(1 + 2 v ) e x p

in the diffraction limit. Here the nucleation parameter B for HON or 3D HEN is specified by (13.49), the practically A/t-independent kinetic factor Kv(s) is defined as 2 6v l/(l+6v) K v - [(1 + 6v) i~d/21raeLsZfe * Covofe, s]

(29.54)

in the Rayleigh limit and as

K v = [(1 + 2v)~d/2zcLs Z fe * Co d2 fe,s zv ] 1/~l+2v)

(29.55)

in the diffraction limit, Co (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 Z fe* = f~,s, eqs (29.52)-(29.55) become

ti = Kle-~U/kT(1

-

e-a~/kT) -6/7 e x p (B/7A/~ 2)

KI = (7~d/2r~aeLsCov~) 1/7 (l/fe,s)

(29.56) (29.57)

in the Rayleigh limit and

ti

=

Kle-~/kr(1

-

e-AP/kV)-z/3 exp (B/3A~uz)

K~ = (3~d/2ZCLsCod~) 1/3 (1/fe,s)

(29.58) (29.59)

in the diffraction limit. Now, comparing eqs (29.18) and (29.56), we see that if instead of (29.56)

Induction time 425

we use eq. (29.18) for a fit to ti(Ap) 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'efof 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 ti 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 ti for isothermal variation of Ap is of the form ti = [(1 + 2v) fld/2TcaeLszf*CoG2cV]l/(l+2V)e (w*-Al~)/(l+2v)kT

(29.60)

in the Rayleigh limit and of the form ti = [(1 + 6v) fld/2JrLszf*CoG6V]l/(l+6V)e (w*-au)/(l+6v)kr

(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 Au/kr and Jf e,s'-",,Au/kVreplaced by f* and fs, 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 ti exhibits a minimum with respect to Ap or T. For example, with f* from (10.64) or (10.65) in lieu of f * e ~xu/kT, for HON or 3D HEN of cap-shaped nuclei eqs (29.20) and (29.21) yield ti = (3n:vor//~' , Z n a* t t C o V k T ) e ( A, -

All )lk T

exp (B/A~/2)

(29.62)

where nat t = [(1 - COS 0w)/2 I//2/3(0w) ] (co " o2/3 n*2/3/d 2) is the number of attachment sites at the nucleus surface. This formula gives ti 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)) fs = (~kT/3zCVorl) exp [(Ap - ~ + E1 + aao)/kT]

(29.63)

so that, upon replacing rd e , s ' -,,Au/kr by this expression for f s, from (29 18) and " (29.19) we get ti = (4ad/CgCooo)l/4(3;rcvorl/~kT) exp [(&- E1 - crao)/kT]

x e-AU/kr(1 -e-A#kT) -3/4 exp (B/4Ap2).

(29.64)

We recall that this formula implies a sufficient accuracy of the approximation zF ---fs. Similarly, in the same approximation and under the same conditions of nucleation and growth, from (29.56)-(29.59) and (29.63) it follows that

426

Nucleation: Basic Theory with Applications

t i = (7fld/27caeLsCoo2)l/7(31rOorl/~kT) exp [(&- E 1 - trao)/kT] • e-a~/~r(1

-e-Al~/kT)

-6/7 exp (B/7A~ e)

(29.65)

when the induction time is determined optically in the Rayleigh limit and that

t i = (3fld/2trLsCod2)l/3(31rVorl/~kT) exp [ ( ~ - E 1 - Crao)/kT] • e-A~/kr(1

-e-Alu/kT)

-2/3 exp (B/3AIu2)

(29.66)

if this determination is in the diffraction limit. When Ap 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 E l + o'a0 = 0), eqs (29.62), (29.64)(29.66) lead to the following dependences of t i on T: t i = (31t'OoFff)~*Znatt*CoVkT)

exp (Ase/k) exp (B'/TAT 2)

(29.67)

for the mononuclear mechanism (((t), Pl(t) or a(t) measuring technique),

t i = (4ad/CgCooo)l/n(3zrcoorl/~kT) exp (Ase/k) • [1 - exp ( - AseAT/kT)] -3/4 exp (B'/4TAT 2)

(29.68)

for the polynuclear mechanism (a(t) measuring technique), ti =

(7~d/27caeLsCov~)l/7(37CVorl/~kT) exp

(Ase/k)

• [1 - exp (- A s e A T ] k T ) ] -6/7 exp (B'/7TAT 2)

(29.69)

for the polynuclear mechanism (It(t) measuring technique, Rayleigh limit) and ti =

(3fld/2~LsCo d~)l/3(31rOorl/~kT) exp (Ase/k) • [1 - exp (- AseAT/kT)] -2/3 exp (B73TAT 2)

(29.70)

for the polynuclear mechanism (It(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 Co is specified by (7.10), (7.11) or (7.12). It should be noted that, plotted in T-vs-ln ti coordinates, the ti(T) dependences (29.67) and (29.68) represent the so-called TTT (time-temperaturetransformation) 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 ti on A/d are perhaps most often done with supersaturated solutions (e.g. Nielsen [1969]; Kirkova and Djarova [1971, 1977]; S6hnel and Mullin [1978, 1979, 1988]; Koutsoukos and Kontoyannis [ 1984]; Kubota et al. [1986]; Wojciechowski and Kibalczyc [1986]; Sangwal [1989]; van der Leeden et al. [1991, 1992, 1993]; Verdoes et al. [1992]; Symeopoulos and Koutsoukos [1992]; S6hnel and Garside

Induction time

427

[1992]; Mullin [ 1993]; Jakubczyk and Sangwal [ 1994]; Hendriksen and Grant [1995]; Sangwal and Polak [1997]). The circles in Fig. 29.3 represent the ti(S) data of Verdoes et al. [ 1992] for precipitation of CaCO 3 in an aqueous solution at T = 298 K. The induction time at a given supersaturation ratio S -" I'I/H e = e 6u/kr (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 Gc o~ ( S - 1)1.8. Using these findings for v d and Gc and the classical formula (4.39) for W* transforms (29.14) into t i = K ' S - 1 / 4 ( S - 1)-135 exp (B74 In 2 S)

8.5

,

,,,

,

,

.

,

,

,

,

.

,

,

,

,

,

.

.

,

,

.

,

(29.71) ,

,

.

8.0 t.~

"-" "7 09

75

v

09

7.0

e,..

"i

6.5

6.0

f

0.3

. . . .

'

0.4

. . . .

'

0.5

. . . .

'

0.6

. . . . .

'

0.7

. . . .

'

0.8

....

"

0.9

1 / (INS) 2 Fig. 29.3 Supersaturation dependence o f the induction time: circles - data f o r precipitation o f C a C 0 3 crystallites in aqueous solution at T = 298 K [Verdoes et al. 1992]; line - best f i t according to eq. (29.71) (ti is in seconds).

where K" (s) is a practically S-independent factor and B'is given by (13.67). This equation predicts a linearization of the ti(S ) dependence in In [ S ~ / 4 ( S 1)l35ti]-vs-(1/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 O'ef= 37.3 mJ/m 2 and n* = 3 to 14 for the number of CaCO3 molecules constituting the nucleus in the S range studied. This relatively low value of O'efis an indication for 3D HEN of the CaCO3 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 ti(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 i with the supersaturation Ap of the metastable old phase is largely determined by the Js(A/~) dependence. This allows the nucleation theorem to be used for a theory-independent determination of the nucleus size n* from experimental ti(A,u) 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 ti(A/z) 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 ti in the unified form ti = (gi/ Gq)l/(1

+ q)

e(W* -

Ap)/(1 + q)kT.

(30.1)

Here q = O, Ki = 1/A'V

(or Ki = 1/A'As)

(30.2)

for t i by the mononuclear mechanism, q = vd,

Ki = (1 + vd)ad/C,r

(30.3)

for volumetrically measured ti (polynuclear mechanism), q = 6v,

Ki = (1 + 6v)fld/2r~a~LsA"

(30.4)

for optically measured t i in the Rayleigh limit (polynuclear mechanism) and q = 2v,

Ki = (1 + 2v)fld/27rLsA"

(30.5)

for optically measured ti in the diffraction limit (polynuclear mechanism). Then, taking the logarithm of both sides of (30.1), differentiating with respect to Ap, applying the nucleation theorem in the form of eq. (5.29) and neglecting the possible relatively weak dependence of Ki on A/z, we obtain n* = - ( 1 + q)kTd(ln ti)/dA/z - 1 -qkTd(ln Gc)/dA/~.

(30.6)

This is the sought general formula for n* at isothermal variation of Ap. 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 ti(A/.t) data we have independent information about the value of q and the dependence of the growth constant (or rate) Gc on Ap.

Fourth application of the nucleation theorem

429

For the mononuclear mechanism, however, knowing the Gc(Ap) function is not necessary. Then q = 0 and the slope of the ti(A/.t ) c u r v e in In t i - v s - A ~ coordinates is directly related to n*: n* =-kTd(ln

ti)/dAl.t

- 1.

(30.7)

For the polynuclear mechanism the contribution of the In Gc 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(ln G c ) / d A p = (1 - e-aU/kr) -1

(30.8)

so that for AIu/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 ti(Ap) data obtained by isothermal variation of A/~ 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 ( l n ti)/d(ln S) - 1

(30.9)

f o r t i by the mononuclear mechanism,

n* = - ( 1 + vd)d(ln ti)/d(ln S ) - 1 - vdd(ln Gc)/d(ln S)

(30.10)

for volumetrically measured t i (polynuclear mechanism), n* = - ( 1 + 6v)d(ln ti)/d(ln S ) - 1 - 6 v d ( l n Gc)/d(ln S)

(30.11)

for optically measured t i in the Rayleigh limit (polynuclear mechanism) and n* = - ( 1 + 2v)d(ln ti)/d(ln S) - 1 - 2vd(ln Gc)/d(ln S)

(30.12)

for optically measured t i 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 + v d = 4 and 1 + 6v = 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 ti(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 i are necessary for the different experimental techniques. This necessity merely reflects the fact that ti p e r 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 A/~ = 0 and Ap = A/~s (see Chapter 1). The spinodal supersaturation A/~s which corresponds to the spinodal point (if this exists) is the thermodynamic limit of metastability, since for A/~ __. A/~s there is no barrier to nucleation (then W* = 0). In reality, however, metastability is lost at a certain critical s u p e r s a t u r a t i o n A/~c < A/~s at which, though not yet vanishing, W* is already small enough for nucleation to occur at a perceptible rate. Historically, the problem of finding Ape is among the first addressed by the theory of nucleation [Volmer 1939]. Since A/~c is rather easily obtainable experimentally, many studies were carried out with the aim of predicting, measuring and/or analysing the A~c 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]; Chemov [ 1980]; Skripov and Koverda [1984]; Hale [1986, 1988]; Dillmann and Meier [1989, 1991]; Kashchiev et al. [1991]; Mullin [1993]; Nuth III and Ferguson [1993]; Jakubczyk and Sangwal [1994]; Strey et al. [1994]; Laaksonen et al. [1994]; Baidakov [1995]; Fisk et al. [1998]). In contrast to A/~s, however, A/~c 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/~c can be determined in some cases of formation of condensed phases. One way to define A/~c is to state that it is that value of A/~ at which the stationary nucleation rate Js takes a sufficiently high value Js,c for nucleation to become detectable [Volmer 1939]. Mathematically, therefore, A/~c is the solution of the algebraic equation Js(Ab~c) = Js,c-

(31.1)

For nucleation in the volume of the old phase or on a substrate Js,c = 1 supernucleus per second per cm 3 or cm 2 is often used [Volmer 1939; Hirth and Pound 1963], but any other Js,c 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

A/~c = [B/In (A/Js,c)] 1/2

431

(31.2)

where A and B are given by (13.40) and (13.49). As seen, A/~c 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/.tc (e.g. Katz [1970a, b]; Katz et al. [1975, 1976]; Dillmann and Meier [1989, 1991]; Strey et al. [1994]). Nonetheless, the weak dependence of A/Ic on A allows setting A = A' and using eq. (31.2) in the form (31.3)

AlUc = CkB1/2

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 (A' Us,c)]- 1/2

(31.4)

where A' is given by eq. (13.42) or (13.43). IfA' depends on the nucleus size n*, 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 1"1and the other T-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)l/2uo(tYef/kT)3/2.

