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V. I. Kalikmanov

Nucleation Theory

123
Dr. V. I. Kalikmanov
Twister Supersonic Gas Solutions BV
Rijswijk
The Netherlands

and

Faculty of Geosciences
Delft University of Technology
Delft
The Netherlands

ISSN 0075-8450 ISSN 1616-6361 (electronic)

ISBN 978-90-481-3642-1 ISBN 978-90-481-3643-8 (eBook)
DOI 10.1007/978-90-481-3643-8
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Preface

One of the most striking phenomena in condensed matter physics is the occurrence
of abrupt transitions in the structure of a substance at certain temperatures or
pressures. These are first-order phase transitions, and examples such as the
freezing of water and the condensation of vapors to form mist in the atmosphere
are familiar in everyday life. A fascinating aspect of these phenomena is that the
conditions at which the transformation takes place can sometimes vary. The
freezing point of water is not always 0 C: the liquid can be supercooled con-
siderably if it is pure enough and treated carefully. Similarly, it is possible to raise
the pressure of a vapor above the so-called saturation vapor pressure, at which
condensation ought to take place according to the thermodynamic properties of the
separate phases. Both these phenomena occur because of the requirement for
nucleation. In practice, the transformation takes place through the creation of small
aggregates, or clusters, of the daughter phase out of the parent phase. In spite of
the familiarity of the phenomena involved, accurate calculation of the rate of
cluster formation for given conditions of the parent phase meets serious difficul-
ties. This is because the properties of the small clusters are insufficiently well
known.
The development from the 1980s onwards of increasingly accurate experimental
measurements of the formation rate of droplets from metastable vapors has driven
renewed interest in the problems of nucleation theory. Existing models, largely
based upon versions of the classical nucleation theory developed in the 1920s–
1940s, have on the whole explained the trends in nucleation behavior correctly, but
have often failed spectacularly to account for this fresh data. The situation is more
dramatic in the case of binary- or, more generally, multi-component nucleation
where the trends predicted by the classical theory can be qualitatively in error
leading to unphysical results.
This book, starting with the classical phenomenological description of nucleation,
gives an overview of recent developments in nucleation theory. It also illustrates
application of these various approaches to experimentally relevant problems
focusing on the nonequilibrium gas–liquid transition, i.e., formation of liquid

v
vi Preface

droplets from a metastable vapor. A monograph on nucleation theory would be

incomplete without presenting the recent advances in computer simulations of
nucleation on a molecular level, which is a powerful research tool complementing
both theory and experiment. I was glad that my colleague and friend Dr. Thomas
Kraska from the University of Cologne accepted my invitation to write the chapter
on Monte Carlo and Molecular Dynamics simulation of nucleation (Chap. 8)—the
field to which he made a number of significant contributions.
Obviously, in view of the modest size of the book it was not possible to cover all
new approaches formulated in recent years. The choice of the topics, therefore,
reflects the background and prejudices of the author.
This monograph is an introduction as well as a compendium to researchers in soft
condensed matter physics and chemical physics, graduate and postgraduate stu-
dents in physics and chemistry starting on research in the area of nucleation, and to
experimentalists wishing to gain a better understanding of the efforts being made
to account for their data.
I am grateful to a number of colleagues who collaborated with me at various stages
of the work. I benefitted greatly from discussions of fundamental problems of
nucleation with Howard Reiss, Joe Katz, and Gerry Wilemski, which advanced my
understanding of the subject. Several years spent in the group of Rini van Dongen
in Eindhoven University will remain an unforgettable experience of a remarkable
scientific atmosphere and friendly environment; special thanks are due to the
former Ph.D. students Carlo Luijten, Geert Hofmans, and Dima Labetski for
numerous discussions at the seminars and help in understanding the subtleties of
nucleation experiments. It is a pleasure to thank Ian Ford, Barbara Wyslouzil,
Judith Wölk, Jan Wedekind, Dennis van Putten, and Anshel Gleyzer for con-
structive criticisms. I am indebted to my colleagues and friends Jos Thijssen, Lev
Goldenberg, Bob Prokofiev, Leonid Neishtadt, Andrey Morozov, Lyudmila
Tsareva, Dmitry Bulahov, Kees Tjeenk Willink, and Marco Betting for encour-
agement and help without which this book would not have been written. But above
all, I am grateful to my family—Esta and Maria—for the constant support during
the almost endless process of thinking, writing, and editing of the manuscript.

Delft, May 2012 V. I. Kalikmanov

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Some Thermodynamic Aspects of Two-Phase Systems . . . . . . . . . 5

2.1 Bulk Equilibrium Properties. . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Thermodynamics of the Interface . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Planar Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Curved Interface. . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Classical Nucleation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Metastable States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Kinetics and Steady-State Nucleation Rate . . . . . . . . . . . . . . 24
3.4 Kelvin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Katz Kinetic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 Consistency of Equilibrium Distributions. . . . . . . . . . . . . . . . 32
3.7 Zeldovich Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.8 Transient Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.9 Phenomenological Modifications of Classical Theory . . . . . . . 40
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Nucleation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 First Nucleation Theorem for Multi-Component Systems . . . . 44
4.3 Second Nucleation Theorem . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 Nucleation Theorems from Hill’s Thermodynamics
of Small Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 51
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 53

vii
viii Contents

5 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . ...... 55

5.1 Nonclassical View on Nucleation . . . . . . . . . . . . . . . ...... 55
5.2 Fundamentals of the Density Functional Approach
in the Theory of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.1 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.2 Intrinsic Free Energy: Perturbation Approach. . . . . . . 60
5.2.3 Planar Surface Tension . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Density Functional Theory of Nucleation . . . . . . . . . . . . . . . 67
5.3.1 Nucleation Barrier and Steady State
Nucleation Rate . . . . . . . . . . . . . . . . . . . . . ...... 67
5.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... 69
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... 70

6 Extended Modified Liquid Drop Model and Dynamic

Nucleation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . ......... 71
6.1 Modified Liquid Drop Model . . . . . . . . . . . . . . . ......... 71
6.2 Dynamic Nucleation Theory and Definition
of the Cluster Volume. . . . . . . . . . . . . . . . . . . . ......... 75
6.3 Nucleation Barrier . . . . . . . . . . . . . . . . . . . . . . ......... 76
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... 77

7 Mean-Field Kinetic Nucleation Theory . . . . . . . . . . . . . . . . . . . . 79

7.1 Semi-Phenomenological Approach to Nucleation . . . . . . . . . . 79
7.2 Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.3 Statistical Thermodynamics of Clusters . . . . . . . . . . . . . . . . . 81
7.4 Configuration Integral of a Cluster: Mean-Field
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.5 Structure of a Cluster: Core and Surface Particles . . . . . . . . . 91
7.6 Coordination Number in the Liquid Phase . . . . . . . . . . . . . . . 95
7.7 Steady State Nucleation Rate . . . . . . . . . . . . . . . . . . . . . . . . 96
7.8 Comparison with Experiment . . . . . . . . . . . . . . . . . . . . . . . . 99
7.8.1 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.8.2 Nitrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.8.3 Mercury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.9.1 Classification of Nucleation Regimes . . . . . . . . . . . . 103
7.9.2 Microscopic Surface Tension: Universal Behavior
for Lennard-Jones Systems . . . . . . . . . . . . . . . . . . . 104
7.9.3 Tolman’s Correction and Beyond . . . . . . . . . . . . . . . 106
7.9.4 Small Nucleating Clusters as Virtual Chains . . . . . . . 110
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8 Computer Simulation of Nucleation . . . . . . . . . . . . . . . . . . . . . . 113

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Contents ix

8.2 Molecular Dynamics Simulation. . . . . . . . . . . . . . . . . . . . . . 114

8.2.1 Basic Concepts and Techniques . . . . . . . . . . . . . . . . 114
8.2.2 System Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.2.3 Thermostating Techniques . . . . . . . . . . . . . . . . . . . . 120
8.2.4 Expansion Simulation . . . . . . . . . . . . . . . . . . . . . . . 124
8.3 Molecular Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . 125
8.4 Cluster Definitions and Detection Methods . . . . . . . . . . . . . . 128
8.5 Evaluation of the Nucleation Rate . . . . . . . . . . . . . . . . . . . . 130
8.5.1 Nucleation Barrier from MC Simulations . . . . . . . . . 130
8.5.2 Nucleation Rate from MD Simulations . . . . . . . . . . . 133
8.6 Comparison of Simulation with Experiment. . . . . . . . . . . . . . 139
8.7 Simulation of Binary Nucleation . . . . . . . . . . . . . . . . . . . . . 140
8.8 Simulation of Heterogeneous Nucleation . . . . . . . . . . . . . . . . 141
8.9 Nucleation Simulation with the Ising Model . . . . . . . . . . . . . 142
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9 Nucleation at High Supersaturations . . . . . . . . . . . . . . . . . . . . . . 145

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 Mean-Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
9.2.1 Landau Expansion for Metastable Equilibrium . . . . . . 146
9.2.2 Nucleation in the Vicinity of the
Thermodynamic Spinodal . . . . . . . . . . . . . . . . . . . . 149
9.3 Role of Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.4 Generalized Kelvin Equation and Pseudospinodal. . . . . . . . . . 154
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

10 Argon Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... 161

10.1 Temperature-Supersaturation Domain: Experiments,
Simulations and Density Functional Theory . . . . . . . . . . . . . . 161
10.2 Simulations and DFT Versus Theory . . . . . . . . . . . . . . . . . . 165
10.3 Experiment Versus Theory . . . . . . . . . . . . . . . . . . . . . . . . . 166
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

11 Binary Nucleation: Classical Theory . . . . . . . . . . . . . . . . . . . . . . 171

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
11.2 Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
11.3 ‘‘Direction of Principal Growth’’ Approximation . . . . . . . . . . 174
11.4 Energetics of Binary Cluster Formation. . . . . . . . . . . . . . . . . 181
11.5 Kelvin Equations for the Mixture . . . . . . . . . . . . . . . . . . . . . 183
11.6 K-Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
11.7 Gibbs Free Energy of Cluster Formation Within
K-Surface Formalism . . . . . . . . . . . . . . . . . . . .......... 189
11.8 Normalization Factor of the Equilibrium Cluster
Distribution Function . . . . . . . . . . . . . . . . . . . .......... 192
x Contents

11.9 Illustrative Results . . . . . . . . . . . . . . . . . . . . . . . ........ 192

11.9.1 Mixture Characterization: Gas-Phase- and
Liquid-Phase Activities . . . . . . . . . . . . . . . . . . . . . . 192
11.9.2 Ethanol/Hexanol System . . . . . . . . . . . . . . . . . . . . . 194
11.9.3 Water/Alcohol Systems . . . . . . . . . . . . . . . . . . . . . . 196
11.9.4 Nonane/Methane System . . . . . . . . . . . . . . . . . . . . . 198
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

12 Binary Nucleation: Density Functional Theory . . . . . . . . . . . ... 205

12.1 DFT Formalism for Binary Systems.
General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
12.2 Non-ideal Mixtures and Surface Enrichment . . . . . . . . . . . . . 209
12.3 Nucleation Barrier and Activity Plots: DFT Versus BCNT . . . 210
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

13 Coarse-Grained Theory of Binary Nucleation . . . . . . . . . . . . . . . 215

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
13.2 Katz Kinetic Approach: Extension to Binary Mixtures . . . . . . 216
13.3 Binary Cluster Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
13.3.1 Binary Vapor as a System
of Noninteracting Clusters . . . . . . . . . . . . . . ...... 222
13.4 Configuration Integral of a Cluster:
A Coarse-Grained Description . . . . . . . . . . . . . . . . . . . . . . . 224
13.4.1 Volume Term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
13.4.2 Coarse-Grained Configuration Integral qCG na . . . . . . . . 227
13.5 Equilibrium Distribution of Binary Clusters . . . . . . . . . . . . . . 228
13.6 Steady State Nucleation Rate . . . . . . . . . . . . . . . . . . . . . . . . 233
13.7 Results: Nonane/Methane Nucleation . . . . . . . . . . . . . . . . . . 235
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

14 Multi-Component Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

14.1 Energetics of N-Component Cluster Formation. . . . . . . . . . . . 239
14.2 Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
14.3 Example: Binary Nucleation . . . . . . . . . . . . . . . . . . . . . . . . 250
14.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

15 Heterogeneous Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
15.2 Energetics of Embryo Formation . . . . . . . . . . . . . . . . . . . . . 254
15.3 Flat Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
15.4 Critical Embryo: The Fletcher Factor . . . . . . . . . . . . . . . . . . 259
15.5 Kinetic Prefactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
15.6 Line Tension Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Contents xi

15.6.1 General Considerations . . . . . . . . . . . . . . . . . . .... 263

15.6.2 Gibbs Formation Energy in the Presence
of Line Tension . . . . . . . . . . . . . . . . . . . . . . . .... 266
15.6.3 Analytical Solution of Modified
Dupre-Young Equation . . . . . . . . . . . . . . . . . . .... 267
15.6.4 Determination of Line Tension. . . . . . . . . . . . . .... 269
15.6.5 Example: Line Tension Effect in Heterogeneous
Water Nucleation . . . . . . . . . . . . . . . . . . . . . . .... 272
15.7 Nucleation Probability. . . . . . . . . . . . . . . . . . . . . . . . . .... 274
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 276

16 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

16.1 Thermal Diffusion Cloud Chamber . . . . . . . . . . . . . . . . . . . . 278
16.2 Expansion Cloud Chamber . . . . . . . . . . . . . . . . . . . . . . . . . 279
16.2.1 Mie Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
16.2.2 Nucleation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
16.3 Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
16.4 Supersonic Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

Appendix A: Thermodynamic Properties . . . . . . . . . . . . . . . . . . . . . . 293

Appendix B: Size of a Chain-Like Molecule . . . . . . . . . . . . . . . . . . . . 297

Appendix C: Spinodal Supersaturation for van der Waals Fluid . . . . . 299

Appendix D: Partial Molecular Volumes . . . . . . . . . . . . . . . . . . . . . . 301

Appendix E: Mixtures of Hard Spheres . . . . . . . . . . . . . . . . . . . . . . . 305

Appendix F: Second Virial Coefficient for Pure Substances

and Mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Appendix G: Saddle Point Calculations . . . . . . . . . . . . . . . . . . . . . . . 309

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Symbols

Avi Vapor phase activity of component i

Ali Liquid phase activity of component i
B2 Second virial coefficient
cp Specific heat at constant pressure
cv Specific heat at constant volume
F Helmholtz free energy of the system
Fint Intrinsic Helmholtz free energy
Fd Helmholtz free energy of hard spheres with diameter d
Fn Helmholtz free energy of the n-cluster
Fconf
n
Configurational Helmholtz free energy of the n-cluster
FðnÞ Helmholtz free energy of the gas of n-clusters
G Gibbs free energy of the system
DGðnÞ Gibbs free energy of n-cluster formation
DG Nucleation barrier

h Planck constant
Jn Net rate of cluster formation (n ! n þ 1)
J Steady-state nucleation rate
J0 Pre-exponential factor for the steady-state nucleation rate
kB Boltzmann constant
m1 Mass of a molecule
n Number of particles in a cluster
nc ; n Number of particles in a critical cluster
N1 Coordination number in the liquid phase
p Pressure
pl Liquid pressure
pv Vapor pressure
pc Critical pressure
pd Pressure of a hard sphere system
psat Saturation pressure
qn Configuration integral of the n-cluster

xiii
xiv Symbols

qn a n b Configuration integral of the binary ðna ; nb Þ-cluster

S Supersaturation
S Entropy
Sn Entropy of the n-cluster
Sconf
n
Configurational entropy of the n-cluster
T Absolute temperature
Tc Critical temperature
uLJ ðrÞ Lennard–Jones interaction potential
UN ðr1 ; . . .; rN Þ Microscopic potential energy of a configuration of N particles
Zl Compressibility factor in the liquid
Zv Compressibility factor in the vapor
Zn Partition function of the n-cluster
Z ðnÞ Partition function of the gas of n-clusters
Zna nb Partition function of the binary ðna ; nb Þ-cluster
Z Zeldovich factor
b ¼ 1=ðkB TÞ Inverse temperature
c1 Surface tension of a flat interface
cmicro Helmholtz free energy per surface particle in the cluster
(microscopic surface tension)
dT Tolman length
j ¼ cp =cv Ratio of specific heats
eLJ ; rLJ Parameters of a Lennard–Jones potential
K de Broglie wavelength of a particle
l Chemical potential
ln Chemical potential of the n-cluster
ld Chemical potential of a hard sphere with a diameter d
lsat Chemical potential of a substance at vapor–liquid equilibrium
(saturation chemical potential)
m Impingement rate per unit surface
mi Impingement rate per unit surface of component i in binary
nucleation
mav Average impingement rate per unit surface in binary nucleation
ql Number density in the bulk liquid
qv Number density in the bulk vapor
qc Critical number density
st Line tension
qðnÞ Number density of n-clusters
qsat ðnÞ Number density of n-clusters at saturation
X Grand potential of the system
h1 Reduced surface tension of a flat interface
hmicro Reduced Helmholtz free energy per surface particle (reduced
microscopic surface tension)
CKE Classical Kelvin equation
CAMS Constant angle Mie scattering
Symbols xv

CGNT Coarse-grained nucleation theory

CNT Classical nucleation theory
BCNT Binary classical nucleation theory
MKNT Mean-field kinetic nucleation theory
EoS Equation of state
EMLD Extended modified liquid drop model
DNT Dynamic nucleation theory
DFT Density functional theory
FPE Fokker-Planck equation
GKE Generalized Kelvin equation
HPS High-pressure section of the shock tube
LPS Low-pressure section of the shock tube
MC Monte Carlo
MD Molecular dynamics
NPC Nucleation pulse chamber
NVT Canonical (NVT) ensemble
NVE Microcanonical (NVE) ensemble
RESS method Rapid expansion of supercritical solution
tWF ten Wolde–Frenkel cluster definition
SAFT Statistical associating fluid theory
SANS Small-angle neutron scattering
SAXS Small-angle X-ray scattering
SSN Laval supersonic nozzle
MFPT Mean first passage time cluster definition
WCA Weeks–Chandler–Anderson theory
Chapter 1
Introduction

Condensation of a vapor, evaporation of a liquid, melting of a crystal, crystalliza-

tion of a liquid are examples of the processes called phase transitions. Generally
speaking, they reflect the ability of physical systems to explore a huge range of
microscopic configurations in accordance with the second law of thermodynamics.
A characteristic feature of a phase transition is an abrupt change of certain properties.
When ice is heated its state first changes continuously up to the moment when the
temperature 0◦ C (at normal pressure) is achieved, at which ice begins transforming
into liquid water with absolutely different properties. Another example is gas cooled
at constant pressure: its state first changes continuously up to a certain temperature at
which condensation begins transforming gas into a liquid. The states of a substance
between which a phase transition takes place are called phases. If the difference
between phases is of quantitative nature, it can in principle be detected through a
microscope. In this case one speaks of a first-order transition. Condensation of gas
is an example of such a transition in which coexisting vapor and liquid phases have
essentially different densities. On the opposite, if the difference between phases is of
qualitative nature it can not be detected by examination of a microscopic sample of
the substance. Phase transition is associated then with a change in symmetry (“sym-
metry breaking”): the two phases are characterized by different internal symmetries
(e.g. structural transitions in crystals result in formation of crystal lattices with differ-
ent symmetries). This change is also abrupt, although the state of the system changes
continuously; a transition in this case is called continuous or second-order.
Close to the two-phase coexistence lines of a first-order phase transition one can
find domains of metastable states. In particular, it is possible to raise the vapor
pressure above the saturation pressure, so that in the domain, where the liquid phase
is thermodynamically stable, a metastable supersaturated vapor can exist. Similarly,
at certain conditions in the domain, where the vapor phase is stable, a metastable
superheated liquid can exist, and in the domain, where the crystalline phase is stable,
a metastable supercooled liquid can exist. The occurrence of these metastable states
originates from the requirement for nucleation. The phenomenon of nucleation is

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 1

DOI: 10.1007/978-90-481-3643-8_1, © Springer Science+Business Media Dordrecht 2013
2 1 Introduction

associated with the nonequilibrium first-order transitions transforming a metastable

parent phase to a thermodynamically stable daughter phase. The transformation takes
place through the creation of small clusters of molecules of the daughter phase out of
the parent phase by thermal fluctuations. Although the parent phase is metastable with
respect to the bulk daughter phase, it can remain stable with respect to small clusters
of the daughter phase. The reason for this is that the thermodynamic properties of the
clusters differ from those of the bulk because of the presence of an interface between
the phases. Whenever there is an abrupt change in the parent and daughter phases,
there is a thermodynamic cost in creating such an interface. Due to this reason the
parent phase can remain present in conditions where the bulk daughter phase should
be more stable.
Nucleation has many practical consequences in science and technology. For exam-
ple, a transition from dry to wet steam in turbines, which proceeds by nucleation of
small droplets due to the presence of pressure and temperature gradients, can lead
to undesirable effects on the performance of the machine and causes erosion of the
turbine blades [1]. Control of nucleation in rocket and jet engines, wind tunnels, and
combustion processes is important for achieving efficient, ecologically sound oper-
ation. Nucleation is the key process in the supersonic gas-liquid separator Twistertm
[2] aimed at removal of water and heavy hydrocarbons from the natural gas without
use of chemicals. In atmospheric science formation of water droplets and ice crystals
in the atmosphere, proceeding by the same mechanism, affects the weather. In the
long-term, these processes play an important role in understanding global warming
(or cooling) [3]. In biology, there is much interest in bypassing nucleation of ice in
the cryopreservation of human tissues [4].
In an attempt to classify the existing theoretical approaches to nucleation one can
conventionally distinguish four groups of models:
• phenomenological models: Classical Nucleation Theory and its modifications
The tool most often used in nucleation studies is the phenomenological Classical
Nucleation Theory (CNT) formulated in the first half of the twentieth century by
Volmer, Weber, Becker, Döring and Zeldovich [5–7] (see also [8, 9]). Its corner-
stone is the capillarity approximation considering a cluster, however small, as a
macroscopic droplet of the condensed phase. Chapter 3 outlines foundations of the
CNT and its most important implications. One of the successful modifications of
the classical approach—the Extended modified liquid drop model developed by
Reiss and co-workers [10–13]—is discussed in Chap. 6
• density functional theory
Density Functional Theory (DFT) of nonuniform fluids [14] was applied to nucle-
ation by Oxtoby and Evans [15] and later developed in a number of publications
by Oxtoby and coworkers. Chapter 5 presents the fundamentals of the DFT and its
application to nucleation studies.
1 Introduction 3

• semi-phenomenological models
The semi-phenomenological approach [16], discussed in Chap. 7, bridges the
microscopic and macroscopic description of nucleation combining statistical
mechanical treatment of clusters and empirical data.
• direct computer simulations
Simulations of nucleation on molecular level by means of Monte Carlo and Mole-
cular Dynamics methods is a technique that complements theoretical and exper-
imental studies and as such may be regarded as a virtual (computer) experiment.
Chapter 8 gives an introduction to molecular simulation methods that are relevant
for modelling of the nucleation process and presents their application to various
nucleation problems.
An important link between theory and nucleation experiment is provided by the so
called nucleation theorems discussed in Chap. 4. Chapter 9 outlines the peculiar fea-
tures of nucleation behavior at deep quenches near the upper limit of metastability.
Chapter 10 is devoted to argon nucleation because of the exceptional role argon plays
in various areas of soft condensed matter physics; here a comparison is presented of
the predictions of theoretical models (outlined in the previous chapters), computer
simulation and available experimental data on argon nucleation. Extensions of the-
oretical models to the case of binary nucleation are discussed in Chaps. 11–13; a
general approach to the multi-component nucleation is outlined in Chap. 14.
Chapters 3–14 refer to homogeneous nucleation (unary, binary, multi-component). If
the nucleation process involves the presence of pre-existing surfaces (foreign bodies,
dust particles, etc.) on which clusters of the new phase are formed, the process is
termed heterogeneous nucleation; it is discussed in Chap. 15.
Though the main aim of the book is to present various theoretical approaches to
nucleation, the general picture would be incomplete without a reference to experi-
mental methods. Chapter 16 gives a short insight into the experimental techniques
used to measure nucleation rates.

References

1. F. Bakhtar, M. Ebrahami, R. Webb, Proc. Instn. Mech. Engrs. 209(C2), 115 (1995)
2. V. Kalikmanov, J. Bruining, M. Betting, D. Smeulders, in SPE Annual Technical Conference
and Exhibition (Anaheim, California, USA, 2007), pp. 11–14. Paper No: SPE 110736
3. P.E. Wagner, G. Vali (eds) Atmospheric Aerosols and Nucleation (Springer, Berlin, 1988)
4. M. Toner, E.G. Cravalho, M. Karel, J. Appl. Phys. 67, 1582 (1990)
5. R. Becker, W. Döring. Ann. Phys. 24, 719 (1935)
6. M. Volmer, Kinetik der Phasenbildung (Steinkopf, Dresden, 1939)
7. Ya. B. Zeldovich, Acta physicochim. URSS 18, 1 (1943)
8. J.E. McDonald, Am. J. Phys. 30, 870 (1962)
9. J.E. McDonald, Am. J. Phys. 31, 31 (1963)
10. H. Reiss, A. Tabazadeh, J. Talbot, J. Chem. Phys. 92, 1266 (1990)
4 1 Introduction

11. H.M. Ellerby, C.L. Weakliem, H. Reiss, J. Chem. Phys. 95, 9209 (1991)
12. H.M. Ellerby, H. Reiss, J. Chem. Phys. 97, 5766 (1992)
13. D. Reguera et al., J. Chem. Phys. 118, 340 (2003)
14. R. Evans, Adv. Phys. 28, 143 (1979)
15. D.W. Oxtoby, R. Evans, J. Chem. Phys. 89, 7521 (1988)
16. V.I. Kalikmanov, J. Chem. Phys. 124, 124505 (2006)
Chapter 2
Some Thermodynamic Aspects
of Two-Phase Systems

In this chapter we briefly recall the basic features of equilibrium thermodynamics of

a two-phase system, i.e. a system consisting of two coexisting bulk phases, which
will serve as ingredients for the nucleation models discussed in this book.

2.1 Bulk Equilibrium Properties

Two phases (1 and 2) can coexist of they are in thermal and mechanical equilibrium.
The former implies that there is no heat flux and therefore T1 = T2 , and the latter
implies that there is no mass flux, which yields equal pressures p1 = p2 . However,
this is not sufficient. Let N be the total number of particles in the two-phase system
N = N1 + N2 . The number of particles in either phase can vary while N is kept fixed.
If the whole system is at equilibrium, its total entropy S = S1 + S2 is maximized,
which means in particular that
∂S
=0
∂ N1

Using the additivity of S , this condition can be expressed as

∂S1 ∂S2
= (2.1)
∂ N1 ∂ N2

The basic thermodynamic relationship reads:

dU = T dS − pdV + μdN (2.2)

Rewriting it in the form

dU p μ
dS = + dV − dN
T T T

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 5

DOI: 10.1007/978-90-481-3643-8_2, © Springer Science+Business Media Dordrecht 2013
6 2 Some Thermodynamic Aspects of Two-Phase Systems

we find
∂S μ
=−
∂N T
From (2.1): μ1 /T1 = μ2 /T2 and since T1 = T2 , the chemical potentials of the
coexisting phases must be equal. Hence, two phases in equilibrium at a temperature
T and pressure p must satisfy the equation

μ1 ( p, T ) = μ2 ( p, T ) (2.3)

which implicitly determines the p(T )-phase equilibrium curve. Thus, T and p cannot
be fixed independently, but have to provide for equality of the chemical potentials
of the two phases. Differentiating this equation with respect to the temperature and
bearing in mind that p = p(T ), we obtain:

∂μ1 ∂μ1 d p ∂μ2 ∂μ2 d p

+ = + (2.4)
∂T ∂ p dT ∂T ∂ p dT

From the Gibbs–Duhem equation (see e.g. [1])

S dT − V d p + N dμ = 0 (2.5)

we find
dμ = −s dT + v d p (2.6)

where s = S
N and v =
V
N are entropy and volume per particle, implying that
   
∂μ ∂μ
= −s, =v
∂T p ∂p T

Using (2.4), we obtain the Clapeyron equation describing the shape of the ( p, T )-
equilibrium curve:
dp s 2 − s1
= (2.7)
dT v2 − v1

In the case of vapor-liquid equilibrium this curve is called a saturation line and the
pressure of the vapor in equilibrium with its liquid is called a saturation pressure,
psat . Liquid and vapor coexist along the saturation line connecting the triple point
corresponding to three-phase coexistence (solid, liquid and vapor) and the critical
point. Below the critical temperature Tc one can discriminate between liquid and
vapor by measuring their density. At Tc the difference between them disappears. The
line of liquid-solid coexistence has no critical point and goes to infinity since the
difference between the symmetric solid phase and the asymmetric liquid phase can
not disappear.
2.1 Bulk Equilibrium Properties 7

Most of the first-order phase transitions are characterized by absorption or release

of the latent heat. According to the first law of thermodynamics (expressing the
principle of conservation of energy), the amount of heat supplied to the system, δ Q,
is equal to the change in its internal energy δU plus the amount of work performed
by the system on its surroundings pδV

δ Q = δU + pδV (2.8)

In the theory of phase transitions the quantity δ Q is called the latent heat L. For
processes at constant pressure the latent heat is given by the change in the enthalpy:

L ≡ δ Q = δ(U + pV ) = δ H (N , p, S ) = T (δS ) p,N

The latent heat per molecule l = L/N is then

l = T (s2 − s1 ) (2.9)

Using (2.9) the Clapeyron equation at T < Tc can be written as

dp l
= (2.10)
dT T (v2 − v1 )

For the gas–liquid transition at temperatures far from Tc the molecular volume in the
liquid phase v1 ≡ vl is much smaller than in the vapor v2 ≡ vv . Neglecting v1 and
applying the ideal gas equation for the vapor

p v2 = kB T (2.11)

we present Eq. (2.10) as

dp lp
= (2.12)
dT kB T 2

where
kB = 1.38 × 10−16 erg/K (2.13)

is the Boltzmann constant. Considering the specific latent heat to be constant, which is
usually true for a wide range of temperatures and various substances,1 and integrating
(2.12) over the temperature, we obtain:

1
psat (T ) = p∞ e−β l , β = (2.14)
kB T

where p∞ is a constant.

1 For example, for water in the temperature interval between 0 and 100 ◦ C, l changes by only 10 %.
8 2 Some Thermodynamic Aspects of Two-Phase Systems

2.2 Thermodynamics of the Interface

Let us discuss the interface between the two bulk phases in equilibrium. For con-
creteness we refer to the coexistence of a liquid with its saturated vapor at the
temperature T . Equilibrium conditions are characterized by equality of temperature,
pressure, and chemical potentials in both bulk phases. The density, however, is not
constant but varies continuously along the interface between two bulk equilibrium
values ρ v (T ) and ρ l (T ). Note, that local fluctuations of density take place even in
homogeneous fluid, where, however, they are small and short-range. In the two-
phase system these fluctuations are macroscopic: for vapor–liquid systems at low
temperatures the bulk densities ρ v and ρ l can differ by 3–4 orders of magnitude.

2.2.1 Planar Interface

Consider a two-phase system contained in a volume V with a planar interface between

the vapor and the liquid. Inhomogeneity is along the z direction; z → +∞ corre-
sponds to bulk vapor, and z → −∞ to bulk liquid (see Fig. 2.1). Variations in the
density give rise to an extra contribution to the thermodynamic functions: they are
modified to include the work γ d A which has to be imposed by external forces in
order to change the interface area A by d A:

dF = − p dV − S dT + γ d A + μ dN (2.15)
dG = V d p − S dT + γ d A + μ dN (2.16)
dΩ = − p dV − S dT + γ d A − N dμ (2.17)

(in (2.17) N is the average number of particles in the system). The coefficient γ is the
surface tension; its thermodynamic definition follows from the above expressions:

Fig. 2.1 Schematic representation of the vapor–liquid system contained in a volume V = L 2 L 1 .

Inhomogeneity is along the z axis (Reprinted with permission from Ref. [1], copyright (2001),
Springer-Verlag.)
2.2 Thermodynamics of the Interface 9

Fig. 2.2 Gibbs dividing

surface bulk vapor Vv

Mv
exc
M
dividing surface
bulk liquid Vl
l
M

 
∂F
γ = (2.18)
∂A N ,V,T
 
∂G
γ = (2.19)
∂A N , p,T
 
∂Ω
γ = (2.20)
∂A μ,V,T

Following Gibbs [2] we introduce a dividing surface, being a mathematical surface

of zero width which establishes a boundary between the bulk phases as shown in
Fig. 2.2. Although its position is arbitrary, it is convenient to locate it somewhere in
the transition zone. Once the position of a dividing surface is chosen, the volumes
of the two phases are fixed, and satisfy

Vv + Vl = V

The idea of Gibbs was that any extensive thermodynamic quantity M (the number
of particles, energy, entropy, etc.) can be written as a sum of bulk contributions M v
and M l and an excess contribution M exc that is assigned to the chosen dividing
surface:
M = M v + M l + M exc (2.21)

Equation (2.21) is in fact a definition of M exc ; its value depends on the location of
the dividing surface, and so do the values of M v and M l (as opposed to M , which
is an actual physical property and as such can not depend on the location of the Gibbs
surface). Several important examples are

N = N v + N l + N exc (2.22)
S = S v + S l + S exc
Ω = Ω v + Ω l + Ω exc
F = F v + F l + F exc
V = Vv + Vl
10 2 Some Thermodynamic Aspects of Two-Phase Systems

By definition the dividing surface has a zero width implying that V exc = 0. Since
the location of the dividing surface is arbitrary, the excess quantities accumulated on
it can be both positive or negative.
One special case that will be useful for future discussions is the equimolar surface
defined through the requirement N exc = 0. The surface density of this quantity

N exc
Γ = (2.23)
A
is called adsorption. Thus, the equimolar surface corresponds to zero adsorption. The
thermodynamic potentials, such as F , Ω, G, are homogeneous functions of the first
order with respect to their extensive variables. We can derive their expressions for
the two-phase system by integrating Eqs. (2.15)–(2.17) using Euler’s theorem for
homogeneous functions (see e.g. [1], Sect. 1.4). In particular, integration of (2.17)
results in
Ω = −p V + γ A (2.24)

whereas in each of the bulk phases Ω v = − p V v , Ω l = − p V l , where we used the

equality of pressures in the coexisting phases. Thus,

Ω exc = γ A (2.25)

irrespective of the choice of the dividing surface. Independence of Ω exc on the loca-
tion of a dividing surface gives rise to the most convenient thermodynamic route
for determination of the surface tension. Equation (2.25) is used in density func-
tional theories of fluids (discussed in Chap. 5) to determine γ from the form of the
intermolecular potential.
By definition
Ω exc = Ω − Ω v − Ω l (2.26)

For each of the bulk phases

dΩ v = − p dV v − S v dT − N v dμ (2.27)
dΩ l = − p dV l − S l dT − N l dμ (2.28)

Differentiating (2.26) using (2.17) and (2.27)–(2.28) yields

dΩ exc = −S exc dT + γ d A − N exc dμ

On the other hand, from (2.25)

dΩ exc = γ d A + A dγ
2.2 Thermodynamics of the Interface 11

Comparison of these two equalities leads to the Gibbs adsorption equation

A dγ + S exc dT + N exc dμ = 0 (2.29)

describing the change of the surface tension resulting from the changes in T and μ.
An important consequence of (2.29) is the expression for adsorption:
 
∂γ
Γ =− (2.30)
∂μ T

where the surface tension refers to a particular dividing surface.