(31.5)

Here Ck is related to A' from (13.42), and the critical supersaturation ratio Sc is given by Sc=Pc/Pe,

Sc=IcH e

(31.6)

for nucleation in vapours (cf. eq. (13.68)) and by Sc = CclCe,

Sc = aclae,

Sc = Hc/He

(31.7)

for nucleation in solutions (cf. eq. (13.69)), Pc, Ic, Cc, ac and Flc being the values of p, I, C, a and H at which Js is equal to Js,c. Equation (31.5) reveals that in a temperature range in which O'ef changes relatively little, the critical supersaturation ratio decreases with increasing temperature according to the law In ScooT -3/2 [Volmer 1939]. Recalling that for HON O'ef-" O" and for 3D H E N O'ef < O', from ( 3 1 . 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 Sc 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 tYef= or, c 3 = 367r (spherical nuclei) and typical A' = 1035 m -3 s-1, from eqs (12.19) (31.4) and (31.5) we find that (flSST = 0.2 to 0.6) In Sc -- 0.5Oo( ty/kT) 3/2 = 0.5(flSST/~,]kT) 3/2

(31.8)

for HON of condensed phases in vapours or liquid solutions at a rate of Js,c = 1 supernucleus per cm 3 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 v0 - 0.05 nm 3 and ty= 10 to 100 mJ/m 2 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 Ap is controlled by T according to (2.23). In this case, upon setting T-- Te in eq. (13.49), from (31.3) it follows that (e.g. Walton [ 1969a]) ATc/T e = Ck(4C3/27)l/2(k/)0]ASe) (tyef/kTe)3/2

(31.9)

where the critical undercooling ATc is given by (cf. eq. (2.22))

ATc = Te- Tc,

(31.10)

and Tc is the value of T at which Js = Js,c. Again for HON, with O'ef= O" and the above values of c 3 and A; from eqs (12.19) (used with ~,= TeAse), (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

ATclTe = 0.5(kvo/Ase)( ty[kTe) 3/2 =

0.5/"SSTR 3/2

(Aselk) 1/2.

(31.11)

With flSST = 0.4 and typical Ase/k = 1 to 5 this yields ATc/Te = 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 Apc is to identify it with that Ap 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 t i, since it corresponds to the moment of detecting the onset of the phase transformation. According to this definition, Apc is thus the solution of the algebraic equation [Kashchiev et al. 1991 ] ti(A/lc) = ti,c

(31.12)

where ti,c (s) is the arbitrarily chosen time mentioned above. Evidently, in manifestation of the kinetic nature of A/~c, the usage of different ti(A/.t) dependences in eq. (31.12) will result in different A/lc values. This will be seen explicitly below with the help of concrete expressions for t i from Chapter 29.

Metastabili~ limit

433

The ti(Ap) dependence for various definitions of the induction time is given in a general form by eq. (30.1) which is valid both when Ap is changed isothermally and when it is varied by means of T. Using this equation with Ap 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, Apc, Sc and ATc are again given by (31.3), (31.5) and (31.9), but with . l+q

ck = [In (Gqti,c /Ki) ]

-1/2

.

(31.13)

Here q and Ki 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 C k --

[In (A'Vti,c)] -1/2

(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 A s of the substrate surface). Comparing eqs (31.4) and (31.14), we see that they are identical when Js,c is chosen to be equal to 1 supernucleus formed in the volume V (or on the area As) within time ti,c. The situation is different, however, for the polynuclear mechanism which is characterized by q > 0. Then the kinetic factor Ck 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 Ki from (30.3)-(30.5), from eq. (31.13) it follows that for the polynuclear mechanism ck = {In [cgA'GVc d'l+vd ./ ( 1 . + v. d ) .a d ] ti, c

} 1/2

(31.15)

for volumetrically measured Apc, Sc or ATc, ck = {In [21rgleLsA'~ ._.~6v'l+6v t~x /(1 + 6v)

~d]}

-1/2

(31.16)

for optically measured Apc, Sc or ATc in the Rayleigh limit and l+2v/(| ck = {ln [2nrLsA,---2v t/c tix

+

2v)

f l d ] } -1/2

(31.17)

for optically measured Apc, Sc or ATc in the diffraction limit. Equations (31.14) and (31.15) show that while for the mononuclear mechanism Ck (and, thereby, Apc) 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 Ls of the path of the light beam through the scattering medium. To get a feeling about the numerical difference between Ck calculated from eq. (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 = 47r/3, d = 3 and v = 1). With V = 1 cm 3 t i c = 1 s, a d -- 10-6, ~ d -- 10-3, ae = 103 pm -4, Ls = 1 cm, the already used A' = 1035 m -3 s-1 and an exemplary Gc = 1 pm/s, eqs (31.14)-(31.17) yield Ck --- 0.122 for the mononuclear

434

Nucleation: Basic Theory with Applications

mechanism and Ck --- 0.137 (volumetrically), Ck = 0.127 (optically, Rayleigh limit) and Ck---0.133 (optically, diffraction limit) for the polynuclear mechanism. The value of Ckfor the mononuclear mechanism coincides with that following from eq. (31.4) with A' = 1035 m -3 s-1 and Js,c = 1 cm -3 s-1. The rather small difference between the above Ck values implies that even though obtained by various experimental techniques, the Apc 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 Ape, Sc or ATc 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

,e, 20

9 1014

14

9 101~ I

101~

10 6

X

15

9 10 6 9 10 6

r

O3 10

220

240

260

280

300

320

T (K) Fig. 31.1 Temperature dependence of the critical supersaturation: symbols- data of Volmer and Flood [1934] (crosses), Katz and Ostermier [1967] (up triangles), Heist and Reiss [19731 (circles), Miller [19761 (squares), Anderson et al. [19801 (down triangles) and Viisanen et al. [1993] (diamonds)for HON of water droplets in vapours at Js.c = 106, 1010 and 1014 m -3 S -1 (as indicated); curves 106, 10 ~~and 1014- eq. (31.18) with Js,c = 106, 1010 and 1014 m -3 s -1, 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 Js,c - 106 to 1014 m -3 s-1 (as indicated). The curves display the approximate Sc(T) dependence In Sc = { 16zd3 In [(~voCoPe/Js,ckT)(2o'/lt'n'to)l/2]}

1/2

Vo(Cr/kT)

3/2

(31.18)

which follows from eqs (13.44), (31.4) and (31.5). The calculation is done

Metastability limit

435

with Co from (7.53), ~'*, v0 and m0 from Table 3.1 and cr and Pe determined from [Dillmann and Meier 1991 ] or(T) = 93.6635 + 0.009133T-0.000275T 2 log [pe(T)] = 21.426045 - 2892.3693/T- 2.892736 log T-4.9369728 x 10-3T

+ 5.506905 x 10-6T 2 -4.645869 x 10-9T 3 + 3.7874 • 10-12T 4 where cr is in mJ/m 2, Pe is in Pa, and T is in K. The different curves correspond to Js,c = 106, 101~ and 1014 m -3 s -1 (as indicated). We see that the Sc(T)

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 e x c l u s i o n zones, are schematized in Fig. 32.1 by the grey circles. The space

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 supernuclei 437

(32.1)

Nm <- Na .

Our task now is to find out how much less Nm is than Na. To that end we shall determine Nm as a function of Na, the nucleation rate Ja (s-l) per active centre and the parameters of growth of the exclusion zones. Before doing that, however, we note that Nm 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 supernuclei in a nucleating system are, e.g. the exhaustion of the supersaturation in the system or the coalescence between the growing supernuclei. 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 et al. 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; Detsik et al. 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 supernuclei 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 t' 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 V z ( t ; t ) =czR3z =Cz

Is:

Gz(t')dt"

]3

(32.2)

where Cz is a numerical shape factor (e.g. Cz = 4n/3 for spherical zones), and Gz(t) - dRz/dt 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 t > 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 Nucleation: 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 Pf we shall have Pf = PnoPni . Realizing that NaP f has the physical significance of Na - N in eq. (25.13), we can use this equation with N a - N replaced by NaP f 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 N(0) = 0 thus yields

N(t) = Na

Ja(t')Pno(t')Pni(t') dt'.

(32.3)

To proceed further we must know the probabilities Pno and Pni- Since the probability Pno for non-occupation is equivalent to the (Na - N)INa ratio in eq. (25.16), we can write Pno(t) = exp [- Nex(t)lN a]

(32.4)

where the extended number Nex 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 w e shall have Pni(t) = exp [- Vex,z(t)/V]

(32.5)

where the extended volume Vex,z of the zones is given by (26.5) with J and Vn replaced by Ja and Vz from (25.10) and (32.2):

Ja(t')Vz(t', t) dt'.

Vex,z(t) = Na

(32.6)

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) = N a

J a ( t ' ) exp

-

Ja (t")[1 + N a V z (t'; t')/V] dt"

dt'.

(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 (V~ = 0) or when Na 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 ~e 0), eq. (32.7) takes the form

N(t) = Na

l

Ja(t') exp {- (Na/V)

i'

Ja(t")Vz(t", t') dt"} dt'.

(32.8)

Maximum number of supernuclei 439

When the zones coincide with the supernuclei themselves, we have Vz - Vn where V, is given by (26.7). Then, in 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 Vz if Na does not satisfy the condition Na >> 1 [Markov and Kashchiev 1972a]. Having found the N(t) dependence, we can now determine the maximum (in fact, the saturation) number Nm - N(oo) of supernuclei formed on the Na active centres in the system. Setting t = ~ in eq. (32.7), we get Nm = Na

Ja(t)

exp -

J a ( t ' ) [ 1 + NaVz (t;

t)/V] dt" dt. (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 Na 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 = Na

Ja(t) exp

-(Na/V)

Ja(t')Vz(t', t) dt" dt

(32.~0)

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 - V,,, i.e. when the supernuclei 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 V and Vz in eqs (32.5)-(32.10) by, respectively, the area As of the substrate surface and the area Az of a zone on the substrate. Analogously to (32.2), Az is given by

Az(t;t) =c~R2z =Cz

G,~(t")dt"

(32.11)

where Cz is a numerical shape factor (e.g. Cz = ~ for circular zones). To see explicitly what is the effect of Na, 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 = (Gc.zt) v.

(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 Gc,z (m 1/v s-l) is the zone growth

440 Nucleation: Basic Theory with Applications

constant. When the zones coincide with the supernuclei themselves (then Rz = R), we have Gc,z - Gc, Gc being the growth constant of the supernuclei.

32.2 Instantaneous nucleation Similar to J(t) from (26.13), in this case Ja(t) can be represented as Ja(t) = 6D(t),

(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 t~D, noted in Section 26.2, from (32.9) we find that in the IN case, for both volume and surface nucleation, Nm = Na

(32.14)

regardless of the growth law of the zones. Physically, this result is obvious: as the supernuclei appear simultaneously on the active centres at t = 0, none of the centres can be ingested by the zones formed around the supernuclei, no matter how fast the zone growth may be.

32.3 Progressive nucleation In the case of PN in stationary regime Ja(t) equals the time-independent stationary nucleation rate Ja, s per active site. Also, for zone growth according to (32.12), from (32.2) and (32.11) we have 3v 2v Vz(t', t) = czGc,z (t - t ,)3v , Az(t , t) = czG~,z (t - t ,)2v ,

(32.15)

so that eq. (32.7) leads to the following N(t) dependence:

N(t) = Na

~~a,st

exp [- x - (Bzx)1 § vd] dx.

(32.16)

Here the zone dimensionality d and the numerical factor Bz > 0 are given by d = 3, B z = [ c z N a G~,~/(1 3v + 3v) vl3v]l/(l+3v) -~a,~,

(32.17)

for volume nucleation (3D zones) and by

d = 2, Bz = [czNaGc2,Vz/(1 + 2V)As Ja,s]2V 1/~1+2v)

(32.18)

for surface nucleation (2D zones). Accordingly, for the maximum number of supernuclei in the considered case of PN, from (32.9) or (32.16) we obtain Nm = Na f o

exp [- x -

(Bzx)1 § vd] dx.

(32.19)

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 supernuclei

441

(32.12), inspection of these equations shows that the character of the N(t) dependence and the departure of Nm from Na is controlled by the parameter B z. 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 supernuclei 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 Bz >> 1,

(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 f Ja,sBzt

N(t) = (Na/Bz)

exp ( - x 1 + vd) dx

(32.22)

,tO

Nm = 1-'[(2 + vd)/(1 + vd)]Na/Bz,

(32.23)

F being the gamma-function defined by (26.21). Equation (32.23) reveals that in the Na ~ oo limit Nm is not any more a linear function of Na: with Bz from (32.17) and (32.18), it follows that Nm = F[(2 + 3v)/(1 + 3v)][(1 + 3v)V/cz] 1/~1+ 3v)(gaJa,s]Gc,z)3V/(l+ 3v) (32.24) for volume nucleation (d = 3) and that Nm = F [ ( 2 + 2 v ) / ( 1 + 2V)][(1 + 2V)Asicz] 1/~1 +

2V)gM ! a,s..~J Ir c,zj,~2v/(l + 2v) k~t, a~,

(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 Ja, s and Gc,z. For example, when the zones grow with time-independent rate (v = 1), we have N m o,: N3a/4 and N m o,: N2a/3for volume and surface nucleation, respectively. We note also that when the role of the zones is played by the supernuclei themselves, we have Cz = Cg, Gc,z = Gc and in view of (25.3) and (32.17), as it should be, Nm from (32.24) becomes identical with Nm from (26.26). Knowing the limiting dependences of Nm on Na, eqs (25.9) and (32.23), we can now employ them to find an interpolation formula for Nm 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 = Na{ 1 + Bz/F[(2 + vd)/(1 + vd)]} -~.

(32.26)

This Nm(Na) dependence is in fact an approximation to the exact one given by eq. (32.19). With Bz from (32.17) and (32.18) it leads to N m = N a {1 + [czNaGc3,Vz](1 + 3v) Vl3V]l/(l+3v)[r'[(2 + 3v)/(1 + 3v)]} -1 --va,s J for volume nucleation (d = 3) and to N m = N a {1 +

[czNaGc2,Vz/(1 + 2V)AsJ2V]l/(l+2v)/F[(2 a,s

+ 2v)/(1 + 2 v ) ] } -1

(32.28) for surface nucleation (d = 2). As seen, for sufficiently small or large Na (or Bz) 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 10 5

,,,

8xlO 4

_. . . .

105. "''. /

6x10 4 Z

~

4x10 4

2x10 4

0

4

0

1

2

3

4

5

Oa,$ t Fig. 32.2

Time dependence of the number of supernuclei on active centres: solid curves - eq. (32.16)for nucleation in the presence of exclusion zones; dashed curves - e q . (25.7)for nucleation in the absence of such zones (Na values indicated).

drawn according to eq. (32.16) atNa = 2 • 104, 6 x 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 = 4 ~ 3 (spherical zones), V = 1 cm 3, Ja,s = 0.1 s-1 and Gc,z = 10/.tm/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 Na ones. Figure 32.3 illustrates the decrease of the Nm/Na ratio with Bz, i.e. with Na.

Maximum number of supernuclei |

1.0~ -

'

. . . . . . . .

i

.

_ . .

"l

"| "|

"

=|

..

~.~

i

"

|

|

|

. . . . . . . . . . . . . .

|

~

|

i

.......

"~\

0.6

Z

E

I .

i

|

|

,

|

|

I T

"~. . . . . . .

"...C&Z

0.8

Z

r.='

443

~

, "

"..

~O ".. "', , ~ 0

0.4

".

~~,~".

." ".

~

m ..

0.2 0 10.2

| i i i iI

101

1

10

Bz Fig. 32.3 Dependence of the Nm/Na ratio on the zone parameter Bz: circlescomputer 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 Nm(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(Na) 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 f ~(l+l/q)N't/Nm

N(t)/Nm = [F(1 + l/q)] -t

exp ( - x q) dx

(32.29)

to analyse N(t) data for electrochemical nucleation under potentiostatic conditions, i.e. at constant supersaturation. Here N" - (dN/dt)t.-o and Nm are obtainable from the initial slope and the plateau of a given experimental N(t)

444

Nucleation: 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 CuSO4 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. 1.0

0.8

Z Z

E 0.6 0.4

0.2 0

0

0.5

1.0

1.5

N't/N

2.0

2.5

3.0

3.5

m

Fig. 32.4 Time dependence o f the number o f supernuclei on active centres: symbols - d a t a f o r electrochemical nucleation of Cu crystallites on a graphite electrode in aqueous solution of CuS04 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 Nm(Na) dependence was studied experimentally by Stenzel and Bethge [ 1976] in nucleation of Au crystallites on the (100) face of an NaC1 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 Nm/Na ratio is expected to decrease with the increase of the concentration Na/As of Ca ions on the substrate surface according to Nm/Na = / o

exp [- x -

(32.30)

(coNa/As)x 2] d x

where Co = czGc,z/2Ja,s is a constant at constant supersaturation. The circles in Fig. 32.5 represent the Nm(Na) data of Stenzel and Bethge [1976] for nucleation at substrate temperature T = 473 K and impingement rate I = 3 x 1016 m -2 s-1. The Nm/Na 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 p e r / l m 2 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 Co = 170 nm 2 and is seen to reproduce the experimentally found drop of Nm/Na with Na/As. The dashed line in Fig. 32.5 indicates the value of Arm in the absence of zones. The departure of the experimental Nm(Na) data from this line at N a / A s - 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

,

.

.

,

.

.

!

.

i , , ,

.

.

.

I

.

.

l

.

.

.

.

'

.

.

'

.

.

|

.

~ ' ' l l

.

.

.

.

,

.

.

.

.

.

,

.

.

I

.

.

,

.

' ' ' ' 1

.

.

.

.

,

.

.

.

.

.

,

.

.

i

.

.

i ' l l | |

.

.

.

0.8

Z Z

(~

E

0.6 0.4 0.2 0

1013

al I

1014

i

,

i

1015

I

1016

i

1017

N a / A s (m "2)

Fig. 32.5 Dependence of the Nm/N a ratio on the concentration o f active centres: circles - data for nucleation in ultra high vacuum of Au crystallites on the (100) face o f NaCl at T = 473 K and I = 3 x 10 ~6 m -2 s -1 [Stenzel and Bethge 1976]; curve - eq. (32.30) with Co = 170 nm2; dashed line - eq. (25.9)for nucleation in the absence of exclusion zones.

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 ((t) of all supernuclei in the system, but also their size distribution function Zn(t) or Z(n, t). In the case of the Szilard model of nucleation Z~(t) 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(n, 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 = n*, i.e. for clusters of nearnucleus 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 n l < n < n2 where the left and right ends n l and ne 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 flux 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 Kn(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(n 2, t) = j(n 2, t)/v(n2, t)

(33.1)

which follows from (9.29) upon neglecting the 'diffusion' f l u x - f(ff-Z/oan). Here j(n2, t) is the flux of supernuclei at the right end n2 of the nucleus region, and u(n 2, t) is the growth rate of the nz-sized (i.e. the smallest stable) supernucleus. While the latter is obtainable from eq. (9.30), j(n 2, 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 < n 2 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) = dR/dt are related to Z(n, t) and v(n, t) = dn/dt according to

F(R, t)dR = Z(n, t) dn G(R, t) = v(n, t) dR/dn.

(33.2) (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 -4 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)

__0 F(R, Ot

t) + 0-~ [G(R, t) F(R, t)] = 0

(33.4)

F(R, 0) = 0

(33.5)

F(R2, t) = J(t)/G(R2, t).

(33.6)

Here R2 is the radius of the n2-sized supernucleus, and RM is the radius of the

448 Nucleation: 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 = oo. 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) = GI(R)G2(t).

(33.7)

This presentation of G as a product of two known functions GI(R) > 0 and G2(t) > 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 GI(R ) = VGcR1-l/v,

G2(t ) = 1.

(33.8)

With these GI(R) and Gz(t) functions, G from (33.7) coincides with G from (26.15). Let us now find the size distribution F(R, t) 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 [ 1975b]). 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 supernucleus and the earlier moment t' < 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 = R 2 at t = t', we get

[1/GI(R")] dR" =

G2(t') dt".

(33.9)

2

This formula represents implicitly the functions R(t', t) and t'(R, t) which relate at any time t the radius R and the moment t' of birth of a given supernucleus when its growth is not disturbed by contacts with neighbouring supernuclei. The supernucleus born at t' = 0 is the biggest in size and its radius Rb is obtainable from

•b

[ 1/GI(R")] dR" = ~i G2( t " ) dt".

(33.10)

2

In some cases eqs (33.9) and (33.10) lead to simple explicit t'(R, t) and R(t', t) dependences. For instance, with the aid of G1 and Ge from (33.8) we find that t'(R, t) = t - (IlGc)(R 1Iv - R~ Iv) = (I/Gc)[ R~/v (t) - R l/v] (33.11)

R(t', t) = [ Rl2/v + G c ( t - t')] v

(33.12)

Size distribution of supernuclei

Rb(t) = ( R~/v + Gct) v

449

(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 ( = - J(t')dt', the minus sign taking into account that a greater radius R corresponds to an earlier moment t' of formation. Hence, we have

F(R, t) dR = -J(t') dt'.

(33.14)

In this relation, due to (33.9),

dR =-GI(R)Gz(t') dt"

(33.15)

so that it takes the form [Roginsky and Todes 1940; Todes 1940, 1949b]

F(R, t) = J[t'(R, t)]/GI(R)Gz[t'(R, t)],

R 2 <_R <_Rb(t)

(33.16)

F(R, t) = O, R>Rb(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) t' and R b a r e 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(R, 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) Ot'/Ot = Gz(t)/Gz(t') and &'IOR = - I/GI(R)Gz(t" ). 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 ofj(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) = JR2 [t'(R, t)]/GI(R)Gz[t'(R, t)], F(R,t)=O,

R 2 _< R __. Rb(t )

R>Rb(t )

where JR2 (t) - j ( n 2, t) is the flux at R - R 2, and the R,t dependence of t' 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 nucleationactive 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 Pno of nonoccupation and Pni of non-ingestion given by (32.4) and (32.5). Understandably,

450

Nucleation: Basic Theory with Applications

like J in (33.16), Pno and Pni must refer to the moment t' of appearance of the R-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)]/GI(R)G2[t'(R, t)]}Pno[t'(R , t)]Pni[t'(R , t)], R2 < R < Rb(t)

F(R, t) = 0,

(33.17)

R > Rb(t)

where

Pno(t')Pni(t') = exp

-

Ja(t')[1 + NaVz(t'; t')/V] dt" . (33.18)

In these expressions the R, t dependence of t' is obtainable from (33.9), J and Ja 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 Vz = V,, where the individual volume V~ of a supernucleus is given by eq. (26.7) in R2 = 0 approximation. In this case eq. (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 Ja 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 Vz = 0) the size distribution (33.17) simplifies to

F(R, t)= {J[t'(R, t)]/GI(R)G2[t'(R, t)]} exp

-

J a ( t ' ) dt" , R2 <- R <_Rb(t)

F(R, t) = 0,

(33.19)

R > Rb(t)

where eqs (25.10) and (25.11) connect J and J~ 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) = {J[t'(R, t)IlGI(R)G2[t'(R, t)]} exp

F(R, t) = 0,

-

J ( t ' ) Vz[t'; t R, t)] dt"

, R2 < R <- Rb(t) 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 Pnoeni 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, t), we can determine the time dependences of a number of quantities of experimental interest. Such quantities are, e.g. the most probable radius R m of stable supernuclei, the value Fm of the maximum of F, the average radius Rav of detectable stable supernuclei, the total concentration ~" (m -3 or m -2) of detectable stable supernuclei and the detectable nucleation rate J' (m -3 s-1 or m -2 s-l). By definition, Fro(t) = F[Rm(t), t],

(33.21)

where Rm(t) is the solution of the equation

[tgF(R, t)[tgg]R=Rm(t )

= O.

(33.22)

Also by definition, Ray(t) = [('(t)] -1

RF(R, t) dR,

(33.23)

dR'

and according to (11.13) and (33.2), R,b(t)

('(t) =

F(R, t) dR.

(33.24)

Here R' _ R 2 is the radius of the smallest detectable stable supernucleus (this supernucleus consists of n' _ 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

('(t) =

f

t~ (t)

J(t')Pno(t')Pni(t" )dt"

(33.25)

,~0

which follows from (33.24) with the aid of (33.9), (33.10), (33.15) and (33.17). Here the PnoPni product is given by (33.18), and t'R,(t), defined by (cL eq. (33.9))

[I/GI(R")] dR" = 2

G2(t") dt",

(33.26)

t~

is the moment of appearance of those stable supernuclei that at time t have the smallest detectable radius R'. For example, for supernucleus growth characterized by GI(R) and Gz(t) from (33.8), eq. (33.26) leads to

452

Nucleation: Basic Theory with Applications

t'R' (t) = t -- tg

(33.27)

tg = (1/Gc)(R "l/v - Rzl/V).

(33.28)

where tg is given by

According to eq. (33.11), tg 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, tg 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 R2-sized supernuclei are detectable (then R' = R2), we have t'R,(t) = t and tg = 0. In this case, in the absence of pre-existing supernuclei (~'0 = 0) and of confinements from active centres (Pno = 1) and exclusion zones (Pni = 1), ~" from (33.25) is identical with ~"from (11.10). Finally, the time dependence of the detectable nucleation rate J' is obtainable from the expression

J'(t) = {G2(t)/G2[t~,(t)]} J[t~,(t)] Pno[t'g'(t)] Pni[t~'(t)]

(33.29)

which results from (11.11) and (33.25), since owing to (33.26), dt'R,/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 J',

J'(t) = G(R', t)F(R', t),

(33.30)

which follows from eqs (11.14), (11.15) (with neglectedf' summand), (33.2) and (33.3). When the growth of the detectable supernuclei is characterized by GI(R) and Ge(t) from (33.8), thanks to (33.27), J' from (33.29) simplifies to (t > tg)

J'(t) = J ( t - tg)Pno(t- tg)Pni(t- tg)

(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 J' 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 = tg 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 6D, i.e.

Size distribution of supernuclei 453 J(t) = (m~(t)

(33.32)

where (m = Nm/V or (m = Nm/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(R, t) = ( m 6 o [ R - Rb(t)]

(33.33)

where it is taken into account that 6D(X) = 0 for x ~ 0, 6D(-- X) = 6D(X), 6D(aX) = (1/a)6D(X) and 6D(Xa - X~) = (1/ax a - 1)~)(X- X0) [Korn and Korn 1961] (the first of these equalities allows omitting the PnoPni 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 ) - (Gct) 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 R b. Using F(R, t) from (33.33) in (33.23) and (33.24), we find that in the IN case the average radius Ray and the total concentration ~' of detectable supernuclei are stepwise functions of time: Ray(t) = 0 and ('(t) = 0 for 0 _ t < tg, and Ray(t) = Rb(t) = (Gct) v ('(t) = (m

(33.34) (33.35)

for t > tg, tg = R'~/V/G c being the growth time of the smallest detectable supernucleus (see eq. (33.28)). Also, from (33.31) and (33.32) it follows that

J'(t) = ~mt~D(t- tg).