2.2.2 Curved Interface

Gibbs’ notion of a dividing surface is a useful concept for thermodynamic description

of an interface. At the same time, as we saw in Sect. 2.2.1, the planar surface tension
is not affected by a particular location of a dividing surface since the surface area A
remains constant at any position of the latter. The situation drastically changes when
we discuss a curved interface. Here the position of the dividing surface determines
not only the volumes of the two bulk phases but also the interfacial area. An arbitrary
curved surface is characterized by two radii of curvature. Consider a liquid droplet
inside a fixed total volume V of the two-phase system containing in total N molecules
at the temperature T . The “radius” of the droplet is smeared out on the microscopic
level since it can be defined to within the width of the interfacial zone, which is of
the order of the correlation length. Let us choose a spherical dividing surface with a
radius R. The sizes of the two phases and the surface area are fully determined by
a set of four variables for which it is convenient to use R, A, V l and V v [3], where
V l and V v are the bulk liquid and vapor volumes and A is the surface area:

4π 3 4π 3
Vl = R , Vv = V − R , A = 4π R 2
3 3
A sketch of a spherical interface is shown in Fig. 2.3. The change of the Helmholtz
free energy F of the two-phase system “droplet + vapor” when its variables change
at isothermal conditions is given by [3]:
 
dγ
(dF )T = − p (dV )T − p (dV )T +μ(dN )T +γ (d A)T + A
l l v v
(d R)T (2.31)
dR

Here by a differential in square brackets we denote a virtual change of a thermody-

namic parameter, corresponding to a change in R. The pressure p l inside the liquid
phase refers to the bulk liquid held at the same chemical potential as the surrounding
vapor with the pressure p v : μv ( p v ) = μl ( p l ). The surface tension γ (R) refers to the
12 2 Some Thermodynamic Aspects of Two-Phase Systems

Rv
zs
Re
ze ~ξ
Rs

l
R

Fig. 2.3 Sketch of a spherical interface. The z axis is perpendicular to the interface pointing away
from the center of curvature. Re and Rs ≡ Rt denote, respectively, the location of the equimolar
surface and the surface of tension (see the text). The width of the transition zone between bulk vapor
and bulk liquid is of the order of the correlation length ξ (Reprinted with permission from Ref. [1],
copyright (2001), Springer-Verlag.)

dividing surface of the radius R; the term in the square brackets gives the change of
γ with respect to a mathematical displacement of the dividing surface. It is impor-
tant to stress that the physical quantities F , p v , p l , μ, N , V , do not depend on the
location of a dividing surface. So they remain unchanged when only R is changed
and from (2.31)
 
dγ
0 = [dF ] = −Δp 4π R 2 [d R] + 8π R γ [d R] + 4π R 2 [d R]
dR

where Δp = p l − p v . Dividing by 4π R 2 [d R] we obtain the generalized Laplace

equation:  
2γ [R] dγ
Δp = + (2.32)
R dR

It is clear that since Δp, as a physical property of the system, is independent of R, the
surface tension must depend on the choice of dividing surface. A particular choice
R = Rt , such that  
dγ
= 0, (2.33)
d R R=Rt

corresponds to the so-called surface of tension; it converts (2.32) into the standard
Laplace equation
2γt
Δp = (2.34)
Rt
2.2 Thermodynamics of the Interface 13

where γt = γ [Rt ]. One can relate the surface tension taken at an arbitrary dividing
surface of a radius R to γt . To this end let us write (2.32) in the form
 
d
Δp R = 2
R 2 γ [R]
dR

and integrate it from Rt to R. Using (2.34) for Δp we obtain the Ono-Kondo equation
[4]  
R 1 1 2
γ [R] = γt f ok , with f ok = 2
+ x (2.35)
Rt 3 x 3

Elementary analysis shows that f ok has a minimum at x = 1 corresponding to

R = Rt . Thus, γt is the minimum surface tension among all possible choices of the
dividing surface:    
R − Rt 2
γ [R] = γt 1 + O (2.36)
Rt

When R differs from Rt by a small value, γ [R] remains constant to within terms of
order 1/Rt2 .
Among various dividing surfaces we distinguished two special cases—the equimolar
surface Re and the surface of tension Rt —which are related to the certain physi-
cal properties of the system. Let us introduce a quantity describing the separation
between them
δ = Re − Rt

The limiting value of δ at the planar limit

δT = lim δ = z e − z t (2.37)
Rt →∞

is called the Tolman length. Its sign can be both positive and negative depending on
the relative location of the two dividing surfaces. By definition δT does not depend
on either radius Rt , or Re (whereas δ does) but can depend on the temperature. Both
dividing surfaces lie in the interfacial zone implying that δT is of the order of the
correlation length. Let Γt be the adsorption at the surface of tension. From the Gibbs
adsorption equation  
∂γt
Γt = −
∂μ T

Using the thermodynamic relationship (2.6) in both phases we rewrite this result as

d pv d pl
dγt = −Γt dμ = −Γt = −Γt l
ρ v ρ
14 2 Some Thermodynamic Aspects of Two-Phase Systems

From the second and the third equations of this chain

ρl
d pl = d pv
ρv

resulting in
d(Δp) = Δρ dμ

Substituting Δp from the Laplace equation (2.34) we obtain

 
Γt 2γt
dγt = − d (2.38)
Δρ Rt

For a curved surface Tolman [5] showed (see also [3]) that
 
Γt δT 1 δT 2
= δT 1+ + (2.39)
Δρ Rt 3 Rt2

but the terms in (δT /Rt ) and (δT /Rt )2 can be omitted to the order of accuracy we
need. This means that in all derivations below we need to keep only the linear terms
in δT . With this in mind Eqs. (2.38) and (2.39) give
 
2γt
dγt = −δT d
Rt

which after simple algebra yields

 
2δT 1 1
d ln γt = d Rt = − d Rt
Rt (Rt + 2δT ) Rt Rt + 2δT

Integrating from the planar limit (Rt → ∞) to Rt we obtain:

γt Rt
= (2.40)
γ∞ Rt + 2δT

where γ∞ is the planar surface tension discussed in Sect. 2.2.1. Keeping the linear
term in δT we finally obtain the Tolman equation
 
2δT
γ t = γ∞ 1− + ... (2.41)
Rt

It is important to emphasize that the second order term in the δT can not be obtained
from (2.40) since this equation is derived to within the linear accuracy in the Tolman
length.
2.2 Thermodynamics of the Interface 15

Equation (2.41) represents the expansion of the surface tension of a curved interface
(droplet) in powers of the curvature. Its looses its validity when the radius of the
droplet becomes of the order of molecular sizes. The concept of a curvature dependent
surface tension frequently emerges in nucleation studies. It is therefore important
to estimate the minimal size of the droplet for which the Tolman equation holds.
For simple fluids (characterized by the Lennard-Jones and Yukawa intermolecular
potentials) near their triple points the density functional calculations [6] reveal that
the Tolman equation is valid for droplets containing more than 106 molecules.

References

1. V.I. Kalikmanov, in Statistical Physics of Fluids. Basic Concepts and Applications (Springer,
Berlin, 2001)
2. J.W. Gibbs, in The Scientific Papers (Ox Bow, Woodbridge, 1993)
3. J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity (Clarendon Press, Oxford, 1982)
4. S. Ono, S. Kondo, in Encyclopedia of Physics, vol. 10, ed. by S. Flugge (Springer, Berlin, 1960),
p. 134
5. R.C. Tolman, J. Chem. Phys. 17(118), 333 (1949)
6. K. Koga, X.C. Zeng, A.K. Schekin, J. Chem. Phys. 109, 4063 (1998)
Chapter 3
Classical Nucleation Theory

3.1 Metastable States

Nucleation refers to the situation when a system (parent phase) is put into a nonequi-
librium metastable state. Experimentally it can be achieved by a number of ways (for
definiteness we refer to the vapor-liquid transition): e.g. by isothermally compress-
ing vapor up to a pressure p v exceeding the saturation vapor pressure at the given
temperature psat (T ). At this state, characterized by p v and T , the chemical potential
in the bulk liquid μl ( p v , T ) is lower than in the bulk vapor at the same conditions
μv ( p v , T ), which makes it thermodynamically favorable to perform a transformation
from the parent phase (vapor) to the daughter phase (liquid). The driving thermody-
namic force for this transformation is the chemical potential difference

Δμ = μv ( p v , T ) − μl ( p v , T ) > 0 (3.1)

Physically a metastable state (supersaturated vapor) corresponds to a local mini-

mum of the free energy (see Fig. 3.1 for a schematic illustration) as a function of
the appropriate order parameter. Metastability means that the system in this state
(state A in Fig. 3.1) is stable to small fluctuations in the thermodynamic variables
but after a certain time will evolve to state B corresponding to the global minimum
of the free energy (bulk liquid). In order to perform this transformation, this system
has to overcome a barrier, representing a local maximum of the free energy. The
latter corresponds to an unstable equilibrium state in which the system is unstable
with respect to small fluctuations of the thermodynamic variables. To overcome the
energy barrier a fluctuation is required which causes a formation of a small quantity
(cluster) of the new phase called nucleus. Such process of homogeneous nucleation is
thermally achieved (the case of heterogeneous nucleation by impurities is discussed
in Sect. 15.1). Let us rewrite (3.1)

Δμ = [μv ( p v , T ) − μsat (T )] + {μsat (T ) − μl ( p v , T )}

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 17

DOI: 10.1007/978-90-481-3643-8_3, © Springer Science+Business Media Dordrecht 2013
18 3 Classical Nucleation Theory

Fig. 3.1 Sketch of the free

energy as a function of the
order parameter. Local min-
imum corresponds to the unstable state

Free energy
metastable state A (super-

barrier
saturated vapor). The global
minimum corresponds to the
stable state B (bulk liquid).
Transition between the states
A and B involves overcoming
the free energy barrier
metastable stable
state A state B

where μsat (T ) is the chemical potential at saturation at the temperature T . Expression

in the curl brackets is the difference in chemical potential in the liquid phase, taken
at saturation (i.e. at the pressure psat ) and at the actual pressure p v . Assuming the
incompressibility of the liquid and applying the thermodynamic relationship (2.6) to
the liquid phase, we rewrite (3.1) as

1
Δμ ≈ [μv ( p v , T ) − μsat (T )] − ( p v − psat ) (3.2)
ρsat
l

where ρsat
l is the liquid number density at saturation. Using the compressibility factor

of the liquid phase at saturation

psat
l
Z sat = (3.3)
ρsat
l k T
B

we present Δμ as
 
pv
Δμ = kB T ln S − kB T Z sat
l
−1
psat

where the dimensionless quantity

 
μv ( p v , T ) − μsat (T )
S = exp (3.4)
kB T

is called the supersaturation ratio, or simply the supersaturation. The chemical

potentials are not directly accessible in experiments. Assuming the ideal gas law for
the vapor
pv
ρ v ( pv ) =
kB T
3.1 Metastable States 19

and using Eq. (2.6) for the vapor phase, the supersaturation can be approximated as

pv
S= (3.5)
psat

so that
Δμ = kB T ln S − kB T Z sat
l
(S − 1) (3.6)

l ∼ 10−6 − 10−3 , and the second term in

At temperatures not too close to Tc , Z sat
(3.6) can be safely neglected resulting in

Δμ = kB T ln S (3.7)

We expressed the driving force to nucleation in terms of the supersaturation S con-

taining the experimentally controllable quantities: pressure and temperature. The
quantity S characterizes the degree of metastability of the system; S ≥ 1, where the
equality sign corresponds to equilibrium.

3.2 Thermodynamics

If the lifetime of the metastable state is much larger than the relaxation time necessary
for the system to settle in this state, we can apply the concept of quasi-equilibrium
treating the metastable state as if it were an equilibrium. Instead of a thermodynamic
probability of occurrence of a nucleus we shall discuss the “equilibrium” (in the
above sense) distribution function of n-clusters, ρeq (n), which is proportional to it.
Considerations based on the thermodynamic fluctuation theory [1] yield:
 
Wmin (n)
ρeq (n) = ρ1 exp − (3.8)
kB T

where ρ1 is a temperature dependent constant and Wmin (n) is a minimum (reversible)

work required to form an n-cluster in the surrounding vapor at the pressure p v and
the temperature T .
To calculate Wmin we use general thermodynamic considerations following
Debenedetti [2]. Consider the system put in contact with a reservoir (heat bath).
The initial state of the system is pure vapor with the internal energy U0v (the sub-
scripts “0” and “f” denote the initial and final conditions, respectively). The final
state (after the droplet was formed) is the “droplet + vapor”. Within the framework
of Gibbs thermodynamics we introduce a dividing surface of a radius R and write
the internal energy at the final state as (cf. Eq. (2.21))

Uf = Ufv + U l + U exc
20 3 Classical Nucleation Theory

where the first term refers to the bulk vapor, the second term refers to the bulk liquid
and the third one gives the surface (excess) contribution. The total change in the
internal energy ΔU = Uf − U0 , caused by the change in the physical state, includes:
• the work W exerted on the system by an external source (creating pressure);
• the work performed by the heat bath to create a droplet, and
• the heat received by the system from the heat bath.
The heat bath is considered to be large enough so that its pressure pr and temperature
Tr remain constant (quantities referring to the heat bath are denoted with a subscript
“r ”). The work performed by the heat bath is pr ΔVr , and the heat given by it is
−Tr ΔSr . Thus,
ΔU = W + pr ΔVr − Tr ΔSr (3.9)

The total volume of heat bath and the system remains unchanged, ΔVr = −ΔV .
According to the second law of thermodynamics the total change of entropy (heat
bath + system) should be nonnegative:

ΔSr + ΔS ≥ 0 (3.10)

From (3.9) and (3.10) we obtain

W ≥ ΔU − Tr ΔS + pr ΔV

Equality sign corresponds to the reversible process (in which the total entropy remains
unchanged) yielding the minimum work

Wmin = ΔU − Tr ΔS + pr ΔV (3.11)

If furthermore we assume that the process of transformation to the final state (i.e.
the droplet formation) takes place at a constant temperature T = Tr and pressure
p v = pr then
Wmin = ΔU − T ΔS + p v ΔV = ΔG (3.12)

where G = U − T S + p v V is the Gibbs free energy of the two-phase system

“droplet + vapor”; V is its volume. Thus, the minimum work to form an n-cluster is
equal to the change of the Gibbs free energy. That is why in nucleation theories it is
common to speak about the “Gibbs free energy of droplet formation”.
Let us calculate ΔG from Eq. (3.12). The change in the internal energy is

ΔU = (Ufv − U0v ) + U l + U exc (3.13)

Integrating (2.2) using Euler’s theorem for homogeneous functions we obtain:

U0v = T S0v − p v V0v + μv N0v (3.14)

3.2 Thermodynamics 21

U vf = T S fv − p v V fv + μv N vf (3.15)
U l = T S l − p l V l + μl N l (3.16)
U exc = T S exc + γ (R) A(R) + μexc N exc (3.17)

Since the vapor pressure and temperature are constant the vapor chemical potential
μv ( p v , T ) does not change during the transformation of vapor from the initial to the
final state. In the last expression γ (R) is a surface tension at the dividing surface R
with a surface area A(R). The change of the system volume is
 
ΔV = V fv + V l − V0v (3.18)

Substituting (3.13)–(3.18) into Eq. (3.12) and taking into account conservation of the
number of molecules N0v = N vf + N l + N exc , we obtain

ΔG = ( p v − p l )V l +γ A + N l μl ( p l ) − μv ( p v ) + N exc μexc − μv ( p v ) (3.19)

This is an exact general expression for the Gibbs free energy of cluster formation.
The location of a dividing surface is not specified. Choosing the equimolar surface
Re (with γ = γe , A = Ae ), the last term in (3.19) vanishes. Considering the liquid
phase to be incompressible we write

μl ( p l ) = μl ( p v ) + vl ( p l − p v ) (3.20)

where vl is a molecular volume of the liquid phase (usually taken at coexistence

vl = vsat
l (T )). Substitution of (3.20) into (3.19) yields

ΔG = −n Δμ + γe Ae (3.21)

where n ≡ N l and Δμ is defined in (3.1). Within the classical approach small

droplets causing nucleation of the bulk liquid from the vapor are treated as macro-
scopic objects. This is the essence of the so called capillarity approximation which
is the fundamental assumption of the Classical Nucleation Theory (CNT) developed
in 1926–1943 by Volmer, Weber, Becker, Döring and Zeldovich [3–5]. Within the
capillarity approximation
• a cluster is viewed as a large homogeneous spherical droplet of a well defined
radius with the bulk liquid properties inside it and the bulk vapor density outside
it;
• the liquid is considered incompressible and
• the surface energy of the cluster containing n molecules (n-cluster) is presented
as the product of the planar interfacial tension at the temperature T , γ∞ (T ), and
the surface area of the cluster A(n).
With these assumptions Eq. (3.21) becomes
22 3 Classical Nucleation Theory

ΔG = −n Δμ + γ∞ A(n) (3.22)

where A(n) = 4π rn2 , rn is the radius of the cluster

rn = r l n 1/3

and  1/3
3 vl
r =
l
(3.23)
4π

is the average intermolecular distance in the bulk liquid. Thus,

A(n) = s1 n 2/3

where  2/3
s1 = (36π )1/3 vl (3.24)

is the “surface area of a monomer”. Using (3.7) the reduced free energy of cluster
formation reads:
βΔG(n) = −n ln S + θ∞ n 2/3 (3.25)

with the reduced surface tension

γ∞ s1
θ∞ = (3.26)
kB T

The function ΔG(n) is schematically shown in Fig. 3.2. At small n the positive
surface term dominates making it energetically unfavorable to create a very small
droplet in view of the large uncompensated surface energy. For large n the negative
bulk contribution prevails. The function has a maximum at
 3
2 θ∞
nc = (3.27)
3 ln S

The cluster containing n c molecules is called a critical cluster; ΔG(n c ) = ΔG ∗

represents an energy barrier which a system has to overcome to form a new stable
(liquid) phase. Droplets with n < n c molecules on average dissociate whereas those
with n > n c on average grow into the new phase. Substituting (3.27) into (3.22) we
obtain the CNT free energy barrier, called also the nucleation barrier:

1 16π (vl )2 γ∞
3
ΔG ∗ = γ∞ A(n c ) = (3.28)
3 3 (kB T ln S)2
3.2 Thermodynamics 23

Fig. 3.2 Gibbs free energy of

cluster formation ΔG(n). n c dissociation growth
is the critical cluster; clusters *
G k T
below n c are on average critical region
dissociating; clusters beyond
n c are on average growing to

G
0
the new bulk phase. Within the
critical region, characterized
by ΔG(n c ) − ΔG(n) ≤ kB T ,
the fluctuation development
of clusters occurs

nc n

or in the dimensionless quantities:

4 θ∞ 3
βΔG ∗ = (3.29)
27 (ln S)2

The critical cluster is in metastable (quasi-) equilibrium with the surrounding vapor
yielding
μv ( p v ) = μl ( p l )

This equality together with Eq. (3.20) leads to an alternative form of the nucleation
barrier:
16π γ∞ 3
ΔG ∗ = (3.30)
3 (Δp)2

where Δp = p l − p v ; note that p l refers to the bulk liquid held at the same temperature
T and the same chemical potential μ as the supersaturated vapor.
Maximum of ΔG(n) corresponds to the exponentially sharp minimum of the distri-
bution function (3.8). Therefore instead of speaking about the critical point n = n c
it would be more correct to discuss the critical region around n c , where ΔG(n) to a
good approximation has the parabolic form:

1 d
ΔG(n) ≈ ΔG ∗ + ΔG (n c )(n − n c )2 , = (3.31)
2 dn

(the term linear in (n − n c ) vanishes). This quadratic expansion yields the Gaussian
form for ρeq (n), centered at n c and having the width
 −1/2
1
Δ = − βΔG (n c ) (3.32)
2
24 3 Classical Nucleation Theory

It follows from (3.31) and (3.32) that the critical region corresponds to the clusters
satisfying
ΔG(n c ) − ΔG(n) ≤ kB T

Recall that the average free energy associated with an independent fluctuation in a
fluid is of order kB T . Therefore, the above expression indicates that the clusters in
the critical region fall within the typical fluctuation range around the critical cluster.
As a result fluctuation development of nuclei in the domain of cluster sizes (n c , n c +
Δ) may with an appreciable probability bring them back to the subcritical region but
nuclei which passed the critical region will irreversibly develop into the new phase.

3.3 Kinetics and Steady-State Nucleation Rate

The equilibrium cluster distribution

 
ΔG(n)
ρeq (n) = ρ1 exp − , (3.33)
kB T

is limited by the stage before the actual phase transition and therefore it can not
predict the development of this process. At large n, the Gibbs formation energy is
dominated by the negative bulk contribution −nΔμ implying that for large clusters
ρeq (n) diverges
ρeq → ∞ as n→∞

The true number density ρ(n, t) (nonequilibrium cluster distribution), as any other
physical quantity, should remain finite for any n and at any moment of time t. To
determine ρ(n, t) it is necessary to discuss kinetics of nucleation. Within the CNT
the following assumptions are made:
• the elementary process which changes the size of a nucleus is the attachment to it
or loss by it of one molecule
• if a monomer collides a cluster it sticks to it with probability unity
• there is no correlation between successive events that change the number of par-
ticles in a cluster.
The last assumptions means that nucleation is a Markov process. Its schematic illus-
tration is presented in Fig. 3.3. Let f (n) be a forward rate of attachment of a molecule
to an n-cluster (condensation) as a result of which it becomes an (n + 1)-cluster, and
b(n) be a backward rate corresponding to loss of a molecule by an n-cluster (evap-
oration) as a result of which it becomes and (n − 1)-cluster. Then the kinetics of the
nucleation process is described by the set of coupled rate equations
3.3 Kinetics and Steady-State Nucleation Rate 25

f(n-1) f(n)

n-1 n n+1

b(n) b(n+1)

Fig. 3.3 Schematic representation of kinetics of homogeneous nucleation; f (n)—forward rate

(condensation), b(n)—backward rate (evaporation)

∂ρ(n, t)
= f (n − 1) ρ(n − 1, t) − b(n) ρ(n, t) − f (n) ρ(n, t) + b(n + 1) ρ(n + 1, t)
∂t
(3.34)
A net rate at which n-clusters become (n + 1)-clusters is defined as

J (n, t) = f (n) ρ(n, t) − b(n + 1) ρ(n + 1, t) (3.35)

implying that
∂ρ(n, t)
= J (n − 1, t) − J (n, t) (3.36)
∂t

The set of equations (3.36) for various n with J (n, t) given by (3.35) was proposed
by Becker and Döring [3] and is called the Becker-Döring equations. The expression
for the forward rate f (n) depends on the nature of the phase transition. For gas-to-
liquid transition f (n) is determined by the rate of collisions of gas monomers with
the surface of the cluster
f (n) = ν A(n) (3.37)

Here the monomer flux to the unit surface ν, called an impingement rate, is found
from the gas kinetics [1]:
pv
ν=√ , (3.38)
2π m 1 kB T

m 1 is the mass of a molecule. Thus, f (n) is proportional to the pressure of the

supersaturated vapor.
The backward (evaporation) rate b(n), at which a cluster looses molecules, a priori is
not known. It is feasible to assume that this quantity is to a large extent determined by
the surface area of the cluster rather than by the properties of the surrounding vapor.
Therefore b(n) can be assumed to be independent on the actual vapor pressure. In
order to find it CNT uses the detailed balance condition at a so called constrained
equilibrium state [6, 7], which would exist for a vapor at the same temperature T and
the supersaturation S > 1 as the vapor in question. In the constrained equilibrium the
net flux is absent J (n, t) = 0 since it corresponds to the stage before the nucleation
process starts, and the cluster distribution is given by ρeq (n). From (3.35) this implies

ρeq (n)
b(n + 1) = f (n) (3.39)
ρeq (n + 1)
26 3 Classical Nucleation Theory

Substituting (3.39) into (3.35) and rearranging the terms we have:

1 ρ(n, t) ρ(n + 1, t)
J (n, t) = − (3.40)
f (n) ρeq (n) ρeq (n) ρeq (n + 1)

The kinetic process described by Eq. (3.34) rapidly reaches a steady state: a character-
istic relaxation time, τtr , is usually ∼1 μs (we briefly discuss the transient nucleation
behavior in Sect. 3.8) which is much smaller than a typical experimental time scale.
In the steady nonequilibrium state the number densities of the clusters no longer
depend on time. This implies that all fluxes are equal

J (n, t) = J for all n, (t → ∞)

The flux J , called the steady-state nucleation rate, is a number of nuclei (of any
size) formed per unit volume per unit time. Summation of both sides of Eq. (3.40)
from n = 1 to a sufficiently large N yields (due to mutual cancelation of successive
terms):
N  
1 ρ(1) ρ(N + 1)
J = − (3.41)
f (n) ρeq (n) ρeq (1) ρeq (N + 1)
n=1

For small clusters the free energy barrier is dominated by the positive surface con-
tribution θ n 2/3 , implying that the number of small clusters continues to have its
equilibrium value in spite of the constant depletion by the flux J :
ρ(n)
→1 as n → 1+ (3.42)
ρeq (n)

For large n the forward rate exceeds the reverse: the system evolves into the new,
thermodynamically stable, phase. As n grows ρeq (n) increases without limit whereas
the true distribution ρ(n) remains finite. Thus,

ρ(n)
→0 as n → ∞ (3.43)
ρeq (n)

By choosing large enough N we can neglect the second term in (3.41). Extending
summation to infinity we rewrite it as

∞
−1
1
J= (3.44)
f (n) ρeq (n)
n=1

Examine the terms of this series. The cluster distribution at constrained equilibrium
is given by Eqs. (3.33) and (3.25)

ρeq (n) = ρ1 S n exp[−θ∞ n 2/3 ] (3.45)

3.3 Kinetics and Steady-State Nucleation Rate 27

In this form it was first discussed by Frenkel [8, 9] and is called the Frenkel distri-
bution. Initially (for small n) ρeq (n) decreases due to the surface contribution, then
reaches a minimum at n = n c beyond which it exponentially grows due to the bulk
contribution S n .
The major contribution to the series (3.44) comes from the terms in the vicinity of
n c . If n c is large enough (and the validity of CNT requires large n c ) the number of
these terms is large while the difference between the successive terms for n and n + 1
is small. This makes it possible to replace summation by an integral:
 ∞ −1
1
J= dn (3.46)
1 f (n)ρeq (n)

The integral can be calculated using the steepest descent method if one takes into
account the sharp exponential minimum of ρeq at n c . Expanding ρeq about n c we
write:
 
1 1
ρeq (n) ≈ ρeq (n c ) exp − ΔG (n c )(n − n c ) , ΔG (n c ) < 0
2
2 kB T

The Gaussian integration in (3.46) results in

J = Z f (n c )ρeq (n c ) (3.47)

where 
1 1
Z = − ΔG (n c ) (3.48)
2π kB T

is called the Zeldovich factor. Comparison of (3.48) with (3.32) shows that Z is
inversely proportional to the width of the critical region:
1
Δ= √ (3.49)
πZ

Using (3.48) and (3.25) the Zeldovich factor takes the form:

1 θ∞ −2/3
Z = nc (3.50)
3 π

or equivalently 
γ∞ 1
Z = (3.51)
kB T 2πρ l rc2

1/3
where rc = r l n c is the radius of the critical cluster.
Summarizing, the main result of the CNT states that the steady state nucleation rate
is an exponential function of the energy barrier
28 3 Classical Nucleation Theory
 
ΔG ∗
J = J0 exp − (3.52)
kB T

16π (vl )2 γ∞
3
ΔG ∗ = (3.53)
3 |Δμ|2

where the pre-exponential factor is

J0 = Z ν A(n c ) ρ v (3.54)

with ρ v ≈ ρ1 being the density of the supersaturated vapor. Here ν A(n c ) = f (n c ) is

the rate at which molecules attach to the critical cluster causing it to grow. However,
for a cluster in the critical region there exist a chance that it will not cross the barrier
but dissociate back the mother phase. The Zeldovich factor stands for the probability
of a critical cluster to cross the energy barrier; therefore, the rate at which the cluster
actually crosses the barrier and grows into the new phase is not f (n c ), but Z f (n c ).
−2/3
From (3.50): Z ∼ = n c , thus for the critical clusters with n c = 10 − 100 molecules
Z is of order 0.1 − 0.01. The pre-exponential factor can be presented in a simple
form containing measurable quantities if we apply the ideal gas model for the vapor.
Then, from (3.37), (3.38), (3.51) and (3.54):

(ρ v )2 2γ∞
J0 ∼
= (3.55)
ρl π m1

It is important to realize that the steady state flux J does not depend on size, but is
observed at the critical cluster size.
Nucleation of water vapor plays an exceptionally important role in a number of
environmental processes and industrial applications. That is why water can be cho-
sen as a first example to analyze the predictions of the CNT. Two experimental
groups, Wölk et al. [10] and Labetski et al. [11] reported the results of nucleation
experiments for water in helium (as a carrier gas) in the wide temperature range:
220–260 K in [10] and 200–235 K in [11]. Though, the two groups used differ-
ent experimental setups—an expansion chamber in [10] and a shock wave tube in
[11]—the results obtained are consistent. Figure 3.4 shows the relative—experiment
to theory (CNT)—nucleation rate of water for the temperature range 200 < T <
260 K. Circles correspond to the experiment of [10], squares—to the experiment
of [11]. The thermodynamic data for water are given in Appendix A. Figure 3.4
demonstrates a clear trend: the CNT underestimates the experiment (up to 4 orders
of magnitude) for lower temperatures, and slightly overestimates it for higher tem-
peratures. The dashed line (“ideal line”) corresponds to Jexp = Jcnt ; the predictions
of CNT coincide with experiment for water at temperatures around 240 K. These
results indicate a general feature of the CNT: while its predictions of the nucleation
rate dependence on S are quantitatively correct, the dependence of J on the tempera-
ture are in many cases in error—as illustrated by the long-dashed line in Fig. 3.4. The
3.3 Kinetics and Steady-State Nucleation Rate 29

Fig. 3.4 Relative nucleation

rate Jrel = Jexp /Jcnt for water. Water
Circles: experiment of Wölk 4
et al. [10]; squares: experiment
of Labetski et al. [11]. The

log 10 (Jexp /Jcnt )

long-dashed line, shown to
guide the eye, indicates the 2
temperature dependence of
the relative nucleation rate.
Also shown is the “ideal line”
(dashed line) Jexp = Jcnt 0

-2
200 220 240 260

T (K)

discrepancy between the CNT and experiment grows as the temperature decreases.
This is not surprising since the critical cluster at 200 < T < 220 K at experimen-
tal conditions (supersaturation) contains only ≈15–20 molecules as follows from
Eq. (3.27), implying that the purely phenomenological approach, based on the capil-
larity approximation becomes fundamentally in error. In the next chapters we discuss
alternative models of nucleation which do not invoke the capillarity approximation.

3.4 Kelvin Equation

Consider in a more detail the metastable equilibrium between the liquid droplet
of a radius R and the surrounding supersaturated vapor at the pressure p v and the
temperature T . We characterize the droplet radius by its value at the surface of tension
R = Rt . Equilibrium implies that the chemical potentials of a molecule outside and
inside the droplet are equal:
μvR ( p v ) = μlR ( p l ) (3.56)

Here p l is the pressure inside the cluster. The same expression written for the bulk
equilibrium at the temperature T yields

μvbulk = μlbulk = μsat (T ) (3.57)

In this case the pressure in both phases is equal to the saturation pressure: psat (T ).
Note that Eq. (3.57) can be viewed as an asymptotic form of (3.56) when the droplet
radius R → ∞. Subtracting (3.56) from (3.57) and using the thermodynamic rela-
tionship (2.6) we obtain
30 3 Classical Nucleation Theory

 pv 1
dp = μlR ( p l ) − μsat (3.58)
psat ρ( p)

For temperatures not close to Tc the vapor density on the left-hand side can be written
in the ideal gas form resulting in
 
pv
kB T ln = μlR ( p l ) − μsat (3.59)
psat

Within the capillarity approximation using the thermodynamic relationship (2.6) for
the liquid phase, the right-hand side of Eq. (3.59) becomes

μlR ( p l ) − μsat = vl ( p l − psat )

Using the Laplace equation this expression gives

    
pv 2γt vl pv
ln = + Z sat
l
−1 (3.60)
psat k B T Rt psat

l
For low temperatures the liquid compressibility factor at saturation Z sat 1 and the
last term in (3.60) can be safely neglected yielding
 
2γ∞ vl
p = psat exp
v
(3.61)
k B T Rt

This result, called the Kelvin equation, was formulated by Sir William Thomson
(Lord Kelvin) in 1871 [12]. It relates the vapor pressure, p v , over a spherical liquid
drop to its radius.
In the original Kelvin’s work nucleation was not discussed. Later on the same equa-
tion naturally appeared in the formulation of the CNT since the critical cluster is
in the metastable equilibrium with the surrounding vapor which can be expressed
in the form of Eq. (3.56). At the same time in nucleation theory this equilibrium
corresponds to the maximum of the Gibbs energy of cluster formation
μvR ( p v ) = μlR ( p l ) ⇔ maxn ΔG(n) (3.62)

Thus, the alternative way to derive the Kelvin equation is to maximize the free energy
of cluster formation. Using this equivalent formulation and applying the capillarity
approximation to ΔG(n), we derive the classical Kelvin equation (3.61). The latter
has long played a very important role in nucleation theory (for a detailed discussion
see [13]). Since the radius of a spherical n-cluster is R = r l n 1/3 , we can rewrite
it in the form containing only dimensionless quantities which is more suitable for
nucleation studies:
3.4 Kelvin Equation 31

 3
2 θ∞
nc = (3.63)
3 ln S

which coincides with Eq. (3.27).

3.5 Katz Kinetic Approach

To find the evaporation rates CNT uses the concept of constrained equilibrium, which
would exist for a vapor at the same temperature T and supersaturation S > 1 as the
vapor in question. Such a fictitious state can be achieved by introducing “Maxwell
demons” which ensure that monomers are continuously replenished by artificial
dissociation of clusters which grow beyond a certain (critical) size. The necessity
of such an artificial construction clearly follows the fact that a supersaturated vapor
is not a true equilibrium state. In an alternative procedure formulated by Katz and
coworkers [14, 15] and called a “kinetic theory of nucleation”, the evaporation rate
is obtained from the detailed balance condition at the (true) stable equilibrium of
the saturated vapor at the same temperature T . Within this procedure no artificial
construction is needed.
Assuming, as in the CNT, that b(n) is independent of the gas pressure, we apply
the detailed balance condition at the saturation point, where J = 0 and the cluster
distribution ρsat (n) is independent of time. From (3.35) this results in

ρsat (n)
b(n + 1) = f sat (n) (3.64)
ρsat (n + 1)

Since the chemical potentials of liquid and vapor at saturation are equal, the
Gibbs formation energy of a cluster at saturation contains only the positive surface
contribution
sat (n) = γ∞ s1 n
ΔG CNT 2/3
(3.65)

implying that
ρsat (n) = ρsat
v
exp[−θ∞ n 2/3 ] (3.66)

Let us divide both sides of Eq. (3.35) by f (n)ρsat (n)S n . Since the forward rate is
proportional to the pressure, we have:

S = f (n)/ f sat (n)

Then from (3.64):

J (n, t) 1 ρ(n, t) 1 ρ(n + 1, t)

= n − n+1
f (n)ρsat (n) S n S ρsat (n) S ρsat (n + 1)
32 3 Classical Nucleation Theory

Summation from n = 1 to an arbitrary large N − 1 yields (due to mutual cancelation

of successive terms):

N −1
J (n, t) ρ(N , t)
=1− N (3.67)
f (n) ρsat (n)S n S ρsat (N )
n=1

Examine the second term on the right-hand side for large N . In the nominator ρ(N , t)
is limited for any N (as any other physical quantity). In the denominator the first
term exponentially diverges as e N ln S while the second term vanishes exponentially,
but slower than the first one—see (3.66). As a whole the last term on the right-hand
side becomes asymptotically small as N → ∞. Extending summation to infinity we
obtain for the steady state nucleation rate:

∞
−1
1
J= (3.68)
f (n) S n ρsat (n)
n=1

This result looks almost similar to the CNT expression (3.46). The difference between
the two expressions is in the prefactor of the cluster distribution function. Nucleation
rates given by the kinetic approach, Jkin,phen and by the CNT, JCNT , differ by a factor
1/S known as the Courtney correction [16]:
 
1
Jkin,phen = JCNT (3.69)
S

Both theories yield the same critical cluster size. Thus, if in the kinetic approach one
uses the same as in the CNT (phenomenological) model for ΔG, two approaches
become identical in all respects except for the 1/S correction in the prefactor J0 .
This conclusion, however, looses its validity if within the kinetic approach a differ-
ent expression for ΔG is chosen. It gives rise to the different form of the cluster
distribution function. The importance of the kinetic approach is, thus, in setting the
methodological framework for nucleation models with other than classical forms of
the Gibbs formation energy.

3.6 Consistency of Equilibrium Distributions

The equilibrium Frenkel distribution employed in the CNT, based on the capillarity
approximation, has the form (3.45)

ρeq (n) = ρ1 exp[n ln S − θ∞ n 2/3 ] (3.70)

Here ρ1 is the monomer concentration of the supersaturated vapor; in terms of the

vapor concentrations the supersaturation can be written as
3.6 Consistency of Equilibrium Distributions 33

pv ρ1
S= = v (3.71)
psat (T ) ρsat (T )

where ρsat
v (T ) is the monomer concentration of the vapor at saturation. The law of

mass action (see e.g. [17], Chap. 6) written for the “chemical reaction” of formation
of the n-cluster E n from n monomers E 1

n E1  En (3.72)

states that the equilibrium cluster distribution function should have the form:

ρeq (n) = (ρ1 )n K n (T ) (3.73)

where K n (T ) is the equilibrium constant for the reaction (3.72) which can depend
on n and T but can not depend on ρ1 , or equivalently on the actual pressure p v . The
Frenkel distribution does not satisfy this requirement:
 n
ρeq (n) 1
= (ρ1 )1−n S n e−θ∞ n = ρ1 e−θ∞ (T ) n
2/3 2/3

(ρ1 ) n ρsat (T )
v

Katz’s kinetic approach replaces the Frenkel distribution (3.70) by the Courtney
distribution
ρeq (n) = ρsat
v
exp[n ln S − θ∞ n 2/3 ] (3.74)

for which the equilibrium constant

(T ))1−n e−θ∞ (T ) n
2/3
K n (T ) = (ρsat
v

satisfies the law of mass action. Weakliem and Reiss [18] showed that the Courtney
distribution is not unique but represents one of the possible corrections to the Frenkel
distribution which converts it to a form compatible with the law of mass action.
Although the Courtney distribution satisfies the law of mass action, it does not satisfy
the limiting consistency requirement [13]: in the limit n → 1 it does not return the
identity
ρ1 = ρ1 , for n = 1

The same refers to the Frenkel distribution. At the same time, the limiting consistency
is not a fundamental property, to which a cluster distribution should obey, but rather a
mathematical convenience to have a single formula which could be valid for all cluster
sizes [13]. However, the fact that the CNT does not satisfy the limiting consistency can
not be considered as its “weakness”, since CNT is valid for relatively large clusters
which can be treated as macroscopic objects. However, for nucleation models which
are constructed to be valid for small clusters the requirement of limiting consistency
deserves special attention.
34 3 Classical Nucleation Theory

3.7 Zeldovich Theory

Nucleation and growth of clusters can be viewed as the flow in the space of cluster
sizes. This space is one-dimensional if the cluster size is determined by the number
of molecules, n, in the cluster. The flow in this space is characterized by the “density”
ρ(n, t) and the “flow rate”
v = dn/dt ≡ ṅ(n)

In the “cluster language”, ρ(n, t) is the cluster distribution function and ṅ(n) is the
cluster growth law. By definition ρ(n, t)dn is the number of clusters (in the unit
physical volume) having the sizes between n and n + dn; thus, the “mass” of the
cluster fluid inside the (one-dimensional) cluster space volume Vn is

ρ(n, t) dn
Vn

Using the analogy with hydrodynamics we can apply general hydrodynamic consid-
erations [19] to calculate the density of the “cluster fluid” ρ(n, t). In the absence of
nucleation ρ(n, t) satisfies the continuity equation
 
∂
ρ(n, t) dn = − ρ v d An (3.75)
∂t Vn

where the zero-dimensional surface An bounds the one-dimensional cluster space

volume Vn . This equation means that the change of the mass of the cluster fluid inside
an arbitrary volume Vn is equal to flow of the cluster fluid through its boundary An .
In the differential form Eq. (3.75) reads

∂ρ ∂
+ (ρ ṅ) = 0 (3.76)
∂t ∂n
Equations (3.75)–(3.76) describe the evolution of the cluster distribution in the
absence of nucleation.
Nucleation introduces an extra, source term in (3.75), describing an additional flux in
the cluster space with the density i. Its role is analogous to diffusion in the physical
space for a real fluid. Using the standard hydrodynamic considerations (see [19],
Chap. 6), Eq. (3.75) is modified to:
  
∂
ρ(n, t) dn = − ρ v d An − i d An (3.77)
∂t Vn

or in the differential form

∂ρ ∂ ∂
=− (ρ ṅ) − i (3.78)
∂t ∂n ∂n
3.7 Zeldovich Theory 35

Similar to hydrodynamics we write the flux i using Fick’s law (now in the cluster
size space):
∂ρ
i = −B
∂n
where B is the diffusion coefficient. Equation (3.78) becomes

∂ρ(n, t) ∂ J (n, t)
=− (3.79)
∂t ∂n

where the flux J (n, t) is:

∂ρ(n, t)
J (n, t) = −B + ṅ ρ(n, t) (3.80)
∂n
This is the Fokker-Planck equation (FPE) [20] for diffusion in the cluster size space.
The first term describes diffusion in the n-space, with B being the corresponding
diffusion coefficient, while the second term describes a drift with the velocity ṅ
under the action of an external force. The necessary input parameters to solve FPE
are:
• the growth law ṅ(n), which takes into account both condensation (growth) of
supercritical clusters (positive drift) and evaporation of subcritical ones (negative
drift) and
• the model for the Gibbs free energy of the cluster formation ΔG(n)
The idea to apply the Fokker-Planck equation to describe kinetics of cluster formation
belongs to Zeldovich [5]. This formalism can be viewed as a continuous analogue
of the set of Becker-Döring equations (3.36) and leads to the alternative formulation
of the CNT, called the Zeldovich theory, which we discuss below. In (constrained)
equilibrium: J (n, t) = 0 for all clusters and ρ(n, t) = ρeq (n), with ρeq (n) given
by the general expression (3.33). This implies that the drift coefficient in (3.80) is
related to the diffusion coefficient by

∂βΔG(n)
ṅ(n) = −B(n) g1 (n), g1 ≡ (3.81)
∂n
∂
This implies that Eq. (3.80) describes diffusion in the field of force − ∂n ΔG(n). As in
the previous sections, we will be interested in the steady state solution J (n, t) = J =
const, ρ(n, t) = ρs (n) of the kinetic equation (3.79). It is convenient to introduce a
new unknown function
y = ρs (n)/ρeq (n)

Using (3.81) and (3.33), Eq. (3.80) takes the form

∂y
− B ρeq =J (3.82)
∂n
36 3 Classical Nucleation Theory

which after integration results in:

 n 1
y = −J dn +C
0 B(n ) ρeq (n )

This equation contains two unknown constants—C and J —which can be found from
the standard boundary conditions in the limit of small and large clusters (3.42)–(3.43):

y → 1, for n → 0, y → 0, for n → ∞

The solution of Eq. (3.82) satisfying these boundary conditions is:

 ∞
ρs (n) 1
=J dn (3.83)
ρeq (n) n B(n ) ρeq (n )

and the steady-state nucleation rate is given by

 ∞ −1
1
J= dn (3.84)
0 B(n ) ρeq (n )

Exploring the exponential dependence of the integrand on n we use the second order
expansion of the Gibbs free energy g(n) ≡ βΔG(n) around the critical cluster:

1 d2 g 
g(n) ≈ g(n c ) + g (n c )(n − n c ) , with g (n c ) =
2
(3.85)
2 dn 2 n c

Following the same steps as in Sect. 3.3 we obtain

J = Z B(n c ) ρeq (n c ) (3.86)

where Z is the Zeldovich factor.

This result looks similar to (3.47); however, we have not yet specified the “diffu-
sion coefficient” for the critical cluster B(n c ). If we were able to determine it from
independent macroscopic considerations, it would give a possibility to estimate the
nucleation rate without using microscopic information—the possibility which could
be very advantageous from experimental point of view. Following Zeldovich [5],
notice that in the supercritical region n  n c the distribution function is practically
constant: a nucleus, after finding itself here, starts monotonically increasing in size,
practically never coming back to the subcritical domain. In view of the considera-
tions presented in Sect. 3.3, one can state that with a high degree of accuracy the
supercritical region corresponds to n > n c + Δ, where Δ is given by Eq. (3.49). In
∂ρ
this domain we can neglect the term with ∂n in the flux (3.80) and set:

J = ṅ ρ, n > n c + Δ (3.87)
3.7 Zeldovich Theory 37

From the physical meaning of J (n, t) as a flux in the n-space, we identify the coef-
ficient ṅ as a velocity in this space:
 
dn
ṅ = (3.88)
dt macro

where the subscript “macro” indicated that the growth of the supercritical nucleus
follows a certain macroscopic equation (e.g. diffusion in the real space). Then from
(3.88) and (3.81) we find
 
1 dn
B(n) = − (3.89)
g1 (n) dt macro

This fundamental Zeldovich relation makes it possible to calculate the nucleation

rate without referring to the microscopic description but using the deterministic
(macroscopic) growth law—ballistic, diffusion, or combination of both. Rigorously
speaking, this result is valid for n > n c , whereas we are interested in B(n c ). However,
since the function B(n) does not have a singularity at n c we can use it there. Indeed,
at n = n c the growth rate becomes zero indicating that the critical cluster is in a
metastable equilibrium (and therefore does not grow). In the vicinity of n c we have
from (3.85)
g1 (n) = g (n c ) (n − n c ) + O(n − n c )2

Similarly for the drift velocity

1
ṅ = (n − n c ) + O(n − n c )2
τ
where the parameter τ , introduced by Zeldovich (Zeldovich time), has a dimension-
ality of time and is defined as 
−1 dṅ 
τ = (3.90)
dn  nc

Using l’Hopital’s rule, we find:

 
ṅ(n) 1
B(n c ) = − lim =−
n→n c + g1 (n) τ g (n c )

At the critical cluster both ṅ and g1 vanish, while B(n c ) remains finite. Using (3.48)
and (3.49) we rewrite this result in terms of the Zeldovich factor

1 1
τ= = Δ2 (3.91)
B(n c ) 2π Z 2 2B(n c )
38 3 Classical Nucleation Theory

The macroscopic growth rate is determined by the mechanism of mass exchange

between the cluster and its surroundings. The widely used growth models are: the
ballistic (surface limited) and the diffusion limited model. Growth rates ṅ(n), refer-
ring to both of these mechanisms can be written in a unified form in terms of the
reduced radius  1/3
n
r=
nc

With neglect of discreteness effects [21]:

 
1 1
ṙ = 1− (3.92)
τ rϑ r

Here ϑ = 0 and ϑ = 1 for the ballistic and diffusion limited cases, respectively;
note that ϑ = −1 corresponds to cavitation [5].

3.8 Transient Nucleation

In the previous section we discussed the steady state regime. The latter is preceded by
the transient non-stationary regime characterized by a characteristic relaxation time
τtr . Strictly speaking the steady regime can be reached only at infinite time when
all transient effects have disappeared. However, one can pose a question: how much
time is required for the flux to reach an appreciable fraction of the steady state value
J ? To answer this question we start with the expression for the time-dependent flux
(3.80) rewritten in the form:
 
∂ ρ(n, t) J (n, t)
=− (3.93)
∂n ρeq (n) B(n)ρeq (n)

Integrating it from some small n 1 to a large n 2 (n 2  n c ) and taking into account

the boundary conditions (3.42)–(3.43) we obtain:
 n2 J (n, t)
dn = 1 (3.94)
n1 B(n)ρeq (n)

We discuss times at which the flux at the point n = n c is a noticeable fraction of the
steady state value J . Due to the sharp maximum of the integrand at n c the following
expansions are plausible:

1
J (n, t) = J (n c , t) + J (n c , t)(n − n c )2
2
ρeq (n) = ρeq (n c ) exp π Z 2 (n − n c )2
3.8 Transient Nucleation 39

Then Eq. (3.94) reads:

 
n2 1 1 J (n c , t) n2
J (n c , t) dn + (n − n c )2 exp −π Z 2 (n − n c )2 dn = 1
n1 B(n)ρeq (n) 2 B(n c )ρeq (n c ) n1

In the first integral one recognizes J −1 whereas the second integral can be calculated
using (3.47) and the Gaussian identity [22]
 +∞ √
π
x 2 e−a
2x2
dx = , a>0
−∞ 2a 3

resulting in
1
J (n c , t) + J (n c , t) = J (3.95)
4π Z 2
Differentiation of the continuity equation (3.79) with respect to n gives:
 
∂ 2 J (n, t) ∂ ∂ρ(n, t)
=− (3.96)
∂n 2 ∂t ∂n

At n = n c the flux contains only the diffusion term:

 
∂ρ
J (n c , t) = −B(n c ) (3.97)
∂n nc

From (3.95)–(3.97) we derive the linear differential equation:

d J (n c , t)
J (n c , t) + τtr =J
dt
with
1
τtr = (3.98)
4π B(n c ) Z 2

whose solution is:  

J (n c , t) = J 1 − e−t/τtr (3.99)

The parameter τtr characterizes the relaxation period to a steady state, or a time-lag.
For times t > τtr one can speak about the steady regime. Comparing (3.91) and
(3.98) one can see that the time-lag is related to the Zeldovich time τ as:

1 τ
τtr = Δ2 = (3.100)
4B(n c ) 2

Typical values of τtr are of the order of 1 ÷ 10 μ s.

40 3 Classical Nucleation Theory

3.9 Phenomenological Modifications of Classical Theory

In a number of physically relevant applications critical clusters predicted by CNT

turn out to be small in contradiction with the assumptions of the classical theory.
Various modifications of CNT were discussed in the literature aiming to propose
an expression for free energy applicable to all cluster sizes. Dufour and Defay [23]
suggested to replace θ∞ (T ) by a size-dependent surface tension θ (n; T ) such that
θ (1; T ) = 0. Girshick and Chiu [24] proposed a model in which the surface energy
of the cluster is reduced by the “surface energy of a monomer”

GC (n) = βΔG CNT (n) − βΔG CNT (n = 1) = θ∞ (n

βΔG surf surf surf 2/3
− 1) (3.101)

One can notice that by writing

GC (n) = θ (n, T ) n
βΔG surf 2/3

with  
θ (n, T ) = θ∞ 1 − n −2/3 (3.102)

the Girshick-Chiu expression reduces to a particular realization of the Dufour-Defay

conjecture. Since the radius of the (spherical) cluster scales as n 1/3 , Eq. (3.102) shows
that the first non-vanishing term in the curvature correction to the plain layer surface
tension scales as ∼1/R 2 implying that the Tolman length δT for all substances and
all temperatures is identically zero (cf. Eq. (2.41)).
Indeed it is known that δT vanishes for symmetric systems, such as lattice-gas models
[25, 26], however for real fluids with asymmetry in vapor-liquid coexistence δT is
nonzero, though of molecular sizes [27–29]. The Girshick-Chiu cluster distribution
reads: 
ρeq (n) = ρsat
v
exp n ln S − θ∞ (n 2/3 − 1) (3.103)

It is straightforward to see that the resulting nucleation model, termed the Internally
Consistent Classical Theory (ICCT) [24] results in
 
eθ∞
JICCT = JCNT (3.104)
S

Compared to the modest Courtney (1/S) correction the ICCT correction to the clas-
sical theory, eθ∞ /S can be very large. Expression (3.101) implicitly suggests that the
CNT form of the free energy barrier is valid down to the cluster containing just one
molecule. That is why ICCT can be viewed as a rather arbitrary choice which may,
however, empirically improve the fit to experiment [30].
References 41

References

1. L.D. Landau, E.M. Lifshitz, Statistical Physics (Pergamon, Oxford, 1969)

2. P.G. Debenedetti, Metastable Liquids (Princeton University Press, Princeton, Concepts and
Principles, 1996)
3. R. Becker, W. Döring. Ann. Phys. 24, 719 (1935)
4. M. Volmer, Kinetik der Phasenbildung (Steinkopf, Dresden, 1939)
5. Ya. B. Zeldovich, Acta Physicochim. URSS 18, 1 (1943)
6. J.E. McDonald, Am. J. Phys. 30, 870 (1962)
7. J.E. McDonald, Am. J. Phys. 31, 31 (1963)
8. J. Frenkel, J. Chem. Phys. 7, 538 (1939)
9. J. Frenkel, Kinetic Theory of Liquids (Clarendon, Oxford, 1946)
10. J. Wölk, R. Strey, J. Phys. Chem. B 105, 11683 (2001)
11. D.G. Labetski, V. Holten, M.E.H. van Dongen, J. Chem. Phys. 120, 6314 (2004)
12. W.T. Thomson, Phil. Mag. 42, 448 (1871)
13. G. Wilemski, J. Chem. Phys. 103, 1119 (1995)
14. J.L. Katz, H. Wiedersich, J. Colloid Interface Sci. 61, 351 (1977)
15. J.L. Katz, M.D. Donohue, Adv. Chem. Phys. 40, 137 (1979)
16. J. Courtney, J. Chem. Phys. 35, 2249 (1961)
17. C. Garrod, Statistical Mechanics and Thermodynamics (Oxfor University Press, New York,
1995)
18. C.L. Weakliem, H. Reiss, J. Phys. Chem. 98, 6408 (1994)
19. L.D. Landau, E.M. Lifshitz, Fluid Dynamics (Pergamon, Oxford, 1986)
20. E.M. Lifshitz, L.P. Pitaevski, Physical Kinetics (Pergamon, Oxford, 1981)
21. V.A. Shneidman, J. Chem. Phys. 115, 8141 (2001)
22. I.S. Gradshtein, I.M. Ryzhik, Tables of Integrals, Series, and Produscts (Academic Press, New
York, 1980)
23. L. Dufour, R. Defay, Thermodynamics of Clouds (Academic, New York, 1963)
24. S. Girshick, C.-P. Chiu, J. Chem. Phys. 93, 1273 (1990)
25. J.S. Rowlinson, J. Phys. Condens. Matter 6, A1 (1994)
26. M.P.A. Fisher, M. Wortis, Phys. Rev. B 29, 6252 (1984)
27. E. Blokhuis, J. Kuipers, J. Chem. Phys. 124, 074701 (2006)
28. J. Barrett, J. Chem. Phys. 124, 144705 (2006)
29. M.A. Anisimov, Phys. Rev. Lett. 98, 035702 (2007)
30. D.W. Oxtoby, J. Phys. Cond. Matt. 4, 7627 (1992)
Chapter 4
Nucleation Theorems

4.1 Introduction

Various nucleation models use their own set of approximations, have their own range
of validity and certain fundamental and technical limitations. Therefore it is desirable
to formulate some general, model-independent statements which would establish the
relationships between the physical quantities characterizing the nucleation behavior.
One of such statements was proposed by Kashchiev [1] later on termed the Nucleation
Theorem (NT). In 1996 Ford [2] derived another general statement which was termed
the Second Nucleation Theorem. Since then Kashchiev’s result and its generalization
is sometimes also referred to as the First Nucleation Theorem.
Following Kashchiev, consider a general form of the Gibbs free energy of n-cluster
formation
ΔG(n,Δμ) = −n Δμ + Fs (n,Δμ) (4.1)

Here Fs (n,Δμ) is the excess beyond the first (bulk) term free energy of cluster
formation. For our present purposes we do not need to specify it. The critical cluster
satisfies: 
∂ΔG 
=0
∂n T,Δμ

resulting in 
∂ Fs 
− Δμ + =0 (4.2)
∂n n c

The work of the critical cluster formation is

W ∗ ≡ ΔG(n c (Δμ), Δμ) = −n c (Δμ) Δμ + Fs (n c (Δμ), Δμ)

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 43

DOI: 10.1007/978-90-481-3643-8_4, © Springer Science+Business Media Dordrecht 2013
44 4 Nucleation Theorems

Taking the full derivative with respect to Δμ

  
dW ∗ ∂ Fs ∂n c ∂ Fs 
= −n c + + −Δμ +
dΔμ ∂Δμ ∂Δμ ∂n n c

and noticing that the expression in the round brackets vanishes in view of (4.2), we
obtain the Nucleation Theorem in the form given in Ref. [1]:

dW ∗ ∂ Fs 
= −n c + (4.3)
dΔμ ∂Δμ n c

This result is particularly useful when Fs is only weakly dependent on the supersatura-
tion; for example in the CNT within the capillarity approximation Fs = γ∞ (T )A(n)
is totally independent of Δμ. In this case one can determine the size of the critical
cluster from the nucleation experiments measuring the nucleation rates and finding
the slope of ln J − ln S curves (cf. Eq. (3.52)).

4.2 First Nucleation Theorem for Multi-Component Systems

In 1994 Oxtoby and Kashchiev [3] extended original Kashchiev’s treatment to a

general form which is valid for multi-component systems and applicable to various
types of nucleation phenomena. Consider an arbitrary first order phase transition
from the mother phase “v” to the new phase “l” (the phases, as mentioned earlier,
should not necessarily be vapor and liquid). The mother phase is at constant pressure
p v and temperature T and contains inside itself a cluster of the new phase. Using
Gibbs thermodynamics, we introduce an arbitrary dividing surface and write the total
volume V of the system “cluster inside a metastable phase” as

V = Vv + Vl

(see Fig. 4.1). Here V l encloses the cluster of the new phase “l” and V v contains the
original phase “v”. In a mixture of q components the total number of molecules of
component i is an extensive quantity and according to (2.22) is given by

Ni = Niv + Nil + Niexc , i = 1, . . . , q

where Niv is the number of molecules of type i in the homogeneous phase “v” occu-
pying the volume V v , Nil is the number of molecules of type i in the homogeneous
phase “l” occupying the volume V l , Niexc is the excess number of molecules of type
i accumulated on the dividing surface. The Gibbs free energy of cluster formation
4.2 First Nucleation Theorem for Multi-Component Systems 45

Fig. 4.1 Sketch of the system

“cluster inside a metastable v
V
phase”

l
V

in a multi-component case is given by the straightforward extension of Eq. (3.19) to

the multi-component case:


q  
ΔG = ( p v − p l )V l + Nil μil ( p l ) − μiv ( p v )
i=1

q

+ Niexc μiexc − μiv ( p v ) + φ(V l , {μiv }, T ) (4.4)
i=1

The last term, φ(V l , {μiv }, T ), is the total surface energy of the cluster. We do not
specify here its functional form (e.g. by introducing the surface tension and the
surface area of the cluster) which implies that the general form (4.4) can be applied
to small clusters (for which the physical meaning of the surface tension looses its
validity). The critical cluster (denoted by the subscript “c”) is in the mechanical and
chemical equilibrium (though metastable) with the mother phase resulting in the
extremum of the Gibbs energy with respect to {Nil }, {Nisurf } and V l :

μil ( pcl ) = μiv ( p v ) = μiexc for all i (4.5)


∂φ 
pc = p +
l v
(4.6)
∂ V l {μv },T
i

Substituting these expressions into (4.4) we find the work of formation of the critical
cluster at the given external conditions—the temperature T and the set of the vapor
phase chemical potentials {μiv }:

W ∗ ≡ ΔG ∗ = ( p v − pcl ) Vcl + φc (4.7)

where φc ≡ φ(Vcl , {μiv }, T ).

46 4 Nucleation Theorems

Now let us study how W ∗ changes if we change the external conditions. The variation
of W ∗ with respect to the variation of the chemical potential of the component i while
keeping the temperature and the rest chemical potentials fixed, reads:

∂W∗ l ∂( p − pc )
v l ∂ Vcl ∂φc ∂φc
= Vc + ( p v − pcl ) + +
∂μiv ∂μiv ∂μiv ∂ Vcl ∂μiv

In view of the equilibrium condition (4.6) the expression in the square brackets
vanishes resulting in
∂W∗ l ∂( p − pc )
v l ∂φc
= Vc + (4.8)
∂μiv ∂μi
v ∂μiv

A straightforward extension of the Gibbs–Duhem Eq. (2.5) for a mixture reads


q
SdT − V d p + Nk dμk = 0 (4.9)
k=1

We write (4.9) for both of the bulk phases at isothermal conditions:

 
− Vcl d pcl + l
Nk,c dμlk ( pcl ) = 0, −V v d p v + Nkv dμvk ( p v ) = 0 (4.10)
k k

Similarly, the Gibbs adsorption equation (2.29) for a mixture at isothermal condi-
tions is:
q
dφc + k =0
Nkexc dμexc (4.11)
k=1

In view of the equality of the chemical potentials (4.5) these relations can be
written as:
⎡ ⎤

Vcl d pcl = Ni,c
l
dμiv ( p v ) + ⎣ l
Nk,c dμvk ( p v )⎦
k=i

⎡ ⎤

V v d p v = Niv dμiv ( p v ) + ⎣ Nkv dμvk ( p v )⎦ (4.12)
k=i

⎡ ⎤

dφc = −Niexc dμiv ( p v ) − ⎣ Nkexc dμvk ( p v )⎦
k=i
4.2 First Nucleation Theorem for Multi-Component Systems 47

In all of these relations the sums in the square brackets has to be set to zero since in
Eq. (4.8) all dμvk = 0 except for k = i. Substituting (4.12) into (4.8), we find:
 l
∂W∗ Vc
= −N l
i,c − N i
exc
+ Niv (4.13)
∂μiv Vv

This result can be simplified if we introduce the number densities of the component
i in the both phases “v” and “l”:
l
Ni,c Niv
ρil = , ρiv =
Vcl Vv

Then  
Vcl
Niv = ρiv Vcl
Vv

is the number of molecules of component i that existed in the volume Vcl before the
critical cluster was formed. Eq. (4.13) becomes
  
∂ W ∗  ρv
= − 1 − l
Ni,c + Niexc ≡ −Δn i,c (4.14)
∂μiv {μv }, j=i ρl
j

The quantity Δn i,c is the excess number of molecules of component i in the cluster
beyond that present in the same volume (V l ) of the mother phase before the cluster
was formed. While the quantities Ni,c l , N v , N exc depend on the location of the
i i
dividing surface the excess number Δn i,c is independent of this choice and of the
∗
cluster shape (which for small clusters can have a fractal structure). Thus, ∂ Wv is also
∂μi
independent of the location of the dividing surface. This is consistent with the fact
that the nucleation barrier W ∗ itself is invariant with respect to the dividing surface
(which is a mathematical abstraction rather than a measurable physical quantity).
Equation (4.14) represents the Nucleation Theorem for multi-component systems.
The most important feature of this result is that it is derived without any assumptions
concerning the size, shape and composition of the critical cluster and thus is of general
validity.1 Although the presented proof is based on thermodynamics, it makes no
assumptions about the size of the critical cluster and thus is valid down to atomic
size critical nuclei.
For a unary system Eq. (4.14) reads:
 
dW ∗ ρv
= − 1 − Ncl + N exc ≡ −Δn c (4.15)
dμv ρl

1Bowles et al. [4] showed that the Nucleation Theorem is a powerful result which is not restricted to
nucleation, as its name suggests, but refers to all equilibrium systems containing local nonuniform
density distributions stabilized by external field (not only a nucleus in nucleation theory).
48 4 Nucleation Theorems

We can rewrite this result in terms of the supersaturation ratio S, recalling that

kB T ln S = μv ( p v , T ) − μsat (T )

The Nucleation Theorem then becomes

d(βW ∗ )
= −Δn c (4.16)
d ln S

If the mother phase is dilute—ρ v /ρ l  1—and the number of molecules in the

critical cluster is defined as n c = Ncl + N exc , Eq. (4.16) reduces to the original
Kashchiev’s expression (4.3) with ∂∂μFs
= 0.
Nucleation Theorem provides a general, model-independent tool for the analysis of
nucleation phenomena. However, in experiments the directly measurable quantity is
not the work of the critical cluster formation but the nucleation rate J . For the steady
state
∗
J = J0 e−βW (4.17)

where the prefactor J0 depends on a particular nucleation model and on the dimen-
sionality of the problem. Taking in both sides of (4.17) the derivative with respect to
ln S, we obtain    
d(βW ∗ ) ∂ ln J ∂ ln J0
=− + (4.18)
d ln S ∂ ln S T ∂ ln S T

where first term on the right-hand side can be directly measured in nucleation exper-
iments. In the CNT the prefactor J0 is given by Eq. (3.55)

J0 = ψ(T ) S 2

where ψ(T ) does not depend on S. This implies that

 
∂ ln J0
=2 (4.19)
∂ ln S T

and the Nucleation Theorem for the single-component systems takes a particularly
simple form:  
∂ ln J
Δn c = −2 (4.20)
∂ ln S T

Within the kinetic approach to nucleation, discussed in Sect. 3.5, the prefactor
J0 contains the Courtney (1/S) correction

J0 = ψ(T ) S
4.2 First Nucleation Theorem for Multi-Component Systems 49

so that the Nucleation Theorem becomes

 
∂ ln J
Δn c = −1 (4.21)
∂ ln S T

For binary nucleation of components a and b the prefactor J0 has a more com-
plex form than in the single-component case. An important feature of J0 is that it
depends on the composition of the critical cluster; a simple expression describing
this dependence is not available. Applying (4.14) to the binary case we find:

∂ ln J ∂ ln J0
Δn i,c = − , i = a, b
∂(βμiv ) T ∂(βμiv ) T

A contribution of the second term to Δn i,c is small and typically ranges from 0 to 1
[3] resulting in
∂ ln J
Δn i,c = − (0 to 1) , i = a, b (4.22)
∂(βμiv ) T

4.3 Second Nucleation Theorem

Nucleation theorem studied in the previous section describes the variation of W ∗

with respect to the variation of the chemical potential of one of the species at a
constant temperature. In [2, 5] Ford derived what is called now the Second Nucleation
Theorem which describes the variation of W ∗ with respect to the temperature at the
fixed chemical potentials. From Eq. (4.7) we obtain:

∂W∗ ∂( p v − pcl ) ∂V l ∂φc ∂ Vcl ∂φc

= Vcl + ( p v − pcl ) c + +
∂T ∂T ∂T ∂ Vc ∂ T
l ∂T

In view of the equilibrium condition (4.6) the expression in the square brackets
vanishes resulting in

∂W∗ l ∂( p − pc )
v l ∂φc 
= Vc + (4.23)
∂T ∂T ∂ T Vcl ,{μv }
i

The Gibbs–Duhem equations for both of the bulk phases at the fixed chemical
potentials read

− Vcl d pcl + S l dT = 0, − V v d p v + S v dT = 0 (4.24)

The Gibbs adsorption equation at the fixed chemical potentials becomes:

dφc + S exc dT = 0 (4.25)

50 4 Nucleation Theorems

From (4.23)–(4.25) we find:

  l
∂ W ∗  Vc
 =− S +S −
l exc
S v ≡ −ΔSc (4.26)
∂ T {μv } Vv
i

where ΔSc is the excess entropy due to the critical cluster formation. (Note that
S v/V v is the entropy per unit volume in the vapor phase). Equation (4.26) represents
the Second Nucleation Theorem.
As in the case of the First Nucleation Theorem, the application of the Second Theorem
to the analysis of experiments requires its representation in the form containing
measurable quantities. Since the nucleation barrier enters the nucleation rate in the
form of the Boltzmann factor, we combine Eq. (4.26) with the identity
 
∂(βW ∗ ) ∂W∗ βW ∗
=β −
∂T ∂T T

which results in

∂(βW ∗ ) βW ∗ (W ∗ + T ΔSc )
= −β ΔSc − =− (4.27)
∂T T kB T 2

According to the thermodynamic consideration of Sect. 3.2 (see Eq. (3.12)) the
expression in the round brackets is the excess internal energy of the critical clus-
ter, i.e. the change in the internal energy beyond that present in the same volume
(V l ) of the mother phase before the cluster was formed:

∂(βW ∗ )  ΔU ∗
= − (4.28)
∂ T {μi } kB T 2

Then, Eq. (4.28) becomes


∂ ln(J/J0 )  ΔU ∗
 = (4.29)
∂T {μi } kB T 2

The contribution of the pre-exponential term can be found easily for the one com-
ponent case. Approximating J0 by the classical expression (3.55) we find for the
leading temperature dependence:

∂ ln J0  d ln psat 2
 =2 −
∂T S dT T

The first term can be worked out using the Clausius–Clapeyron equation (2.14)

d ln psat l
=
dT kB T 2
4.3 Second Nucleation Theorem 51

where l is the latent heat of evaporation per molecule. Thus, for a unary system the
Second Nucleation Theorem reads:

∂ ln J  2(l − kB T ) + ΔU ∗
= (4.30)
∂ T S kB T 2

With its help one can find the excess internal energy of the critical cluster from the
nucleation rate measurements and the known specific latent heat.

4.4 Nucleation Theorems from Hill’s Thermodynamics

of Small Systems

An alternative derivation of both Nucleation Theorems (as well as several other

useful forms of NT) can be obtained within the framework of thermodynamics of
small systems developed by Hill [6, 7]. Hill’s fundamental result states that for a
q-component system the change of the work of critical cluster formation in terms of
q + 1 independent variables T ; μv1 , . . . , μqv can be presented in the following exact
form:
q
dW ∗ (T ; μv1 , . . . , μqv ) = −ΔSc dT − Δn i,c dμiv (4.31)
i=1

where ΔSc is the excess entropy of the critical nucleus and Δn i,c is the excess number
of molecules of component i in it. This result does not depend on the choice of a
dividing surface.
It is straightforward to see that both the First and the Second Nucleation theorems
follow from Eq. (4.31). In particular, fixing the temperature and all but one chemical
potential in the mother phase, we obtain

∂ W ∗ 
= −Δn i,c (4.32)
∂μiv T ; {μv }, j=i
j

which is the First Nucleation Theorem (cf. (4.14)). Varying T and keeping all chem-
ical potentials in (4.31) fixed results in the Second Nucleation Theorem:

∂ W ∗ 
= −ΔSc (4.33)
∂ T {μv }
j

(cf. (4.26)). There exist various other forms of NT resulting from Hill’s theory (see
[8] for the detailed discussion). Below we consider one such form which can be
particularly useful for the analysis of the effect of total pressure on multi-component
nucleation. Let us choose the following set of independent variables: the temperature
52 4 Nucleation Theorems

T , the total pressure p v and the molar fractions of components in the mother phase
{y j }, j = 1, . . . , q. All these parameters are measurable in experiments. In view of
normalization
q
yj = 1
j=1

the number of independent variables is still q + 1. Presenting

μiv = μiv ( p v , T ; {y j } j=i )

we write its full differential as

  ∂μv 
i
dμiv = −siv dT + viv d p v + dy j (4.34)
∂yj
j=i

where the second term results from the Maxwell relation


∂μiv 
= viv (4.35)
∂ p v {y j },T

and viv and siv are, respectively, the partial molecular volume and entropy in the
mother phase

∂ V v 
viv ≡ , i = 1, 2, . . . (4.36)
∂ Niv  pv ,T, N v
j, j =i

∂S v 
si ≡
v
, i = 1, 2, . . . (4.37)
∂ Niv  pv ,T, N v
j, j =i

Substituting (4.34) into Hill’s fundamental equation (4.31) we obtain its alternative
form in the p v , T, {y j } variables:
   

q 
q
∗
dW = − ΔSc − siv Δn i,c dT − viv Δn i,c d pv
i=1 i=1
⎛ ⎞

q  ∂μv
− ⎝ i
dy j ⎠ Δn i,c (4.38)
∂yj
i=1 j=i

(for a single-component case summation over j in the last term should be omitted).
Let us differentiate the general expression for the steady-state nucleation rate
 
J
ln = −β W ∗
J0
4.4 Nucleation Theorems from Hill’s Thermodynamics of Small Systems 53

with respect to ln p v :
 
∂ ln(J/J0 )  ∂ W ∗ 
= −β
∂ ln p v {yk },T ln p v {yk },T

Applying (4.38), we find


∂ ln(J/J0 )  pv  v
= vi Δn i,c (4.39)
∂ ln p v  {yk },T kB T
i

This result, termed Pressure Nucleation Theorem for multi-component systems [9],
makes it possible to study the effect of total pressure on the nucleation rate in mixtures.

References

1. D. Kashchiev, J. Chem. Phys. 76, 5098 (1982)

2. I.J. Ford, J. Chem. Phys. 105, 8324 (1996)
3. D.W. Oxtoby, D. Kashchiev, J. Chem. Phys. 100, 7665 (1994)
4. R.K. Bowles, D. Reguera, Y. Djikaev, H. Reiss, J. Chem. Phys. 115, 1853 (2001)
5. I.J. Ford, Phys. Rev. E 56, 5615 (1997)
6. T.L. Hill, J. Chem. Phys. 36, 3182 (1962)
7. T.L. Hill, Thermodynamics of Small Systems (Dover, New York, 1994)
8. D. Kashchiev, J. Chem. Phys. 125, 014502 (2006)
9. V.I. Kalikmanov, D.G. Labetski, Phys. Rev. Lett. 98, 085701 (2007)
Chapter 5
Density Functional Theory

5.1 Nonclassical View on Nucleation

Classical phenomenological description of nucleation is based on the capillarity

approximation treating all droplets (clusters) as if they were macroscopic objects
characterized by a well defined rigid boundary of radius R with a bulk liquid density
inside R and bulk vapor density outside R. Moreover, the surface free energy of the
cluster is the same as for the planar interface at the same temperature, and there-
fore is characterized by the planar surface tension. Making a step from the purely
macroscopic to a microscopic view, it is plausible to consider the system “a droplet
in a supersaturated vapor” as an inhomogeneous fluid with some (unknown) den-
sity profile ρ(r ) continuously changing from the liquid-like value in the center of
the droplet to the bulk vapor density far from it as shown in Fig. 5.1. This density
profile gives rise to the free energy functional F [ρ(r )] which can be expressed in
terms of molecular interactions and therefore does not invoke information about the
macroscopic properties. As a result the capillarity approximation is avoided. The
critical cluster in this picture is given by the density profile ρc (r ) being an extremum
(saddle point) of the corresponding free energy (grand potential) functional over all
admissible functions ρ(r ), while the nucleation barrier is the value of the functional
at ρc (r ).
In the theory of nonuniform fluids the approach based on the concept of function-
als of arbitrary distribution functions is called the density functional theory (DFT).
It was proposed by Ebner et al. [1] and developed by Evans [2] (for an excellent
review see the paper of Evans [3]).1 Application of DFT to nucleation was formu-
lated by Oxtoby and Evans [6] and later developed in a number of publications by

1 Historically the DFT in the theory of fluids originates from the quantum mechanical ideas

formulated by Hohenberg and Kohn [4] and Kohn and Sham [5]; these authors showed that the
intrinsic part of the ground state energy of an inhomogeneous electron liquid can be cast in the
form of a unique functional of the electron density ρe (r). By doing so the quantum many-body
problem—the solution of the many-electron Shrödinger equation—is replaced by a variational
one-body problem for an electron in an effective potential field.

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 55

DOI: 10.1007/978-90-481-3643-8_5, © Springer Science+Business Media Dordrecht 2013
56 5 Density Functional Theory

Fig. 5.1 Representation of

a cluster in the density func-
tional theory: continuous liquid
density profile ρ(r ), changing

equimolar surface
between the liquid-like value
in the center of a cluster to
the bulk vapor density far
from it. Re indicates the Gibbs
vapor
equimolar dividing surface

0 Re
r

Oxtoby and coworkers [7–11]. The general feature of all density functional models
is the assumption that the thermodynamic potential of a nonuniform system can be
approximated using the knowledge of structural and thermodynamic properties of
the corresponding uniform system. Various DFT models differ from each other in the
way this approximation is formulated [12]. Although the results of DFT calculations
can not be presented in a closed form, these calculations are much faster than the
purely microscopic computer simulations (Monte Carlo, molecular dynamics). One
can thus classify the DFT as a semi-microscopic approach.
An important feature of DFT in nucleation is that calculation of the nucleation bar-
rier does not invoke the a priori information about the surface tension. The curvature
effects of the surface free energy are incorporated into the DFT so that no ad hoc
assumptions are necessary. This is the consequence of the fact that DFT uses the
microscopic (interaction potential) rather the macroscopic input. The DFT approach
naturally recovers the CNT when the system is close to equilibrium (low supersat-
urations, large droplets). However, it considerably deviates from CNT at higher S.
In particular, the DFT predicts vanishing of the nucleation barrier at some finite S
(which signals the spinodal) while the CNT barrier remains finite even as the spinodal
is approached.
Before studying the application of DFT to nucleation we briefly formulate its fun-
damentals for the theory of nonhomogeneous fluids.