(33.36)

Thus, in the IN case the detectable rate J' is the same pulse function of time as the nucleation rate J, but the pulse is at t = tg rather than at t = 0, because tg is the moment of appearance of the smallest detectable supernuclei. The F(R, t), Rav(t) and ~'(t) dependences (33.33)-(33.35) are illustrated in Figs 33.1-33.3. The size distribution is shown at times tl and t2 > tl at which the supernuclei are, respectively, undetectable and detectable. 2. Progressive nucleation (a) Stationary regime In this case both J 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 v in 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 l/v-1 exp [-(Ja.s/Gc)(R~/v - R 1Iv) - (BzJa.s/Gc)l+Vd(R~/v _ R1/V)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

undetectable

detectable

Fm

1

U_

,,,

0

Rb(tl)

I

R'

Rb(t2) R

Fig. 33.1

Size distribution of supernuclei in the case of IN according to eq. (33.33) (at time t2 > t/ the supernuclei are already detectable).

n;

n," R

!

i

i. t I" f f o~~

~

o

I"

t"1"11

"1

t~

"

I

!

tg

t2 g

Fig. 33.2

.

Time dependence of the average radius Ray of detectable supernuclei (the solid line) and of the radius Rb of the biggest supernucleus (the dot-dashed line) in the case of IN according to eq. (33.34).

Size distribution of supernuclei 455

T __._._1

0

t~

t

tg

t2 t

,,

ram-

Fig. 33.3

Time dependence of the concentration of detectable supernuclei in the case of IN according to eq. (33.35). 1~llv-I

Fm (t) = ( J s / V G c ) " ' b

= (Js/VGc 1/v ) t 1/v-1

(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)[F m = (R[Rb) l/v- 1 exp {- Ja,st[ 1 _

-

(R[Rb) l/v]

(BzJa,st) 1 + vd[ 1 - (R/Rb) 1/v]l + 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 t through Rb 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 < Rb(t)) F(R, t) = (Js/VGc)R l/v- 1 = Fm(R/Rb)l/v-1

(33.40)

for both volume and surface nucleation. The absence of Ja,~ in this formula indicates that it is applicable also to HON. In the particular case of v = 1 (constant growth rate Gc of the supemuclei), according to (33.40), the differently sized supernuclei have the same concentration: F(R, t) = Js/Gc = constant [Roginsky and Todes 1940; Todes 1940]. At a later stage, if Bz 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) = (Js/VGc)R l/v- 1 exp [- (Ja,s/Gc)(Rb l / v - R1/V)] = Fm(R/Rb)l/v-1 exp {- Ja, st[1 - (R/Rb)l/v]}.

(33.41)

This expression is obtainable also from eq. (33.19) and in it Fm and Rb depend on t as noted above. In the often considered case of v = 1 we have [Toschev and Gutzow 1967b] (0 < R < Rb(t)) F(R, t) = (Js/Gc) exp [- (Ja,s/Gc)(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 _
the Fm(t) 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 _ Rb(t)) F(R, t) -- (Js/Gc) exp [- (BzJa,s/Gc) l + d(R b - R) 1 + d].

(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 = Cg, Gc,z = Gc and from eq. (33.43) it follows that [Kashchiev 1975b] (0 < R < Rb(t)) F(R, t) = (Js/VGc)R l/v- 1 exp {- [CgJs/(1 + vd)Gc] (Rlb/v - R1/V) 1+ vd} = Fm(R/Rb)l/v- 1 exp {- [Cg RdJst/(1 + vd)][1 - (R/Rb)l/v] 1 § va}.

(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 _
(33.46)

for volume nucleation (either HON or HEN) and growth of spherical (Cg "4zr/3), cubic (Cg = 8), etc. supernuclei and that (0 < R < Rb(t)) F(R, t) = (Js/Gc) exp [- (CgJs/3Gc)(Rb- R) 3]

(33.47)

for surface nucleation and growth of circular (Cg -- /E), 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 of supernuclei

457

between the supernuclei at later times, eqs (33.45)-(33.47) can be used only until some 30% of the initial volume V or substrate area As is occupied by the supernuclei [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 supernuclei, their size distribution is controlled by a single p a r a m e t e r - the Js/Gc 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 t~ 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 supernuclei grow faster than the bigger ones, the maximum value Fm of the size distribution increases linearly with Rb (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 supernuclei of a given size are less numerous than at the earlier moment tl. The size distribution function in stationary PN is most notable perhaps with its stepwise rise at

undetectable

Fm

.

.

.

.

.

.

detectable

.

T

ii

0

Rb(t 1)

R'

Rb(t2)

R Fig. 33.4 Size distribution of supernuclei 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 te > tl the supernuclei are already detectable).

458

Nucleation: Basic Theory with Applications

l

undetectable

detectable

Fro(t2)

..

T ii

~

Fro(t1) Rb(t ~)

R'

Rb(t2) 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) f o r confinements from both centres and zones (at time t2 > t I the supernuclei are already detectable). Rm(t) = Rb(t) and its 'tail' towards the smaller sizes. As already mentioned above, this 'tail' shortens by increasing the JJG~ ratio and can be quite short at high enough values of Js/Gc. This conclusion is of practical value, for it tells us what should be done with the Js/Gc 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), ~"(t) and J'(t) dependences. We first note that in this case the enoPni product (33.18) is given by the exponential factor in (33.37) so that, in view of (33.11), Pno(t)Pni(t)

= exp [- Ja, s t - (BzJa,st) 1 § vd].

(33.48)

Then, from (33.13), (33.23), (33.25), (33.27), (33.28), (33.31) and (33.37) we find that, in general, -tg

exp [-Ja,~t'- (BzJa,st') l§

Ray(t) = G~'

( t - t') v exp [- Ja, s t ' -

dt"

}

(BzJa, st') 1 + vd] dt'

(33.49)

t-tg

('(t) = Js

f,tO

exp [- Ja, st" - (BzJa, st') 1 + vd] dt'

(33.50)

Size distribution of supernuclei

J'(t) = Js exp {- Ja,s(t- tg) - [BzJa, s ( t - tg)] 1 § vd}

459

(33.51)

for t > tg, and Rav(t) = 0, ~'(t) = 0, J'(t) = 0 for 0 < t < tg where tg = R'I/V/G c 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 Ray, ~' and J' are simple explicit functions of time (t > tg)"

Ray(t)

= R" [(t/tg) v+ 1

~'(t) = J s ( t -

1]/(v + l)[(t/tg) - 1]

(33.52)

tg)

(33.53) (33.54)

J'(t) = Js.

Equations (33.52) and (33.53) show that the average radius Ray of the detectable supernuclei increases gradually with time from R' at t = tg to Rb(t)/(V + 1) for t >> tg and that their total concentration (' is a delayed linear function of time. This behaviour of Rav from (33.52) at v = 1 and of (' from (33.53) is seen in Figs 33.6 a n d 33.7, respectively. For illustration of the delay of (', the dash-dotted line in Fig. 33.7 represents the ((t) 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,

b i "/"

l n~ n,

R

!

d.1 "t p./'t" o/'"

0 Fig. 33.6

t

f"

I

I

t~

tg

t2

Time dependence of the average radius Ray of detectable supernuclei in the case of stationary IN at v = 1: dotted line - eq. (33.52)for no confinements from active centres and exclusion zones; solid l i n e - eq. (33.58)for confinements from centres only. The dot-dashed line represents the time dependence of the radius R b of the biggest supernucleus.

460

Nucleation:

Basic Theory with Applications

! I

~a

! I !

m

.--: .....

!

!

,."

9

!

T

,

I

,,

/

/

i i

/

! I

, I

i / It

i' .ii' !, !, 1' t i. I

f 0

I t~

,,

!,, t2 ---

tg t

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

detectable

curve correspond

to the c a s e w h e n all s u p e r n u c l e i

are

(then tg = 0).

eqs (33.53) and (33.54) are important, because they tell us that ('(t) data for nucleation in stationary regime can give the same information about Js as the respective ((t) 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 > tg) Ray(t) = Ja,sGc {1 - exp [- Ja~s(t - tg)]} -1 (t- t

~0 -'g

v exp(-Ja,st

,)

dt"

(33.55)

')

~'(t) = ~a{1 - exp [- Ja,s(t- tg)]}

(33.56)

J'(t) = Js exp [- Ja,s(t- tg)].

(33.57)

Here, according to (25.3) and (25.4), Js and Ja,s are related by Js = Ja,s~'a, and ~a = Na/V or ~a = Na/As 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 (t > tg)

Size distribution of supernuclei 461

Rav(t) = R" {t/tg - 1/Ja,stg - (1 - 1/Ja,stg) exp [- Ja,s(t- tg)]} (33.58)

{ 1 - exp [- Ja,s(t- tg)] }-1.

This equation reveals that Rav 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 Rav from (33.52), Rav from (33.58) is controlled not only by Gc (through tg), but also by the stationary nucleation rate Ja,s 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 = tg to Rb(t) - Gc/Ja, s for t >> tg. As to the ('(t) dependence (33.56) (the solid curve in Fig. 33.7), it parallels the ((t) 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 tg and this results in a delay of ('. This delay is reflected also in the detectable nucleation rate J' (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 Rav(t), ('(t) and J'(t) functions which we find from eqs (33.49)-(33.51) for t > tg: Rav(t) = G~'

~'(t) = Js

{s:

-tg

exp [-(BzJa,st

p)l+vd

] dt

t

}1

~i -tg ( t - t')v exp [- (BzJa,st') 1 + vd] dt"

(33.59)

~

(33.60)

-tg

exp [- (BzJa, st') 1 + va] dt"

J'(t) - Js exp {-[BzJa,s(t- tg)] 1 + vd}.

(33.61)

These formulae apply to volume (d = 3) or surface (d = 2) nucleation with Bz from (32.17) or (32.18) and are valid also for HON (then d = 3). Qualitatively, the above Ray(t) and ~"(t) dependences are similar to those illustrated by the solid curves in Figs 33.6 and 33.7, and the detectable rate J" from (33.61) is again a delayed vanishing function of time. At tg = 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 have Cz = Cg and Gc,z = Gc so that eqs (33.59)-(33.61) describe this case, too, but with BzJa,s replaced by [Cg GVcd Js/(1 + v d ) ] 1/(l§ . We note also that if the Rav(t) and ('(t) 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 J. 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 Rav, ~' and J' 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) = (Js/VGc)R I/v- 1 oo

{1 + 2 Z (-1) i exp [-(izzc2/6GcO)(R~ I v - R1/V)]} i=1

(33.62)

for 0 < R < Rb(t) and F(R, t) = 0 for R > Rb(t). This formula says that in nonstationary PN (either HON or HEN) the size distribution of supernuclei is governed not only by Js and Go, but also by the nucleation time lag r, since this is related to the nucleation delay time 0by 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 tl at which none of the supernuclei is detectable and at a later time t2 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

undetectable ,:~

detectable

..................................................

. ..

..

". .. .. ".

:

LL

Fm

Rm(tl) Rb(tl)

R'

Rm(t2) Rb(t2) R

-

Fig. 33.8 Size distribution of supernuclei in the case of non-stationary PN at v = 1: dotted curves - eq. (33.62)for no confinements from active centres and exclusion zones; solid c u r v e s - eq. (33.70)for confinements from centres only (at time t2 > tl the supernuclei are already detectable).

Size distribution of supernuclei

undetectable

It.

463

detectable

Frn(t2)

~

Frn(tl)

~ 1

Rm(tl) Rb(tl)

~

R'

I

Rm(t2) Rb(t2)

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 c u r v e s - eq. (33.70)for confinements from centres only (at time t2 > tl 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, J 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 z and delay time 0 and can be used for determination of these quantities from experimentally obtained size distributions of supernuclei in non-stationary PN [Toschev and Gutzow 1967b; Krster 1984; Blanke and Krster 1985; Krster and BlankBewersdorff 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, t) from (33.62), which is valid with an error of less than 1% for the bigger supernuclei, namely those with radii in the (R~/v - 24GcO/Zc 2) v < R < R b range" F( R, t) = (24 j20/zrV2 Gc)l/2

p b1iv _ R uv)-l/2 R1/V-l(..

x exp [-6Gc O/(R~/v - R1/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(t), Rav(t), ~"(t) and J'(t) dependences in unconfined non-stationary PN. Because of the mathematical difficulties in the calculation of the position R m 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 Pno(t) = Pni(t) = 1 when nucleation is not confined by active centres and exclusion zones, we find that Rav(t) = G v { t - t g - 0 - (12/n2)0 oo

]~ [(- 1)i/i 2] exp [-i27r2(t- tg)/60] }-1 i=1

x

(t - t')v [1 + 2 ~ (-1)i exp (-i2~:2t760)] dt'

(33.64)

i=1 oo

('(t) = Js {t - tg - 0 - (12/~)0 ]~ [(- 1)i/i2] exp [- i2zt2(t - tg)/60] } i=1

(33.65) oo

J'(t) = Js {1 + 2 ]~ (- 1)i exp [- i2n2(t- tg)/60]}

(33.66)

i=l

for t _> tg and that Ray(t) = 0, ('(t) = 0, J'(t) = 0 for 0 _< t < tg. As seen, the only difference between the J(t) and ~'(t)dependences (15.118) and (15.119) and the above J'(t) and ('(t) dependences is that the latter are 'shifted' towards longer times with respect to the former, the 'shift' being equal to the growth time tg 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 tg = 0 (then all stable supernuclei are detectable) and at tg > 0, respectively. We observe that in the t --> oo limit ('(t) becomes a straight line (shown by double dots and dashes) with time intercept 0 or 0'. Physically, 0'(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 0 of nucleation, 0' is defined by ('(t) = Js ( t - 0') at t --) oo.

(33.67)

Hence, with the aid of ('(t) from (33.65), it follows that 0 ' = 0 + tg

(33.68)

where 0 is related to the nucleation time lag r through 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 0' is accessible to a direct experimental determination. According to this equation, when we measure the detectable delay time 8', we can interpret it as the actual delay time 0 of nucleation only

Size distribution of supernuclei i:

..!

q9ii

.:..;

..t

I: 9t I:

~a

-

;

.i:

..[ ..

.

.

"~

.

.,

i:i i ,,'''"""""" i" i: i; i, ," i: z i 9:i z z

T

465

i:'i::

i!z ill

/ ' /

9 !/

.'~

/z

//! i'!

." ;

i/i

." i

i" i

/,';

i

,iS

// .

..

; i'

i F

I

I

0

t1

tg

O'

t2 t

-

Fig. 33.10 Time dependence of the concentration of detectable supernuclei in the case of non-stationary PN: dotted curve - eq. (33.65)for no confinements from active centres and exclusion zones; solid c u r v e - 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 ~ oo asymptotes of ('(t) in the absence of confinements from centres and zones and define the respective delay times.

if tg << 0. We note also that (33.68) is an approximation to the exact formula O" = 02 + tg of Shneidman and Weinberg [1992a], in which 02 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 n* (see eq. (15.1)). The nucleation rate becomes stationary earlier than the flux j(n2, t), because n* < n2. Hence, 0 < 02 and eq. (33.68) underestimates 0'. Nonetheless, since typically 02 = 1.50 to 20 [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 (supemuclei growing radially at constant rate Gc), with the help of (15.104) and the equality [Gradshtein and Ryzhik 1962] oo

[(- 1)i/i 4 =-7/r4/720, i=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 (t > tg) oo

Rav(t ) = R" {t]tg- 1 - O/tg- (120]lr2tg) ~, [(- 1)i/i 2] i=1

• exp [- izzrz(t - tg)/60] }-1 {(l/2)(t 2/t~ - 1) - Ot/t 2 oo

-(120[rC2tg) Z [(-1)i[i 2] exp [-i21t'2(t- tg)[60] i=1

+ (702/lOt2g) + (7202/lr4t 2) oo

~, [(-1)i/i 4] exp [-i2/t2(t- tg)]60] }.

(33.69)

i=1

This cumbersome formula shows that Rav(t) data obtained in non-stationary regime can give a reliable information about the growth time tg or rate Gc of the detectable supernuclei only at longer times ( t - tg >> 0) when, as it should be, eq. (33.69) turns into eq. (33.52) with v - 1. Alternatively, however, if we know tg independently, we can use (33.69) for determination of the nucleation delay time 0 and, thereby, of the time lag 7:of nucleation provided the Rav(t) data are obtained at sufficiently short times ( t - tg < 0). 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, t) is given by eq. (33.19) which, in view of (15.118), (25.12), (33.8) and (33.11), leads to oo

F(R, t) = (Js/VGc)R l/v- 1{ 1 + 2 ~, ( - 1)i exp [- (i2n2/6Gc O) ( R 1Iv - R1/V)] } i=1 oo

• exp (- (Ja,s/Gc){ R~/v - R1/V- GcO- (12/~)GcO

~, [(- 1)i/i 2] i=1

• exp [- (i2~/6Gc O) (g~/v - R1/V)]})

(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 fight 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, t) in Figs 33.8 and 33.9, it is analogous to that of F(R, t) 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 Rm(t) and of

Size distribution of supernuclei

467

the average radius Rav(t) of the detectable supernuclei. We shall obtain only the corresponding time dependences of the total concentration ~" of detectable supernuclei and of the detectable nucleation rate J'. 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(t') is given by the exponential factor in (33.19) and that Pni(t') = 1, from (33.25) and (33.31) we find that oo

~'(t) = ~a{1 - exp (- Ja,s{t- tg - 0 - (12//t'2)0 ]~ [(-

1)i]i 2]

i=I

• exp [ -

i2n2(t - tg)/60] }) }

(33.71)

oo

J'(t) = Js {1 + 2 ]~ (- 1) i exp [ - i 2 n 2 ( t - tg)/60]} i=1 oo

• exp (- Ja,s{t- tg - 0 - (12/n2)0 ]~ [(-

1)i/i 2]

i=1

• exp [-

i2~(t - tg)/60] })

(33.72)

for t > tg and that ~"(t) = 0 and J'(t) = 0 for 0 < t < tg. We recall that here Js and Ja,s are related by Js = Ja, s~'a and that (a = Na/V or (a = Na/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 0 = 0 which corresponds to stationary PN. Also, for t - tg << 0 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 supernuclei are detectable (then tg = 0), eq. (33.71) turns into eq. (25.15) provided in the latter "r is replaced by 0 from (15.105). The ('(t) dependence (33.71) at tg = 0 and tg > 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 supernuclei.

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], Kern et al. [1979], Stoyanov and Kashchiev [ 1981], Vook [1982], Venables et al. [ 1984], Zinke-Allmang et al. [ 1992]. On the other hand, the nucleationmediated 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 et al. [ 1977]; Kashchiev [ 1978]; Lewis and Anderson [ 1978]; Stoyanov and Kashchiev [ 1981 ]; Barth~s 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 t = 0 a 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 Jl(t) (m -2 S-1) and grow laterally with a radial velocity v1(0 (m/s). This process of overall 2D crystallization results in a total area Al(t) occupied by the first

Growth of thin films

469

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 al(t) - A l ( t ) / A s . Once a sufficiently large a r e a A 1 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) -- AE(t)/As appears on the first one, offering its surface with total area A2(t) 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 Gf(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 -~ oo 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, Gf 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 Ji and u i of 2D nucleation and growth within the ith 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 the theory differences, i.e. when J1 = J2 - J3 - 9 9 9and u 1 = I')2 = U 3 " - . . . . 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 Gf (m/s) which are defined by [Kashchiev 1977] oo

hf(t)-- ~, hiai(t) i=l

(34.1)

470 Nucleation: Basic Theory with Applications

Gf(t) = dhf(t)/dt.

(34.2)

Here h/is the thickness of the ith layer of the film, oti(t) = Ai(t)[A s 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 h i = d o where do is the molecular diameter. Thus, the problem of finding hf(t) reduces to the determination of all txi's on the basis of concrete model considerations and to the summation of these quantities in conformity with eq. (3411). 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 al(t) = A l ( t ) / A s = 1 - e x p

[--(t[Ol) ql]

(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 ql > 0 and t~l (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 q2, q3, etc. and ~2, t~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 areaAi_ l(t) of this layer and the exact determination of the coverage ai(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 dAi[dAi- 1 ratio. Indeed, if dAi_ 1 is a small area occupied by layer i - 1 between the moments t" and t" + dt', at a later time t >_ t" the ith layer will occupy an a r e a d A i o f dA i_ l" Hence, similar to (34.3), we shall have d A i "- (1

- exp {- [ ( t - t')[19 i ]qi }) dAi_ 1

(34.4)

which, upon dividing by As and integrating under the initial condition ai(0) = 0, yields [Kashchiev 1977] (i - 2, 3, 4 . . . . ) a~(t) =

(1 - exp {- [(t - t')lOi ]q~})[dai _ l(t')ldt']dt'.

(34.5)

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 ql = q2 = q3 = 9 9 9and L~1 = /-~2 "- /-~3 -" 9 (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 a r e a A i_ I of layer i - 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 ai_ 1 and ai. 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, qi and /~i 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 qi and Oi 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 g(t)

(34.6)

= (Us,it) vi ,

qi and Oi are given by qi = 2vi,

for IN of

Nm, i

L~i =

(34.7)

(As]cgNm,i) 1/2vi (1]Os,i)

crystallites in the ith layer and by qi = 1 + 2Vi,

L~i =

[(1

+ 2Vi)[CgO2ViJs,i s,i]l/(l+2vi)

(34.8)

for PN at stationary r a t e Js,i. Here the numerical shape factor Cg is considered as being the same for the 2D crystallites in all layers (e.g. Cg = zr for disks, Cg = 4 for square prisms, etc.). From (34.7) and (34.8) we thus see that qi = 2 in IN and qi = 3 in PN of crystallites with radii growing linearly with time (then vi = 1, and the growth constant Vs,i 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 p

p

qi = 1,

1~i = l [ ~ s , i - gs,i)

(34.9)

where fs,i (s-l) and gs,i ( S - l ) a r e , 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,i = 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 ai(t) from (34.3) and (34.5) we can now determine the time dependence of the mean thickness hf 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 ai in (34.1) leads to [Kashchiev 1977] hf(t) =

hlal(t) +

Z h i a i _ l ( t ' ) ( d exp { - [ ( t - t')[l~i] qi }/dt')dt'

i=2

(34.10) where al 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 hf(t) 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 h l ~ h2 = h3 = . . .

= do

q l r q2 = q3 = 9 9 9= q

O1 r ~92= / ~ 3 = . . . =

L,q

where do is the molecular diameter, and q and O are the kinetic index and the time constant of polylayer growth of the corresponding crystal face. Using these relations and the identity

i=2

hioti_ 1 = hf + (do - h l ) o q

resulting from the first of them and (34.1) simplifies essentially eq. (34.10) to the integral equation [Kashchiev 1977] hf(t) = h l a l ( t ) +

[hf(t') + ( d o - h l ) a l ( t ' ) ] ( d

exp {- [ ( t - t ' ) l o ] q } l d t ") dt" (34.11)

in which al(t) is given by (34.3). When the solution hf(t) 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) =

[Gf(t') + ( d o -

hl)dal(t')ldt']

exp {- [ ( t - t')lt~] q } dt"

(34.12) which follows from (34.11) after integration by parts. We note that in the particular case of ha = do, ql = q and t~l = 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 hf(t) and Gf(t) dependencies for arbitrary ql > 0 provided q = 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 al is specified by (34.3), useful approximate solutions of eqs (34.11) and (34.12) for arbitrary ql and q values are the functions

hf(t)=Gct+

hi{1 -exp[-(t/191)qi]}

-GcO1

f

tlO1

exp (-xql) dx

.;0

(34.13) Gf(t) = Gc{ 1 - exp [--(t]L~l) qi ]}

+ (qlhl/O1)(t]1~l) qi-1 exp [--(t]L~l) ql ]

(34.14)

where Gc is given by (F is the gamma-function (26.21)) Gc = d0/1-'(1 + l/q)t~.

(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 al from (34.3) and that at q r 1 they describe correctly the asymptotic behaviour of hf and Gf both for t --~ 0 and for t ~ co. 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]

he(t) = Gc(t - tint)

(34.16)

tint = F(1 + 1/ql)Ol- (hi~do)F(1 + 1/q)O.

(34.17)

where 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 do, q and t~ and is thus the rate of polylayer growth of crystals. The fact that Gc from (34.15) with q and ~ from (34.8) coincides with Gc 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 tint of the asymptotic hf(t) dependence (34.16) is sensitive to the properties of the substrate, which are characterized by hi, q l and OlThe solid curves in Fig. 34.2 depict the hf(t) dependence (34.13) at ql = 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 ql 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 Ol/O = 0.25, 1 and 4 (as indicated) and hi = do. The dashed lines in Fig. 34.2 represent the corresponding hf(t) asymptotes (34.16). As seen, at 01/0= 0.25 the first layer is filled up quickly and after that the film grows in stationary regime. We can therefore say that for ~1/~ << 1 the substrate is hospitable to the deposited material. Conversely, at 01/~ = 4 the filling of the first layer is delayed considerably (the substrate

474

Nucleation: Basic Theory with Applications

8

6

"~ t,....