5.2 Fundamentals of the Density Functional Approach

in the Theory of Liquids

5.2.1 General Principles

The cornerstone of DFT is the statement that the free energy of an inhomogeneous
fluid is a functional of the density profile ρ(r). On the basis of the knowledge of
5.2 Fundamentals of the Density Functional Approach in the Theory of Liquids 57

this functional one can calculate interfacial tensions, properties of confined sys-
tems, adsorption properties, determine depletion forces, study phase transitions, etc.
From these introductory remarks it is clear that DFT represents an alternative—
variational—formulation of statistical mechanics. Determination of the exact form
of the free energy functional is equivalent to calculating the partition function, which,
as it is well known, is not possible for realistic potentials. Therefore, one has to for-
mulate approximations which could lead to computationally tractable results, and
at the same time be applicable to a number of practical problems. However, even
without knowing the exact form of the functional one can formulate several rigorous
statement about it.
Let us consider the N -particle system in the volume V at the temperature T . Being
concerned with classical systems, we can always consider the momentum part of the
Hamiltonian to be described by the equilibrium Maxwellian distribution. This implies
that the arbitrariness in the distribution function refers only to the configurational
part of the Hamiltonian. We require that an arbitrary N -body distribution function,
ρ̂ N (r N ), should be positive and satisfy the normalization

ρ̂ (N ) (r N ) dr N = N !, ρ̂ (N ) (r N ) > 0 (5.1)

The “hat” indicates that the distribution does not necessarily have its equilibrium
form; the corresponding equilibrium function will be denoted without the “hat”. The
requirement (5.1) shows that functions ρ̂ (N ) are subject to the same normalization
as the equilibrium function ρ (N ) (r N ) (see e.g. [13] Chap. 2). To limit the class of
functions ρ̂ (N ) , we employ the following considerations [14]. For a given interatomic
interaction potential u(r ) every fixed external field u ext (r) gives rise to a certain
equilibrium one-particle distribution function

ρ (1) (r)[u ext (r)] ≡ ρ(r)[u ext (r)]

i.e. only one u ext (r) can determine a given ρ(r). Here the square brackets denote
that ρ(r) is a functional of u ext (r). Keeping the form of this functional and varying
u ext (r) we can generate a set of singlet density functions ρ̂(r) so that each of them
will be an equilibrium density corresponding to some other external field, û ext (r),

ρ̂(r) = ρ[û ext (r)]

but is nonequilibrium with respect to the original field u ext (r). For every ρ̂(r) there
exists a unique N -particle distribution function ρ̂ (N ) (for a proof see [2]). Thus, any
functional of ρ̂ (N ) (r N ) can be equally considered a functional of the one-particle
distribution function ρ̂(r).
Let us formally define the functional of intrinsic free energy

1
Fint [ρ̂] = dr N ρ̂ (N ) [U N + kB T ln(Λ3N ρ̂ (N ) )] (5.2)
N!
58 5 Density Functional Theory

where 
2π 2
Λ=
m 1 kB T

is the thermal de Broglie wavelength of a particle, m 1 is its mass,  is the

Planck constant, U N (r N ) is the potential energy of a N -particle configuration
r N = (r1 , . . . , r N ). As we have established, Fint [ρ̂] is a unique functional of ρ̂(r)
for a given interaction potential u(r). This means that Fint [ρ̂] has the same depen-
dence on ρ̂(r) for all systems with the same u(r) irrespective of the external field
u ext producing inhomogeneity. As an implication of this statement Fint will look
the same for the vapor–liquid or liquid–solid interface as soon as the substances
are characterized by the same interaction potential. The term intrinsic free energy
becomes clear if we apply the functional (5.2) to the equilibrium function ρ (N )(r )
N

given by the Boltzmann distribution. In the absence of external fields it reads

e−βU N
ρ (N ) = (5.3)
Λ3N Z N

where Z N is the canonical partition function. Substitution of (5.3) into (5.2) results in

Fint [ρ] = −kB T ln Z N = F (5.4)

which is the Helmholtz free energy of the system in the absence of external fields
(intrinsic free energy). In the presence of an external field the free energy functional
can be defined as a straightforward extension of Eq. (5.2):

1
F [ρ̂] = dr N ρ̂ (N ) [U N + U N ,ext + kB T ln(Λ3N ρ̂ (N ) )] (5.5)
N!

where

N
U N ,ext = u ext (ri ) (5.6)
i=1

is the total external field energy of the configuration r N . Substituting (5.6) into (5.5)
we obtain

ρ̂ (N ) 
N
F [ρ̂] = Fint [ρ̂] + dr N u ext (ri )
N!
i=1

It is easy to see that the integral contains N equal terms

 
ρ̂ (N ) (r N )
F [ρ̂] = Fint [ρ̂] + N dr1 u ext (r1 ) dr2 . . . dr N
N!
5.2 Fundamentals of the Density Functional Approach in the Theory of Liquids 59

Integration of the N -particle probability density function ρ̂ (N ) /N ! over all possible

positions of (N − 1) particles results in the singlet distribution function:

ρ̂ (N ) (r N ) ρ̂ (1) (r1 )
dr2 . . . dr N =
N! N

implying that 
F [ρ̂] = Fint [ρ̂] + dr u ext (r)ρ̂(r) (5.7)

Thus, the free energy functional in the presence of an external field is a sum of the
intrinsic free energy functional and the (average) energy of the system in an external
field. Similarly to Eq. (5.4), for equilibrium conditions F [ρ] recovers the Helmholtz
free energy of the system in an external field.
Finally, we define the grand potential functional:

Ω[ρ̂; u ext ] = F [ρ̂] − μ dr ρ̂(r) (5.8)

where μ is the chemical potential. Obviously, for equilibrium conditions it reduces

to the grand potential of the system:

Ω[ρ; Uext ] = F − μN = Ω

The functionals of arbitrary distribution functions possess two important properties:

• they reach extrema when the distribution functions are those of the equilibrium
state, and
• those extremal values are the equilibrium values of the corresponding thermody-
namic potentials.
The are summarized in the following:
Theorem. Among all density profiles with the normalization

ρ̂(r) dr = N (5.9)

the equilibrium profile ρ(r) minimizes the functional of the free energy. The proof
is presented elsewhere (see e.g. [13], Chap. 9). Using the Lagrange multipliers the
minimizing property for F under the condition (5.9) can be cast in the form of the
unconditional minimum of the grand potential functional:

δΩ 
=0 (5.10)
δ ρ̂(r) ρ(r)
60 5 Density Functional Theory

The variational equation (5.10) using (5.8) and (5.9) yields:

μ = μint (r) + u ext (r) (5.11)

where 
δFint [ρ̂] 
μint (r) ≡ (5.12)
δ ρ̂(r) ρ(r)

is the intrinsic chemical potential. The spatial dependence of μint must be exactly can-
celed by the radial dependence of u ext (r ), since the “full” chemical potential μ (which
is the Lagrange parameter in this variational problem) is constant. Equations (5.11)–
(5.12) represent the fundamental result of DFT. If we had means to determine Fint ,
then (5.11) would be an exact equation for the equilibrium density.

5.2.2 Intrinsic Free Energy: Perturbation Approach

For realistic interactions the exact expression for the functional Fint is not available,
and one has to invoke approximations. To this end let us study the response of the
system to a small change in the pair potentials δu(ri j ) that alters the total interaction
energy (assumed to be pairwise additive)

δU N (r N ) = δu(ri j )
i< j

We describe this macroscopic reaction by the change in the grand potential

Ω(μ, V, T ) = −kB T ln Ξ (μ, V, T )

where 
Ξ (μ, V, T ) = λN Z N
N ≥0

is the grand partition function of the system, λ = eβμ is the activity. We have

δΞ 1  N 1
δΩ = −kB T = λ dr N e−β(U N +U N ,ext ) δU N (r N )
Ξ Ξ Λ3N N !
N ≥0

Examination of the right-hand side reveals that it represents the thermal average (in
the grand canonical ensemble) of δU N
δΩ = δU N

In view of the pairwise additivity of δU N , this result can be transformed by means

of the standard argument (theorem of averaging [13]) to give
5.2 Fundamentals of the Density Functional Approach in the Theory of Liquids 61

1
δΩ = dr1 dr2 ρ (2) (r1 , r2 ) δu(r12 )
2

where ρ (2) (r1 , r2 ) is the pair distribution function. Using the definition of a varia-
tional derivative, this result can be expressed as

δΩ 1
= ρ (2) (r1 , r2 ) (5.13)
δu(r12 ) 2

From the definition of Ω[ρ] it follows that the same expression is valid for Fint :

δFint [ρ] 1
= ρ (2) (r1 , r2 ) (5.14)
δu(r12 ) 2

Let us decompose the interaction potential

u = u0 + u1 (5.15)

where u 0 is some reference interaction and u 1 is a perturbation, and introduce a

family of “test systems” characterized by potentials

u α (r12 ) = u 0 (r ) + α u 1 (r ), 0 ≤ α ≤ 1 (5.16)

which gradually change from u 0 to u when the formal parameter α changes from
zero to unity. The functional integration of (5.14) then gives:
 1 
1
Fint [ρ] = Fint,0 [ρ] + dα dr1 dr2 ρα(2) (r1 , r2 )u 1 (r12 ) (5.17)
2 0

where we have expressed δu α as

∂u α
δu α = dα = u 1 dα
∂α
The first term in (5.17) is the reference contribution—the intrinsic free energy of the
system with the interaction potential u 0 (r ). The second term refers to the perturbative
(2)
part. The pair distribution function ρα is that of a system with the density ρ and
the interaction potential u α . Equation (5.17) is the second fundamental equation of
the DFT which gives the exact (though intractable!) expression for the intrinsic free
energy. To make the theory work we imply the perturbation approach in which u 1 is
considered a small perturbation. Expanding (5.17) in u 1 to the first order we obtain

1
Fint [ρ] = Fint,0 [ρ] + dr1 dr2 ρ0(2) (r1 , r2 ) u 1 (r12 ) + O(u 21 )
2
62 5 Density Functional Theory

u 0(r)
u(r)

rm rm
r r
- u1 (r)
-

Fig. 5.2 Weeks–Chandler–Andersen decomposition of the interaction potential u(r ); u 0 (r ) is the

reference interaction, u 1 (r ) is the perturbation

(2)
where the distribution function ρ0 is now that of the reference system with the
density ρ(r) and interaction potential u 0 (r ). One can go further and treat the reference
part in the local density approximation (LDA):

Fint,0 [ρ] ≈ dr ψ0 (ρ(r)) (5.18)

and
(2)
ρ0 (r1 , r2 ) ≈ ρ(r1 ) ρ(r2 ) g0 (ρ̄; r12 ) (5.19)

where ψ0 (ρ) is the free energy density of the uniform reference system with number
density ρ; g0 (ρ̄; r12 ) is the pair correlation function of the uniform reference system
evaluated at some mean density ρ̄, e.g. ρ̄ = [ρ(r1 ) + ρ(r2 )]/2. The LDA is valid
for weakly inhomogeneous systems, such as a liquid–vapor interface. For strongly
inhomogeneous systems, e.g. liquid at a wall, it becomes too crude and one has to
use a nonlocal approximation, such as the weighted density [15] or the modified
weighted-density approximation [12].
The most widely used decomposition of the interaction potential, (5.15) is given by
the Weeks–Chandler–Anderson theory (WCA) [16] (see Fig. 5.2):

u(r ) + ε for r < rm
u 0 (r ) = (5.20)
0 for r ≥ rm

−ε for r < rm
u 1 (r ) = (5.21)
u(r ) for r ≥ rm

where ε is the depth of the potential u(r ) and rm is the corresponding value of r :
u(rm ) = −ε. The advantage of the WCA scheme is that all strongly varying parts of
the potential are subsumed by the reference model describing the harshly repulsive
interaction, whereas u 1 (r ) varies slowly and therefore the importance of fluctuations
in the free energy expansion (represented by the second order term) is reduced.
5.2 Fundamentals of the Density Functional Approach in the Theory of Liquids 63

The free energy of the reference model can be expressed as the free energy of a hard-
sphere system with a suitably defined effective diameter d at the same temperature T
and the number density ρ as the original system. The thermodynamic properties of
a hard-sphere system are readily available from the Carnahan–Starling theory [17].
In particular the pressure and the chemical potential are (see e.g. [13]):

pd 1 + φd + φd2 − φd3
= (5.22)
ρkB T (1 − φd )3
μd 8φd − 9φd2 + 3φd3
= ln(ρΛ3 ) + (5.23)
kB T (1 − φd )3

where φd = (π/6)ρd 3 is the volume fraction of effective hard spheres. Then, from
the standard thermodynamic relationship the Helmholtz free energy density of the
hard-sphere system reads
ψd = ρμd − pd (5.24)

In the WCA theory it equals the free energy density of the reference model: ψ0 (ρ) =
ψd (ρ).
One can go further and ignore all correlations between particles in the perturbative
term which results in setting g0 = 1 in (5.19); this is the random phase approximation
(RPA), equivalent to the mean field (van der Waals-like) theory. A number of studies
showed that for systems with weak inhomogeneities, the RPA is sufficient when the
system is not too close to the critical point. Combining LDA and RPA, we obtain the
intrinsic free energy in its simplest form:
 
1
Fint [ρ] = dr ψd (ρ(r)) + dr1 dr2 ρ(r1 ) ρ(r2 ) u 1 (r12 ) (5.25)
2

In the same approximation the DFT equation (5.11) becomes


μd (ρ(r)) = μ − dr ρ(r ) u 1 (|r − r |) − u ext (r) (5.26)

where μd (ρ(r)) is the local chemical potential of the hard-sphere fluid. The integral
equation (5.26) can be solved iteratively for ρ(r) starting with some initial profile
satisfying the boundary conditions corresponding to bulk equilibrium:

ρ(z) → ρsat
v
in the bulk vapor
ρ(z) → ρsat
l
in the bulk liquid

In turn, the bulk equilibrium properties μ = μsat (T ), ρsat

v (T ), ρ l (T ), p (T ) can
sat sat
be found from the DFT applied to a uniform system (ρ = const). Equations (5.25)–
(5.26) (with u ext (r) ≡ 0) in this case become
64 5 Density Functional Theory

F [ρ] = Fd [ρ] − ρ 2 aV (5.27)

μ = μd (ρ) − 2ρa, (5.28)

where 
1
a=− dr u 1 (r ) (5.29)
2

is the background interaction parameter. Bulk equilibrium properties satisfy the

equations
μl (ρsat
l
, T ) = μv (ρsat
v
, T ) ≡ μsat (T ) (5.30)
p (ρsat , T ) = p (ρsat , T ) ≡ psat (T )
l l v v
(5.31)

Given μsat one can perform an iteration process for the density profile ρ(z) starting
l for z < 0 and ρ(z) =
with an initial guess, say a step-function ρ(z) with ρ(z) = ρsat
ρsat for z > 0. This profile is put into the rhs of Eq. (5.26) and the latter is solved
v

by inversion of the function μd (ρ) (given by the Carnahan-Starling approximation)

for each point z of the interface. The density is then put back into the rhs of (5.26)
and the process continues. One can be sure that iterations will eventually converge
to the equilibrium profile describing the gas-liquid interface since it corresponds to
the minimum of the grand potential functional:

δ2 Ω
>0
δρ(z)δρ(z  )

Differentiation of F with respect to the volume yields the virial equation of state:

p = pd − ρ 2 a (5.32)

where pd is the pressure of the hard-sphere system.

For a Lennard–Jones fluid with the interaction potential
 σ
σ 12 6
u LJ (r ) = 4ε − (5.33)
r r

the WCA decomposition of u LJ (r ) yields for the parameter a

√
16π 2 3
WCA
aLJ = εσ (5.34)
9
There is an obvious resemblance between (5.32) and the van der Waals equation:

ρkB T
p= − ρ 2 a vdW
1 − bvdW
5.2 Fundamentals of the Density Functional Approach in the Theory of Liquids 65

where for the Lennard–Jones fluid the van der Waals parameters are:

16π 2π 3
vdW
aLJ = εσ 3 , bLJ
vdW
= σ
9 3
At the same time the DFT uses a more sophisticated approach to describe the repul-
sive part of the potential than the free volume considerations of van der Waals. Fur-
thermore, due to the different decomposition
√ schemes, the background interaction
parameters are different: aLJWCA = 2 a vdW .
LJ

5.2.3 Planar Surface Tension

Consider an inhomogeneous (vapor-liquid) system characterized by a density profile

ρ(z) (inhomogeneity is in the z direction). The thermodynamic relationship (2.25)
for the surface tension reads:

Ω exc Ω[ρ] + pV
γ = = (5.35)
A A
where A is the surface area. The grand potential functional (5.8) with u ext = 0 then
reads
 
Ω[ρ] = − dr pd (ρ) + dr ρ μd (ρ)
  
1
+ dr ρ(r) dr ρ(r ) u 1 (|r − r |) − μ dr ρ(r)
2

where we used the intrinsic free energy functional in the RPA-form (5.25). Using the
DFT equation (5.26) and taking into account that dr = A dz we find
  
1
γ =− dz pd (ρ(z)) + ρ(z) dr ρ(z  ) u 1 (|r − r |) − psat (5.36)
2

where the equilibrium vapor–liquid density profile ρ(z) satisfies Eq. (5.26).
Let us apply the DFT to a Lennard–Jones fluid [9, 18] characterized by the interaction
potential u LJ (r ). We begin by searching for the bulk equilibrium conditions at a
given temperature T < Tc . Performing the WCA decomposition, we determine the
effective hard-sphere diameter d for the reference model. Densities in the bulk phases,
together with the equilibrium chemical potential and pressure, are found from the
coupled nonlinear equations (5.28)–(5.32)

μ = μd (ρ v ) − 2ρ v a = μd (ρ l ) − 2ρ l a
p = pd (ρ v ) − (ρ v )2 a = pd (ρ l ) − (ρ l )2 a
66 5 Density Functional Theory

Fig. 5.3 Density profiles for 1 t=0.55

the two-phase Lennard–Jones
fluid at various dimensionless 0.8
t=1
temperatures t = kB T /ε
0.6 t=1.2

3
0.4

0.2

0
0 5 10 15 20
z/

Fig. 5.4 Surface tension of 2

the Lennard–Jones fluid. Solid
line: DFT predictions; stars:
1.5
simulation results of Chapela
et al. [19]
/

1
2

0.5

0
0.6 0.8 1 1.2
kBT/

The density profiles for different dimensionless temperatures t ≡ kB T /ε are shown

in Fig. 5.3. The higher the temperature, the smaller the difference between the two
bulk densities and the broader the transition zone: for t = 0.55 it is ≈ 2σ , while
for t = 1.2 it is about 7σ . Using the equilibrium ρ(z) for each temperature, we find
the surface tension by integration in (5.36) (note that the integrand in this expression
vanishes outside the transition zone). The DFT predictions of the surface tension are
shown in Fig. 5.4 together with the simulation results of Chapela et al. [19]. The latter
are located somewhat lower then the DFT line in view of truncation of the interaction
potential in computer simulations, while in DFT the untruncated potential is used.
One important remark concerning these results must be made. The presented approxi-
mate theory is strictly mean-field in character, and therefore does not take into account
fluctuations, which become increasingly important at high temperatures close to the
critical point of the gas–liquid transition. This means that this treatment is not valid
in the critical region. Several models have been proposed to improve this approach
(for a review see [3]). In spite of this difficulty, the perturbation DFT turns out to be
very productive in giving an insight into various problems of the liquid state, such
as adsorption and wetting phenomena [20], phase transitions in confined fluids [21],
depletion interactions [22], etc.
5.3 Density Functional Theory of Nucleation 67

5.3 Density Functional Theory of Nucleation

5.3.1 Nucleation Barrier and Steady State Nucleation Rate

It is convenient to reformulate the nucleation problem in terms of the grand poten-

tial Ω. In CNT the system is considered to be at constant pressure p v , number of
molecules N and temperature T and therefore the natural potential is the Gibbs free
energy G. The latter is related to Ω through the Legendre transformation

G = Ω + p v V + μv N (5.37)

where V is the volume of the system “droplet + vapor” and μv ( p v , T ) is the chemical
potential of a vapor molecule. In the absence of a droplet the Gibbs free energy and
the grand potential are

G 0 = μv N , and Ω0 = − p v V

Then from (5.37) the change in the Gibbs energy due to the droplet formation is

ΔG = Ω − Ω0 ≡ ΔΩ (5.38)

and therefore the energy barrier to nucleation can be calculated in the grand ensemble.
The grand potential functional is related to the intrinsic free energy (in the absence
of external field) via

Ω[ρ(r )] = Fint [ρ(r )] − μ drρ(r ) (5.39)

For Fint [ρ(r )] we take the mean field form (5.25) which consists of the local den-
sity approximation for the (effective) hard-sphere part of the interaction potential
and the random phase approximation for the attractive part u 1 (r ) considered as a
perturbation:
 
1
Fint [ρ] = drψd (ρ(r)) + dr1 dr2 ρ(r1 )ρ(r2 )u 1 (r12 ) (5.40)
2

The uniform hard-sphere system is described by the Carnahan-Starling approxima-

tion. Hence, Ω[ρ] reads:
  
1
Ω[ρ(r )] = drψd (ρ(r))+ dr1 dr2 ρ(r1 )ρ(r2 )u 1 (r12 )−μ drρ(r) (5.41)
2

Critical droplet refers to a metastable state of the system “droplet + supersaturated

vapor”. The chemical potential μ in (5.41) is away from its coexistence value μsat .
68 5 Density Functional Theory

This implies that the DFT equation

δΩ
=0
δρ(r )

resulting in: 
μd (ρ(r)) = μ − dr ρ(r )u 1 (|r − r |) (5.42)

refers to a local (rather than the global) minimum of the free energy. Still there
is a nontrivial solution of Eq. (5.42) which corresponds to a saddle point of the
functional Ω[ρ] in the functional space. This solution describes a critical nucleus.
The iteration process is now unstable. Nevertheless the solution can be found once
an appropriate initial guess is chosen. As an initial guess for a radial droplet profile
a step-function can be taken with a range parameter Rinit . If Rinit is small enough
the droplet will shrink in the process of iteration giving rise to a metastable vapor
density solution. If Rinit is large the droplet will grow into a stable liquid. There
∗ which in the process of iteration will give rise to the
exist an intermediate value Rinit
critical droplet neither growing, nor shrinking over a large number of iteration steps
n. So the function Ω(n) will exhibit a long plateau staying at a constant value Ω ∗ .
The energy barrier for nucleation is given by

ΔΩ ∗ = Ω ∗ − Ω0

Finally, the steady state nucleation rate can be written as

∗
J = J0 e−βΔΩ (5.43)

The pre-exponential factor can be taken from CNT (see (3.55))


(ρ v )2 2γ∞
J0 = (5.44)
ρl π m1

since the nucleation rate is far less sensitive to J0 than to the value of the energy
barrier.
The great advantage of the DFT over the purely phenomenological models is that
in the DFT one does not have to invoke the macroscopic equilibrium properties
and equation of state. Yet, calculations of nucleation behavior are much faster than
direct computer simulations using Monte Carlo or Molecular Dynamics methods
(discussed in Chap. 8).
5.3 Density Functional Theory of Nucleation 69

5.3.2 Results

Oxtoby and Evans [6] calculated density profiles for a critical droplet with a Yukawa
attractive potential
e−κr
u 1 (r ) = −ακ 3
4π λr
According to CNT one expects that the density in the center of the droplet is equal
l (T ). However, the results of [6] show that the
to the liquid density at coexistence ρsat
density in the center of the droplet is lower than ρsat
l and is lower than the density

of the bulk liquid at the same chemical potential as the supersaturated vapor ρ l (μv ).
Another important feature of this approach is that the barrier to nucleation, ΔΩ ∗ ,
vanishes as spinodal is approached whereas in the classical theory it remains finite.
Zeng and Oxtoby [9] applied the DFT to predict nucleation rates for a Lennard-
Jones fluid. Following the preceding discussion the free energy functional is written
using the hard-sphere perturbation analysis based on the WCA decomposition of the
non-truncated Lennard-Jones potential.
Figure 5.5 based on the results of Ref. [9] shows the comparison of the nucleation rates
predicted by the DFT and the CNT. To make such a comparison consistent the macro-
scopic surface tension used in the CNT was obtained by means of Eq. (5.35). The
results in Fig. 5.5 correspond to the fixed classical nucleation rate JCNT = 1 cm−3 s−1 .
At each temperature this condition determines the chemical potential difference (and
consequently the supersaturation) for which the DFT calculation is carried out. The
results of Ref. [9] demonstrate that CNT and DFT predict the same dependence of
nucleation rate on the supersaturation but show the essentially different temperature
dependence. Due to the latter the disagreement between the two approaches is up
to 5 orders of magnitude in J . The difference between two theories becomes pro-
nounced when the nucleation temperature is away from kB T /ε ≈ 1.1. As one can see

Fig. 5.5 Ratio of the nucle- 2

ation rates (CNT to DFT) for a J CNT =1cm
-3 -1
s
Lennard-Jones fluid [9]. DFT
calculations at each temper- 0
log 10 (J CNT /J DFT )

ature are carried out for the

supersaturation corresponding
to the fixed classical nucle-
ation rate JCNT = 1 cm−3 s−1 . -2
The predictions of both
models become equal at
kB T /ε ≈ 1.08. The solid line -4
is shown to guide the eye

-6
0.6 0.7 0.8 0.9 1 1.1 1.2

k BT/
70 5 Density Functional Theory

from Fig. 5.5, at lower temperatures CNT underestimates the nucleation rate (com-
pared to DFT) while at higher temperatures it overestimates it. The same trend is
demonstrated by the CNT when it is compared to experimental nucleation rates: e.g.
for water CNT underestimates experimental rates at T < 230 K, and overestimates
experimental rates at T > 230 K [23, 24].

References

1. C. Ebner, W.F. Saam, D. Stroud, Phys. Rev. A 14, 2264 (1976)

2. R. Evans, Adv. Phys. 28, 143 (1979)
3. R. Evans, Density functionals in the theory of nonuniform fluids. in Fundamentals of Inhomo-
geneous Fluids, ed. by D. Henderson (Marcel Dekker, New York 1992), p. 85
4. P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964)
5. W. Kohn, L.J. Sham, Phys. Rev. 140, A1133 (1965)
6. D.W. Oxtoby, R. Evans, J. Chem. Phys. 89, 7521 (1988)
7. D.W. Oxtoby, in Fundamentals of Inhomogeneous Fluids, ed. by D. Henderson (Marcel Dekker,
New York, 1992), Chap. 10
8. D.W. Oxtoby, J. Phys. Cond. Matt. 4, 7627 (1992)
9. X.C. Zeng, D.W. Oxtoby, J. Chem. Phys. 94, 4472 (1991)
10. V. Talanquer, D.W. Oxtoby, J. Chem. Phys. 99, 4670 (1993)
11. V. Talanquer, D.W. Oxtoby, J. Chem. Phys. 100, 5190 (1994)
12. A.R. Denton, N.W. Ashcroft, Phys. Rev. A 39, 4701 (1989)
13. V.I. Kalikmanov, Statistical Physics of Fluids. Basic Concepts and Applications (Springer,
Berlin, 2001)
14. J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity (Clarendon Press, Oxford, 1982)
15. W.A. Curtin, N.W. Ashcroft, Phys. Rev. A 32, 2909 (1985)
16. D. Weeks, D. Chandler, H.C. Andersen, J. Chem. Phys. 54, 5237 (1971)
17. N.F. Carnahan, K.E. Starling, J. Chem. Phys. 51, 635 (1969)
18. C.C.M. Luijten, Ph.D. Thesis, Eindhoven University, 1999
19. A.G. Chapela, G. Saville, S.M. Thompson, J.S. Rowlinson, J. Chem. Soc. Faraday Trans. II
73, 1133 (1977)
20. S. Dietrich, in Phase Transitions and Critical Phenomena, vol. 12, ed. by C. Domb, J.L.
Lebowitz (Academic Press, New York 1988), p. 1
21. R. Evans, J. Phys. Condens. Matter 2, 8989 (1990)
22. B. Götzelmann et al., Europhys. Lett. 47, 398 (1999)
23. J. Wölk, R. Strey, J. Phys. Chem. B 105, 11683 (2001)
24. D.G. Labetski, V. Holten, M.E.H. van Dongen, J. Chem. Phys. 120, 6314 (2004)
Chapter 6
Extended Modified Liquid Drop Model
and Dynamic Nucleation Theory

In the classical theory and its modifications an arbitrary cluster is characterized by

one parameter—the number of molecules in it. In a series of papers [1–3] Reiss and
co-workers discussed an alternative form of cluster characterization. It was suggested
that a cluster should be characterized not only by the particle number, i, but also by
its volume v. As a result dynamics of such an i, v-cluster becomes two-dimensional
(as opposed to the CNT, where it is one-dimensional) resembling nucleation in binary
systems. Using these arguments Weakliem and Reiss [4] put forward the modified
liquid drop model and performed extensive Monte Carlo simulations to calculate free
energy of the i, v-clusters. Based on these ideas Reguera et al. [5] put forward the
“extended modified liquid drop” model (EMLD), taking into account the effect of
fluctuations which are important for the formation of tiny droplets in a small N V T -
system. More recently Reguera and Reiss [6] combined EMLD with the Dynamic
Nucleation Theory (DNT) of Shenter et al. [7, 8]. The new model, called “Extended
Modified Liquid Drop Model-Dynamical Nucleation Theory” (EMLD-DNT), is dis-
cussed in the next sections.

6.1 Modified Liquid Drop Model

The CNT studies formation of droplets in an open system. The main feature of the
modified liquid drop model of Ref. [4] is that it considers the closed system containing
N molecules confined within a small spherical volume V at a temperature T . This
small N V T system is called an EMLD-cluster. Within the volume V various sharp
n-clusters can form, n = 1, . . . , N . The important difference between the closed an
open system is that in the system with the fixed total amount of molecules N the
formation and growth of droplets is accompanied by the depletion of the vapor—the
effect neglected in CNT, which assumes the existence of an infinite source of vapor
molecules. The depletion of the vapor molecules in EMLD results in the decrease
of supersaturation so that the droplet can not become arbitrarily large. Following

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 71

DOI: 10.1007/978-90-481-3643-8_6, © Springer Science+Business Media Dordrecht 2013
72 6 Extended Modified Liquid Drop Model and Dynamic Nucleation Theory

Fig. 6.1 A schematic rep- V, T

resentation of the EMLD-
cluster: a closed system of N
molecules confined inside the
volume V of the radius R. N-n
n out of N molecules form R
a liquid drop of the radius r ,
while the rest N −n molecules
remain in the vapor phase

n r

Ref. [6] we analyze the properties of the EMLD-cluster using purely thermodynamic
considerations.
The spherical volume V of the radius R is assumed to have impermeable hard walls.
Under certain conditions a liquid drop with n molecules, n ≤ N − 1, can be formed
inside V . The rest N − n molecules remain in the vapor phase, occupying the volume
V − n vl , where vl is the volume per molecule in the bulk liquid phase (see Fig. 6.1).
This vapor has the pressure described within the ideal gas approximation

(N − n) kB T
p1 = (6.1)
V − nvl

The sharp n-cluster is treated within the capillarity approximation, i.e. it is assumed
that it has a sharp interface, characterized by the macroscopic surface tension γ∞ ,
and the bulk liquid properties inside it. We stress that within this model the entire
N V T -system is considered as the EMLD-cluster, and not just one (sharp) n-droplet.
Since we discuss the closed system, the appropriate thermodynamic potential is the
Helmholtz free energy
F = U − TS

where U is the internal energy and S is the entropy of the EMLD-cluster. Denoting
the vapor and liquid subsystems inside the EMLD-cluster by subscripts 1 and 2,
respectively, we write the differentials of U1 and U2 as:

dU1 = T dS1 − p1 dV1 + μ1 dN1 (6.2)

dU2 = T dS2 − p2 dV2 + μ2 dN2 + γ∞ d A (6.3)

whereA = 4πr 2 is the surface area of the n-cluster. The free energy change associated
with the formation of a sharp n-droplet inside the EMLD-cluster is:

(dF ) N V T = dU1 + dU2 − T dS1 − T dS2

6.1 Modified Liquid Drop Model 73

For the closed system

dV1 = −dV2 , dN1 = −dN2 = −n

which using (6.2)–(6.3) gives

 
2γ∞
(dF )NVT = − p2 − p1 − dV2 + (μ2 − μ1 ) dN2 (6.4)
r

Equilibrium of the sharp n-droplet with the surrounding vapor corresponds to

(dF)NVT = 0 resulting in
μ2 = μ1 (6.5)

2γ∞
p2 − p1 = (6.6)
r
The second equality is the Laplace equation. We can rewrite these results applying the
ideal gas approximation for the vapor and considering liquid to be incompressible.
Using the thermodynamic relationship

1
(dμ)T = dp
ρ

for the vapor and liquid phases, we relate μi to the bulk vapor-liquid equilibrium
properties
p1
μ1 ( p1 ) − μsat = kB T ln (6.7)
psat

μ2 ( p2 ) − μsat = vl ( p2 − psat ) (6.8)

Then, the equilibrium conditions (6.5)–(6.6) result in

p1 2γ∞ l
kB T ln = v + vl ( p1 − psat ) (6.9)
psat r

The last term is usually very small and can be neglected. The result is the classical
Kelvin equation ( 3.61) relating the pressure inside the n-droplet to its radius
 
2γ∞ vl
p1 = psat exp (6.10)
r kB T

Using Eq. (6.1) for the vapor pressure inside the EMLD-cluster, we can solve (6.10)
for the size of the coexisting droplet. In CNT, dealing with the open μV T system,
the solution of the Kelvin equation determines the critical cluster radius at the given
temperature and supersaturation. In the closed system the supersaturation p1/psat is
not fixed but depends on the amount of molecules in the n-cluster since the total
74 6 Extended Modified Liquid Drop Model and Dynamic Nucleation Theory

amount N is conserved. Another difference is that while in the open system the
solution of the coexistence equations is unique and corresponds to the critical cluster,
in the closed system Eqs. (6.5)–(6.6) have two solutions: one corresponding to the
critical cluster, i.e. the cluster in the metastable equilibrium with the vapor, and the
other one—corresponding to the stable cluster.
For an arbitrary n-cluster (i.e. not necessarily a critical one), which is not in equi-
librium with the surrounding vapor, the free energy change (dF ) N V T = 0 and the
chemical potential difference μ2 ( p2 ) − μ1 ( p1 ) reads:
p1
μ2 ( p2 ) − μ1 ( p1 ) = vl ( p2 − psat ) − kB T ln (6.11)
psat

Substituting (6.11) into (6.4) and using the incompressibility of the liquid phase
(dV2 = vl dn) we obtain:
 
2γ∞ l p1 (n)
(dF ) N V T = v + v ( p1 (n) − psat ) − kB T ln
l
dn (6.12)
r (n) psat

We can perform thermodynamic integration of this equation from n = 0, correspond-

ing to the state of pure vapor, to an arbitrary n. Then
 n
(dF ) N V T = F (n) − F (0) ≡ ΔF (n)
0

is the free energy difference between the state in which EMLD-cluster contains the
n-droplet and the state of pure vapor; thus, ΔF(n) is the Helmholtz free energy
of the n-droplet formation. Taking into account that r (n) = r l n 1/3 (where r l =
(3vl /4π )1/3 ) and using (6.1) we find after the integration
     
p1 vl psat p1
ΔF (n) = −n kB T ln psat + γ∞ A + n k B T 1 − kB T + N kB T ln p0

(6.13)

where
N kB T
p0 =
V
is the pressure corresponding to the pure vapor. In Eq. (6.13) the first and the second
terms are, respectively, the standard bulk and surface contributions to the free energy
of cluster formation (recall, that here the supersaturation is S(n) = p1 (n)/ psat ); the
third term is the volume work (usually small and is commonly neglected); the last
term originates from depletion of the vapor molecules in the EMLD-cluster when an
n-droplet is formed. In the thermodynamic limit p1 = p0 recovering the CNT result
for the free energy of the cluster formation. Note, that in this model the state of pure
vapor corresponds to n = 0 and not to n = 1; the value n = 1 corresponds to a
hypothetical liquid cluster of size 1, which is not the same as a molecule of the vapor.
6.1 Modified Liquid Drop Model 75

The EMLD-cluster can contain sharp droplets of various sizes n. The total free energy
of the cluster ΔFtot is found by accounting of all possible fluctuations of the droplet
size. Each such fluctuation enters the configuration integral Q of the cluster with the
Boltzmann factor e−βΔF (n) resulting in

N
Q= e−βΔF (n)
n=0

Then,
N
ΔFtot = −kB T ln Q = −kB T ln e−βΔF (n) (6.14)
n=0

The probability of finding a sharp cluster with exactly n molecules inside is

e−βΔF (n)
N
f (n) = , f (n) = 1
Q
n=0

The total pressure Ptot of the EMLD-cluster is the weighted sum of the vapor pressure
in the confinement sphere over all possible n. The vapor pressure consists of p1 and
the pressure exerted by the n-drop (for n = 0), modelled as a single ideal gas molecule
moving within the container

N  
kB T
Ptot = f (n) p1 + Ξ (n)
Vc
n=0

Here Vc (n) = 4π(R − r (n))3 /3 is the volume accessible for the center of mass of
the n-droplet with the radius r (n), R is the radius of the EMLD-cluster
 1/3
3V
R=
4π

and Ξ (n) is the unit step-function.

6.2 Dynamic Nucleation Theory and Definition

of the Cluster Volume

Considerations put forward in the previous section, have not yet answered the ques-
tion: how to choose the volume V of the EMLD-cluster which would be physically
relevant for nucleation at the given external conditions? To address it we notice that
76 6 Extended Modified Liquid Drop Model and Dynamic Nucleation Theory

since nucleation is a kinetic nonequilibrium process, it is plausible to search for the

suitable criterion for V in the theory of rate processes. An important development in
this direction is the Dynamic Nucleation Theory (DNT) formulated by Shenter et al.
[7]. The key issue of DNT is the determination of the evaporation rate for an isolated
cluster.
DNT is a molecular based theory: for a given intermolecular interaction potential
one derives the Helmholtz free energy of the cluster using an appropriate statistical
mechanical sampling. The detailed balance condition is used to derive the relation-
ship between the condensation and evaporation rates which is expressed in terms of
the differences in Helmholtz free energies between the N - and (N + 1)-clusters. An
important quantity in these calculations is the volume V of the N -cluster, or, equiva-
lently, the size of the configurational space of the cluster. In the DNT the ambiguity
in the choice of V is removed by means of the variational transition state theory
[9–11]. At each stage of the nucleation process the DNT defines the dividing surface
of the radius rcut in the phase space that separates reactant states from the producing
states. From this dividing surface an unambiguous cluster definition emerges which
is consistent with the detailed balance between the condensation and evaporation
rates [8]. This dividing surface leads to the evaporation rate which is proportional to
the Helmholtz free energy of the cluster. Using the variational transition state theory
Shenter et al. [12] showed that the proper kinetic definition of V is the one that min-
imizes the evaporation rate. From the above discussion it follows that the proper V
corresponds to the minimum of the free energy change of the cluster with respect to
its volume, or in other words the physically relevant cluster volume Vm corresponds
to the minimum of the pressure:

∂ Ptot
=0
∂V Vm

The combined (EMLD and DNT) model is called the EMLD-DNT theory.

6.3 Nucleation Barrier

In the open (μV T )-system the work of formation of the physical (N , Vm )-cluster at
a temperature T is given by

ΔG = ΔΩ = ΔF (N , Vm ) − Vm ( p0 − Ptot ) + N Δμ0 (6.15)

where Δμ0 = μ0 − μ. The role of the cluster size in this expression is played by N .
By construction the EMLD-DNT cluster represents the diffuse interface system. That
is why N is not the molecular excess quantity determined by the nucleation theorem
6.3 Nucleation Barrier 77

(discussed in Chap. 4). The nucleation barrier corresponds to the maximum of ΔG:

∂ΔG
=0 (6.16)
∂N Vm ,T

Having determined the critical cluster N ∗ from this equation, we substitute it into
Eq. (6.15) to obtain the nucleation barrier

ΔG ∗EMLD−DNT = ΔF (N ∗ , Vm ) − Vm ( p0 − Ptot ) + N ∗ kB T ln( p0 /Ptot ) (6.17)

In the thermodynamic limit p0 coincides with the actual vapor pressure p v . The
EMLD-DNT nucleation rate reads:
 
ΔG ∗EMLD−DNT
JEMLD−DNT = J0 exp − (6.18)
kB T

with the CNT kinetic prefactor, J0 .

The advantage of EMLD-DNT is that it does not require information about the
microscopic interactions (as DNT or DFT). It uses the same set of the macroscopic
parameters as the CNT: psat (T ), ρ l (T ), γ∞ (T ). At the same time the model is able
to predict the vanishing of the nucleation barrier at some finite S which signals the
thermodynamic spinodal; recall that in the CNT the nucleation barrier remains finite
for all values of S. The important conceptual feature of EMLD-DNT is that instead
of a sharp density profile it allows for a diffusive cluster. Within this approach every
sharp n-droplet inside the EMLD-cluster enters the free energy with its Boltzmann
weight. Note, that describing a sharp droplet of any size 1 ≤ n ≤ N − 1 this model
uses the capillarity approximation with the planar surface tension γ∞ which becomes
dubious for small n. The latter, however, is smeared out between all admissible cluster
sizes. In Chap. 10 we show the predictions of EMLD-DNT for argon nucleation and
compare them to experiments and other theoretical models as well as to the DFT and
computer simulations.

References

1. H. Reiss, A. Tabazadeh, J. Talbot, J. Chem. Phys. 92, 1266 (1990)

2. H.M. Ellerby, C.L. Weakliem, H. Reiss, J. Chem. Phys. 95, 9209 (1991)
3. H.M. Ellerby, H. Reiss, J. Chem. Phys. 97, 5766 (1992)
4. C.L. Weakliem, H. Reiss, J. Chem. Phys. 99, 5374 (1993)
5. D. Reguera et al., J. Chem. Phys. 118, 340 (2003)
6. D. Reguera, H. Reiss, Phys. Rev. Lett. 93, 165701 (2004)
7. G.K. Shenter, S.M. Kathmann, B.C. Garrett, Phys. Rev. Lett. 82, 3484 (1999)
8. S.M. Kathmann, G.K. Shenter, B.C. Garrett, J. Chem. Phys. 116, 5046 (2002)
78 6 Extended Modified Liquid Drop Model and Dynamic Nucleation Theory

9. E. Wigner, Trans. Faraday Soc. 34, 29 (1938)

10. J.C. Keck, J. Chem. Phys. 32, 1035 (1960)
11. J.C. Keck, Adv. Chem. Phys. 13, 85 (1967)
12. G.K. Shenter, S.M. Kathmann, B.C. Garrett, J. Chem. Phys. 110, 7951 (1999)
Chapter 7
Mean-Field Kinetic Nucleation Theory

7.1 Semi-Phenomenological Approach to Nucleation

On the microscopic level nucleation behavior is determined by intermolecular inter-

actions in the substance. This is clearly demonstrated by the Density Functional
Theory. Its applicability, however, is limited by relatively simple types of interac-
tions. For the substances with highly nonsymmetric molecules the applicability of
the standard DFT scheme becomes increasingly difficult. It is therefore desirable to
propose a compromise between the microscopic and phenomenological descriptions.
One can classify it as a semi-phenomenological approach to nucleation. It was pio-
neered by Dillmann and Meier [1] and developed by Ford, Laaksonen and Kulmala
[2], Delale and Meier [3] and Kalikmanov and van Dongen [4]. The main idea of the
semi-phenomenological approach is a combination of statistical thermodynamics of
clusters with available data on the equilibrium material properties.
The statistical thermodynamic part of all these models is based on the seminal Fisher
droplet model of condensation [5], in which the real gas is considered as a system of
noninteracting clusters and the cluster distribution function is expressed in terms of
the cluster configuration integral. The latter in the Fisher theory contains an undeter-
mined quantity—the Helmholtz free energy per unit surface of the cluster- termed
in [5] a microscopic surface tension. A common feature of the above-mentioned
models is that Fisher’s microscopic surface tension is presented in the form of the
expansion of the cluster surface tension in powers of the curvature. The coefficient
at the first order term is known as the Tolman length (cf. Sect. 2.2.2); the expansion
itself is frequently called the Tolman expansion. In [2–4] the Tolman expansion is
truncated at the first or second order term.
By doing so one has to realize that the range of validity of this expansion is a matter of
great importance. Clearly, since the expansion is in the cluster curvature, it is applica-
ble for sufficiently big clusters. In a large number of experiments, however, the clus-
ters that dominate nucleation behavior (those close to the critical size), are relatively
small, containing tens or hundreds of molecules, for them the cluster curvature is not

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 79

DOI: 10.1007/978-90-481-3643-8_7, © Springer Science+Business Media Dordrecht 2013
80 7 Mean-Field Kinetic Nucleation Theory

a small parameter implying that the Tolman expansion becomes dubious. Therefore,
it is desirable to formulate a nonperturbative semi-phenomenological approach valid
for all cluster sizes. In this chapter we discuss such a model—a Mean-field Kinetic
Nucleation Theory (MKNT) of Ref. [6].

7.2 Kinetics

Kinetics of cluster formation is governed by the standard CNT assumptions:

• cluster growth and decay are dominated by monomer addition and monomer
extraction;
• if a monomer collides a cluster it sticks to it with probability unity (the sticking
coefficient is unity); and
• there is no correlation between successive events that change the number of par-
ticles in a cluster.
Using Katz’s “kinetic approach” of Sect. 3.5, we write the steady state nucleation
rate in the form given by Eq. (3.68):
 ∞
−1
 1
J= (7.1)
f (n) S n ρsat (n)
n=1

where S is the supersaturation, f (n) is the forward rate of n-cluster formation in the
supersaturated vapor, and ρsat (n) is the equilibrium cluster distribution at saturation
(corresponding to S = 1). The purely phenomenological considerations adopted
in the CNT and in Katz’s kinetic version of the CNT are valid when the major
contribution to the nucleation rate comes from big clusters (the notion of a “big
cluster” will be specified below). In this case the free energy of cluster formation
reads

sat (n) = γ∞ s1 n
ΔG CNT 2/3

yielding for the cluster distribution function at saturation

ρsat (n) ∼ e−β γ∞ s1 n

2/3

The formal extension of these expressions to all cluster sizes does not pose the
problem since the contribution of small clusters to J is negligible.
On the other hand, if nucleation behavior is primarily driven by the formation of
small clusters, one has to apply microscopic considerations. In this regime the very
notion of a surface tension of a small cluster looses its physical meaning. In the next
section we consider the model for ρsat (n) which is valid for arbitrary n.
7.3 Statistical Thermodynamics of Clusters 81

7.3 Statistical Thermodynamics of Clusters

A typical interaction between gas molecules consists of a harshly repulsive core and
a short-range attraction. The most probable configurations of the gas at low densi-
ties and temperatures will be isolated clusters of n = 1, 2, 3, . . . molecules. Hence,
to a reasonable approximation one can describe a real gas as a system of nonin-
teracting clusters.1 At the same time intracluster interactions are important—they
are responsible for the formation of a cluster All clusters are in statistical equilib-
rium, associating and dissociating. Even large clusters have a certain probability of
appearing. The partition function of an n-cluster at a temperature T is:

1
Zn = qn (7.2)
Λ3n
where Λ is the thermal de Broglie wavelength of a particle (being an atom or a
molecule); qn is the configuration integral of the n-cluster in a physical domain of
volume V :

1
qn (T ) = drn e−βUn , (7.3)
n! cl
1
Un is the potential energy of the n-particle configuration in the cluster; the factor n!
takes into account the indistinguishability of particles inside the cluster. The sym-
bol cl indicates that integration is performed only over those atomic configurations
that belong to the cluster. At this point it is important to emphasize the difference
between the n-particle configuration integral Q n and the n-cluster configuration inte-
gral qn . The latter includes only those configurations in the volume V that form the n-
cluster, while Q n contains all different configurations of n particles in the volume V ;
therefore Q n ≥ qn . The cluster as a whole can move through the entire volume V of
the system, while the particles inside the cluster are restricted to the configurations
about cluster’s center of mass that are consistent with a chosen definition of the clus-
ter. For that one can adopt, e.g., Stillinger cluster [7]: an atom belongs to a cluster
if there exists at least one atom of the same cluster separated from the given one by
a distance r < rb , where rb is some characteristic distance describing the range of
interparticle interactions. In other words, an atom belongs to the cluster if inside a
sphere of radius rb there is at least one atom belonging to the same cluster. Early
Monte Carlo studies of Lee et al. [8] showed that a cluster’s free energy is almost
independent of a cluster definition provided that the definition is reasonable and the
temperature is sufficiently low. For the present model a particular type of a cluster
definition is not important. What matters is that a cluster is a compact object around
its center of mass.

1 Note that at high temperatures interactions between clusters can not be neglected.
82 7 Mean-Field Kinetic Nucleation Theory

The partition function Z (n) of the gas of Nn noninteracting n-clusters in the volume
V at the temperature T is factorized:

1
Z (n) = Z Nn (7.4)
Nn ! n

where the prefactor 1/Nn ! takes into account the indistinguishability of clusters (recall
that indistinguishability of atoms inside the cluster is taken into account in qn ). The
Helmholtz free energy of the gas of n-clusters is: F (n) = −kB T ln Z (n) , which
using Stirling’s formula becomes:
 
(n) Nn
F = Nn kB T ln
Zn e

The chemical potential of the n-cluster in the gas is

 
∂F (n) Nn
μn = = kB T ln (7.5)
∂ Nn Zn

Introducing the number density of n-clusters ρ(n) = Nn/V and substituting (7.2)
into (7.5), we obtain  
V Λ3n
μn = kB T ln ρ(n) (7.6)
qn

Equilibrium between the cluster and surrounding vapor molecules requires

μn = nμv (7.7)

where μv is the chemical potential of a molecule in the vapor phase. Combining (7.6)
and (7.7) we find:
qn
ρ(n) = zn (7.8)
V
where
z = eβμ /Λ3
v
(7.9)

is the fugacity of a vapor molecule. From the definition of qn it is clear that the
quantity qn /V involves only the degrees of freedom relative to the center of mass
of the cluster and remains finite in thermodynamic limit (V → ∞). The pressure
equation of state for the vapor is given by Dalton’s law
∞
p v (μv , T ) 
= ρ(n)μv ,T (7.10)
kB T
n=1
7.3 Statistical Thermodynamics of Clusters 83

and the overall number density of the gas is:

∞

ρv = nρ(n) (7.11)
n=1

Equation (7.8) holds for every point of the gaseous isotherm. In particular for the
saturation point it reads

qn n eβμsat
ρsat (n) = z sat , z sat = (7.12)
V Λ3

where the chemical potential at saturation μsat (T ) can be found from the suitable
equation of state. The problem of finding the equilibrium cluster distribution is
reduced to the determination of the cluster configuration integral. Up to this point
all results were exact. To proceed with calculation of qn it is necessary to introduce
approximations.

7.4 Configuration Integral of a Cluster: Mean-Field

Approximation

An n-cluster is an object containing n particles satisfying a certain cluster definition.

A geometrical form of the cluster can be quite different. Big clusters tend to a compact
spherical shape with a well defined surface area scaling with the cluster size as n 2/3 .
This is definitely not true for small clusters: they look more like fractal objects—one
can think here about binary-, ternary clusters, etc. It is not clear how to define the
surface area of such an object. It is therefore reasonable to replace the concept of
the surface area of an arbitrary cluster by the number of suitably defined surface
particles.
With this in mind, following Zhukhovitskii [9], we decompose n into two groups:
the core n core and the surface particles n s

n = n core + n s (7.13)

The physical idea behind this distinction is that the core of the cluster, if present,
should possess the liquid-like structure which can be characterized by a certain
property typical for the liquid phase, e.g. by the liquid coordination number N1 . The
surface molecules can then be viewed as an adsorption layer covering the core.2 By
definition the integer numbers n core and n s satisfy

n core ≥ 0, ns ≥ 1

2 This decomposition should not be confused with the Gibbs construction involving a dividing
surface discussed in Sect. 2.2.
84 7 Mean-Field Kinetic Nucleation Theory

Both quantities can fluctuate around their mean values

n core = n core (n; T ) + δn core , n s = n s (n; T ) + δn s

so that δn core + δn s = 0. For the present purposes it is not necessary to specify the
form of these quantities; we postpone this discussion till Sect. 7.5.
Similar to the seminal Fisher droplet model of condensation [5], we write the internal
potential energy of an n-cluster as a sum of the bulk and surface contributions

Un = −n E 0 + Wn (7.14)

Here −E 0 (E 0 > 0) is the binding energy per particle related to the depth of inter-
particle attraction; Wn is the surface energy of the n-cluster. In the Fisher model Wn
has the form Wn = w A(n), with w being the energy per unit surface and A(n) is the
cluster surface area. In view of the previous discussion, we present Wn in a different
way making use of the concept of surface particles:

W n = w1 n s , w1 > 0 (7.15)

where w1 is the surface energy per surface particle. Both E 0 and w1 are material con-
stants independent of temperature. The difference between the two models becomes
increasingly important for small clusters, for which the surface area of a cluster can
not be properly defined.
Let us place the origin of the coordinate system in Eq. (7.3) into the center of mass
of the cluster. Then the configuration integral can be written as

qn = V enβ E 0 G n (β) (7.16)

where 
1
drn−1 e−βw1 n
s (rn−1 )
G n (β) = (7.17)
n! cl

describes the “surface part” of qn and has the dimensionality of (volume)n−1 . The
notation n s (rn−1 ) indicates that n s depends on a particular configuration of cluster
particles. The configurational Helmholtz free energy of the cluster reads

Fnconf = −kB T ln qn

from which the cluster configurational entropy is:

∂Fnconf ∂ ln qn
Snconf = − = kB ln qn − β (7.18)
∂T ∂β
7.4 Configuration Integral of a Cluster: Mean-Field Approximation 85

From (7.16)
ln qn = ln V + nβ E 0 + ln G n (β) (7.19)

yielding
∂ ln qn 1 ∂G n
= n E0 +
∂β G n ∂β

Let us discuss the last term of this expression. From (7.17):

 
−w1 n s (rn−1 ) e−βw1 n (r )
n−1 s n−1
1 ∂G n cl dr
=  ≡ −w1 n s (β) (7.20)
G n ∂β cl dr
n−1 e−βw1 n s (rn−1 )

where n s (β) is the thermal average of n s . Substituting (7.19) and (7.20) into (7.18)
we find

Snconf (β) = kB [ln G n + βw1 n s + ln V ]

The bulk entropy per molecule can be identified with the entropy per molecule in the
bulk liquid, or equivalently, in the infinitely large cluster

Snconf
S0 (β) = lim
n→∞ n
When n → ∞ most of the particles belong to the core while the relative number of
surface molecules vanishes:

ns
→ 0 as n → ∞
n
(the rigorous proof of this statement is presented in Sect. 7.5). Then

1
S0 (β) = kB lim ln G n (β) (7.21)
n→∞ n

By virtue of the cluster definition, mutual distances between molecules in the cluster
can not exceed some maximum value, therefore the integral (7.17) remains finite.
Let us introduce a temperature dependent parameter v0 with the dimensionality of
volume which can be understood as an average volume per molecule in the cluster.
1/3
Scaling all distances with v0 , we rewrite G n as

1
rn−1 e−βw1 n
s (
rn−1 )
G n (β) = v0n−1 d
n! cl

whereri are the dimensionless positions of the cluster molecules; the integral in the
square brackets is now dimensionless (and finite). The number of surface molecules
86 7 Mean-Field Kinetic Nucleation Theory

in the n-cluster depends on a particular configuration, but can have the values in
the range 1 ≤ n s ≤ n. Replacing integration over 3(n − 1) configuration space by
summation over all possible values of n s , we obtain

g(n, n s ) e−βw1 n
s
G n (β) = v0n−1 (7.22)
1≤n s ≤n

where the degeneracy factor g(n, n s ) gives the number of different molecular con-
figurations in the n-cluster having the same number n s of the surface molecules.
We calculate the positive definite series G n (β) using the mean-field approximation.
We assume that the sum in Eq. (7.22) is dominated by its largest term to the extent that
it is possible replace the entire sum by this largest term while completely neglecting
the others:  
s −βw1 n s
G n (β) ≈ v0n−1 max
s
g(n, n ) e (7.23)
n

The similar approximation is used in the theory of phase transitions giving rise to
the Landau theory [10]. The maximum in (7.23) is attained at the most probable
number of surface particle in the n-cluster, n s (n; β), corresponding to the parti-
cle configuration with the maximum statistical weight. Thus, within the mean-field
approximation
G n (β) = v0n−1 g(n, n s ) e−βw1 n
s
(7.24)

Recalling the physical meaning of g(n, n s ), one can expect that ln g(n, n s ) is related
to the configurational entropy of the n-cluster. To verify this conjecture we use Eqs.
(7.21) and (7.24) to obtain:

1
S0 = kB ln v0 + lim ln g(n, n s ) (7.25)
n→∞ n

Similar to the decomposition of Un , we decompose the configurational entropy of

the cluster as
Snconf = n S0 + ν1 n s (7.26)

where the temperature independent material parameter ν1 is the configurational

surface entropy per particle; the term ν1 n s characterizes the number of distinct
cluster configurations of n s surface particles.3 Comparing Eqs. (7.25) and (7.26) it
is plausible to assume that the surface entropy satisfies

ν1 n s = kB n ln v0 + kB ln g(n, n s (β)) − nS0 (7.27)

3 Note that as a thermodynamic quantity S conf depends on the average number of surface molecules
n
n s , whereas Un as a microscopic quantity depends on n s itself.
7.4 Configuration Integral of a Cluster: Mean-Field Approximation 87

yielding:
n S0 ν1 n s
g(n, n s ) = v0−n exp + (7.28)
kB kB

Combining (7.28), (7.16) and (7.24), we obtain

 
qn S0 n s
=C exp β E 0 + {exp [−β(w1 − ν1 T )]}n (7.29)
V kB

where
C = v0−1 (7.30)

Substituting (7.29) into (7.8), the number density of n-clusters takes the form
s
ρ(n) = C y n x n (7.31)

where for convenience we introduced the following notations

S0
y ≡ z exp β E 0 + (7.32)
kB
 
w1 − ν1 T
x ≡ exp − (7.33)
kB T

The quantity x > 0 measures the temperature; at low temperatures x is small. The
quantity y measures the fugacity, or, equivalently, the chemical potential of the vapor.
A big cluster can be associated with a liquid droplet in the vapor. The growth of a
macroscopic droplet corresponds in this picture to condensation. Following [5], let
us discuss the probability of finding an n-cluster in the vapor at the temperature T
and the chemical potential μv . This probability is proportional to ρ(n).
If in (7.31) y < 1, which corresponds to a small z, or equivalently to a large and
negative μv , then ρ(n) exponentially decays as exp[−const × n]. As y approaches
unity this decrease becomes slower. When y = 1, ρ(n) still decays but only as
exp[−const × n s (n)]. Finally, if y slightly exceeds unity, then ρ(n) first decreases,
reaching a minimum at some n = n 0 , and then increases without bounds (see
Fig. 7.1). The large (divergent) probability of finding a very large cluster signals
the condensation. Thus, we identify

ysat = 1 (7.34)

with the saturation point. Applying Eq. (7.32) to saturation and using (7.34) we find

S0 1
exp β E 0 + = (7.35)
kB z sat
88 7 Mean-Field Kinetic Nucleation Theory

Fig. 7.1 The number density

of n-clusters ρ(n) for various
values of the fugacity, or
equivalently, parameter y. For y <1
y > 1 ρ(n) attains a minimum y =1
at n = n 0 and for n > n 0 it

(n)
y >1
diverges

n0 n

where z sat (T ) is the fugacity at saturation. From (7.31) and (7.34) the cluster distri-
bution at saturation reads:

(w1 − ν1 T ) n s
ρsat (n) = C exp −
kB T

The quantity
γmicro = w1 − ν1 T (7.36)

is the Helmholtz free energy per surface particle of the cluster, it includes both
energy and entropy contributions and depends on the temperature but not on the
cluster size. The size-dependence of the surface energy is contained in n s (n). By
analogy with the fluctuation theory γmicro can be termed a microscopic surface tension
per particle (the combination of the terms “microscopic” and “surface tension” is
purely terminological and should not cause confusion). It is convenient to introduce
the dimensionless quantity
γmicro
θmicro = (7.37)
kB T

which we term the “reduced microscopic surface tension”, being the Helmholtz
free energy per surface particle in kB T units. From (7.29), (7.35) and (7.37) the
configuration integral of the n-cluster takes the form
qn −n −θmicro n s (n)
= C z sat e (7.38)
V
This is the central result of the model. Substitution of (7.38) into (7.8) yields the
cluster distribution function in supersaturated vapor

ρ(n) = C enβ (μ
v −μ )
e−θmicro n
s (n)
sat , 1≤n<∞ (7.39)
7.4 Configuration Integral of a Cluster: Mean-Field Approximation 89

At saturation μv = μsat (T ) yielding

ρsat (n) = C e−θmicro n

s (n)
(7.40)

The unknown parameter C can be found from (7.11) and (7.40):

ρsat
v
C = ∞ , where h ≡ eθmicro (7.41)
−n s (n)
n=1 n h

Without going into details of the behavior of n s (n) for arbitrary n we make use of
the physically obvious fact that small clusters (with n ≤ N1 ) do not have the liquid
core implying that:
n s (n) = n, for n ≤ N1 (7.42)

At low temperatures we expect that

h 1 (7.43)

or equivalently:
θmicro (T ) > 2 (7.44)

The above constraint is the domain of validity of the model. From (7.41) to the
leading order in 1/ h
v θmicro
C = ρsat e (7.45)

Thus, the cluster distribution at saturation (7.40) reads:


v −θmicro n s (n)−1
ρsat (n) = ρsat e (7.46)

Recalling that C = 1/v0 we conclude from (7.45) that

v0 = vsat
v
e−θmicro (7.47)

v = 1/ρ v is the volume per molecule in the bulk vapor at saturation.

where vsat sat
Equation (7.47) is the manifestation of the fact that molecules in the cluster are
on average more densely packed than in the vapor at the same temperature. To
accomplish the theory it is necessary to present approximations for θmicro and n s (n).
Consider the vapor compressibility factor at saturation
psat
v
Z sat =
ρsat kB T
v
90 7 Mean-Field Kinetic Nucleation Theory

Using Eqs. (7.11), (7.10) and (7.40) it can be written as

∞ ∞ −n s
n=1 ρsat (n) n=1 h (n)
v
Z sat = ∞ = ∞
n=1 nρ sat (n) n=1 n h −n s (n)

Truncating both series at N1 and using (7.42) we have

 N1 −n
h h −N1 (−1 + h) (−1 + h N1 )
v
Z sat =  Nn=1 =−
1 −n −h + h −N1 (h − N1 + h N1 )
n=1 n h

Expanding the right-hand side in 1/ h, we obtain:

  N1   N1 +1
1 1 1
v
Z sat =1− + N1 +O
h h h
 
With the high degree of accuracy ∼O h −N1 we can set

1
v
Z sat =1− (7.48)
h
On the other hand,
v
Z sat = 1 + Z exc,sat
v
(7.49)

v
where the first term is the ideal gas part and Z exc,sat is the excess (over ideal) contri-
bution. Comparing (7.48) and (7.49) we find

1
h=− v (7.50)
Z exc,sat

This result shows that the microscopic surface tension originates from the nonideality
of the vapor and can be determined from a suitable equation of state; its simplest
form is the second order virial expansion [11]:

B2 psat
v
Z sat =1+ (7.51)
kB T
  
v
Z exc,sat

where B2 (T ) is the second virial coefficient. From (7.50) and (7.51)

(−B2 ) psat
θmicro = − ln (7.52)
kB T

This feature of the model makes it especially attractive for applications: in order to
find θmicro one does not need to solve the two-phase equilibrium equations but can
7.4 Configuration Integral of a Cluster: Mean-Field Approximation 91

Fig. 7.2 Vapor and liquid 1

compressibility factors at Zvsat
saturation. The vapor com-
v (T )
pressibility factor Z sat
decreases with T , while liquid
l (T )

Zsat
compressibility factor Z sat
increases with T . Both curves
meet at the critical point Tc
Zc

Z lsat

0 1
T/Tc

use the experimental or tabulated data on B2 (T ) and psat (T ) which are available for
a large amount of substances in a broad temperature range (see [11]). From (7.43)
and (7.52) the range of validity of the theory is given by:
 
 B2 (T ) psat (T ) 
  1 (7.53)
 kB T 

−8 ÷ 10−5 . At higher
exc,sat ≈ 10
v is close to unity, so that Z v
At low temperatures Z sat
v
temperatures Z sat decreases (see Fig. 7.2). If we formally apply (7.50) at the critical
point we would find
1
h(Tc ) =
1 − Zc

The critical compressibility factor lies in the limits Z c ≈ 0.2 ÷ 0.4 [11] indicating
that the constraint (7.43) is violated. For example, for van der Waals fluids Z c = 3/8
[12] yielding
8
h vdW
c =
5
The failure of the model at high temperatures manifests the fact that in this domain one
has to take into account intercluster interactions which are completely neglected in the
present model. Besides, in the close vicinity of Tc fluctuations become increasingly
important and the mean-field approach can be in error.

7.5 Structure of a Cluster: Core and Surface Particles

Equation (7.13) written for the average quantities is

n = n core + n s (7.54)
92 7 Mean-Field Kinetic Nucleation Theory

Fig. 7.3 Sketch of a cluster

surface
rl

core

n core core
R
l
R

l s
n

By definition the core possesses the liquid-like structure characterized by the coor-
dination number N1 in the liquid phase; N1 gives the average number of nearest
neighbors for a molecule in the bulk liquid. This quantity influences a number of
physical properties: density, viscosity, diffusivity, etc. The determination of N1 is a
nontrivial problem in its own right (it is discussed in Sect. 7.6). In the present section
we assume N1 to be known. By construction the clusters with n ≤ N1 do not have
the core—all their particles belong to the surface:

n s (n) = n, n core (n) = 0, for n ≤ N1 (7.55)

Consider now a cluster with n ≥ N1 + 1. It has core, which contains on average

n core particles and can be characterized by a radius R core and the number density
ρ l (we discuss spherical clusters for simplicity). Following Zhukhovitskii [9] we
characterize the surface layer, containing n s particles, by a thickness λ r l , λ ≥ 1
and the constant density ξρ l , lower than the bulk liquid density: ξ < 1; r l is the
average intermolecular distance in the bulk liquid (see Fig. 7.3). One can view the
surface molecules as an “adsorption layer for the core” separating it from the bulk
vapor surrounding the cluster. This analogy suggests that one can expect λ to vary
in a narrow range: 1 ≤ λ < 2, with the left boundary corresponding to a monolayer
and the right boundary—to a double layer of surface molecules. We stress that this
division is purely schematic and serves the purposes of the model.
Physically a cluster can be viewed as a density fluctuation in the vapor, characterized
by a smooth profile ρm (r ) asymptotically tending to the bulk vapor density at r → ∞.
It is convenient to define a cluster radius R at the location of the equimolar dividing
surface Re which by definition is characterized by the zeroth (physical) adsorp-
tion [13]. Outside Re The model construction illustrated in Fig. 7.3 replaces the
smooth profile ρm (r ) by a two-step function ρ(r ) with the width of the middle step
7.5 Structure of a Cluster: Core and Surface Particles 93

Fig. 7.4 Two-step density

profile ρ(r ) discussed in the (r)
n > N1
l
model. Also shown is the
true smooth thermodynamic
l
profile ρm (r ) (Reprinted with
permission from Ref. [6],
copyright (2006), American
Institute of Physics.) m(r)

core surface
v

0 Rcore Re
r

λr l = Re − R core ; outside Re : ρ(r ) = ρ v , as shown in Fig. 7.4. The core radius is

then
R core = Re − λr l , n ≥ N1 + 1

Recall that CNT replaces ρm (r ) by a single-step function

ρCNT (r ) = ρ l (1 − Ξ (r − Re )) + ρ v Ξ (r − Re )

where Ξ (x) is the Heaviside unit step function. Clearly, for large clusters the
relative width of the adsorption layer λr l /R core → 0 and both approaches become
asymptotically identical.
Within the two-step approximation Eq. (7.54) reads:

4π 4π
n = ρl (R core )3 + ξρ l [(R core + λr l )3 − (R core )3 ] (7.56)
 3    3  
n core ns

Let us introduce a dimensionless core radius

R core
X=
rl
From (3.23)
X 3 = n core (7.57)

It is convenient to use the pair of material parameters λ and ω ≡ ξ λ instead of the

pair (λ, ξ ). Then Eq. (7.56) reads:

X 3 = −3ωX 2 − 3ωλX + (n − ωλ2 ) (7.58)

This cubic equation for X (n) has the unique real positive root if n − ωλ2 > 0.
For n ≥ N1 + 1: n core ≥ 1, implying that X ≥ 1. The minimum value X = 1 is
94 7 Mean-Field Kinetic Nucleation Theory

achieved when
n = 1 + N1

In this case the core contains just one particle while the rest N1 particles belong to
the surface. Eq. (7.58) then results in the relation between ω and λ:

N1
ω=
3 + 3λ + λ2

Solving for λ we find: 

N1 3 3
λ= − − (7.59)
ω 4 2

Since λ > 1, ω should lie in the limits

0 < ω < N1 /7

The average number of surface particles is found from (7.57)

n s (n) = n − [X (n)]3

where X (n) is the solution of Eq. (7.58). It is convenient to introduce two dimen-
sionless quantities

ns X3
α= =1− and (7.60)
n n
ζ = n −1/3 (7.61)

Then
X = ζ −1 (1 − α)1/3

In these variables (7.58) becomes the equation for α(ζ ):

α = 3ωζ (1 − α)2/3 + 3ωλζ 2 (1 − α)1/3 + ζ 3 ωλ2 (7.62)

Equation (7.62) can be used to determine the parameter ω. To this end let us consider
the behavior of the model for large clusters: n → ∞. In this case almost all cluster
particles belong to the core:

n core ns
→ 1, → 0 for n → ∞
n n
The problem has two small parameters:

0<α 1, 0<ζ 1
7.5 Structure of a Cluster: Core and Surface Particles 95

Keeping in Eq. (7.62) the first order terms in α and ζ , we obtain:

α = 3ωζ

Then from (7.60)–(7.61)

n s = 3ω n 2/3 , n→∞ (7.63)

This result comes as no surprise, since at large n the droplet is a compact spherical
object, its radius scales as n 1/3 and the number of surface atoms scales as the surface
area ∼n 2/3 . From (7.46) to the same order in n

ρsat (n) ∼ exp[− 3ω θmicro n 2/3 ], n→∞ (7.64)

On the other hand, for big clusters the CNT description is valid:

ρsat (n) ∼ exp[− θ∞ n 2/3 ], n→∞ (7.65)

Comparing (7.64) and (7.65) we find

1 θ∞
ω= (7.66)
3 θmicro

7.6 Coordination Number in the Liquid Phase

The remaining unknown parameter of the model—the coordination number in the

liquid phase N1 —can be derived from X-ray diffraction experiments. In these exper-
iments one measures the static structure factor whose Fourier transform gives the
pair correlation function g(r ; ρ, T ) [12]. N1 can be obtained by integration of the
area under the first peak of g(r ). However, there is an ambiguity in the way this
integration is carried out, which sometimes results in a substantial discrepancy in
the values of N1 . Moreover, an error in numerical integration may lead to the values
N1 > 12 which is impossible since N1 = 12 corresponds to the closed packing
η = 0.74 (face-centered cubic structure). Typically for various liquids close to the
melting point: N1 (ρ, T ) = 4 ÷ 8.
Cahoon [14] proposed a simple alternative method for calculation of N1 for a liquid.
It is based on the observation that in a solid there is one-to-one correspondence
between the coordination number and the packing fraction

π d3
η≡ (7.67)
6va
96 7 Mean-Field Kinetic Nucleation Theory

Table 7.1 Coordination

Unit cell N1 va η
numbers and volume √
fractions for different cubic Diamond cubic 4 8d 3 /3 3 0.34009
crystal structures Simple cubic 6 d3 √ 0.52360
Body-centered cubic 8 4d 3 /3
√ 3 0.68018
Face-centered cubic 12 d3/ 2 0.74048

characterizing a given crystal structure—face-centered cubic, body-centered cubic,

simple cubic, diamond cubic. In (7.67) d is the atomic diameter and va is the atomic
specific volume. The relation between N1 and η in a solid is given in Table 7.1.
The main idea of Ref. [14] is that for any pure isotropic liquid (or amorphous) material
the function relating N1 to η will be similar to that for the isotropic crystal solid with
the exception that noninteger values are permissible. Using Table 7.1 the dependence
N1 (η) for 4 ≤ N1 ≤ 8 can be well approximated by

N1 = 5.5116η2 + 6.1383η + 1.275 (7.68)

For the case of a liquid the atomic volume va should be replaced by 1/ρ l and the
atomic diameter d should be replaced by the position dg of the first peak of the pair
correlation function g(r ; ρ l , T ). Thus,
π l 3
η= ρ dg (7.69)
6
If the diffraction data is not available one can find dg using the ideas of perturbation
approach in the theory of liquids applying the Weeks–Chandler–Anderson decom-
position scheme of the interaction potential given by Eqs. (5.20), (5.21) and shown in
Fig. 5.2. Within the WCA dg is approximated by the effective hard-sphere diameter
 rm  
3
dhs =3 1 − e−βu 0 (r ) r 2 dr (7.70)
0

where u 0 (r ) is the WCA reference potential (5.20).4

7.7 Steady State Nucleation Rate

Combining the results of the previous sections the steady state nucleation rate is
 ∞
−1

−H (n)
J = K0 e (7.71)
n=1

4 Equation (7.70) is the mean-field approximation to the original WCA expression, where the cavity
function of the hard sphere system is set to unity (for details see e.g. [12] Chap. 5).
7.7 Steady State Nucleation Rate 97

where psat s1
K 0 = ρsat
v
f 1,sat S, f 1,sat = √ (7.72)
2π m 1 kB T

and
2 
H (n) = ln n + n ln S − θmicro n s (n) − 1 (7.73)
3

Here θmicro is given by Eq. (7.52); n s (n) = n for n ≤ N1 , and

n s (n) = n − [X (n)]3 , for n ≥ N1 + 1 (7.74)

X (n) is the real positive root of Eq. (7.58) in which the parameters λ and ω are found
from 
1 θ∞ N1 3 3
ω= , λ= − − (7.75)
3 θmicro ω 4 2

The coordination number N1 is expressed in terms of the molecular packing fraction

in the liquid phase
η = (π/6) ρ l dhs
3
(7.76)

(where dhs (T ) is the effective hard sphere diameter in the theory of liquids) by means
of Eq. (7.68). We refer to this model as a Mean-field Kinetic Nucleation Theory
(MKNT) [6].
It is easy to see that −H (n) is the free energy of the cluster formation in kB T units

−H (n) = βΔG(n)

Apart from the small logarithmic corrections (which can safely be set to a constant
2
3 ln n c , n c being the critical cluster) the free energy reads

βΔG(n) = −n ln S + θmicro n s (n) − 1 (7.77)

At small n the surface part of ΔG(n) is

βΔG surf (n) = θmicro (n − 1), n ≤ N1 (7.78)

yielding ΔG surf (n = 1) = 0. Equation (7.78) implies that the limiting consistency

(cf. Sect. 3.6) is an intrinsic property of MKNT. By virtue of the kinetic approach
MKNT satisfies also the law of mass action.
Let us define a critical cluster as the one that makes the major contribution to the
series in (7.71). The latter corresponds to the minimum of H (n), or equivalently—to
the maximum of ΔG(n). A close inspection of Eq. (7.77) shows that the func-
tion ΔG(n) has two maxima. For small n the Gibbs energy is an increasing linear
function of n
98 7 Mean-Field Kinetic Nucleation Theory

βΔG(n) = (θmicro − ln S) n − θmicro , n ≤ N1

Expression in the round brackets is positive: the supersaturation can not exceed some
maximum value given by the pseudospinodal corresponding to the nucleation barrier
≈ kB T . From the pseudospinodal condition, discussed in Chap. 9, it follows that

ln S < θmicro

Hence, ΔG(N1 − 1) < ΔG(N1 ). Due to the model construction when the clus-
ter size is increased from N1 to N1 + 1, the number of surface particles does not
change

n s (N1 ) = n s (N1 + 1) = N1

leading to ΔG(N1 ) > ΔG(N1 + 1). Thus, ΔG(n) has a maximum at n = N1 which
is an artifact of the model and has to be ignored. The second maximum corresponds
to the critical cluster n c :

dΔG dn s
ΔG ≡ = − ln S + θmicro =0 (7.79)
dn dn

Expanding ΔG(n) to the second order around n c

1
ΔG(n) ≈ ΔG ∗ + ΔG (n c ) (n − n c )2 , ΔG ∗ ≡ ΔG(n c )
2
we have:
∞
  ∞ ∗
βΔG(n) βΔG ∗ 1 eβΔG
e ≈e dx exp βΔG (n c ) x 2 =
−∞ 2 Z
n=1

where 
−βΔG (n c )
Z =
2π

is the Zeldovich factor. The steady-state nucleation rate reads:

∗
J = J0 e−βΔG (7.80)

where
J0 = Z ρsat
v
f (n c ) (7.81)

2/3
is the kinetic prefactor, f (n c ) = f 1,sat S n c , and

βΔG ∗ = −n c ln S + θmicro [n s (n c ) − 1] (7.82)

7.7 Steady State Nucleation Rate 99

is the nucleation barrier. Apparently, the notion of a critical cluster is a convenient

concept but not a necessity: Eqs. (7.81)–(7.82) are the approximation to the exact
result (7.71)–(7.73).

7.8 Comparison with Experiment

7.8.1 Water

In Chap. 3 we discussed nucleation of water vapor comparing predictions of CNT

with various experimental data available in the literature [15–17]—see Fig. 3.4. Now
we can supplement this comparison by adding the predictions of MKNT for the same
experimental conditions. Figure 7.5 shows the relative nucleation rate

Jrel = Jexp /Jth

with the closed symbols referring to Jth = JMKNT and open symbols referring
to Jth = JCNT . Circles (open and closed) correspond to the experiment of Wölk

Water

4
C
N
T
log 10(Jexp/Jth)

MKNT

Open symbols: CNT

Closed symbols: MKNT

-2
200 220 240 260
T(K)

Fig. 7.5 Relative nucleation rate Jrel = Jexp /Jth for water; Log(Jrel ) ≡ log10 Jrel . Closed symbols:
MKNT, open symbols: CNT. Circles: experiment of Wölk et al. [15]; squares experiment of Labetski
et al. [17]. The lines labelled ‘CNT’ and ‘MKNT’, shown to guide the eye, illustrate the temperature
dependence of the relative nucleation rate for the CNT and MKNT, respectively. Also shown is the
“ideal line” (dashed): Jexp = Jth
100 7 Mean-Field Kinetic Nucleation Theory

Fig. 7.6 Nucleation rate for 3

water: theory (MKNT, CNT) 320 310 300 290
versus experiment of Brus
et al. [18]. Solid lines: MKNT,
dashed lines: CNT; closed 2

log10J (cm-3s-1)
symbols: experiment [18].
Labels: nucleation tempera-
ture in K; horizontal labels 1
refer to the theory, inclined
(italicized) labels refer to
experiment. The CNT line
0

320
for 300 K is almost coincid-

310

290
300
ing with the MKNT line for
310 K; and the CNT line for H2O/He
310 K is almost coinciding -1
with the MKNT line for 320 K 3 3.5 4 4.5
S

et al. [15]; squares (open and closed)—to the experiment of Peeters et al. [16] and
Labetski et al. [17]. The thermodynamic data for water used in both models are
given in Appendix A. In the whole temperature range the MKNT predictions are
1–2 orders of magnitude off the experimental data, while the CNT demonstrates
much larger deviation. The most important observation, however, is that MKNT
predicts the temperature dependence of the nucleation rate correctly. Meanwhile,
the discrepancy between CNT and experiment depends on the temperature: at low T
CNT underestimates the experimental data, reaching about 4 orders of magnitude at
the lowest temperature T = 201 K, while at high T CNT slightly overestimates the
experiment. At 200 < T < 220 K critical clusters, corresponding to experimental
conditions, contain ≈15–20 molecules; for such small objects the dominant role in
the cluster formation is played by the microscopic (rather than the macroscopic)
surface tension which explains the success of MKNT.
Brus et al. [18] measured water nucleation in helium in the thermal diffusion cloud
chamber. It is important to note that the measurements were performed for temper-
atures T = 290, 300, 310, 320 K which are beyond the freezing point implying
that all macroscopic properties of water are well known from experiment. Figure 7.6
shows the experimental J − S curves together with MKNT and CNT predictions.
At these relatively high temperatures CNT systematically overestimates experiment
(in qualitative agreement with the previously shown results) by 2–4 orders of magni-
tude. MKNT predictions are in perfect agreement with experiment (within one order
of magnitude). There is a regular temperature shift of CNT curves with respect to
experiment by about 10◦ K; as a result they practically overlap the MKNT curves
related to the nucleation temperatures which are 10◦ K higher. In particular, the 290 K
CNT line overlaps the 300 K MKNT line; the 300 K CNT line overlaps the 310 K
MKNT line, etc.
7.8 Comparison with Experiment 101

Fig. 7.7 Relative nucleation

rate log10 Jrel for nitrogen. Jexp/JMKNT
Closed circles: MKNT, open Jexp/JCNT
20
diamonds: CNT. Experiment:
[19]; Jexp = 7 ± 2 cm−3 s−1 .
Also shown is the “ideal line”

log (J rel)
(dashed): Jexp = Jth
10

0
Nitrogen

1.8 1.9 2 2.1 2.2 2.3 2.4

100 T-1 (K-1)

7.8.2 Nitrogen

Figure 7.7 shows the comparison of experimental nucleation rate Jexp of Ref. [19]
with predictions of the CNT JCNT and MKNT JMKNT . Thermodynamic data used
in the analysis is given in Appendix A. As in Fig. 7.5 the relative nucleation rate is:
Jrel = Jexp /Jth with Jth = JCNT or JMKNT .
The dashed line corresponds to the “ideal case”: Jexp = Jth . As one can see, MKNT
predictions for J deviate on average from the experimental data by 2–7 orders of
magnitude while the CNT predictions are 10–20 orders of magnitude lower then the
experimental values.