4

2

0

0

2

4

6

8

10

tie Fig. 34.2

Time dependence of the average film thickness (the solid curves) according to eq. (34.13) at 01/0 = 0.25, 1 and 4 (as indicated) in the case of ql = 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 tin t. Thus, tin t appears as a direct indicator of the differences in the kinetics of filling of the first and next film layers. Qualitatively, the hf(t) 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 Gf(t) dependence (34.14) (the solid curves) and the stationary regime of growth (the dashed line) again at q l = 3, q = 1, hi = do and Ol/t~ = 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/99 at which it reaches continuity. Let h99 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 t99 a s the moment at which the substrate becomes 99% covered by the first layer, we have al(t99 ) = 0.99 so that in view of (34.3) t99 -- 4"61/ql ~ql.

(34.18)

For example, t99/01 = 4.6, 2.1 and 1.7 at ql - l, 2 and 3, respectively. In accordance with (34.16)-(34.18), the mean thickness h99 = hf(t99) at which the film reaches continuity is therefore given approximately by [Kashchiev 1977] h99 = h i + {[4.6 l/q~- F(1 + 1/ql)]/1-'(1 + 1/q)}dot~l/t~.

(34.19)

Growth of thin films

475

6 5 4

(.'9

3

0.25

2 1

1 0

0

1

2

3

4

5

6

tie Fig. 34.3

Time dependence of the film growth rate (the solid curves) according to eq. (34.14) at t~l/O = 0.25, 1 and 4 (as indicated) in the case of ql = 3 and q = 1. The dashed line indicates the stationary regime of growth.

Here the bracketed factor in front of do is a number close to unity, e.g. it is equal to 0.9 at ql = 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 Ol by the first layer, during this period 01/0 more successive layers will heap up on the first one provided each of them takes time ~ for its filling. Like eqs (34.13)-(34.18), with hi = do, ql = q and t~l = ~ eq. (34.19) is applicable to polylayer growth of crystals, i.e. to deposition on own substrate. Denoting by h99,own the value of h99 in this case, from (34.19) we get h99,own = [4.61/q/I-'(1 + 1/q)]d o.

(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,own/d0 = 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 ---) hi at O1/t~ --~ 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 (01 ~ ~) and/ or very rapid filling of the second, third, etc. layers (O ~ 0), for t < t99 high island-like crystallites form on the substrate and at t = t99 the film on it is macroscopically thick (h99 ----) oo). This is the island mode of growth of the film. Hence, the conclusion: h99 or the tgl/O 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 01/0 ratio diminishes with increasing the supersaturation Ap, and 01 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 h99 values, i.e. control the film growth mode. But how does h99 depend on Ap? 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 h99 on A/.t [Kashchiev 1977, 1978; Kashchiev et al. 1977; 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 ql = q = 3 and under the assumption hi = do, from eqs (34.8) and (34.19) it follows that [Kashchiev 1977]

h99/d 0 = 1 + 0.9 (V2Js/V21Js,1) 1/3.

(34.21)

Here Js,1 and Vs,1 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,1 and Js 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 h99(Ap) dependence is of the form (Ap > a0Acr)

h99]d0 = 1 + 0.9Kf(Ap) exp [B/3(Ap - a0Acr) - B/3Ap]

(34.22)

where Act and B are specified by (3.72) and (13.55). The dimensionless factor Kf is given by Kf = [v2(A]2) A ( A l . t ) / v 2 1 ( A p ) AI(AI.t)] 1/3 ,

(34.23)

A1 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 Ap, the above dependence of h99 on Ap is dominated by that of the exponential factor when the supersaturation is not too high. Figure 34.4 displays the h99(A//) dependence (34.22) at Kf = 1 and B/aoAcr = 30. The dashed line indicates the number h99,own/d0 of monolayers which would have been deposited on the substrate till time t99 if it were the own one (i.e. with Act = 0). According to eq. (34.20), h99,own]d 0 = 1.9 monolayers. As seen from Fig. 34.4, h99 is a strong function of the supersaturation in the range of smaller Ap 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 Ap, h99 diminishes steeply and above a certain supersaturation Apch (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 Ap leads to a change from island (h99 >> h99,own) to layer (h99-- h99,own) mode of growth, which occurs at Ap = APch.

Growth of thin films

477

100000

10000

~176 looo .c::

40

J

-

30 A,U,ch

I

20 10 c 0

,

0

,

,

I

2

,

,

i

I

i

,

4

,

I

6

Akt/tZ0At~ Fig. 34.4 Supersaturation dependence of the mean thickness at which a thin film reaches continuity: solid l i n e - eq. (34.22); dashed l i n e - eq. (34.20) at q = 3. The arrow indicates the supersaturation for change between the island (for zip < Ajuch) and the layer (for Ap > ZS~ch) modes of film growth.

It turns out that the decrease of the film thickness with increasing A/~, which is illustrated in Fig. 34.4, is a general property of h99 for deposition on substrates with At~ > 0 (these substrates are energetically inhospitable, because Act > 0 corresponds to incomplete w e t t i n g - 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 Act = 0) in the sense that t~l = #. 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 Ap-independent value h99,owngiven by eq. (34.20).

478

Nucleation: Basic Theory with Applications

Thus, the h99(Ap) dependence contains essential information about the film growth mode and, with the help of the definition equality [Kashchiev 1977] h99(A//ch ) - h99,own = 1,

(34.24)

can be used for determination of the supersaturation A/.tch for change between the island and layer modes of film growth. Physically, eq. (34.24) means that at APch, 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 Ol and ~ are practically equal. This equality of L~1 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/~chas the solution of eq. (34.24) we have the possibility to take account of various mechanisms of layer filling by making use of different h99(A,t/) dependences resulting from (34.19) with different time constants t~ and O. 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/.tch is readily obtained from (34.20), (34.22) and (34.24) when Kf = 1 [Kashchiev 1977]: Apch = (1/2)[1 + (1 + 5.2B/3aoAcr)l/2]aoACr.

(34.25)

This formula applies when A/~ch --- a0Ao'. At B/aoAt7 = 30 it yields A/.tch/a0Ao" - 4.1 which is the value indicated in Fig. 34.4. The usage of eq. (34.24) for determination of A/~ch in other cases of thin film growth is exemplified elsewhere [Kashchiev et al. 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 ~3 >> ~91 = v~2= t~4 = ~95 = . . .), 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 qi and the time constants ~9i 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 and Kanter 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 1974] 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 nm 2. 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. 35.1 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).

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 A~ defined by eq. (2.1) in which now ~told =/~b and ]-/new= fla, ]-/b and ]-/a being the chemical potentials of the amphiphiles in, respectively, the bilayer and the ambient phase. Thus [Kashchiev 1987], A~/

= ,t/b -- ]./a.

(35.1)

If P3D 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,/~b can be expressed as [Kashchiev 1987, 1989b, 1990] /~b =/~3D + a00"b

(35.2)

where a0 (m 2) is the area occupied by an amphiphile molecule at the bilayer surface, and Orb (J m -e) 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 ,Ha requires specification of the ambient phase. In the often encountered case when this phase is a solution of the amphiphiles

482 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 /-/a = /-/e,3D 4-

kT In (C]Ce,3D)

(35.3)

provided the solution is sufficiently dilute. Here C (m -3) is the actual concentration of amphiphiles in the solution, Ce,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/t/e,3D is the value of/t/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/-/a is given approximately by [Kashchiev and Exerowa 1998] /3a -- /-/e,3D q"

a0(Trs-

(35.4)

~e,3D)

if the monolayer compressibility is negligible (accounting for this compressibility leads to a quadratic ~rs term in ,t/a [Kashchiev and Exerowa 1998]). In eq. (35.4) Zrs (N/m) is the actual surface pressure of the monolayer, and 7re,3D is the ~rs value at which the reference 3D lamellar phase is in equilibrium with the monolayer in respect to growth and dissolution. Now, since/~3D in eq. (35.2) is practically independent of C or 7rs, we have /t/3D =/-/e,3D" Thus, combining eqs (35.1)-(35.4) leads to Ap(C) = kT In (Ce,3D/C) -I- a0tYb(C)

(35.5)

for a bilayer in contact with a solution of amphiphiles [Kashchiev 1987, 1990] and to A/-/(~s) = a0(/17e,3D- /17s) +

a0crb(Trs)

(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 ~re of the monolayer, which are defined by C e = Ce,3D

exp [aotrb(Ce)/kT ]

(35.7) (35.8)

/t"e = /t'e,3D -I- O'b(7~e) ,

eq. (35.5) can be given the equivalent form [Kashchiev 1987, 1990] A/J(C) = kT In (CJC) + a0[crb(C)- trb(Ce)],

(35.9)

and eq. (35.6) becomes [Kashchiev and Exerowa 1998] A/~(Zrs) = 2a0(Tre- 7rs)

(35.10)

because of the relation Crb(ZCs) - O'b(Tre)--- Crm(Trs) - trm(Tre) = 7re - 7rs (O'm is the specific surface energy of the monolayer). We note that eq. (35.9) can be approximated by [Kashchiev and Exerowa 1980, 1983] A/~(C) = kT In (Ce/C)

(35.11)

when the last term in it is negligible, e.g. when Orb 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 Jre in eqs (35.7)-(35.11) are 2D analogues of Ce,3D and Jre,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 7rs = Zre there is no driving force for bilayer rupture" A/~ = 0. Rupture is thermodynamically favoured only when C < Ce or Zrs< Zre,as then A/~ > 0. The practical significance of eqs (35.5), (35.6), (35.9)-(35.11) is that they reveal how it is possible to change Ap and, hence, affect bilayer rupture by varying C or rCs which are experimentally controllable parameters. Knowing A/~, we can now determine the work W(n) for formation of an nsized 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 W we can use eq. (3.75) with a0 replaced by a0/2. Thus, in the scope of the classical nucleation theory, it follows that [Kashchiev and Exerowa 1980, 1983]

W(n) = - n A p + (2rCao)l/2lcn1/2

(35.12)

where tr (J m -l) 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 (2~ra0n) 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 W* [Kashchiev and Exerowa 1980, 1983]" n* = /r.a0K'2/2A/.t2

(35.13)

W* = zra0tc2/2Ap.

(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 Ap from (35.5), (35.10) and (35.11) in eqs (35.12)-(35.14) yields the dependence of W, n* and W* on C or Zrs 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(m -2 s-l) of hole nucleation is given by (C < Ce) Js = A exp {- B/[ln (Ce,3D/C) -I- aoCrb(C)/kT]}

(35.15)

or, approximately, by Js = A exp [- B/ln (Ce/C)]

(35.16)

484

Nucleation: Basic Theory with Applications

for a bilayer in contact with a solution and by (Zrs < Zre) Js = A exp [- B/(~re - ~s)]

(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 = xa0tc2/2(kT) 2

(35.18)

B = tctc2/4kT

(35.19)

for a bilayer contacting a solution or an insoluble monolayer, respectively. The kinetic parameter A (m -2 s-1) is specified by eq. (13.40), A = zf'Co,

(35.20)

and to a certain approximation can be treated as independent of C or rCs. Here, as usual, the Zeldovich factor z is a number between 0.01 and 1 and is given by eq. (13.37) with n* and W* from (35.13) and (35.14). For a bilayer free of nucleation-active centres, according to (7.8), Co = 1/ao ~ 1018 m -2. As to the frequency f* (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 = 1, f* is of the form [Kashchiev and Exerowa 1980] f* = c*DvZl

(35.21)

where c* --- 1 to 5 is the capture number, Zl (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 f * 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 Zrs,we can now determine the average lifetime tb(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 supemucleus 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 Ab (m 2) is the area of the bilayer surface, using (26.35) with d = 2, v = 1, Cg = ~ and Gc and V replaced by oh 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 < (Zcv2/3J2) 1/3.

(35.22)

In this case tb is identical with tav from eq. (26.39) with A b in lieu of V. Hence, with the help of eqs (35.15)-(35.17) it follows that for rupture by the mononuclear mechanism tb is given by (C < Ce) tb(C) = Am exp {B/[ln (Ce,3D/C) + aoCrb(C)/kT]}

(35.23)

or, approximately, by [Kashchiev and Exerowa 1980, 1983] tb(C) = A m exp [B/In (Ce/C)]

(35.24)

for a bilayer in contact with a solution and by [Kashchiev and Exerowa 1998] (Zrs < tee) tb(~'s) = A m exp [B/(zce- Zrs)]

(35.25)

for a bilayer in contact with an insoluble monolayer. Since the factor Am (s) is defined as

A m = 1/AAb,

(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 tb is equal to tav from eqs (26.24) and (26.28) also with d = 2, v = 1, Cg = tc and Gc replaced by Oh. Employing Js from (35.15)-(35.17), we thus find that (C < Ce) tb(C) = Ap exp {B/3[ln (Ce,3o/C) + aoab(C)/kT]}

(35.27)

or, approximately, tb(C) = Ap exp [B/3 In (Ce/C)]

(35.28)

when a bilayer contacts a solution and that (Zrs < tee) tb(Trs) = Ap exp [B/3(nre - ZCs)]

(35.29)

when the contacting phase is an insoluble monolayer. Here the factor AD (s) is given by

Ap = F(4/3)(3/zcv 2 A)1/3

(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 A m, to a certain approximation, the kinetic parameter Ap can be treated as independent of C or Zrs,because A and On 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 tb on C or tcs has a threshold character. Namely, tb diminishes monotonously with decreasing C or trs, but remains greater than an arbitrarily chosen value tb,c (e.g. tb,c = 1 S) if C or 7Cs is higher than a certain critical amphiphile

486

Nucleation: Basic Theory with Applications

concentration Cc or critical surface pressure ZCc.When C or ZCsis below this critical concentration or pressure, however, the bilayer ruptures virtually instantaneously. The critical concentration Cc or pressure Zrc 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 rupture by hole nucleation [Exerowa and Kashchiev 1986; Exerowa et al. 1992]. Analogously to A/.tc in (31.12), Cc and ZCc can be defined as the solutions of the algebraic equations tb(Cc) = tb,c

(35.31 )

tb(Zrc) = tb,c.