7.8.3 Mercury

Vapor–liquid nucleation in mercury is a somewhat extreme example of nucleation

studies requiring very high supersaturations. The reason for that is that the surface
tension of mercury is more than ten times higher than that of molecular fluids,
implying that a high supersaturation is necessary to compensate for the energy cost
to build the cluster surface. An important feature of mercury, typical for fluid metals, is
that its electronic structure strongly depends on the thermodynamic state of the system
[20] implying that the interatomic interaction also depends on the thermodynamic
state. Mercury vapor is a simple rare-gas system with only van der Waals dispersive
interactions. Being combined in clusters, mercury atoms can behave differently: in
small clusters interactions between them are purely dispersive (as in the vapor);
however, beyond a certain cluster size the energy gap becomes smaller than kB T
resulting in the nonmetal-to-metal transition. This transition was estimated to appear
at the cluster size n ≈ 70 according [21], n ≈ 80 according to [22], n ≈ 135
102 7 Mean-Field Kinetic Nucleation Theory

Fig. 7.8 Critical supersatura-

tion S (for J = 1 cm−3 s−1 ) 20
CN
Hg
as a function of nucleation T J=1 cm -3s -1
temperature. Closed circles: 18
experiment of [25]; solid
line: MKNT, dashed line: 16
CNT (Reprinted with permis-
sion from Ref. [6], copyright 14

ln S
(2006), American Institute of
Physics.) 12 MK
NT

Expt
10

6
260 280 300 320
T [K]

according [23]. Recently the cluster size dependent interaction potential of mercury
was proposed by Moyano et al. [24].
Experimental study of mercury nucleation in helium as a carrier gas was carried out
by Martens et al. [25]. The measurements were made in an upward diffusion cloud
chamber. This technique makes it possible to detect the onset of nucleation rather
than to measure directly the nucleation rates (as e.g. in shock wave tube experiments).
The onset corresponds to the nucleation rate

Jexp = 1 cm−3 s−1 (7.83)

The supersaturation giving rise to the onset of nucleation is referred to as the critical
supersaturation. Figure 7.8 shows the experimental results (closed circles) for the
critical supersaturation S as a function of the nucleation temperature. Also shown
are predictions of the MKNT (solid line), CNT (dashed line). The thermodynamic
data for mercury is presented in Appendix A. As stated in [25] the measured values
of S are about 3 orders of magnitude lower than the CNT predictions; the MKNT
results are in good agreement with experiment.
Recalling how sensitive the nucleation rate is to the value of S it is instructive to
illustrate the difference between the two models in terms of J . For that purpose we
choose an experimental point T = 284 K, ln S = 9.35 and compare experimental
and theoretical results corresponding to these conditions. Experimental rate is given
by the onset condition (7.83) while theoretical predictions are:

JCNT (T = 284 K, ln S = 9.35) = 4.7 × 10−67 cm−3 s−1

JMKNT (T = 284 K, ln S = 9.35) = 6.1 × 10−3 cm−3 s−1
7.8 Comparison with Experiment 103

The CNT predictions deviate from experiment by about 67 orders of magnitude (!)
while MKNT predictions are within 3 orders of magnitude.5

7.9 Discussion

7.9.1 Classification of Nucleation Regimes

The general MKNT result for the nucleation rate can be written as
⎡ ⎤−1
⎢N ⎥
⎢ 1 Nclass ∞ ⎥
⎢ ⎥
J = K0 ⎢ e−H (n) + e−H (n) + e−H (n) ⎥ (7.84)
⎢ ⎥
⎣n=1   n=N1 n=Nclass ⎦
     
small intermediate large

where we divided all clusters into 3 groups: (i) small, containing 1 ≤ n ≤ N1

particles; (ii) intermediate with N1 ≤ n ≤ Nclass ; and (iii) large, containing n ≥
Nclass particles. Using considerations of Sect. 7.5 we can estimate Nclass from the
constraint
1
1/3
∼ (1 ÷ 1.5) × 10−1
Nclass

yielding Nclass  (3 ÷ 10) × 102 .

For large clusters the distribution function can be approximated by the classical form
yielding:
∞
 ∞
 1
e−H (n) ≈ (7.85)
S n e−θ∞ n
2/3
n=Nclass n=Nclass
n 2/3
  
large

If the critical cluster falls into this domain the contribution of the other two groups
is negligible and we recover the kinetic CNT result (3.69). This case can be called
a macroscopic nucleation regime; here the surface part of the free energy is entirely
determined by the macroscopic surface tension θ∞ .

5 Note, that the nucleation rate is very sensitive to the surface tension: if γ∞ is measured within
the 10 % relative accuracy (common for most of the liquid metals), the accuracy of the predicted
nucleation rate for mercury lies within 4 orders of magnitude.
104 7 Mean-Field Kinetic Nucleation Theory

For small clusters MKNT gives:


N1 
N1
1
e−H (n) = (7.86)
n 2/3 S n e−θmicro (n−1)
n=1 n=1
  
small

The free energy is determined solely by the microscopic surface tension while the
macroscopic surface tension does not play a role.
In majority of nucleation experiments the critical cluster contains ∼10 − 102 mole-
cules, falling into the domain of intermediate clusters where the influence of both
micro- and macroscopic surface tension is important. This intermediate nucleation
regime represents a challenging problem for a nucleation theory. MKNT solves it
by providing a smooth interpolation between the two limits—of large and small
clusters—for which it becomes “exact” by construction. Such interpolation is pos-
sible because MKNT is not based on a perturbation in the cluster curvature making
it possible to describe all nucleation regimes within one model.

7.9.2 Microscopic Surface Tension: Universal Behavior

for Lennard-Jones Systems

The microscopic surface tension of a substance is determined solely by the vapor

v (cf. Eq. (7.50)). Consider the substances
compressibility factor at coexistence Z sat
for which the intermolecular interaction potential u(r ), where r is the separation
between the molecules, has a two-parametric form,
r
u(r ) = ε Φ
σ
Here ε describes the depth of the interaction and σ describes the molecular size. An
example of such substances are Lennard-Jones fluids characterized by the potential

r −12 r −6
u LJ (r ) = 4ε −
σ σ

Scaling the distance with σ and the energy with ε we introduce the dimensionless
quantities
kB T pσ 3
tLJ = , ρLJ = ρσ 3 , pLJ =
ε ε
The compressibility factor then reads
p p
Z= = LJ
ρkB T ρLJ tLJ
7.9 Discussion 105

At coexistence v
pLJ,
v
Z sat = sat

ρLJ,
v t
sat LJ

is a universal function of tLJ which is a manifestation of the law of corresponding

states. This implies using (7.37) and (7.50) that
 
γmicro = −ε [tLJ ln −Z exc,sat
v
] (7.87)

The expression in the square brackets is also a universal function of tLJ . Introducing
the reduced temperature t = T /Tc we present tLJ as

tLJ = tLJ ,c t (7.88)

where the critical temperature for Lennard-Jones fluids tLJ,c can be determined from
Monte Carlo simulations (see e.g. [26]): tLJ,c = kB Tc /ε ≈ 1.34. Dividing both sides
of Eq. (7.87) by kB Tc , we find
γmicro
= Ψ (t)
kB Tc

where Ψ (t) is again a universal function. From Eq. (7.36) we expect that at low
temperatures this function is linear
   
γmicro w1 ν1
= − t (7.89)
kB Tc kB Tc kB

implying that for Lennard-Jones fluids at low temperatures the dimensionless surface
entropy per particle, ν/kB , and the surface energy per particle, w1 /kB Tc , are universal
parameters.
This conjecture can be verified using the mean-field density functional calculations
described in Chap. 5. Bulk equilibrium follows Eqs. (5.30)–(5.31) with

μ = μd (ρ) − 2ρa (7.90)

p = pd (ρ) − ρ a 2
(7.91)

Here the hard-sphere pressure, pd , and chemical potential, μd , follow the Carnahan-
Starling theory [27] and the background interaction parameter for a Lennard-Jones
fluids is given by Eq. (5.34): √
16π 2 3
a= εσ (7.92)
9
Figure 7.9 shows the results of the DFT calculations (the dotted line). Indeed, to the
high degree of accuracy the temperature dependence of γmicro turns out to be linear,
106 7 Mean-Field Kinetic Nucleation Theory

4
DFT
micro = w1 - 1T

3.5

3 Ar

/kBTc
N2
CH
2.5 4

micro
2
DFT for LJ fluids:
DF
T
1.5 w1/kBTc=3.62
1/kB=2.90

1
0.2 0.3 0.4 0.5 0.6 0.7 0.8
t=T/Tc

Fig. 7.9 Microscopic surface tension γmicro for Lennard-Jones systems. Dotted line: DFT cal-
culation for a Lennard-Jones system; solid lines: MKNT results for argon (Ar), nitrogen (N2 ) and
methane (C H4 ) obtained using Eq. (7.52) and empirical fit for the second virial coefficient [11]. The
MKNT curves for argon and methane are practically indistinguishable (Reprinted with permission
from Ref. [6], copyright (2006), American Institute of Physics.)

supporting the conjecture (7.89), and

w1
≈ 3.62 (7.93)
kB Tc
ν1
≈ 2.90 (7.94)
kB

In Fig. 7.9 we compare the DFT result with predictions of MKNT for various simple
fluids—argon, nitrogen, methane— using Eq. (7.52) and empirical fit to the second
virial coefficient for nonpolar substances [11] (see Eq. (F.2)). All MKNT curves are
close to the DFT line (the curves for argon and methane are practically indistin-
guishable) confirming the validity of the MKNT conjecture for γmicro (T ) and the
numerical values of its parameters (7.93)–(7.94).

7.9.3 Tolman’s Correction and Beyond

In order to treat small clusters within the classical approach, some nucleation models
replace γ∞ in the CNT free energy of cluster formation by the curvature dependent
form, representing the expansion of the surface tension of a cluster in powers of its
curvature. To discuss the range of validity of such an expansion we study the general
Eq. (7.62), valid for all cluster sizes.
7.9 Discussion 107

For large clusters we solve the cubic equation (7.62) for α(ζ ) keeping the terms up
to the 3rd order in ζ , which results in
   2 
λ λ
α = 3ωζ 1 − 2 ω − ζ+ − 3λω + 3ω ζ 2 , ζ → 0
2
(7.95)
2 3

Multiplying (7.95) by n and using (7.60)–(7.61) we obtain

   
λ −1/3 λ2
ns = 3ωn 2/3
1−2 ω− n + − 3λω + 3ω2 n −2/3 (7.96)
2 3

To the same order of accuracy the cluster distribution function at saturation reads:
 
γn A(n)
ρsat (n) = ρsat
v
exp −
kB T

where A(n) = s1 n 2/3 and

   
λ −1/3 λ2
γn = γ∞ 1−2 ω− n + − 3λω + 3ω2 n −2/3 , n → ∞
2 3
(7.97)
is the curvature dependent surface tension of the cluster. Its radius is given by the
radius of the equimolar surface Re . Within the framework of the Gibbs thermody-
namics γn = γ [Re ] ≡ γe is the surface tension measured at the equimolar surface
(see Sect. 2.2.2). Using the relationship Re = r l n 1/3 we rewrite (7.97) as
⎡   ⎤
λ2
2 r l ω − λ2 (r l )2 3 − 3λω + 3ω2
γe = γ∞ ⎣1 − + ⎦, large Re (7.98)
Re Re2

Recall that for sufficiently large droplets the surface tension at the surface of tension,
γt = γ [Rt ], can be expressed using the Tolman formula (2.41)
 
2δT
γ t = γ∞ 1− + ... (7.99)
Rt

In view of the Ono-Kondo equation (2.35) the surface tension is at minimum at Rt .

Hence, to the first order in δT Eq. (7.98) reads:
   
2 r l ω − λ2
γt = γ∞ 1− + ... (7.100)
Rt
108 7 Mean-Field Kinetic Nucleation Theory

Comparing (7.100) with (7.99) we identify the Tolman length

 
λ
δT = r ω −
l
(7.101)
2

Since both ω and λ are of order 1, δT is of the order of molecular size. It is tempting
to apply the Tolman equation (7.100) truncated at the first order term for relatively
small n-clusters once the Tolman correction is small
  
 
2 ω − λ  n −1/3 1 (7.102)
 2 

This condition, however, is not sufficient. By definition (see Eq. (2.37)) the Tolman
length is independent of the radius; meanwhile, for sufficiently small droplets
δ = Re − Rt is a strong function of Rt [28, 29]. Koga et al. [30] obtained δ(Rt )
for simple fluids (Lennard-Jones and Yukawa) using the DFT. They showed that
for these systems the Tolman equation becomes valid for clusters containing more
than 105 ÷ 106 molecules. This result implies that in contrast with conventional
expectation one can not apply the Tolman equation down to droplets containing few
tens or hundreds of molecules. For the nucleation theory it means that even in the
macroscopic nucleation regime one can use Tolman’s correction only for extremely
large clusters, containing n ∼ 105 ÷ 106 Nclass molecules which are practically
irrelevant for experimental conditions (critical clusters contain usually ∼101 ÷ 102
molecules).
MKNT allows to find an independent criterion of the applicability of the Tolman
equation to nucleation problems by analyzing the next-to-Tolman term in the curva-
ture expansion (7.97). The following condition should be satisfied:
 
 λ2
− 3λω + 3ω2 
 3  −1/3
   n 1 (7.103)
 2 ω − λ2 
 

Combining (7.102) and (7.103) we conclude that the Tolman correction is applicable
for the clusters satisfying
⎧  ⎫
⎨   
λ  

λ2
3 − 3λω + 3ω2 ⎬

n 1/3 max 2 ω − ,    (7.104)
⎩ 2   2 ω − λ2 ⎭


For typical experimental conditions the second requirement (Eq. (7.103)) is much
stronger than the first one (Eq. (7.102)). For the cases studied in Sect. 7.8 the Tolman
Eq. (7.100) is valid for clusters containing ∼(4 ÷ 5) × 104 ÷ 105 particles which is
close to the result of Ref. [30] for simple fluids. Note, that although δT is not used
7.9 Discussion 109

15 -0.2
Water

-0.4
, m icro 10 m icro

(A)
T

T
m icro/kBTc,

-0.6

m icro/kBTc

-0.8

0
0.3 0.4 0.5 0.6 0.7 0.8

t=T/Tc

Fig. 7.10 Equilibrium properties of water used in MKNT: θ∞ , θmicro , γmicro /kB Tc (left y-axis)
and the Tolman length δT [Å] according to Eq. (7.101) (right y-axis) (Reprinted with permission
from Ref. [6], copyright (2006), American Institute of Physics.)

in MKNT, its behavior is important for understanding the asymptotic properties of

big clusters.
The temperature dependence of various equilibrium properties of water and nitrogen,
used in MKNT analysis, is shown in Figs. 7.10 and 7.11. The temperature range is
limited from above by the MKNT validity criterion (7.43). For both substances
θmicro is lower than its macroscopic counterpart θ∞ for all T , however the difference
between them decreases with the temperature. The microscopic surface tension γmicro
is approximately linear in t supporting the MKNT assumption. The dashed line in
Fig. 7.11, labelled “LJ”, is the universal form of γmicro /kB Tc for Lennard-Jones fluids
given by (7.93)–(7.94).
For water the Tolman length demonstrates a rather unusual behavior. For all temper-
atures δT is negative; at low T it decreases reaching a weak minimum at T ≈ 308 K.
For higher temperatures δT slowly increases reaching a maximum at T ≈ 394 K
and then decreases again. For nitrogen at experimental temperatures δT is positive
and close to zero: 0.04 < δT < 0.06 Å. At high enough temperatures δT becomes
negative and monotonously decreases. Such a behavior suggests the divergence of δT
near Tc . However, the definitive conclusion and the corresponding critical exponent
can not be drawn from the present model since the critical region is beyond its range
of validity.
110 7 Mean-Field Kinetic Nucleation Theory

20
T Nitrogen
0

15

m icro
-0.2
,

(A)
10

T
m icro/kBTc,

m icro
-0.4

5
LJ m icro/kBTc
-0.6

0
0.3 0.4 0.5 0.6 0.7 0.8

t=T/Tc

Fig. 7.11 MKNT: equilibrium properties of nitrogen.: θ∞ , θmicro , γmicro /kB Tc (left y-axis) and
the Tolman length δT [Å] according to Eq. (7.101) (right y-axis). The dashed line labelled “LJ”
shows γmicro /kB Tc for Lennard-Jones fluids according to Eq. (7.89) with the universal parameters
given by (7.93)–(7.93) (for details see the text)

7.9.4 Small Nucleating Clusters as Virtual Chains

In the limit of small (nano-sized) clusters the behavior of the system (the free energy
barrier and the distribution function) as a function of the cluster size n will be essen-
tially different from that given by the phenomenological CNT. It is instructive to
consider the physical picture emerging from the MKNT in the case of small n. For
n ≤ N1 the equilibrium cluster distribution is
v −θmicro (n−1)
ρsat (n) = ρsat e , 1 ≤ n ≤ N1 (7.105)

Using the relation (7.12) we find for the configuration integral of an n-cluster:

   Λ3 e−βμsat n−1
qn = ρsat
v
V e−θmicro (n−1) e−nβμsat Λ3n = ρsat
v
Λ3 e−βμsat V
h
(7.106)

The chemical potential (not close to Tc ) can be approximated by

μsat = kB T ln(ρsat
v
Λ3 )
7.9 Discussion 111

Fig. 7.12 Virtual chain

cluster

which after substitution into (7.106) results in:

qn = V K n−1 , 1 ≤ n ≤ N1 (7.107)

where
1
K = v eθmicro (7.108)
ρsat

The factorized form (7.107) of the configuration integral suggests that the n-cluster
in this case is characterized by short range nearest neighbor interactions between
particles and contains n − 1 bonds, each bond contributing the same quantity K
to qn . This is the minimum possible number of bonds for an n-cluster (a spherical
droplet represents the opposite limit of maximum number of bonds). The latter means
that a small cluster is not a compact object but rather reminds a polymer chain of
atoms with nearest neighbor interactions [31]. Each particle of such cluster is bonded
to the two neighboring particles belonging to the same chain with exception of the
end-point particles having one neighbor. The simplest example of such a cluster is a
linear chain of molecules. A somewhat more complex cluster structure satisfying Eq.
(7.107) can have branch points where a particle of the given chain has contacts with
particles belonging to another chain; however, loops are prohibited (see Fig. 7.12).
This structure was studied in Ref. [32] where it was termed a “system of virtual
chains” indicating that the sequence of atoms in the chains is not fixed: they are
associating and dissociating.
The value of K depends on the interatomic potential u(r ) and temperature. For
the nearest neighbor pairwise additive interaction we obtain from the definition
of qn :
q2 n−1
qn = V , 1 ≤ n ≤ N1 (7.109)
V
Comparing (7.107) and (7.109) we identify

q2 1
K = = dr e−βu(r ) (7.110)
V 2 cl
112 7 Mean-Field Kinetic Nucleation Theory

From (7.109) and (7.12) it is seen that K represents the dimer association constant:

ρsat (2)
K = (7.111)
[ρsat (1)]2

Hence the microscopic surface tension can be related to the dimer association
constant:  v 
θmicro = − ln ρsat K

References

1. A. Dillmann, G.E.A. Meier, J. Chem. Phys. 94, 3872 (1991)

2. I.J. Ford, A. Laaksonen, M. Kulmala, J. Chem. Phys. 99, 764 (1993)
3. C.F. Delale, G.E.A. Meier, J. Chem. Phys. 98, 9850 (1993)
4. V.I. Kalikmanov, M.E.H. van Dongen, J. Chem. Phys. 103, 4250 (1995)
5. M.E. Fisher, Physics 3, 255 (1967)
6. V.I. Kalikmanov, J. Chem. Phys. 124, 124505 (2006)
7. F.H. Stillinger, J. Chem. Phys. 38, 1486 (1963)
8. J.K. Lee, J.A. Baker, F.F. Abraham, J. Chem. Phys. 58, 3166 (1973)
9. D.I. Zhukhovitskii, J. Chem. Phys. 101, 5076 (1994)
10. J.J. Binney, N.J. Dowrick, A.J. Fisher, M.E.J. Newman, The Theory of Critical Phenomena
(Clarendon Press, Oxford, 1995)
11. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, 4th edn. (McGraw-
Hill, New York, 1987)
12. V.I. Kalikmanov, Statistical Physics of Fluids. Basic Concepts and Applications(Springer,
Berlin, 2001)
13. J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity (Clarendon Press, Oxford, 1982)
14. J.R. Cahoon, Can. J. Phys. 82, 291 (2004)
15. J. Wölk, R. Strey, J. Phys. Chem. B 105, 11683 (2001)
16. P. Peeters, J.J.H. Giels, M.E.H. van Dongen, J. Chem. Phys. 117, 5467 (2002)
17. D.G. Labetski, V. Holten, M.E.H. van Dongen, J. Chem. Phys. 120, 6314 (2004)
18. D. Brus, V. Ždimal, J. Smolik, J. Chem. Phys. 129, 174501 (2008)
19. K. Iland, Ph.D. Thesis, University of Cologne, 2004
20. F. Hensel, Phil. Trans. R. Soc. London A 356, 97 (1998)
21. K. Rademann, Z. Phys, D 19, 161 (1991)
22. P.P. Singh, Phys. Rev. B 49, 4954 (1994)
23. G.M. Pastor, P. Stampfli, K.H. Benemann, Phys. Scr. 38, 623 (1988)
24. G. Moyano et al., Phys. Rev. Lett. 89, 103401 (2002)
25. J. Martens, H. Uchtmann, F. Hensel, J. Phys. Chem. 91, 2489 (1987)
26. D. Frenkel, B. Smit, Understanding Molecular Simulaton (Academic Press, London, 1996)
27. N.F. Carnahan, K.E. Starling, J. Chem. Phys. 51, 635 (1969)
28. A.H. Falls, L.E. Scriven, H.T. Davis, J. Chem. Phys. 75, 3986 (1981)
29. V. Talanquer, D.W. Oxtoby, J. Chem. Phys. 99, 2865 (1995)
30. K. Koga, X.C. Zeng, A.K. Schekin, J. Chem. Phys. 109, 4063 (1998)
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32. D.I. Zhukhovitskii, J. Chem. Phys. 110, 7770 (1999)
Chapter 8
Computer Simulation of Nucleation

8.1 Introduction

Besides the development of analytical theories describing the nucleation process,

molecular simulation is a powerful tool for investigating nucleation on a microscopic
level. In the best case analytical solutions of theoretical approaches require various
approximations to simplify the problem. Often approximations have to be made with
respect to the molecular details of a substance. It is not always obvious how such
approximations affect the performance of the theory. On the other hand, it is very
difficult to get such information on the molecular-level processes from nucleation
experiments. In this context molecular simulation is a technique that complements the
theoretical and experimental methods. To a certain extent molecular simulation may
be regarded as a computer experiment. Interactions between atoms and molecules
are mapped on a potential model that determines intermolecular forces. One may
distinguish two simulation techniques: molecular dynamics (MD) simulation and
molecular Monte Carlo (MC) methods.
MD simulations represent numerical integration of classical equations of motion for
the system of interacting particles (atoms, molecules, etc.) while tracking the system
in time. Taking the time averages of the physical quantities of interest during its
evolution and relying on the ergodicity hypothesis, one can state that these averages
are equivalent to the ensemble averages of these quantities in the micro-canonical
(NVE) ensemble.
In Monte Carlo simulations the physical time is not present: the motion of a molecule
does not correspond to a real trajectory of a molecule; instead, one simulates the
behavior of the system in the configurational space. Therefore, the results of MC
simulations are ensemble averages. MC allows moves that lead fast towards the
equilibrium state making it possible to overcome high energy barriers; this would
not be possible to achieve within a reasonable computational time using MD methods.
Hence, both approaches and their further developments have advantages that can
be employed in a complementary way to address molecular level investigations of

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 113

DOI: 10.1007/978-90-481-3643-8_8, © Springer Science+Business Media Dordrecht 2013
114 8 Computer Simulation of Nucleation

nucleation processes. From the above mentioned general observations it is clear

that MD simulations have the advantage of gaining insights into the kinetics of the
nucleation process, while MC typically gives information on the thermodynamics
of the process. In this chapter we introduce molecular simulation methods that are
relevant for modelling of the nucleation process and discuss their application to
various nucleation problems.

8.2 Molecular Dynamics Simulation

8.2.1 Basic Concepts and Techniques

Classical molecular dynamics simulation is based on the numerical integration of

Newton’s equations of motion. A principal scheme of MD can be summarized as
follows:
1. One introduces a potential model (interaction potential) describing interactions
between particles (atoms or molecules) in the system.
2. The net force acting on a specific particle by all other particles is given by the nega-
tive gradient of the intermolecular potential with respect to the particle separation.
3. From the Newton law one calculates the acceleration caused by the force acting
on the particle.
4. Finally, particles are moved according to the direction and magnitude of the
resulting force using a numerical technique.
In the context of MD simulations it is useful to generalize Newton’s dynamics by
Lagrange dynamics. In classical mechanics, the equations of motion are contained
in the fundamental Lagrange equation [1]
   
d ∂L ∂L
− = 0, k = 1, 2, . . . , s (8.1)
dt ∂ q˙k ∂qk

where s is the number of degrees of freedom of the system containing N particles;

the “upper dot” denotes the time derivative. The Lagrangian L of the system is
the function of the generalized coordinates q = (q1 , q2 , . . . , qN ) and their time
derivatives q̇ defined in terms of the kinetic energy Ekin and potential energy U:

L (q, q̇) = Ekin − U.

For a system of N molecules the kinetic energy in Cartesian coordinates reads


N
p2i
Ekin = . (8.2)
2 mi
i=1
8.2 Molecular Dynamics Simulation 115

where mi is the mass of a particle i and pi is its momentum. The generalized momenta
pk are given by
∂L
pk = , k = 1, 2, . . . , s
∂ q̇k

The potential energy may be divided into terms which depend on external potentials,
particle interactions of pairs, triplets etc.:


N 
N 
N 
N 
N 
N
U= u1 (ri ) + u2 (ri , rj ) + u3 (ri , rj , rk ) + . . . (8.3)
i=1 i=1 j>i i=1 j>i k>j>i

The first term in Eq. (8.3) contains the external potential u1 acting on the individual
molecules. The second term contains the potential u2 of the interaction between any
pair of particles in the system. We could continue to include the interactions between
triplets, quadruplets etc. The second term, however, is typically the most important
one. Neglecting any external fields, the first term in Eq. (8.3) equals zero. In addition,
the contributions from triplet or higher order interactions are typically much smaller
and are often neglected, such as for the widely used Lennard-Jones potential. Thus,
the total potential of the system reduces to a good accuracy to the sum of all pairwise
interactions:


N 
N
U= u2 (ri , rj ) (8.4)
i=1 j>i

The second summation j > i provides the exclusion of double counting. If we insert
the expressions for potential and kinetic energy into Eq. (8.1), the Lagrange equation
reduces to
mi r̈i = fi , (8.5)

which is the Newton second law in terms of the force fi acting on particle i. For time
and velocity-independent interaction potentials, the force is given by

fi = ∇ri L = −∇ri U. (8.6)

Within Lagrangian dynamics equations of motion can be solved while conserving

the energy and momentum.
Application of Lagrangian dynamics to the simple problem of one particle with a
mass m interacting with the environment by a potential U(x) gives:

1 2
L= mv − U(x)
2
116 8 Computer Simulation of Nucleation

where v is the velocity of the particle. The Lagrange equation (8.1) now reads:
   
d ∂L ∂L
=
dt ∂ ẋ ∂x
     
mv − dU
dx

On the left-hand side the derivative of the momentum mv with respect to the time
t yields ma, where a is the acceleration of the particle, while the right-hand side is
equal to the force f . Lagrangian dynamics does not only recover the Newton law
but can also be applied to more complicated problems. For example it is useful for
the proper derivation of the equations of motion for a system including a thermostat.
In this case the Lagrange equation has to be extended by a term representing the
energy of the thermostat. The application of the Lagrange differential equation to
this approach gives the desired proper equation of motion for the NVT ensemble.
Besides Lagrangian dynamics one can also describe the evolution of the system using
the Hamiltonian dynamics. Let us introduce the Hamiltonian of the system [1]

H (p, q) = q̇k pk − L (q, q̇)
k

With its help the equations of motion can be written in the Hamiltonian form:
∂H
q̇k = , (8.7)
∂pk
∂H
ṗk = − . (8.8)
∂qk

For a time- and velocity-independent interaction potential the Hamiltonian of the

system simply reduces to its total energy, which in this case must be conserved:

H(p, q) = Ekin (p) + U(q).

Finally, we can write down the Hamiltonian equations (8.7)–(8.8) in Cartesian form:

ṙi = pi /m, (8.9)

ṗi = −∇ri U = fi . (8.10)

Computing of the trajectories in the phase space involves solving either a system of
3N second-order differential equation (8.5) or an equivalent set of 6N first-order dif-
ferential equations (8.9)–(8.10). The classical equations of motion are deterministic
and invariant to time reversal. This means that if we change the sign of the velocities,
the particles will trace back on exactly the same trajectories. In computer simula-
tions, exact reversibility usually is not observed because of the limited accuracy of
the numerical calculations and the chaotic behavior of the dynamics of a many-body
system.
8.2 Molecular Dynamics Simulation 117

The “natural” ensemble for MD simulation is the constant energy (micro-canonical)

NVE ensemble. For this ensemble the equations of motion can be solved numerically
by calculating the motion of the molecules caused by the forces in small time steps Δt.
How small these time steps have to be, depends on the type of interaction between
the molecules. For a very steep potential a very short time step is required while
for a rather flat potential the time step may be larger. A typical time step for MD
simulations is typically in the order of a femto-second (10−15 s). In order to calculate
the position of a molecule at the time t + Δt one expands the position of the molecule
in a Taylor series around the present position at time t:

dx 1 d2 x 1 d3 x 3
x(t + Δt) = x(t) + Δt + 2
Δt 2
+ 3
Δt + O(Δt 4 )
dt
 2 dt
 6 dt
v(t) a(t)

From Newton’s law we can calculate the acceleration a at the moment t for the given
force f (t):
f (t)
a(t) =
m
yielding

f (t) 2 1 d3 x 3
x(t + Δt) = x(t) + v(t) Δt + Δt + Δt + O(Δt 4 ) (8.11)
2m 6 dt 3
In order to solve the above equation, we need to know the velocity at time t. However,
it can be eliminated by expanding the position of the molecules backward in time.
Replacing in (8.11) Δt by −Δt, we obtain

f (t) 2 1 d3 x 3
x(t − Δt) = x(t) − v(t) Δt + Δt − Δt + O(Δt 4 ) (8.12)
2m 6 dt 3
Adding expansions (8.11) and (8.12), the odd terms cancel resulting in

f (t) 2
x(t + Δt) = 2x(t) − x(t − Δt) + Δt + O(Δt 4 ) (8.13)
2m
This way of numerically integrating the equations of motion is called the Verlet
algorithm [2]. Though it is a rather simple approach, its error is just of the order
of O(Δt 4 ). For calculation of positions of the particles the velocity is not needed.
Meanwhile, it is required for calculation of the kinetic energy of the system, instan-
taneous temperature, transport properties, to mention just a few. One can calculate v
by subtracting the expansions (8.11) and (8.12):

x(t + Δt) − x(t − Δt)

v(t) = + O(Δt 3 ) (8.14)
2Δt
118 8 Computer Simulation of Nucleation

Thus, velocity at time t is calculated after the positions at time (t + Δt) and (t − Δt)
have been already found. The positions in Eq. (8.13) are exact up to errors of the
order of O(Δt 4 ), while the velocities are exact up to an error of the order of O(Δt 3 ).
The Verlet algorithm is properly centered on the respective positions at (t ± Δt) and,
thus, is time reversible. However, the difference between the accuracies in positions
and velocities may result in deviations from the classical trajectories. Consequently,
most Verlet-type algorithms exhibit a drift in the energy of the system on short and
long time-scales, which is strongly influenced by the length of the time-step Δt.
Since calculation of the velocities in the original Verlet method contains a compar-
atively large error, a more accurate form, known as the velocity-Verlet algorithm,
was proposed [3]:

1
x(t + Δt) = x(t) + Δt v(t) + f (t) Δt 2 (8.15)
2m
Δt
v(t + Δt) = v(t) + f (t) + f (t + Δt) (8.16)
2m
Calculation using the velocity-Verlet algorithm typically proceeds through the two
steps: first, the new positions x(t + Δt) and the velocities are calculated at an inter-
mediate time interval:
 
1 1
v t + Δt = v(t) + f (t) Δt
2 2m

Then, the acceleration at t + Δt is calculated and the velocities are updated:

 
1 1
v(t + Δt) = v t + Δt + f (t + Δt) Δt
2 2m

The original Verlet algorithm may be recovered by eliminating the velocity from
Eq. (8.16). In the velocity-Verlet method, the new velocities are obtained with the
same accuracy and at the same time as the positions and accelerations of the mole-
cules. Therefore, the kinetic and potential energy are known at each time. The
velocity-Verlet algorithm requires a minimum storage memory and its numerical
stability, time reversibility, and simplicity in both form and implementation make it
by far the most preferred method in MD simulations to date.
A number of other algorithms for solving the equations of motion have been devel-
oped. For example, the leap-frog algorithm improves the accuracy by using half-steps
between the time steps. Other methods are based on higher order terms of the Taylor
expansions. In general, the more accurate the algorithm—the larger the time step
can be used. For a complete survey of various algorithms the reader is referred to the
literature [4, 5].
8.2 Molecular Dynamics Simulation 119

8.2.2 System Size

Imagine that we would like to simulate a macroscopic system containing 1 mol of

gas. The number of molecules in this system, given by the Avogadro number, is
of the order of 1023 . Molecular simulation of such an immensely large number of
particles is impossible to accomplish because of the limitations of computer power.
Although computer power rises continuously it will not be possible to simulate a
macroscopic system for a very long time, if ever. Therefore it is necessary to reduce
the size of the simulation system. Usually molecular simulations are performed with
N ∼ 103 − 104 particles (atoms or molecules). The required computational effort
depends on the kind of interaction potential between the atoms and molecules. For
example, short range van der Waals forces can be evaluated rapidly because not all
interactions have to be taken into account but only those between molecules separated
by a short distance. On the other hand, electrostatic interactions are long range and
require long range corrections leading to considerable computational efforts.
In order to simulate a molecular system one has to cut out a small piece of the complete
macroscopic system. The question is whether such a small piece actually represents
the total system properly. An obvious problem is that a small system has a large
surface-to-volume ratio. If we take a system of 1,000 molecules in a cubic box, we
can easily estimate that roughly half of the molecules will be in the surface region of
the box. As a result, simulation of such a system will give results strongly influenced
by the surface effects rather than describing the bulk properties. These difficulties can
be overcome by implementing periodic boundary conditions. The idea of periodic
boundary conditions is that the simulated cubic box is replicated indefinitely in each
direction, thus forming and infinite lattice. Any molecule moving in the simulation
volume has an infinite number of copies in the neighboring boxes, which move
in exactly the same way. If a molecule leaves the system, a copy of it enters the
simulation volume from the opposite side. The box itself appears without any walls
and the number of molecules in it is conserved. This construction is illustrated in
Fig. 8.1. Note, that it is not necessary (and indeed impossible) to store the coordinates
and velocities of the particles and all of their copies in neighboring systems. For
calculation of the force the closest copy of a molecule is considered, which is called
the minimum image convention.
This approach has been proven to yield accurate results for many properties of the
system. However, one has to keep in mind that periodic boundary conditions generate
a system with periodicity. Whenever a property or an effect is short range, i.e. its
correlation length is smaller than half the box length, it can be calculated using
periodic boundary conditions. On the other hand there are long range phenomena that
may not fit into a given simulation box. A famous example is the density fluctuations
in a substance close to the critical point. In the critical region the density fluctuations
rise beyond the box dimensions. If periodic boundary conditions are applied, these
long range fluctuations are cut off leading to the mean-field behavior. The latter yields
critical exponents that deviate from the correct ones. Other simulation techniques,
such as finite size scaling [6], allow the treatment of such a problem.
120 8 Computer Simulation of Nucleation

Fig. 8.1 Periodic boundary

conditions. The central box is
the simulation system while
the other boxes and all their
molecules are replicated in all
spatial directions

Fig. 8.2 Kinetic and potential

energy during the collision
of two Lennard-Jones argon
atoms in a constant energy
ensemble (NVE)

Finite size effects, i.e. the influence of the system size on the simulated properties,
are also important for calculation of the surface tension [7]. Surface fluctuations of
a vapor–liquid interface lead to the so-called capillary waves with the wavelength
related to the surface tension. Besides the wavelength, also the amplitude of the
capillary waves varies with the system size leading to a widening of the interface
with increasing system size.

8.2.3 Thermostating Techniques

Nucleation is in principle a non-isothermal process. When vapor condenses its poten-

tial energy is lowered because intermolecular separations become smaller approach-
ing the minimum of the interaction potential energy (corresponding to maximum
attraction). In a closed adiabatic system the loss in potential energy has to be com-
pensated by a gain in kinetic energy. In Fig. 8.2 this exchange of energy is depicted for
a two-atom collision. If a cluster grows the exchange of energy is similar, although,
due to many-body processes, the curves look more chaotic. Thus, in adiabatic systems
8.2 Molecular Dynamics Simulation 121

the process of condensation is accompanied by heating of the system. Therefore, to

stabilize the embryo of the new phase this latent heat has to be removed. There are
several approaches to perform MD simulations at constant temperature. In the sim-
plest case one can make use of the relation between the velocities of the molecules
and the instantaneous temperature T of the system. The mean kinetic energy per
molecule of the system reads

1 
N
Eat,kin  = mi vi2
2N
i=1

The equipartition theorem of statistical mechanics states that each degree of freedom
contributes kB T /2 to the total kinetic energy of the system (see e.g. [8]) yielding

Eat,kin  = Nf kB T /2 (8.17)

where Nf is the number of degrees of freedom of a particle; for a particle with trans-
lational motion in 3D Nf = 3. In general, after one (or several) steps of the dynamics,
the instantaneous temperature T will be different from the desired temperature Tset .
By rescaling the velocities of the molecules one can set the system to the temperature
Tset :
Tset
vi,new = vi
T

While such velocity scaling approach is suitable in some cases and for some prop-
erties, especially in equilibrium, in the context of nucleation this method of ther-
mostating will give wrong results. Rescaling the velocities is actually a method to
keep the kinetic energy of a particle constant while all temperature fluctuations are
completely eliminated. At the same time if one wants to simulate the system in the
canonical NVT ensemble, one has to realize that this ensemble exhibits thermal fluc-
tuations. These fluctuations are the origin of the heat capacity. Thus, the velocity
scaling method is by construction unable to predict the heat capacity. Furthermore,
in nucleating systems the velocity scaling can lead to an artificial cooling down of
the remaining monomers in the vapor phase [9].
Several thermostating techniques were proposed which are able to correctly realize
the canonical ensemble. One example is the stochastic Andersen thermostat [10].
Instead of rescaling the velocities of all molecules in every time step, one or a few
molecules are randomly chosen from the vapor and their new velocities are calculated
from the Maxwell distribution function at the desired temperature. In this way, the
Andersen thermostat mimics a collision of a molecule with a carrier gas particle. The
frequency at which a molecule is picked from the vapor, determines the effectiveness
of this method. In the limit of very high frequencies one recovers a procedure similar
to velocity scaling but with a certain temperature distribution. On the other hand, a
low frequency may not be sufficient to keep the temperature constant.
122 8 Computer Simulation of Nucleation

Another widely used method is the Nosé-Hoover thermostat [11, 12]. It includes
the thermostat in the equations of motion. Adding an energy term, which describes
the thermostat, to the Lagrange (or Hamilton) function and applying the Lagrangian
(or Hamiltonian) dynamics leads to equations of motion including a friction term,
that affects the acceleration of the molecules. The Nosé-Hoover thermostat correctly
realizes the canonical ensemble allowing for fluctuations in the system temperature.
If the thermostat itself is coupled to another thermostat, one obtains a so-called Nosé-
Hoover chain thermostat [13], which represents an improved thermostat realizing
the canonical NVT ensemble.
In physical experiments the exchange of energy between the system and the environ-
ment is usually provided by a carrier gas. Such carrier gas usually does not affect the
phase transition beyond its function as a latent heat transfer agent. It is also possible
to include a carrier gas in MD simulations. To keep the computational effort low, it
is useful to employ for this purpose a mono-atomic gas, such as the Lennard-Jones
argon. It is added into the simulation box, typically in abundance. When a cluster
is formed by nucleation, the latent heat is removed from the cluster by collisions of
the cluster particles with the carrier gas atoms. In real physical experiment the heat
is removed from the carrier gas either by expanding the system or by collisions with
the container walls. In MD simulations it is possible to simulate the expansion [14]
directly or model the heat removal from the carrier gas by coupling it to one of the
regular MD thermostats as described above. This is possible because the carrier gas
remains in the gas phase and does not condense. Special care should be taken in the
cases in which the carrier gas is present in the interior of the cluster or is adsorbed at
the cluster surface. It should be checked whether the application of a MD-thermostat
to the carrier affects the simulation results.
Westergren et al. [15] analyzed several effects on the heat exchange between a cluster
and the carrier gas in MD simulation. They found an increasing energy transfer with
rising the atomic mass of the noble gas (acting as a carrier gas). By using different
forms of cluster-gas interaction potentials they also found that soft interactions are
more efficient in the heat transfer from the cluster to the gas. While Westergren et
al. employed “real” noble gas atoms, each having a distinct set of parameters of the
interaction potential and the atomic mass, one can also use a pseudo-noble gas, having
the same interaction parameters but different atomic masses [16]. This allows one to
separate the effect of the atomic mass from the effect caused by different interaction
parameters of the different noble gases. It is useful for the fundamental analysis of the
heat exchange to optimize the heat transfer in a simulation, however one should be
careful to directly draw conclusions for experimental systems. Figure 8.3 shows the
effect of the atomic mass, given in the diagrams in atomic units, on the cooling of the
largest cluster in the simulation system during a nucleation simulation. Temperature
8.2 Molecular Dynamics Simulation 123

Fig. 8.3 Heat exchange of

the zinc cluster with car-
rier gases having differ-
ent atomic masses but the
same interaction parameters.
(Reprinted with permission
from Ref. [16], copyright
(2009), American Institute of
Physics.)

Fig. 8.4 Temperature devel-

opment of the zinc subsystem
as a function of time for dif-
ferent amount of carrier gas
argon. The numbers above
the plots are the ratio Zn:Ar
atoms. The horizontal line
gives the temperature of the
argon subsystem

jumps in these graphs are related to cluster-cluster collisions. In principle one can
identify two effects:

• The heavy atoms are slower than the light ones which means that the heavy atoms
move a smaller distance and, hence, less likely collide with a cluster than the light
ones. Hence, the heat exchange should become less efficient.
• On the other hand a collision with a heavy atoms itself is more efficient, i.e. more
heat is transferred.

As Fig. 8.3 shows, the effect of the mass on the velocity dominates, because the lighter
the carrier gas the more heat is removed from the clusters. In order to steer the
heat transfer during particle formation processes, one can also vary the amount of
the carrier gas. The more carrier gas is present, the more heat can be removed from the
forming clusters. Figure 8.4 shows the effect of the amount of carrier gas (Ar) on the
temperature of the zinc subsystem, which includes all zinc clusters in the simulation
box [17]. The horizontal line gives the temperature of the argon subsystem. The labels
are the ratio of Zn to Ar atoms in the system. The higher this ratio, the faster is the
convergence of the zinc temperature to the carrier gas temperature.
124 8 Computer Simulation of Nucleation

8.2.4 Expansion Simulation

In physical experiments supersaturation is often initiated by the temperature drop

caused by a pressure drop. This is the case for an expansion chamber experiment as
well as for an expansion in a nozzle. In simulations it is more difficult to actually
mimic the expansion using an expanding simulation. It is computationally much less
costly to set the carrier gas by a thermostat to a low temperature. Still in some cases
the expansion of the simulation system is inevitable. An example is the simulation of
a process called a “rapid expansion of a supercritical solution” (RESS) [18]. In RESS
a solute is dissolved in a supercritical solvent which is then expanded in the nozzle.
In this process the supersaturation is achieved by two effects: first the solubility
of the supercritical solvent decreases drastically with decreasing density during the
expansion, and secondly the expansion leads to a temperature drop which further
rises the supersaturation.
To mimic the expansion, a simulation box filled with the supercritical solution is set
up at a pressure and temperature significantly above the critical values of the solvent.
In this initial state a simulation run is performed until the system reaches equilibrium.
In case of RESS simulation equilibration runs are necessary to make sure that the
solute does not precipitate at the initial supercritical conditions. In the next step the
box size is increased in a small step followed by a constant energy simulation (NVE).
The enlargement step and the NVE simulation steps are repeated several times until
the vapor phase density is reached. In has been found that an optimum with respect
to computational effort and the accuracy of the simulation can be achieved by around
100 expansion steps [14].
In Fig. 8.5 the expansion path of such simulation for carbon dioxide is plotted in
all three coordinates: temperature, pressure and density. Comparison with a highly
accurate Span-Wagner equation of state [19] shows that the expansion path of the

(a) (b)

Fig. 8.5 Comparison of the path during an expansion simulation for pure CO2 with the Span-
Wagner EoS (reference equation)[19]. In addition the expansion path of the CO2 /naphthalene solu-
tion is plotted. (Reprinted with permission from Ref. [14], copyright (2009) American Chemical
Society.)
8.2 Molecular Dynamics Simulation 125

simulation system agrees very well with the adiabatic expansion calculated with the
Span-Wagner EoS in all three coordinates. In both figures also the expansion path of
a dilute naphthalene solution in carbon dioxide is plotted. There is no such accurate
reference equation for this mixture but the expansion path is very close to that of
pure carbon dioxide. The small shift is related to the small amount of naphthalene
in the solution that affects the properties through its molecular interactions.

8.3 Molecular Monte Carlo Simulation

While in MD the actual equations of motion are numerically integrated, Monte Carlo
simulations sample the configurational space of the system. To be more specific, let
us consider a canonical (NVT ) ensemble of interacting particles (molecules). The
potential energy of a given configuration rN ≡ (r1 , . . . , rN ) is U(rN ). The average
value of an arbitrary function of coordinates X(rN ) is given by the integral

X(r ) =
N
X(rN ) w(rN ) drN (8.18)

where w(rN ) is the Boltzmann probability density function of a given state rN :

1 −β U(rN )
w(rN ) = e (8.19)
QN

and 
e−β U(r
N)
QN = drN (8.20)

is the configuration integral

 ofNthe system. Obviously, the function w(rN ) is positive
and normalized to unity: w(r ) dr = 1. From the standpoint of probability theory
N

Eq. (8.18) defines the mathematical expectation of X(rN ). Imagine that we have a
digital camera that can instantaneously take photos of the system so that we can
use this camera to scan and memorize the 3D coordinates of all N molecules in the
volume V . We can repeat these actions M times per second. Then, the computer
memory will contain the set of coordinates (rN )1 , (rN )2 , . . . , (rN )M , where M is a
number of configurations. The average observed value of X


M
AVRG(X) = (1/M) X[(rN )k ] (8.21)
k=1

gives an estimate of the true (exact) value X(rN ) given by (8.18), which we are
actually not able to calculate. The mean-square deviation
126 8 Computer Simulation of Nucleation


M
σ 2 = (1/M) X 2 [(rN )k ] − [AVRG(X)]2 (8.22)
k=1

characterizes the accuracy of our statistical averaging. In MC the integral (8.18) is

approximated by [20]
 
σ
X(rN ) ≈ AVRG(X) 1 ± √ (8.23)
M

Note, that σ becomes independent of the number of observations for large M, imply-
ing that the error of approximation (8.23) is inversely proportional to the square root
of the number of observations, which is typical for mathematical statistics.
MC simulation are based on the random generation of atom coordinates in a
simulation box. In view of the large number 3N of space coordinates of the
N-particle system it is clear, that calculations employing solely random choice of
these coordinates is a hopeless task. The introduction of the so-called importance
sampling by Metropolis et al. [21] allowed the sampling of the system with sufficient
accuracy within reasonable simulation time. As indicated by the term “importance
sampling”, not all states of the system are sampled with equal probability, but pre-
dominantly those that bring significant contribution to the configuration integral. The
criterion indicating the importance (statistical weight) of a state is its Boltzmann fac-
tor e−β U(r ) . Clearly, a very large positive U results is a very low statistical weight
N

of the state. Generating such term is hence a waist of time and should be avoided. In
molecular simulation the energy of the system can become very high if two atoms
overlap sufficiently. The repulsive part of the interaction potential is usually very
steep yielding high energies resulting in low probabilities of such states. Based on
these observations, instead of generating all coordinates of all atoms each time at
random, one can modify only those configurations that are already very likely. To do
so, one picks an atom and moves it at random throughout the simulation box. Then
the change of the configurational energy ΔU due to this (virtual) move is computed.
If the move leads to overlapping of atoms, it is clear that the energy would become
very high and the move should be rejected with high probability. On the other hand,
if the move lowers the energy of the system, it should be accepted. The resulting
Metropolis algorithm [21] is hence given by the following set of instructions:

if ΔU < 0
accept move
else
q = exp(−ΔU/kB T )
x = Random[0,1]
if x < q
accept move
else
reject move
8.3 Molecular Monte Carlo Simulation 127

endif
endif
Here Random [0,1] means a random number at the interval [0,1].
This is the core of the Metropolis scheme but, of course, a lot of additional numerical
procedures are needed to set up such an MC simulation. For example, the parameters
of the random atom move should be optimized during the simulation in order to
increase the efficiency. Periodic boundary conditions are required as in MD simula-
tions. A possible cutoff of the potential has to be corrected for. A difference between
MD and MC is that MC does not require calculation of the forces. Furthermore, the
“natural” ensemble for MC is the canonical NVT ensemble, (while in MD it is the
micro-canonical NVE ensemble). In order to switch to another ensemble in MC, one
has to modify the simulation procedure in connection with a modified Boltzmann
factor. For example, simulation of the NpT ensemble requires modification of the
simulation box at random and an additional term pV in the Boltzmann factor.
An advantage of MC compared to MD simulation is that in MC it is not necessary
to follow a real trajectory of an atom. The movement of an atom is in principle
random and hence it is possible to overcome high energy barriers that would be
impossible to overcome in MD simulations. For example, in MD an atom would not
be able to pass through two atoms which are close to each other, because that would
require a significant overlap. In MC a jump on the other side of the two atoms is
possible. However, even Metropolis sampling is in a number of cases not sufficient
to overcome high energy barriers most efficiently. A further improvement of MC
for such cases is the so-called umbrella sampling introduced by Torrie and Valleau
[22]. In this method the Boltzmann factor in the acceptance criterion is extended by
a weighting function w1 (rN ). The resulting transition probability is then

w1 (rN )e−βU(r
N)

π(rN ) =  (8.24)
w1 (sN )e−βU(s ) dsN
N

The weighting function w1 (rN ) > 0, normalized to unity, is chosen in a way provid-
ing the reduction of the energy barrier which the original system has to overcome.
In order to calculate the desired properties in the canonical ensemble, it is necessary
to eliminate afterwards the effect of the weighting function. The thermodynamic
average X of a quantity X is then found from the relationship

 wX1 w1
X =
 w11 w1

where  w1 denotes averaging over the probability distribution w1 .

128 8 Computer Simulation of Nucleation

8.4 Cluster Definitions and Detection Methods

One of the fundamental questions in nucleation is the definition of a cluster. While it

intuitively may appear as a simple question, its physical definition turns out to be a
difficult task. In general a cluster is defined as a condensed phase, i.e. liquid or solid
but due to the large surface to volume ratio one cannot simply apply the criteria for
a condensed bulk phase. First of all, the relatively large contribution of the surface
compared to the bulk phase affects the properties of the cluster. This is partly related
to a difference in the atomic structure of the cluster and the bulk phase. For the
definition of a cluster and the development of numerical methods for its detection it
is especially important to decide which atoms belong to a cluster and which do not.
Here various criteria are possible.
Within Stillinger’s definition [23] a molecule belongs to the cluster if a separation
between one of its atoms and at least one of the atoms of the cluster is smaller
than a certain bonding distance rb . The latter is typically in the range between one
and two atomic diameters σ . A very common value is rb = 1.5 σ . This definition is
appealing, because the corresponding cluster detection algorithm is relatively simple
to implement. On the other hand, there may exist certain drawbacks, depending on
the system under study. For example, in the case of a liquid cluster in a metastable
equilibrium with a relatively dense vapor phase, there are many vapor-phase atoms
surrounding the cluster and forming a corona. The use of the Stillinger criterion can
lead to inclusion of many of these atoms into the cluster even though they are not
physically bonded. Also atoms passing by the cluster are counted to the cluster for
a short period of time.
It is possible to exclude the atoms that pass by the cluster or those, which remain
a part of the cluster during a very short time. This can be done by supplementing
the Stillinger criterion with a live-time criterion [14]. The idea of this method is the
following: (i) for each connection between two atoms the duration of this contact is
determined; (ii) only those atoms are considered to belong to the cluster for which
the separation remains smaller than rb during a certain time, called the live-time.
To suggest a characteristic value of the live-time, one should calculate the time
required for the atom
• to enter the Stillinger sphere of radius rb ,
• collide with other atoms of the cluster and
• leave the Stillinger sphere again.
The knowledge of the average atomic velocity for the given temperature makes it
possible to calculate the time required for a regular collision. A typical value for the
collision time defined in this way is in the range of 1–2 ps.
Besides solely geometric criteria one can also formulate an energy-based criterion of
a cluster. The two competing energies here are the kinetic energy of molecules, related
to the motion and separation of the molecules, and the potential energy keeping the
atoms together. If the relative kinetic energy is smaller than the potential energy, the
8.4 Cluster Definitions and Detection Methods 129

Fig. 8.6 Schematic representation of the tWF cluster detection method

(a) (b)
Fig. 8.7 Detection of clusters using the tWF- and Stillinger definitions. a Argon: there are 4
Stillinger clusters, within each of them dark spheres comprise a tWF-cluster; for example, the 5/22
cluster contains 22 particles according to Stillinger criterion from which only 5 comprise a tWF-
cluster. (Reprinted with permission from Ref. [26], copyright (2007), American Institute of Physics).
b Zinc: there are 2 Stillinger clusters, within each of them dark spheres comprise a tWF-cluster
(Reprinted with permission from Ref. [17], copyright (2007), American Institute of Physics.)

atoms are likely to be physically bonded in a cluster [24]. Further developments of

this approach have been proposed by several authors. Harris and Ford, for example,
[25] proposed a cluster criterion that includes the subsequent dynamics to judge
whether a molecules belongs to a cluster or not.
Another approach focuses on the number of nearest neighbors of an atom, N1 , called
also the coordination number, discussed in Chap. 7). Depending on the state con-
ditions, N1 in the liquid phase lies in the range N1 ≈ 4 ∼ 8 atoms. Based on this
knowledge, ten Wolde and Frenkel (tWF) [27] defined a cluster as a set of atoms
having at least 5 nearest neighbors shown in Fig. 8.6. In the analysis of a simulation
system an algorithm, similar to the calculation of a pair correlation function, can be
used. This approach eliminates the corona of the cluster and is, thus, useful to detect
clusters of loosely bonded atoms or molecules. An example is the argon system:
argon atoms are not strongly bonded at the liquid state due to weak van der Waals
130 8 Computer Simulation of Nucleation

interactions. In this case the nearest-neighbor definition of a cluster is suitable for a

proper analysis of the simulation data. The snapshots referring to argon nucleation
depicted in Fig. 8.7a clearly show that the tWF definition of a cluster is more suit-
able. However, in case of a strongly bonded system, such as a metallic system, the
tWF criterion leads to an underestimation of the cluster size. In metallic systems the
atoms are strongly bonded and a corona does not exist as in the case of zinc shown
in Fig. 8.7b. If the nearest-neighbor criterion is applied to such a system, the surface
shell is removed from the cluster simply because the surface atoms do not necessarily
have the required number of nearest neighbors [14]. In summary, the proper cluster
definition depends on the specific system, as well on the properties calculated from
the simulation system.

8.5 Evaluation of the Nucleation Rate

One of the most important properties, which can be obtained from molecular sim-
ulation of nucleation, is the nucleation rate. It is defined as the number of clusters
formed per unit time and unit volume, which continue to grow to the stable bulk
phase. In experiments the droplets or particles are usually counted by optical or scat-
tering methods. These methods require droplets which after the nucleation stage have
grown to a relatively large size, comparable to the wavelength of a laser beam. To
obtain a reliable nucleation rate estimate one has to perform the experiment under
conditions at which coagulation can be avoided because that would change the num-
ber of droplets. This can for example be accomplished with a low droplet density. In
molecular simulation the determination of the nucleation rate depends strongly on
the chosen simulation method.

8.5.1 Nucleation Barrier from MC Simulations

The key quantity determining the nucleation behavior of a substance is the free energy
of an n-cluster formation, ΔG(n). Various theoretical models, discussed in this book,
invoke various approximations to derive this quantity; the most widely used one—
is the capillarity approximation of the classical nucleation theory. Calculation of
the free energy of cluster formation in Monte Carlo simulations, pioneered by Lee
et al. [28], is based on the analysis of cluster statistics, emerging in simulations,
without referring to a particular model for ΔG(n). Below we follow the procedure
outlined by Reiss and Bowles [29]. N molecules of the NVT -system can be grouped
in various clusters. We will consider the system configuration containing exactly Nn
clusters with n particles. Each n-cluster generates and exclusion volume vn which
is unaccessible for other N − n molecules of the system. Using the assumption of
non-interacting clusters (which is usually a good approximation for vapor–liquid
nucleation), we present the partition function of the NVT -system as
8.5 Evaluation of the Nucleation Rate 131

Z(N, V , T ) = Z(N − nNn , V − vn Nn , T ) Z (n) (8.25)

where Z is the partition function of the vapor which is forbidden to contain n-

clusters—all such clusters are contained in the partition function Z (n) of the gas of
n-clusters, given by Eq. (7.4):
1 Nn
Z (n) = Z (8.26)
Nn ! n

Here Zn (n, V , T ) is the partition function of one n-cluster in the volume V (see
Eq.(7.2)). Note, that Zn depends on the size of the system through the translational
degree of freedom of the center of mass of the cluster: Zn (n, V , T ) ∼ V /Λ3 . From
(8.25)–(8.26) using Stirling’s formula we have

ln Z = Nn ln Zn − Nn ln(Nn /e) + ln Z(N − nNn , V − vn Nn , T )

For each cluster size n a variety of Nn is possible; we will be interested in the most
probable value. The latter maximizes ln Z with respect to Nn :
     
∂ ln Z Zn ∂ ln Z
= ln + =0 (8.27)
∂Nn N,V Nn ∂Nn
N,V

Consider the second term of this equation

     
∂ ln Z ∂ ln Z ∂ ln Z
= −n − vn
∂Nn ∂(N − Nn ) ∂(V − vn Nn )
N,V N,V ,V −vn Nn N,V ,N−nNn

Using the standard thermodynamic relationships we write this expression as

 
∂ ln Z n μv pv vn
= − (8.28)
∂Nn kB T kB T
N,V

where μv and pv are the chemical potential and pressure of the vapor of volume
V − vn Nn containing N − nNn molecules. In general, μv = μv and pv = pv (μv is
the chemical potential of a molecule in the supersaturated vapor and pv is the vapor
pressure), however for rare clusters (recall the assumption of noninteracting clusters)
the difference between the barred and non-barred quantities is negligible. Therefore,
Eqs. (8.27) and (8.28) yield:

Nn = Zn (n, V , T ) e−β(p vn −nμv )

v
(8.29)

The ratio Zn (n, V , T )/V does not depend on V and remains constant in thermody-
namic limit. This means, that if we chose another volume of the system V , we would
have
132 8 Computer Simulation of Nucleation

Zn (n, V , T )/V = Zn (n, V , T )/V

It is convenient to choose V = V /N = vv , where vv is the volume per molecule in

the vapor. Then, Eq. (8.29) takes the form

Nn = N Zn (n, vv , T ) e−β(p vn −nμv )

v
(8.30)

Introducing the Helmholtz free energy of the n-cluster, confined to the volume vv

Fn (n, vv , T ) = −kB T ln Zn (n, vv , T )

we define the cluster size probability

Nn  
P(n) ≡ = exp −β (Fn (n, vv , T ) + pv vn ) − n μv (8.31)
N
The expression in the square brackets is

(Fn + pv vn ) − n μv = G(n) − G(n)bulk

where G(n)bulk = n μv is the Gibbs free energy of n molecules in the bulk super-
saturated vapor prior to the formation of the cluster; G(n) is the same quantity after
the n-cluster was formed. Hence, ΔG(n) = G(n) − G(n)bulk is the “intensive Gibbs
free energy” of n-cluster formation and

P(n) = e−βΔG(n)

From these considerations we can schematically interpret the process of cluster for-
mation as consisting of two steps [29]:
1. n molecules are picked up anywhere in volume V of the system and gathered in
the volume equal to the molecular volume in the vapor phase vv ;
2. within the volume vv the cluster is formed with the volume vn < vv .
Thus, measuring the cluster-size probability distribution P(n) in MC simulation, we
can determine the free energy of cluster formation ΔG(n) from the relationship:

βΔG(n) = − ln P(n) (8.32)

Its maximum gives the anticipated nucleation barrier ΔG∗ . This barrier can be com-
pared to the corresponding quantity resulting from nucleation theory. From Eq. (8.32)
one can determine the nucleation barrier, but not the kinetic prefactor, which deter-
mines the flux over this barrier and can be obtained from MD simulations.
In principle, Eq. (8.32) opens a possibility of calculating the nucleation barrier by
simulating the metastable vapor and counting clusters of various sizes. The total
number of particles used in modern MC simulations is of the order of N ∼ 105 −106 .
8.5 Evaluation of the Nucleation Rate 133

With this number of particles one can detect clusters from reliable statistics when
ΔG(n) < 10 kB T . Those are small n-mers, having the energy of formation of several
kB T . Only for extremely high supersaturations, close to pseudo-spinodal (discussed
in Chap. 9) will the height of the nucleation barrier be in this range (such high S
and therefore high nucleation rates are realized, e.g., in the supersonic Laval nozzle,
where J is in the range of 1016 − 1018 cm−3 s−1 [30]). For moderate supersaturations
nucleation barriers are in the range of ∼40 − 60 kB T and thus the clusters formed
in simulations are much smaller than the critical cluster. E.g. for ΔG∗ = 53 kB T the
chance to find a critical cluster is P = e−53 ≈ 10−23 , which means that the simulated
system should contain > 1023 particles which is equal to the Avogadro number. The
previously discussed umbrella sampling technique makes it possible to overcome
this difficulty. For a given n-cluster we introduce a weighting function w1 (rn ) which
according to Eq. (8.24) replaces the internal energy of the cluster U(rn ) by

U (rn ) = U(rn ) + W1 (rn )

where the biasing potential W1 is

W1 (rn ) = −kB T ln w1 (rn )

The simplest form of W1 is a harmonic function [31]:

1
W1 = kn (n − n0 )2 , kn > 0 (8.33)
2
It ensures that the formation of n-clusters with the sizes outside a certain range (char-
acterized by kn ) around n0 becomes highly improbable: the probability of finding the
n-cluster becomes proportional to
 
1
w1 = exp −β kn (n − n0 )2
2

This function forms a Gaussian umbrella in the space of cluster sizes, centered at n0 .
Only those clusters, which find themselves under this umbrella, will be sampled.
Thus, introduction of the biasing potential opens a “window” of the cluster sizes,
located at n0 , with a width of kn , which are sampled in simulations. By changing n0
one “opens consecutive windows” performing simulation runs within the windows,
thereby consecutively scanning the cluster size space.

8.5.2 Nucleation Rate from MD Simulations

For MD simulations the nucleation process is actually mapped on a simulation

system. From the time resolved simulation of the formation of a certain amount
134 8 Computer Simulation of Nucleation

of clusters one can draw a time dependent cluster statistics. This cluster statistics can
then be analyzed yielding the nucleation rate (besides other properties). For such an
analysis of the cluster statistics, several methods have been proposed in the literature.
Here we focus on two approaches which are commonly used in simulation studies
on nucleation.

8.5.2.1 Yasuoka-Matsumoto Method

The method of Yasuoka and Matsumoto [32], which also can be called the threshold
method, requires MD simulation system large enough to generate a significant amount
of clusters. This method can be obtained from the continuity equation in the space
of cluster sizes:
∂ ∂
N(n, t) = − j(n, t) (8.34)
∂t ∂n
Here N(n, t) is the number of clusters of size n in the simulation box at time t and
j(n, t) is the rate of the formation of clusters of size n in the box. In the steady state
∂
∂t N(n, t) = 0 yielding
∂
j(n, t) = 0
∂n
showing that j(n, t) is constant. If V is the volume of the simulation box, then the
steady-state nucleation rate is given by J = j(n, t)/V . In order to determine J from
the cluster statistics let us choose a certain, threshold, cluster size nthres and integrate
(8.34) over n from nthres to ∞:
 ∞  ∞
∂ ∂
N(n , t)dn = − j(n , t)dn (8.35)
∂t nthres nthres ∂n

The integral on the left-hand side is the total number of clusters with sizes larger
than nthres :  ∞
N (nthres , t) = N(n , t) dn
nthres

while the right-hand side is

j(nthres ,t) − j(∞, t) = j(nthres ,t)

where we took into account that the rate of formation of infinitely large clusters is
zero. Thus, Eq. (8.35) becomes

∂
N (nthres , t) = j(nthres , t)
∂t
8.5 Evaluation of the Nucleation Rate 135

(a) (b)

Fig. 8.8 Derivation of nucleation rate by means of the threshold method. a Cluster size dis-
tribution at time t: N(n, t); the shaded area under the curve gives the total number of clusters
N (nthres , t) larger than nthres at time t. b Time evolution of N (nthres , t). The slope of the domain
II ∂t∂ N (nthres , t) is proportional to the nucleation rate

yielding for the nucleation rate:

1 ∂
J= N (nthres , t) (8.36)
V ∂t
In the threshold method the number of clusters larger than a threshold value,
N (nthres , t), is plotted as a function of the simulation time for different values
of nthres . As a result one obtains curves with four domains schematically depicted
in Fig. 8.8.
The slope of the linear domain II is ∂t∂ N (nthres ,t). Hence, the nucleation rate is
this value, divided by the simulation box volume V . The plateau-like domain III
and also the following descending part IV of the curve result from the finite size of
the system. The steady state situation is only possible as long as sufficient number
of monomers is present in the box to deliver the clusters of size nthres . At some
point, due to depletion of the vapor, the monomer concentration becomes too small
to provide further nucleation and at the same time clusters grow by collision. This
leads to stagnation and then decreasing of the number of clusters. Therefore, from the
analysis of the plateau domain of the data one can not derive the nucleation properties.
In practice one finds that linear parts of the curves are not necessarily all parallel.
One may use this fact for a rough estimate of the critical cluster size by calculating
the curve for each single threshold value and detect the threshold value of the cluster
size beyond which the slope of domain II does not change any more. Application of
Yasuoka-Matsumoto method to vapor-liquid nucleation of zinc is shown in Fig. 8.9.
Though the critical cluster is not known a priori, the threshold method can safely
be applied. By choosing various threshold values it is possible to detect the linear
domain of steady-state nucleation. If nthres > n∗ , each simulation curve exhibits a
linear domain where all N (t) lines are parallel (cf. Fig. 8.9). The average of the
slopes in the linear domain can be used to calculate the nucleation rate.
136 8 Computer Simulation of Nucleation

Fig. 8.9 Number of clusters larger than a certain size, indicated above each curve, for vapor-liquid
nucleation of zinc (for explanation see Fig. 8.8). The plot is used to derive the nucleation rate by
means of the threshold method. Zinc vapor density is 0.0315 mol/dm3 , temperature T = 400 K.
The resulting nucleation rate is J = 25.5 × 1028 dm−3 s−1 . (Reprinted with permission from Ref.
[17], copyright (2007), American Institute of Physics.)

8.5.2.2 Mean First-Passage Time Method

The Mean First-Passage Time method (MFPT) provides an instruction to analyze the
stochastic dynamics of the nucleation process. In contrast to the threshold method, in
the MFPT a relatively small simulation system is sufficient to obtain the nucleation
rate. However, to get good statistics a large number of simulations is required. The
stochastic dynamics of a system with an activation barrier is governed by the Fokker-
Planck equation, describing the evolution of the cluster distribution function ρ(n, t)
caused by diffusion and drift in the space of cluster sizes, discussed in Sect. 3.7 (see
Eqs. (3.79)–(3.81)):
 
∂ρ(n, t) ∂ ∂ρ B(n) ∂ΔG(n)
= B(n) +ρ (8.37)
∂t ∂n ∂n kB T ∂n

Here B(n) is a diffusion coefficient in the space of cluster sizes. In Fig. 8.10 the work
of cluster formation ΔG(n) is plotted versus the cluster size. The solution of the
Fokker-Planck equation requires boundary conditions. In case of nucleation the left
boundary na is the monomer na = 1. It is called a reflecting boundary, since there
are no clusters smaller than a monomer. The right boundary nb is a size large enough
so that the cluster > nb has a negligibly small chance to evaporate. In view of this
feature, nb is called an absorbing boundary.
Let us fix na = 1 and an initial value n0 and follow the time evolution of the system
for various values of the absorbing boundary nb . For each nb we can identify the
mean first passage time τ (nb ) which is the average time necessary for the system,
starting at n0 , to leave the domain of cluster sizes (na , nb ) for the first time. Clearly,
8.5 Evaluation of the Nucleation Rate 137

Fig. 8.10 Gibbs free energy

G
of a n-cluster formation.
n0 is the initial value of n
(a starting point for MFPT
analysis) located between
the reflecting—na —and G*
absorbing—nb —boundaries;
n∗ is the critical cluster

na n0 n nb n

τ (nb = n∗ ) = τ ∗ is the average time necessary to reach the critical size. Since the
nucleation rate is the flux through the critical cluster, we may write

1 1
J= (8.38)
2 τ∗ V
where V is the volume of the simulation box; the factor 1/2 stands for the fact that
the critical cluster, corresponding to the maximum of ΔG(n), has a 50 % chance
of either growing to the new bulk phase or decaying, i.e. evaporating. Solving the
Fokker-Planck equation (8.37), Wedekind et al. [33] showed that for reasonably high
nucleation barriers the behavior of τ (nb ) in the vicinity of the critical size can be
approximated by the function
τJ   
τ (nb ) = 1 + erf c nb − n∗ (8.39)
2
shown schematically in Fig. 8.11. Here
 x
2
e−x dx
2
erf(x) =
π 0

is the error function;

τJ = 1/(J V )

and c is the inverse width of the critical region given by Eq. (3.49):
√
c = 1/Δ = πZ

The critical cluster n∗ is the inflection point of τ (nb ). Note, that τJ = 2τ ∗ .

In practice the MFPT method is applied to MD simulation in the following way: the
time for the system to pass for the first time a certain cluster size nb is averaged over a
large number of simulation runs. The procedure is repeated for various values of nb .
Then, the resulting function τ MD (nb ) is fitted to the 3-parametric expression (8.39),
138 8 Computer Simulation of Nucleation

Fig. 8.11 Mean first passage

time as a function of the
cluster size nb given by Eq. J
(8.39). The inflection point of
the curve corresponds to the
critical cluster n∗
*

na n0 n nb

providing the 3 fit parameters: τJ , n∗ and c. The nucleation rate being the reciprocal
of τJ reads:
1
J=
τJ V

An example of the MFPT curve obtained from simulation of zinc vapor–liquid nucle-
ation is shown in Fig. 8.12. Figures 8.10 and 8.11 demonstrate an advantage of the
MFPT method: one can recognize whether or not the nucleation process is coupled
to growth. If only the nucleation process takes place in the system, the simulation
data reach a plateau prescribed by the error function. Deviation from this shape is
related to the influence of particle growth on nucleation. In case of argon modelled
by the Lennard-Jones potential and small system sizes of approximately 300 atoms
it is possible to perform hundreds of simulations for averaging [33]. For larger sys-
tems with more complex interaction potentials the results have to be obtained from
a small number of simulation runs: e.g., for zinc nucleation (see Fig. 8.12) MFPT
results were obtained from 10 simulation runs.
While there are other methods to determine the nucleation rate from MD simula-
tions, the two methods described in this chapter are most frequently employed. Both
methods have advantages and drawbacks. The threshold method requires a large
simulation system in order to provide a significant amount of clusters to obtain good
statistics. If the systems are large enough possible depletion effects can be minimized.
The MFPT method requires a large number of simulation runs. In order to reach such
a large number each run should be sufficiently fast. This can be accomplished by ter-
minating a simulation at the point when the chosen cluster size is passed for the first
time. Another method for optimization is to choose a relatively small system since
only clusters not larger than few times n∗ are required for MFPT. On the other hand,
the smaller a simulation system is, the more important become finite sized effects.
8.6 Comparison of Simulation with Experiment 139

Fig. 8.12 MFPT analysis of a series of simulation runs for zinc at T = 800 K and log10 S = 2.79.
The critical cluster size, indicated by the dashed vertical line, is n∗ = 9. Because of the large system
size only 10 simulations were performed (Reprinted with permission from Ref. [17], copyright
(2007), American Institute of Physics.)

8.6 Comparison of Simulation with Experiment

MD simulation is limited by the available computational power. This means that

the system size as well as the duration of simulation is limited. In order to observe
nucleation within these limits, the supersaturation has to be sufficiently high yielding
high nucleation rates which for MD are typically in the range of 1025 to 1030 cm−3 s−1 .
Meanwhile, supersaturations which can be realized in nucleation experiments are
usually lower resulting in lower nucleation rates: in an expansion chamber J ranges
from 105 to 1010 cm−3 s−1 ; nozzle experiments make it possible to reach higher
supersaturations leading to nucleation rates in the order of 1018 − 1020 cm−3 s−1 .
These values are, however, still lower than the typical MD values.1
That is why usually a direct comparison of MD results with experimental data can
not been done in a straightforward way. Nevertheless, some important conclusions
can be drawn. To illustrate this statement we show in Fig. 8.13 the results for zinc
vapor-liquid nucleation obtained by MD simulation, using both the threshold and the
MFPT methods [17], along with CNT predictions and available experimental data.
Both MD methods are close to each other within the range of their accuracy. At the
same time MD data and experiment lead to nucleation rates which are considerably
higher than the CNT predictions.