(35.32)

With the help of (35.18), (35.24) and (35.28), from (35.31) it thus follows that Cc = Ce exp [- zr~a0x'2/2(kT)2 In (tb,c/Am)]

(35.33)

for rupture by the mononuclear mechanism [Kashchiev and Exerowa 1980, 1983] and that Cc = Ce exp [- ~.aolC2/6(kT) 2 In (tb,c/Ap)]

(35.34)

for rupture by the polynuclear mechanism. Likewise, using (35.19), (35.25) and (35.29) in eq. (35.32) leads to

ZCc= tee- zc~/4kT In (tb,c/Am)

(35.35)

when rupture occurs by the mononuclear mechanism [Kashchiev and Exerowa 1998] and to 7~c -" ~ e -

zc~/12kT In (tb,c/Ap)

(35.36)

when it occurs by the polynuclear mechanism. As seen from the above formulae, while Cc and ZCcare quite sensitive to the magnitude of the hole specific edge energy to, they are affected relatively weakly by changes in the kinetic parameters A m and Ap. Finally, we can obtain a unified formula for the bilayer lifetime tb, which is valid for whatever number of supernucleus holes causing the bilayer rupture. Similar to the induction time t i 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) tb(C )

= A m

exp {B/[ln (Ce,3D/C) -!- aoCrb(C)/kT]}(1 + (Ap/Am)

• exp {- 2B/3[ln (Ce,3D/C) + aoCrb(C)/kT]})

(35.37)

or, approximately, tb(C ) = A m exp [B/In (Ce/C)]{ 1 + (Ap/Am) exp [- 2B/3 In (CJC)] } (35.38) for a bilayer contacting a solution and that (ZCs< Zre)

Rupture of amphiphile bilayers

487

tb(Zrs) = Am exp [B/(Tre - Its)]{ 1 + (Ap/Am) exp [- 2B/3(]re- rCs)]} (35.39) for a bilayer in contact with an insoluble monolayer. As it has to be, for C or ZCsvalues smaller than, but close enough to, Ce and ]re eqs (35.37)-(35.39) pass into eqs (35.23)-(35.25) of the mononuclear mechanism. Conversely, when the C f l C ratio or the ]re - Its difference is sufficiently large, eqs (35.37)(35.39) turn into eqs (35.27)-(35.29) of the polynuclear mechanism. The tb(~rs) 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 tb(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 tb(C) dependence (35.24) with, respectively, Am = 8.5 • 10 -9, 2.5 X 10 -7 and 8.1 x 10 -6 s, B = 49, 39 and 33 and Ce = 8.8 • 10-4, 1.33 x 10-3 and 1.53 x 10-3 mol/dm 3 for the above temperatures [Nikolova et al. 1989]. Using these B values in eq. (35.18) and the independently known a 0 - 0.42 nm 2 results in to= 31 pJ/m [Nikolova et al. 1989]. According to eq. (35.13), it turns out that the nucleus 500

400

,~

300 200

100

0 1.0

1.5

2.0

2.5

C (10 .4 mol/dm 3 ) 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 at T = 283.15 K (the circles), 295.15 K (the squares) and 303.15 K (the triangles); c u r v e s - best fit according to eq. (35.24).

488

Nucleation: 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/tbAb 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 mm 2 we find that J~ = 1.2 • 104 t o 5 • 105 m -2 s-1 under the experimental conditions. Employing the latter Js 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 Oh = 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 tr and n* 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 Yn be the Laplace-Carson transform [Korn and Korn 1961 ] of the flux j ~ ( n = l, 2 . . . . . M - l ) : Yn(A) = ~

(AI.1)

~ : jn(t)e -At dt.

Multiplying each equation in the set (15.68) by 2e -~ dr, integrating from t = 0 to t = oo and accounting for (15.69) and (AI.1) transforms the set into (A + g2) YI - g2Y2 = $fl C1 - f n r n - 1 + (~ + fn + g n + l ) Y n - gn+lYn+l = O,

(n = 2, 3 . . . .

, M - 2) (A1.2)

-- f M - 1 Y M - 2

+ (r + f M - 1 ) Y M

- 1 = O.

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) Yn(~) = flf2.

. . fnC1/q'Pg - n- I(~)/PM-1

(A1.3)

(/]') 9

Here Pn(A) is a polynomial of A,of degree n, which is subject to determination from the recursion formulae P0 = 1 P1 = ~ +fM- 1 Pn = (~ + f M - n + g M - n + 1)Pn- 1 - f M - n + l g M - n + 1Pn - 2,

(n = 2, 3 . . . . .

M-

2)

(A1.4)

P M - 1 = (/]" + g 2 ) P M - 2 - f 2 g 2 P M - 3"

We can now make use of the theorem according to which the inverse Laplace-Carson transform (i.e. the original) of the function $ P M - n - I($)/PM- 1(~) is the function [Rodigin and Rodigina 1960]

(A1.5)

490 Nucleation: Basic Theory with Applications M-1

Z [PM_n_l(-/~i)]P~t_l(-~,i)] exp (--/],i t)

(A1.6)

i=1

where - ~,i < 0 is the ith of the M - 1 simple roots of the polynomial PM- 1(~,). Since this polynomial can be written down as M-1

(A1.7)

PM-1 (~,) = I'I (,~, + ~i), i=l

for the derivative P~/_I(-~,i) - (dPM_l/d~,)z=_z~ we have (k ~ i) M-1 P~_I(-,~,i)

=

l'-I ('~'k k=l

i['i).

(A1.8)

Hence, applying the inverse Laplace-Carson transform to both sides of (A1.3) and accounting for (AI.1), (A1.5), (A1.6) and (A1.8), we find (k ~ i) M-1

j , ( t ) = f l f 2 "" "f~Cl

Z i=1

M-1

[PM_n_l(--~,i)/ I'I (~k

-

'~'i)] exp

k=l

(-/],it). (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 t oo, jn(t) tends to the n-independent stationary nucleation rate Js, because one of the non-positive roots - ,~i of the polynomial PM- 1(~,) is equal to zero. Denoting by - ~,l this root, i.e. setting Al = 0, in the limit of t ~ oo (then only the first summand survives) from (A1.9) we get (n = 1, 2 . . . . . M - 1) Js =flf2...fnCIPM-n-1(0)//~2~3

9 9 9'~M-1"

(AI.10)

We note that since Js is the same at any n, its connection with the M - 2 positive quantities 22, ~3 . . . . . AM- 1 is most simply expressed by the equation Js = f t f 2 . . . f M - 1 C 1 / ~ ' 2 ' ~ 3

9 9 9AM-1

(AI.ll)

which results from (A1.10) at n = M - 1 (then PM- ~- 1 = P o = 1). Finally, from (A1.9) and (AI.10), for the non-stationary nucleation rate J(t) - j n , ( t ) we obtain the following exact formula (k ~e i, n* = 1, 2 . . . . . M -1) M-1

J(t) = Js - f l f 2 "" " f ' C 1

Z

i=2

[PM_n,_l(-~i)[/~ i

M-1

1-I (~,k - ~i)] exp (-~,it)

(Al.12)

k=2

where f* - f,,. Comparing eq. (15.59) or (15.60) and eq. (Al.12), we see that the &i's in the latter are nothing else but the ~,~'s in the former. The equivalence between eq. (15.59) or (15.60) and eq. (Al.12) is easily demonstrated in the particular

Appendices 491 case of M = Mef = 4 and n* = 3, considered in Section 15.2. In this case, from (A1.4) we find that the polynomial PM-1(~) is of the form

PM-1 = P3 = ~[/]2 +/~(f2 + f3 + g2 + g3) + f2f3 + g2f3 + gzg3] so that its roots are ~1 = 0 and - ~ = - (1/2){ (f2 + f3 + g2 + g3) - [(f2 + f3 + g2 + g3) 2 -4(f2f3 +

g2f3 + g2g3)] 1/2}

- 23 = - (1/2){(f2 +f3 + g2 + g3) + [(f2 +f3 + g2 + g3) 2 -4(f2f3 + g2f3 + g2g3)]l/2} 9 As seen, these A2 and ~3 coincide with A2 and ~3 from (15.20) and (15.21). Also, according to (A1.4) and ( A I . l l ) , e M - n * - 1 = P0 = 1 and Js =ftf2f*C1/ ~ 3 , since f3 =f*. Using these relations in (Al.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~ and replace (15.70)-(15.72) by ( n l ' -< n < _ n 2')

Oj(n, t)/cgt = f * OZj(n, t)/On 2

j(n, O)= (A'/A*)f*C*t~o(n- n() Oj(n, t)/On = 0 at n = n;,

Oj(n, t)/On = 0 at n = n~

(A2.1)

(A2.2) (A2.3)

where 6D is the Dirac delta-function. Physically, eq. (A2.3) implies instantaneous establishment of steady state for the subnuclei of size n __. n~, i.e. on the left of the size region with centre n*, width A' and left and right ends n~ and n~ defined by (15.48) and (15.49). This size region corresponds to the nucleus region so that in it, approximately, f(n) =f* 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 f'X(n~) =f*(A'/A*)C* (the factor A'/A* > 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~ _
492

Nucleation: Basic Theory with Applications

j(n, t) = (1/A')

f it

n~

(A'/A*)f*C* 6D(n - n~) dn !l~ oo

+ (2/A') Z cos [i~n - n~')/A'] exp (- i21r2f*t]A "2) i=1

~

"~ (A'IA*)f*C*6D(n- n~)cos [ i ~ n - n;)/A'] dn (A2.4) f where A'= n 2' - n I' . Owing to the properties of the Dirac delta-function [Korn and Korn 1961 ], we have

~ ~

"2 gD(n -- n;) d n = 1,

zf

n~ ~ D ( n - n~ ) cos [iz(n - n~ )/A'] dn = cos 0 = 1. if 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 oo

J(t) = Js [1 + 2 ]~ cos (izr/2) exp (- i2t/4"c)] i=1

which is equivalent to eq. (15.64).

A3 Initial concentration of supernuclei in previously supersaturated systems The initial concentration ~'0 of supernuclei is given exactly by the first integral in (24.53) when the system is presupersaturated (then Ap0 > 0) before the imposition of the working supersaturation All. Applying (24.54) to this integral, we thus find that, approximately, ~'o = X2(n*, Apo,

7"o)/I [dX(n,

Apo, To)/dn],, = ,,,

I.

(A3.1)

To a good accuracy, the stationary size distribution X(n, Apo, To) of the pre-existing clusters is expressed by eq. (13.21). Using this equation along with eq. (7.4) yields

X(n, Apo, To) = (1/2)Co exp [- W(n, Alto, To)lkTo]{1 - erf [/3o(n - no )]} (A3.2)

dX(n, Apo, To)ldn = - (1/2)Co exp [-W(n, Apo, To)lkTo] x {(l/kTo){[dW(n, Apo, To)/dn],,=,,}{ 1 - eft [flo(n- no )]} + (2flo/ZC 1/2) exp [-fl~(n - no,)2 ]}. (A3.3)

Appendices

493

Combining eqs (A3.1)-(A3.3), taking into account that C* = Co exp

[- W(n*, Ap, T)/kT] and recalling (3.86), we arrive at the sought general formula for the concentration of all supernuclei existing initially in a presupersaturated system (0 < Ap0 < Ap):

~o = zoC* exp [- n*(Ap/kT- Apo/kTo) + qr*(Ap, T)/kT- qr*(Apo, To)/kTo].

(A3.4)

Here, for brevity, Zo - ({ 2[dW(n, Apo,To)/dn]n =n*/kTo}{ 1 - erf [flo(n* - n~ )]} + (4/3o/Jr1/2) exp [-fl~ (n* - no )2])-1{ 1 - erf [flo(n* - no )]}2, (A3.5) the concentration C* of nuclei is specified by (7.44) with W* from (4.6), n~ - n*(Apo,To) is the nucleus size at Ap0 and To, and 130 - fl(Apo, To) is given in conformity with (7.38) as /~0 = {[- d2W( n, Ap0, To)/dn2 ]n=,,~/2kTo }1/2.

(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*Z with Z from (24.21)-(24.27), and the dW/dn derivative in (A3.5) is equal to Ap Apo. In this case we thus have ~'o = ([2(Ap - Apo)/kTo]{ 1 - erf[/3o(n* - n~ )]} + (4/3o/Zr1/2) exp [- /3o2 (n* - n~ )2])-1 x { 1 - erf[/3o(n* - n~ )] }2C* exp (- n'Z).

(A3.7)

We observe that this equation passes into (24.56) or (24.57) when Apo is sufficiently less than Ap. Indeed, then/30(no* - n * ) > 1 and, as erf ( - x ) - 1 and exp (- x 2) = 0 for x > 1 [Korn and Korn 1961 ], the product in front of C* in eq. (A3.7) becomes equal to the bracketed factor in front of C* in (24.56) or (24.57). In the opposite limit of Ap0 approaching Ap (then 0 < /3o( no - n ) < 1) eq. (A3.7) can also be simplified, but with the help of the approximations erf (x) --- (2/zcl/2)x and exp (- x 2) --- 1 holding for x < 1 [Korn and Korn 1961 ]. We thus find that in this limit ~0 = {[2(Ap - Apo)/kTo][1 + (2f10]~l/2)(n~ - n*)] + 4fl0]~1/2} -1 • [1 + (2flo/Zrl/2)( n o*

-

n*)]2C * exp (- n'Z).

(A3.8)

In particular, when Apo = Ap and To = T (then n~ = n*, 130 =/3 and Z = 0), eq. (A3.8) leads to

~o = (zc~/2/4~)C* = (1/4z)C* = (A*/4)C*

(A3.9)

494

Nucleation: 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.

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Author index

Abidor I.G., 315, 481 Abraham F.E, 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 Ahn T.M., 135 Aksenov A.A., 59, 228 Aleksandrov 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, 11, 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 Aubauer H.P., 128, 134 Avrami M., 366-368, 373, 375, 436, 437, 439, 446, 448, 450 Avramov I., 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 Barrett 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 Barth~s 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 Bennema P., 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 Boudart M., 115, 118, 119, 171,231,260, 262, 346 Brailsford A.D., 135 Brainin M.I., 315 Brandes H., 38, 52 Brice J.C., 391 Brick V.B., 270 Brown I.W.M., 315

516

Author index

Brtickner R., 256 Budevski E. x, 202, 220, 221, 286, 386, 398, 406, 408, 409 Budurov S., 382 Buff EP., 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 Carlier C.C., 231,262, 346 Carlson G.A., 258, 287 Carslaw H.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 Chakraverty 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.I., 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., 315, 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., 83, 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 E.G., 14, 153

Dadyburjor D.B., 126, 135 Debenedetti P.G., x de Boer 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 Dissado L.A., 315, 329 Djarova M., 426 Dobreva A., 382 Donohue M.D., 35 Doremus R.H., x, 373,378, 391,417 D~ring W., 18, 20, 22, 115, 121,158, 184, 185, 192-194, 200 Dufour L., x, 88 Dunning W.J., 24 Dupr6 A., 37

Ebner C., 202 Eggington A., 28 Egry I., 97 Ehrenfest P., 6 Einstein A., 143 El-Shall M.S., 330, 336 Entin I.A., 468, 471 Ernst M.H., 133 Erre R., 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 E, 437, 446 Farkas L., 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

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 Flynn P.C., 135 Fokin V.M., 256, 261,269, 270, 366, 369, 370 Ford I.J., 28, 59, 83, 84, 330, 335,430 Frank EC., 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 Fuchs N.A., 132 Fujita K., 115 Fujiwara S., 147 Fulcher H., 153 Furu 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 Gattef E., 315, 329 Gauch M., 168 Gauthier 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 I.I., x Glasstone S., 7, 87, 154, 160, 196, 332 Goldbeck-Wood G., 387 Golubovic L., ix

517

Goodrich EC., 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 Grassi 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 Gtinther S., 437, 446 Gutop Yu.V., 481 Gutzow I., 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 I., 79, 81,109 Hagemann J.W., 387 Hagen D.E., 430, 434 Haken H., 119 Hale B.N., 430 Halpern V., x, 129, 147, 149, 161, 162, 178 Ham ES., 144 Hamerton R.G., 115 Hammel J.J., 270 Hanbticken 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 Hill R.M., 315, 329 Hilliard J.E., 18, 69, 97, 100-102, 112

518

Author index

Hillig W.B., 398 Hirth 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 Hoyt J.J., 97, 102 Hruska S.J., 147 Huang K., 7, 119 Hung C.H., 215 Hutchinson T.E., 437

Ickert L., x, 468 Ievlev V.M., 437 Isard J.O., 315, 324-326, 328 Ishibashi Y., 373

Jaeger J.G., 142, 167, 241,246, 264, 491 Jakubczyk M., x, 414, 427, 430 Jalaluddin A.K., 315, 329 James P.F., 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 Kahlweit M., x, 135, 174 Kai S., ix Kaischew R., x, 18, 21, 22, 26, 27, 32, 3537, 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 K~impfer B., ix Kane D., 330, 336 Kaneko N., 386 Kanne-Dannetschek I., 231 Kanter Yu.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, 166168, 171,172, 174, 179-182, 191,224227, 231,237,240, 241,245, 334, 338, 339, 343, 346-350, 354, 357-359, 362, 366, 373, 380, 381,386, 388-390, 398400, 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 Katz J.L., ix, 28, 83, 84, 88, 115, 118, 120, 140, 171,199, 209, 215,330, 336, 430432, 434 Kaverin A.M., 59, 226 Kawakami T., 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 Khadir A., 102 Khrushchev V.V., 135 Kiang C.S., 28 Kibalczyc W., 426 Kikuchi R., 84 Kirkova E., 140, 426 Kirkwood J.G., 79, 80 Kitamura M., 387 Klapka V., 286

Author index

Klein W., ix, 102 Knudsen M., 391-393 Kodenev G.G., 59, 215, 226, 229 Kolmogorov A.N., 373,375,376, 378, 380, 437, 439 Komatsu H., 387 Kondo 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 Ktister U., 247, 256, 271,463 Kostrovskii V.G., 58, 59, 226, 228 Kotake S., x Kotsev E.I., 132 Kotzeva A., 386, 409 Koutsoukos 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 Kozlovskii M.I., 315, 329 Krasnopoler M.J., 215 Krastanov L., x, 32, 293,296 Kretzschmar G., 482, 487 Krishna-Murthy M., 382 Krohn M., 228 Kruglova T.I., 446, 468 Kruglyakov P.M., 480 Kubota N., 413, 426 Kukushkin S.A., x, 135, 437,446, 448 Kulmala M., 28, 58, 65, 67, 227, 228, 430 Kumar N.G., 430-432 Kuni EM., 315, 317, 318, 446 Kunz K.M., 446 Kupenova T., 267, 386 Kurihara K., 387 Kusaka I., 20, 69, 79, 97, 102, 112 La D., ix

519

Laaksonen A., x, 28, 58, 59, 65, 67, 69, 79, 97, 108, 227, 228, 330, 335, 430 Laidler K.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 Lifshitz E.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 E-P., 125 Luijten C.C.M., 59, 228, 330, 333, 336 Lutrus C.K., 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 MacKenzie K.J.D., 315 Mahnke R., 79, 310, 311 Marder M., 391,396, 397 Marinov M., 256, 265,267, 270 Markov I., x, 42, 54, 366, 391,437-439, 443 Markov T., 382 Markworth A.J., 135 Marthinsen K., 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.E, 373, 375,446 Meier G.E.A., 28, 79, 430-432, 434 Melentev I.I., 315, 329 Mendell G., ix Mersmann A., 183 M6tois J.J., x, 133, 148, 168, 183,202, 468, 478 Meunier M., 437, 446 Meyer H.J., 228 Mie G., 420 Mikheev 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-Hughes E.A., 87, 137, 143, 166 Monnette 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 ER.N., 311 Nagel K., 15 Nakada T., 387 Nancollas G.H., 42 Nasteva V., 301 Navascu6s G., 300 Nechaev Yu.I., x, 47, 49 Nedyalkov M., 301 Neizvestnyi I.G., 315, 329 Nenow D., 394, 395 Nes E., 373 Nesladek M., 366 Neu J.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 Parmar D.S., 315, 329 Paskova R., 208, 271 Pastushenko V.E, 315, 481 Patterson E.M., 28 Paunov M., 54, 228 Penkov I., 261,267, 270, 271 Petrov A.G., 481 Pilinis C., 133 Pimpinelli A., 437, 446 Platikanov D., ix, 300, 301, 481,482, 486488 Pocker D.J., 147 Polak W., 414, 427 Polchinski J., ix Popov E., 256, 265, 267, 270 Poppa H., 446 Postnikov 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

Price G.L., x, 56, 88, 119, 132, 147, 167, 170, 437, 468 Prigogine I., x, 79, 81 Probstein R.E, 231 Prokhorov A.V., 481,484 Pronin I.P., 437 Pulvermacher B., 133, 166, 168 Puri O.P., 28 Purushotaman S., 135 Ptischl W., 128, 134 Qiu H., 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 Ratke L., 97, 133, 135 Ratsch C., 437, 446 Rayleigh, 419 Ree EH., 119, 122, 232, 233 Ree T., 119, 122, 232, 233 Ree T.S., 119, 122, 232, 233 Rees G.J., 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 Rostrup 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

521

Rudek M.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 Ryzhik I.M., 261,465 Sakurai K., 387 Saltsburg H., 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 Scherer G.W., 373 Schiffner U., 256, 270 Schlesinger M.E., 270 Schmelzer J., x, 14, 79, 81, 125, 126, 248, 270, 309-311, 391,426 Schmelzer J., 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 Sedunov 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 Shablakh M., 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 Shi G., 115, 131, 132, 232, 237, 249, 380 Shichiri T., 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.Yu., 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 Sinha D.B., 315,329 Sirota 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 Smolyak B.M., 315, 327 S6nel O., x, 12, 13, 79, 89, 183, 378, 387, 413,414, 416, 417, 426 Spassov T., 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 Steinhardt P.J., ix Stenzel H., 228, 366, 444, 445 Sternitzke M., 256 Stoinov Z., 386, 409 Stoldt C.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 Sutugin 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 Temkin D.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 J.J., 78, 315, 317 Thomson W., 74 Tien H.T., 480 Tien J.K., 135 Toda A., 387 Todes O.M., 133-135, 398, 420, 437, 446, 448, 449, 455 Tohmfor G., 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 Trusov L.I., 315 Tsindergozen A.N., 398, 468 Tunitskii N.N., 125, 126, 279, 286, 373 Turnbull D., 154, 160, 183, 231,309, 311, 346, 384, 386, 387, 426 Tzuparska S., 382

Uchtmann H., 215, 430 Ueno S., 387 Uhlmann D.R., 373, 426 Ulbricht H., 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

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 G.M., 338, 339, 343, 413418, 426, 427, 430, 432 Vassileva E., 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 Virkler T.L., 430-432 Vitanov T., 386, 409 Vogel H., 153 Volmer A., 468 Volmer M., 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 G.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

523

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 Wynblatt P., 135 Wyslouzil B.E., 20, 115, 330, 336

Yang C.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 Zangwill A., 437, 446 Zanotto E.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 Zeng X.C., 97, 103, 109 Zettlemoyer A.C., x, 10, 21, 25, 27, 28, 3234, 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

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Subject index

Active centres for nucleation, 86, 366, 436, 444, 445, 449 Activity factor of nucleation, 55, 295, 305 Adsorption equilibrium, 148 AgNO3, 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

CaCO3, 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-mediated, 396 spiral, 404 Crystallization: overall, 373, 396, 397, 468 two-stage, 387 CuSO4, 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 Direct-impingement control, 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 Dupr6, 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 Stokes-Einstein, 143, 153 Tunitskii, 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 rate, 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

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 LiO2.2SiO2, 14, 242, 244, 269, 288, 359, 361-363

Master equation: general, 115-123 of coalescence, 131-133 of nucleation, 125-128, 130 of Ostwald ripening, 133-135 Maximum number of supernuclei, 367,376380, 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, 428,433,484488"

NaC1, 444 Naphthalene, 338 NaPO3, 266, 267 n-butanol, 217, 229 Nucleation: at pre-existing clusters, 346, 492 at variable supersaturation, 279 atomistic, 26, 39, 42, 46, 55, 202, 205209, 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, 219222, 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,

527

218, 219, 226, 227, 250, 255-257, 265-267, 269, 272, 273, 275-278, 287, 360-364, 425, 426, 432 in solutions, 11, 57, 77, 141-145, 151155, 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, 137140, 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 on ions, 315, 317 on seeds, 293 progressive, 377, 379, 397, 415, 416, 418, 421,440, 453, 471 stationary, 184 1D, 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 supernuclei, 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, 401404, 410, 415-419, 421-428, 433, 434, 468, 484-487 Pores, 343 Pressure tensor, 98, 105-107 Probability for: 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 limit, 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 dodecyl 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, 399401,406, 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, 381,384386, 414-416, 462

Subject index

Time-temperature-transformation (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--442, 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,

529

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 2Na20.CaO.3SiO2, 269, 270

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