1 It must be noted that performing extensive simulations with up-to-date computers allows to
approach the region of supersonic nozzle experiments.
140 8 Computer Simulation of Nucleation

Fig. 8.13 Comparison of MD simulations, experimental data and CNT calculations for zinc vapor-
liquid nucleation. Filled symbols: calculations with the Yasuoka-Matsumoto method (“Yas”), open
symbols: calculations with MFPT method [17]. The dashed curve is the CNT prediction. An exper-
imental point from Ref. [34] is indicated by “exp”

8.7 Simulation of Binary Nucleation

Most molecular simulations of nucleation have been performed for pure substances;
much fewer computational studies have been devoted to binary systems.
MD simulation of nucleation in the binary vapor of iron and platinum performed
in Ref. [35] revealed that besides the mixed clusters, in which both components
are present, there are also single-component clusters containing pure iron and pure
platinum. Due to the difference in attraction strength between the two substances,
one observes a large amount of big platinum clusters compared to a relatively small
number of small iron clusters. The iron atoms have a weaker attraction compared
to platinum and either do not condense on hot platinum cluster or rapidly evaporate
after condensation.
Compared to metals, the binary mixture of n-nonane and methane is characterized
by much weaker van der Waals interactions. Nucleation in this mixture was studied
in MD simulations of Braun [36]. The peculiar feature of this system is a very low
vapor molar fraction of nonane: ynonane ≈ 10−4 ; meanwhile it is nonane that ensures
nucleation in the system. In order to tackle this problem, a simulation box containing
around 105 methane molecules was chosen and expansion simulation method was
used. From the bulk phase behavior one would expect a mole fraction of methane
in the liquid-like clusters to be around 0.2–0.4 at the given nucleation conditions
(high pressure and ambient temperature). At the same time simulation results show
that even for weakly interacting van der Waals systems the critical clusters are very
different from what one would expect from the bulk equilibrium. Clusters have
a peculiar structure resulting from minimization of the surface energy. The system
tends to lower its energy by phase separation and moving the more volatile component
(methane) towards the shell region of the cluster.
8.7 Simulation of Binary Nucleation 141

In MC simulation of binary nucleation the free energy of formation of the (na , nb )-

cluster, containing na molecules of component a and nb molecules of component b,
in a supersaturated vapor can be found from the cluster statistics accumulated in
simulation runs:  
Nna nb
βΔG(na , nb ) = − ln
N

where Nna nb is the number of (na , nb )-clusters, N is the number of molecules in the
system. This expression is a generalization of the corresponding result (8.32) for the
unary case [37].
Kusaka et al. [38] performed simulations of nucleation in water-sulfuric acid sys-
tem which plays an important role in atmospheric processes. Simulations employed
water, hydronium ion, sulfuric acid and the bisulfate ion as species modeled by force
field combining the Lennard-Jones potential with electrostatic potential by means
of partial charges. From MC simulations the free energy of cluster formation was
analyzed along with the cluster structures. It was found that the shape of most of the
clusters differs from the spherical one. Different conformations of the clusters turn
out to be very close in energy and have a fairly long lifetime. This observation leads
to a conclusion that various clusters contribute to the nucleation rate.
Chen et al. [39] performed MC simulation of the nucleation in binary water/ethanol
systems. They found that ethanol is enriched in the cluster surface leading to a lower
surface tension than that of the ethanol/water mixture of the given mole fraction.
They argue that the shortcomings of the CNT in such cases might be related to this
surface enrichment.

8.8 Simulation of Heterogeneous Nucleation

Heterogeneous nucleation requires a surface at which the supersaturated vapor can

nucleate. In general such surface can be implemented in molecular simulations by
adding solid particles in the supersaturated vapor or by setting up a solid film in the
middle of the simulation box. The heterogeneous surface lowers the activation barrier
for nucleation. Therefore, the supersaturation that is necessary to observe nucleation
on the time scale of MD simulations is lower than for homogeneous nucleation. The
extent of this effect depends on the attraction between the nucleating substance and
the substrate. This attraction strength is in turn related to the wetting behavior of a
liquid on a substrate. The stronger the attraction—the smaller the contact angle of
the droplet on the surface. On the other hand, for a completely repelling surface one
would expect the contact angle around 180◦ recovering homogeneous nucleation.
Toxvaerd [40] investigated heterogenous nucleation by MD simulations using a r −9
potential for the wall interactions of the molecules. Depending on the interaction
between the wall and the molecules, he observed either nucleation at the surface
for weak attraction or prewetting, i.e. the formation of a molecular layer on the sur-
face for strong interactions. In this investigation the wall was perfectly flat, i.e. it
142 8 Computer Simulation of Nucleation

did not contain inhomogeneities. Kimura and Maruyama [41] instead used a sur-
face composed by harmonically vibrating molecules coupled to a heat bath. They
investigated the nucleation of argon vapor for various vapor phase temperatures and
pressures. The nucleation rate was analyzed using the Yasuoka-Matsumoto threshold
method. Simulation results showed good agreement with the classical heterogenous
nucleation theory—the Fletcher model, discussed in Sect. 15.1.
In Ref. [42] the nucleation of argon on a polyethylene substrate was investigated. In
this work the polyethylene film in the center of the simulation box was coupled to a
thermostat. All latent heat of condensation was hence withdrawn from the system via
the substrate. In this way the process of heterogeneous nucleation can be modelled
realistically. In the transient stage of nucleation a temperature gradient develops,
which after condensation is complete vanishes again. Depending on the supersatu-
ration of the argon vapor, different types of growth take place. Comparison with the
classical heterogeneous nucleation theory exhibits good agreement.

8.9 Nucleation Simulation with the Ising Model

The simplest possible model of interacting particles is the Ising model. It consists of a
lattice of spins that can have two values: either s = +1 or s = −1. Each spin interacts
with its nearest neighbors. Their number N1 depends on the type of the lattice. In the
simplest case of a 2D square lattice N1 = 4, while for a 3D cubic lattice N1 = 6.
Interaction between spins is given by the coupling constant K. If K is positive the
model describes ferromagnetic behavior favoring the alignment of neighboring spins
parallel to each other, while a negative K mimics antiferromagnetic behavior with
the preference for the anti-parallel alignment of neighboring spins. The Hamiltonian
of the Ising model reads:
 
H = −K si sj − H sk (8.40)
(i,j)nn k

Here H is the magnitude of an external magnetic field. In Monte Carlo simulations

of the Ising model a lattice site is chosen at random and the spin at this site is flipped
while all other spins of the system remain unchanged. The energy difference ΔU
resulting from this spin flip is calculated. To judge whether or not the spin flip (which
in molecular MC terminology represents a MC move) can be accepted the Metropolis
algorithm described above is employed.
The Ising model, which has been developed for investigation of magnetism, can
be also used as a model of a fluid. A typical intermolecular interaction potential
in a fluid is characterized by a strong repulsion at short distances, a potential well,
and a relatively fast decaying attractive tail (cf. the Lennard-Jones potential). To
a good approximation such a potential can be approximated by a square well. In
turn a fluid with a square well potential can be mapped on a lattice-gas model.
8.9 Nucleation Simulation with the Ising Model 143

Within this model molecules are only allowed to occupy the sites of a regular lattice
instead of continuous distribution in space. This requirement mimics a short-range
repulsion in a real fluid: molecules can not be closer than the lattice spacing. Attractive
interactions, i.e. the attractive well, is modelled by a nearest-neighbor potential , so
that the potential energy of a certain configuration takes the form

U = −ε ρi ρj
(i,j)nn

where ρk = 1 if the site k is occupied and ρk = 0 in the opposite case. Let us set

si = 2 ρi − 1

Then, in the Ising model si = −1 if the site i in the lattice gas model is free and
si = +1 if it is occupied. This transformation thus maps the fully occupied lattice
of spins si = ±1 on the partially occupied lattice of fluid molecules. The two
systems—lattice gas and Ising model—become thermodynamically equivalent, i.e.
their partition functions are the same, if we set [43]:

K = ε/4, H = (2μ + N1 ε)/4 (8.41)

Thus, the Ising model can be used for simulation of a vapor–liquid system and
hence also for vapor–liquid nucleation. A metastable state of the spin system can be
achieved by varying the external field H which from (8.41) is equivalent to varying
of the chemical potential of a fluid molecule resulting in a supersaturation.
Glauber [44] was the first to study the kinetics of the 1D Ising model which is solvable
analytically, but does not exhibit a phase separation. Stoll et al. [45] performed
Monte Carlo simulations of the 2D spin-flip Ising model and analyzed its relaxation
towards equilibrium. Simulations demonstrated consistency with the dynamic scaling
hypothesis. Stauffer et al. [46] analyzed nucleation in 3D Ising lattice gas by Monte
Carlo simulations. They observed that the results obtained from the lattice model are
roughly in agreement with CNT.
The Ising model is very useful for the investigation of fundamental concepts of
nucleation. It can be employed to the analysis of the scaling behavior expressed in
the form of power laws. Since the pioneering works [45, 46] the model was employed
for various other systems—e.g. for heterogeneous nucleation [47], to mention just
one example. Meanwhile, it must be noted that simulation of real substances goes
beyond the scaling behavior and requires explicit force fields acting between the
molecules.
144 8 Computer Simulation of Nucleation

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Chapter 9
Nucleation at High Supersaturations

9.1 Introduction

At high supersaturations (deep quenches) the system from being metastable becomes
unstable; in the theory of phase transitions the boundary between the metastable
and unstable regions is given by a thermodynamic spinodal being a locus of points
corresponding to a divergent compressibility. Rigorously speaking the transition from
metastable to unstable states does not reduce to a sharp line but rather represents a
region of a certain width which depends on the range of interparticle interactions [1].
Within the spinodal region the fluid becomes unstable giving rise to the phenomenon
of spinodal decomposition [2], characterized by vanishing of the free energy barrier
of cluster formation at some finite value of the supersaturation. The classical theory
does not signal the spinodal: the nucleation barrier decreases with S but remains
finite for all values of S (see Eq. (3.28)). Therefore, nucleation in the spinodal region
can not be described by CNT and a more general formalism is needed.
Such a formalism, the field theoretical approach, was pioneered by Cahn and Hilliard
[3] and developed by Langer [4, 5], Klein and Unger [6, 7]. It is based on the
mean-field Ginzburg–Landau theory of phase transitions. Cahn-Hilliard’s approach
(usually termed a “gradient theory of nucleation”) leads to the existence of a well-
defined mean-field spinodal characterized by a supersaturation Ssp . A mean-field
theory becomes asymptotically accurate in the limit of infinite-range intermolecular
interactions, hence a spinodal line exists in the same limit. At the spinodal the barrier
vanishes which means that the capillary forces can no longer sustain the compact form
of a droplet, clusters in the vicinity of a spinodal are ramified fractal objects [6, 7].
In this chapter we formulate the mean-field (Cahn-Hilliard) gradient theory consid-
ering nucleation at high supersaturations. Special attention in this domain should be
paid to the role of fluctuations giving rise to the concept of pseudospinodal. Analysis
of nucleation near the pseudospinodal results in a generalized form of the classical
Kelvin equation (3.61) relating the size of the critical cluster to the supersaturation.

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 145

DOI: 10.1007/978-90-481-3643-8_9, © Springer Science+Business Media Dordrecht 2013
146 9 Nucleation at High Supersaturations

9.2 Mean-Field Theory

9.2.1 Landau Expansion for Metastable Equilibrium

The starting point for the mean-field analysis of nucleation in the vicinity of the
thermodynamic spinodal is the Landau expansion of the free energy density in powers
of the order parameter m [8]

a 2 b 4
g = g0 + m + m −mh (9.1)
2 4
where
a = a0 t, t ≡ (T − Tc )/Tc , a0 > 0, b > 0

h is the external field conjugate to m. For the gas-liquid transition the order parameter
can be defined as
m = ρ − ρc

where ρc is the critical density, and

h = Δμ = μv ( p v ) − μl ( p v ) (9.2)

is the external field. At the spinodal h = h sp , while at the binodal the chemical poten-
tials of the phases are equal yielding h = 0. Below the critical point a < 0 and the
free energy density has a double-well structure shown in Fig. 9.1. In thermodynamic
equilibrium one should have

∂g ∂2g
= 0, >0
∂m ∂m 2
yielding

a m + b m3 − h = 0 (9.3)
a + 3b m 2 > 0 (9.4)

At h = 0, g(m) has two equal minima corresponding to the two coexisting phases.
For h = 0 the cubic equation (9.3) has a single real root if h 2 > h 2sp [9], where

4 a3
h 2sp = − (9.5)
27 b

This root refers to the single, stable, phase (liquid). If h 2 ≤ h 2sp , there are three real
roots; for h 2 = h 2sp two of them are equal. The left local minimum of the free energy
at m = m ∗ corresponds to the metastable state (supersaturated vapor), while the
9.2 Mean-Field Theory 147

h sp
h
h=0

metastable

g
stable

T< Tc

m * m sp 0 mglob

Fig. 9.1 Schematic plot of Landau free energy density g for T < Tc . At h = 0 (long dashed line)
there are two equal minima corresponding to the coexisting states. At 0 < h 2 < h 2sp (solid line)
the left, local, minimum at m ∗ corresponds to a metastable state (supersaturated vapor), while the
right, global, minimum refers to a stable state (liquid); the two states are separated by the energy
barrier. At h = h sp (short dashed line) the local minimum disappears—this is the case of spinodal
decomposition

right—global—minimum corresponds to the stable state (liquid); the two states are
separated by the energy barrier (these features are illustrated in Fig. 9.1). Expres-
sion (9.5) gives the maximum supersaturation corresponding to h = h sp , where
the local minimum of g becomes an inflection point—this is the case of spinodal
decomposition [2].
Nucleation takes place for 0 < h 2 < h 2sp . The discussion below refers to this case.
Since a < 0, it is convenient to introduce s = −a. Then the solutions of the cubic
equation (9.3) are [9, 10]:
  
s 2π
m(h) = 2 cos α + k , k = 0, 1, 2 (9.6)
3b 3

where √
3 3 b1/2 π
cos 3α = 3/2
h, 0 ≤ α ≤ (9.7)
2 s 6
Substituting (9.6) into the minimization condition (9.4) we obtain
  
 
cos α + 2π k  > 1
 3  2

which is satisfied for k = 0, 1 and is not satisfied for k = 2. The latter case cor-
responds to a maximum of g while the other two solutions correspond to the two
minima.
148 9 Nucleation at High Supersaturations

In order to determine which of these two roots refers to a local minimum (a super-
saturated state) we substitute (9.6) with k = 0 and k = 1 into (9.1) and after some
algebra obtain:

s2
g|k=0 = g0 + (1 + cos 2α)(1 − 3 cos 2α)
6b
     
s2 2π 2π
g|k=1 = g0 + 1 + cos 2 α + 1 − 3 cos 2 α +
6b 3 3
π
It is easy to check that for 0 ≤ α ≤ 6 : g|k=0 ≤ g|k=1 . Thus, the solution with k = 1,
  
s 2π
m ∗ (h) = 2 cos α + (9.8)
3b 3

corresponds to a metastable state of the system in an external field h, the free energy of
this state being g∗ = g|k=1 . The solution with k = 0 gives the global minimum cor-
responding to the thermodynamically stable liquid state to which the system evolves.
For the states close to m ∗ the free energy density can be expanded in powers of
φ = (m − m ∗ )/m ∗ :
b2 2 b3 3
g = g∗ + φ − φ + O(φ)4 (9.9)
2 3
where  
∂ 2 g  1 ∂ 3 g 
b2 (h) = m 2∗ , b3 (h) = −m 3∗ (9.10)
∂m 2 m=m ∗ 2 ∂m 3 m=m ∗

The term linear in φ vanishes since m = m ∗ is a local minimum of g. At the spinodal

b2 (h = h sp ) = 0, b3 (h = h sp ) > 0 (9.11)

The quantities b2 and b3 can be calculated from the equation of state:


∂μ 
b2 = ρ 2 (9.12)
∂ρ ρ=ρ v
 
1 3 ∂ 2 μ  1 ∂b2 
b3 = − ρ = b − ρ (9.13)
∂ρ 2 ρ=ρ v 2 ∂ρ ρ=ρ v
2
2

Using the thermodynamic relationship

∂p ∂μ
=ρ
∂ρ ∂ρ
9.2 Mean-Field Theory 149

we have 
∂ p 
b2 = ρ (9.14)
∂ρ ρ=ρ v

showing that b2 is the inverse isothermal compressibility of the vapor at the given
metastable state.

9.2.2 Nucleation in the Vicinity of the Thermodynamic Spinodal

Consider the behavior of the system near the mean-field spinodal. For this purpose
we construct an appropriate Ginzburg–Landau free energy functional which should
describe the state of the system undergoing a first order phase transition, characterized
by a scalar order parameter [11]

φ(r) = [m(r) − m ∗ ]/m ∗

Now we allow for its spatial variations. Using (9.9), this functional reads:
  
c0 b2 2 b3 3
F [φ(r )] = F∗ + dr |∇φ|2 + φ − φ (9.15)
2 2 3

where F∗ is the free energy of the metastable state m = m ∗ (the local minimum of
the free energy), out of which nucleation starts. The square-gradient term in (9.15)
is an energy cost to create an interface between the phases; c0 > 0 is related to the
correlation length in the system [12] and can be well approximated by [13]

c0 ∼
1/3
= kB Tρc (9.16)

Following Unger [7], we associate the critical cluster with the saddle point of the
functional F [φ(r )]. If the saddle point is found, its substitution into (9.15) yields
the nucleation barrier
  
c0 b2 2 b3 3
W = F − F∗ = dr |∇φ|2 + φ − φ (9.17)
2 2 3

To analyze this expression we proceed by performing a set of scaling transformation

of the variables. Rescaling the order parameter

φ1 = (b3 /c0 )1/3 φ

and denoting
ε = b2 (b32 c0 )−1/3 (9.18)
150 9 Nucleation at High Supersaturations

we rewrite (9.17) as
  2/3
1 c0 ε 2 1 3
W = c0 dr |∇φ1 |2 + φ − φ
2 b3 2 1 3 1

The next transformation rescales the spatial coordinates

r1 = (b3 /c0 )1/3 r

yielding
  
c02 1 ε 1
W = dr1 |∇1 φ1 |2 + φ12 − φ13
b3 2 2 3

∂
where ∇1 = ∂r1 . And finally, further rescaling is useful:

φ1 = εφ, r1 = ε−1/2 r (9.19)

with the help of which W takes the form:

  
c2 1 1 1
W = ε3/2 0 dr |∇ φ|2 + φ 2 − φ 3 (9.20)
b3 2 2 3

∂
where ∇ = ∂r .

The saddle point of W [∇(r)] is given by the Euler-Lagrange equation:

∇ 2φ = φ − φ2 (9.21)

The critical cluster is the nontrivial solution of (9.21) vanishing at infinity. The
existence of such solutions was proved for sufficiently large bounded domains [14].
Without presenting its full form it is instructive to study its behavior at large r , i.e.
far from the center of mass of the cluster. In this domain the amplitude of the droplet
is small and we can neglect the second term in (9.21) which leads to the equation

∇ 2 φ = φ, large r (9.22)

The spherically symmetric solution of (9.22) vanishing at infinity is the screened

Coulomb function:
e−r
φ = C> , large r
r
(where C> is a constant); in the units of (9.19):
√
√ e− ε r1
φ1 (r1 ) = C> ε , large r1
r1
9.2 Mean-Field Theory 151

The spinodal corresponds to ε = b2 = 0. Since φ1 (r1 ) is the density fluctuation

associated with a nucleus, its decay length

R∗ ≈ ε−1/2 , ε→0 (9.23)

characterizes the size of the critical cluster. Equation (9.23) shows that within the
mean-field analysis the critical cluster size diverges as the spinodal is approached.
In the same limit the nucleation barrier (9.20) vanishes as

W ∼ ε3/2 , ε→0 (9.24)

Finally, we must relate ε to the physical parameters of the system. To be more precise
we will determine scaling of ε near the spinodal to the leading order in (h − h sp ).
Obviously
ε(h = h sp ) = 0

From (9.10) and (9.18) it follows that ε is proportional to the curvature of the Landau
free energy at the metastable state:

∂ 2 g 
ε ∼ b2 ∼ g2 (h) ≡ = −s + 3b m 2∗
∂m 2 m=m ∗ (h)

Substituting h = h sp − u into (9.8), where u is a (small) deviation of h from its value

at the spinodal, we obtain to the leading order in u:

ε ∼ g2 = 2(3bs)1/4 h sp − h (9.25)

Expressing h in terms of the supersaturation, we have from Eq. (9.2)

h sp − h = kB T (ln Ssp − ln S) = kB T ln Ssp (1 − η)

where Ssp refers to the spinodal and

ln S
η= , 0≤η≤1 (9.26)
ln Ssp

Ssp (T ) is the upper boundary of S for nucleation at the temperature T ; its value
depends on the equation of state. For van der Waals fluids calculation of Ssp is
presented in Appendix C.
From (9.25)
ε ∼ (1 − η)1/2 (9.27)

Substituting (9.27) into (9.24) and (9.23), we find that in the vicinity of the spinodal
the nucleation barrier vanishes as
152 9 Nucleation at High Supersaturations

Wsp = csp (T ) (1 − η)3/4 , csp (T ) > 0, η → 1− (9.28)

while the radial extent of the critical cluster diverges as

R∗ = R0 (1 − η)−1/4 , R0 (T ) > 0, η → 1− (9.29)

The excess number of molecules in the critical cluster is found from (9.28) using the
nucleation theorem (4.15):

Δn c ∼ (1 − η)−1/4 , η → 1− (9.30)

Comparison of (9.29) and (9.30) gives the scaling:

Δn c ∼ R∗ (9.31)

This is distinctly different from the scaling Δn c ∼ R 3 corresponding to compact

spherical droplets discussed in CNT. Equation (9.31) supports the conjecture of Klein
[6] that the critical cluster near the spinodal is a ramified chain-like object.

9.3 Role of Fluctuations

Previous discussions avoided an important conceptual question: how deep the quench
can be so that the concept of quasi-equilibrium (of the metastable state) can be
considered valid? In other words: what is the limit of validity of the mean-field
gradient theory, which completely neglects the effect of fluctuations? The answer to
this question can be found using the Ginzburg criterion for the breakdown of Landau
theory of phase transitions [12]. Very close to the spinodal fluctuations become
increasingly important and the mean-field theory of Sect. 9.2.2 breaks down. The
Ginzburg criterion determines the width of the domain near the spinodal, inside
which the mean-field considerations are violated. Such a thermodynamic analysis
was carried out by Wilemski and Li [15], who showed that for real fluids the Ginzburg
criterion is violated in the entire spinodal region, where the Landau expansion is
used. Having stated this, Wilemski and Li suggested that the concept of the mean-
field spinodal should be replaced by the concept of a pseudospinodal, introduced
earlier by Wang [16] in the study of polymer phase separation, which is associated
with a nucleation barrier ∼kB T .
The applicability of the mean-field approach can also be considered on the basis
of kinetic considerations. To do this let us compare two characteristic times: (i) the
time tM necessary to form a critical cluster which is a lifetime of the metastable state,
and (ii) the relaxation time tR during which the system settles in this state. The first
quantity can be related to the nucleation rate by using its definition: tM = 1/(J V ).
To find tR one must study the dynamics of the metastable state. Since the order
9.3 Role of Fluctuations 153

parameter φ(r) in the Ginzburg–Landau functional (9.15) is a conserved variable,

its evolution is governed by the Cahn-Hilliard dissipative dynamics [17]:

∂φ δF
= Γ0 ∇ 2 +ζ (9.32)
∂t δφ

where Γ0 is a transport coefficient and ζ (r, t) is a noise source (which models thermal
fluctuations) satisfying

ζ (r, t) ζ (r , t ) = −2T Γ0 ∇ 2 δ(r − r ) δ(t − r )

to ensure that the equilibrium distribution associated with (9.32) is given by the
Boltzmann statistics. From the solution of (9.32) and (9.15), obtained by Patashinskii
and Shumilo [18, 19] (see also [13]), it follows that

16c0
tR =
Γ0 b22

implying that when the system approaches the thermodynamic spinodal (b2 → 0) its
relaxation time diverges. The relation between t M and t R established in [18, 19] is:
   
4π χ χW
t M = tR exp (9.33)
λ0 kB T

where
(b2 c0 )3/2
χ= (9.34)
kB T b32

and λ0 ≈ 8.25. Clearly, the concept of quasi-equilibrium is meaningful for the

metastable states characterized by t M  t R . In the opposite case this concept
becomes irrelevant. The boundary between these two domains is called a kinetic
spinodal [18–20] and can be defined by the condition t M ∼
= t R yielding

χW ∼
=1 (9.35)
kB T

Beyond the kinetic spinodal the phase separation proceeds not via nucleation but via
the mechanism of spinodal nucleation [21], which differs both from nucleation and
spinodal decomposition. As one can see, the kinetic considerations are in agreement
with the thermodynamic analysis of [15]. Hence, in terms of the nucleation barrier
the pseudospinodal is similar to the kinetic spinodal.
The preceding discussion shows that the spinodal limit is hard to achieve in practice:
gradual quenching of the supersaturated vapor results in the barrier becoming equal
to the characteristic value of natural thermal fluctuations of the free energy, which
in a fluid is of the order of kB T ; at these conditions the time necessary to form the
154 9 Nucleation at High Supersaturations

critical cluster becomes comparable to the relaxation time during which the system
settles in the metastable state.

9.4 Generalized Kelvin Equation and Pseudospinodal

The mean-field gradient theory predicts the divergence of the critical cluster as S
approaches the spinodal. Meanwhile, experiments in the supersonic Laval nozzle
[22–26] with nucleation rates as high as 1017 − 1018 cm−3 s−1 do not support this
statement: the critical cluster, determined from the experimental J − S curves by
means of the nucleation theorem, continuously decreases with the supersaturation,
showing no signs of divergence up to the highest values of S. For those values
the critical cluster is a nano-sized object containing 5–10 molecules. A possible
explanation of this qualitative discrepancy was mentioned in the previous section:
the mean-field considerations fail in the entire spinodal region and the physically
relevant limit of the supersaturation is not the spinodal, but the pseudospinodal.
Therefore, we need to study nucleation in the vicinity of the pseudospinodal.
Can the CNT be useful for this study? Recall that in the CNT the critical cluster is
related to the supersaturation by the Classical Kelvin Equation (CKE) (3.63). For
every finite S it predicts a certain critical cluster yielding a certain finite nucleation
barrier (3.29). In other words the CNT does not signal either spinodal or pseudospin-
odal. This is not surprising since at high S the critical clusters become small and
obviously do not obey the capillarity approximation. For the analysis of the system
behavior in the vicinity of the pseudospinodal we need a generalization of the CKE
which extends the limit of its validity down to the clusters of molecular sizes; such
a generalization was proposed in Ref. [27].
The thermodynamic basis for the Kelvin equation is the metastable equilibrium
between the critical cluster and the surrounding supersaturated vapor and according
to (3.62) can be found from maximization of the Gibbs energy of cluster formation.
This is a general statement which holds irrespective of a particular form of the Gibbs
energy. Let us adopt the MKNT form for ΔG given by Eq. (7.77). Its maximum
leads to 
dn s (n) 
ln S = θmicro (9.36)
dn n c

which can be termed the Generalized Kelvin Equation (GKE). Taking into account
(7.63) and (7.66)), it is straightforward to see that in the limit of big clusters the GKE
recovers the Classical Kelvin Equation.
In order to illustrate the important features of the GKE let us consider the profiles
of ΔG(n) for various values of S at a fixed temperature T . To make the illustration
practically relevant, we refer to the experimental conditions for argon nucleation in
the supersonic Laval nozzle studied in Ref. [26].
9.4 Generalized Kelvin Equation and Pseudospinodal 155

30
Argon T=37.5 K

20
ln S=7

10

β ΔG(n) 0

8
-10
8.82

-20
0 10 20 30 40
n

Fig. 9.2 Gibbs free energy profiles ΔG(n) for argon at T = 37.5 K. Labels indicate the value of
ln S. Arrows indicate the critical cluster for each curve (corresponding to the second maximum of
ΔG(n). At ln S = 8.82 the second maximum disappears; the dashed red line gives the nucleation
barrier ΔG ∗ ≈ kB T )

Figure 9.2 shows the free energy profiles for T = 37.5 K and three different values
of ln S. Each curve ΔG(n) has two maxima (cf. Sect. 7.7): the first (left) one is
always at n = N1 (coordination number in the liquid phase) and is an artifact of the
MKNT, while the second (right) maximum corresponds to the critical cluster n c : in
Fig. 9.2 the critical cluster is indicated by the vertical error and the ‘ball’ on the top
of the curve. As S increases, both n c and the nucleation barrier ΔG ∗ = ΔG(n c )
decrease. At a certain supersaturation, ln S = 8.82, the second maximum disappears
which means that for supersaturations higher than this value the critical cluster does
not exist. In other words, there is an upper limit of S, beyond which there is no
nucleation. As one can see from the dashed red line, the nucleation barrier at this
value of S turns out to be ΔG ∗ ≈ kB T which corresponds to the pseudospinodal.
Below we will show that this is not a pure coincidence.
The free energy of formation of the critical cluster in MKNT takes the form (7.82)
which at the pseudospinodal gives

− n c ln S + θmicro [n s (n c ) − 1] = 1 (9.37)

Combination of (9.37) with the GKE Eq. (9.36) determines both n c and ln S at the
pseudospinodal. Excluding ln S, we find that n c satisfies the equation:

dn s  1
nc − n s (n c ) + 1 + υ = 0, where υ ≡ <1 (9.38)
dn n c θmicro
156 9 Nucleation at High Supersaturations

Before solving it let us discuss the domain of admissible values of n c . Suppose that
n c is large, then using the asymptotics (7.63) in Eq. (9.38) we find:

n c = [(1 + υ)/ω]3/2

which is the quantity of order 1, contradicting the assumption of large n.

In the opposite limit of small (n ≤ N1 ) clusters: n s (n) = n and Eq. (9.38) has no
solutions. Thus, the solution of (9.38) belongs to the intermediate cluster range. It is
convenient to present
n s (n) = n − [X (n)]3

where X (n) is the solution of Eq. (7.58). Then (9.38) can be rewritten as:

dX 
[X (n c )] − 3 n c [X (n c )]
3 2
+1+υ =0 (9.39)
dn n c

From the previous discussion we can expect (the assumption to be verified later)
that near the pseudospinodal n c is small and close to the lower boundary of the
intermediate cluster range, i.e. it lies in the vicinity of N1 + 1. Correspondingly,
X (n) is close to unity. Presenting

X (n) = 1 + δ(n)

and linearizing (7.58) in δ(n) we find:

1
δ(n) = (n − N1 − 1) (9.40)
3q
q ≡ 1 + 2ω + ωλ (9.41)

Equation (9.39) now reads


dδ 
3δ(n c ) − 3n c [1 + 2δ(n c )] + (2 + υ) = 0 (9.42)
dn n c

Substituting (9.40) into (9.42), we find the critical cluster at the pseudospinodal:
√ 
1+ 1+τ
n c,psp = (N1 + 1) (9.43)
2
6q [(2 + υ)q − (N1 + 1)]
τ =
(N1 + 1)2

This result supports the assumption that n c is close to N1 + 1 (usually |τ |  1); the
cluster shows the liquid-like features only when it has a core, i.e. when n > N1 (T ).
9.4 Generalized Kelvin Equation and Pseudospinodal 157

Setting
n s (n) = n − [1 + 3δ(n)]

in Eq. (9.36) and using (9.40) we obtain the supersaturation at the pseudospinodal:
 
1
ln Spsp (T ) = θmicro 1− (9.44)
q

This is the maximum value of the supersaturation at the temperature T . To a large

extent it is determined by the microscopic surface tension describing the nonideality
of the vapor (expressed in terms of the second virial coefficient); the temperature-
dependent quantity q(T ) is in the range q(T ) ≈ 2 ÷ 4.
States with S > Spsp are not realizable. With this in mind we rewrite the GKE in the
form
⎧ 
⎨ θmicro dn s  , for n c ≥ N1 + 1
dn
ln S =  nc  (9.45)
⎩ θmicro 1 − 1 , for n c ≤ N1 + 1
q

showing the physical meaning of the pseudospinodal: it corresponds to the disap-

pearance of the liquid-like structure of the critical nucleus.
The derivation of GKE assumes that the supersaturated vapor is weakly non-ideal
implying that
ζ (T, S) ≡ |B2 p v/kB T | = S |B2 psat/kB T |  1 (9.46)

This criterion determines the limit of validity of the GKE.

Setting S = Spsp , we find that Eq. (9.44) for the pseudospinodal is applicable for the
temperatures satisfying
ζpsp (T ) ≡ e−θmicro /q  1 (9.47)

Figure 9.3 shows the classical (CKE) and generalized (GKE) Kelvin equation for
argon at T = 45 K and water at T = 220 K. The horizontal arrow points to
the pseudospinodal. For both substances the criterion (9.46) is satisfied up to the
pseudospinodal:

ζpsp,argon (T = 45 K) ≈ 0.070, ζpsp,water (T = 220 K) ≈ 0.034

CKE and GKE become indistinguishable for clusters exceeding ∼200 molecules. In
terms of the cluster radius it corresponds to R ≈ 1.1−1.5 nm. This rather unexpected
result shows that the classical Kelvin equation may be still valid down to the clusters
containing ∼200 molecules. At large cluster sizes GKE approaches CKE: for argon—
from below, whereas for water—from above. The reason for this difference is the
sign of the Tolman length (7.101) which is negative for water at 220 K and positive
for argon at 45 K.
158 9 Nucleation at High Supersaturations

(a) 15 (b) 7
Argon Water

CK
T=45 K 6

E
T=220 K

C
ln Spsp

KE
10 5

ln S
ln S

ln S ps p 4

G
KE
5 GK 3
E

N1+1 2
N1+1
0 1
1 2 10 20 100 200 1 2 10 20 100 200
nc n

Fig. 9.3 Classical Kelvin equation (CKE) (dashed lines) and Generalized Kelvin Equation (GKE)
(solid lines). a Argon at T = 45 K; b Water at T = 220 K. The horizontal arrow indicates the value
of ln S at the pseudospinodal

Fig. 9.4 Nucleation barrier 40

βW ∗ = βΔG ∗ (lines) and Water
critical cluster size (closed and
T/Tc =0.340
open symbols) as a function of β
30 (T=220 K)
ln S for water at T = 220 K
WC

as predicted by MKNT and

NT
βW
*

CNT. The dashed horizontal

β W , nc

MK
N

line corresponds to the barrier

T

20
W ∗ = kB T characteristic of
the pseudospinodal conditions
10 nc nc, MKNT
,C
N T

βW =1
0
3 3.5 4 4.5 5 5.5 6
ln S
MKNT pseudospin.
ln Spsp =5.27

Figure 9.4 illustrates the behavior of the nucleation barrier W ∗ = ΔG ∗ and the
critical cluster size of water at T = 220 K (predicted by CNT and MKNT) as the
vapor approaches the pseudospinodal

ln Spsp,water (T = 220 K) ≈ 5.27

Although CNT predicts smaller critical clusters than MKNT, the CNT barrier is
larger than the MKNT one, which is the manifestation of the fact that the formation
of small clusters is dominated by the microscopic surface tension rather than by the
macroscopic one.
References 159

References

1. H. Gould, W. Klein, Physica D 66, 61 (1993)

2. P.G. Debenedetti, Metastable Liquids (Princeton University Press, Princeton, Concepts and
Principles, 1996)
3. J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28, 258 (1958)
4. J.S. Langer, Ann. Phys. 41, 108 (1967)
5. J.S. Langer, Ann. Phys. 54, 258 (1969)
6. W. Klein, Phys. Rev. Letters 47, 1569 (1981)
7. C. Unger, W. Klein, Phys. Rev. B 29, 2698 (1984)
8. L.D. Landau, E.M. Lifshitz, Statistical Physics (Pergamon, Oxford, 1969)
9. G.A. Korn, T.M. Korn, Mathematical Handbook (McGraw-Hill, New York, 1968)
10. Yu.B. Rumer, M.S. Rivkin, Thermodynamics, Statistical Physics and Kinetics (Nauka, Moscow,
1977) (in Russian)
11. A.J. Bray, Adv. Phys. 43, 357 (1994)
12. J.J. Binney, N.J. Dowrick, A.J. Fisher, M.E.J. Newman, The Theory of Critical Phenomena
(Clarendon Press, Oxford, 1995)
13. S.B. Kiselev, Physica A 269, 252 (1999)
14. M. Struwe, Variational Methods: Application to Nonlinear Partial Differential Equations and
Hamiltonian Systems (Springer, Berlin, 2000)
15. G. Wilemski, J.-S. Li, J. Chem. Phys. 121, 7821 (2004)
16. Z.-G. Wang, J. Chem. Phys. 117, 481 (2002)
17. P.M. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University
Press, Cambridge, 1995)
18. A.Z. Patashinskii, B.I. Shumilo, Sov. Phys. JETP 50, 712 (1979)
19. A.Z. Patashinskii, B.I. Shumilo, Sov. Phys. Solid State 22, 655 (1980)
20. V.K. Schen, P.G. Debenedetti, J. Chem. Phys. 118, 768 (2003)
21. K. Binder, Phys. Rev. A 29, 341 (1984)
22. K.A. Streletzky, Yu. Zvinevich, B.E. Wyslouzil, J. Chem. Phys. 116, 4058 (2002)
23. A. Khan, C.H. Heath, U.M. Dieregsweiler, B.E. Wyslouzil, R. Strey, J. Chem. Phys. 119, 3138
(2003)
24. C.H. Heath, K.A. Streletzky, B.E. Wyslouzil, J. Wölk, R. Strey, J. Chem. Phys. 118, 5465
(2003)
25. Y.J. Kim, B.E. Wyslouzil, G. Wilemski, J. Wölk, R. Strey, J. Phys. Chem. A 108, 4365 (2004)
26. S. Sinha, A. Bhabbe, H. Laksmono, J. Wölk, R. Strey, B. Wyslouzil, J. Chem. Phys. 132,
064304 (2010)
27. V.I. Kalikmanov, J. Chem. Phys. 129, 044510 (2008)
Chapter 10
Argon Nucleation

Argon belongs to the class of so called simple fluids whose behavior on molecular
level can be adequately described by the Lennard-Jones interaction potential
 
σLJ 12  σLJ 6
uLJ (r) = 4εLJ − (10.1)
r r

where εLJ is the depth of the potential and σLJ is the molecular diameter; for argon
εLJ /kB = 119.8 K, σLJ = 3.40 Å [1]. Since argon plays an exceptional role in
various areas of soft condensed matter physics, its equilibrium properties have been
extensively studied experimentally [2], theoretically [3], in computer simulations—
Monte Carlo and molecular dynamics—[4, 5] and by means of the density functional
theory [6, 7].
Among various other issues, argon represents an important reference system for
non-equilibrium studies. In this context the phenomenon of nucleation is of special
significance. In the situation when no theoretical model can claim to be quantitatively
correct in describing nucleation in all substances under various external conditions,
argon can play a role of the test substance for which experimental, theoretical and
simulation efforts can be combined in order to obtain a better insight into the nucle-
ation phenomenon and abilities of various approaches to adequately describe it. This
chapter is aimed at obtaining a unified picture of argon nucleation combining theory,
simulation and experiment.

10.1 Temperature-Supersaturation Domain: Experiments,

Simulations and Density Functional Theory

Early experimental studies of argon nucleation were carried out using various tech-
niques: cryogenic supersonic [8] and hypersonic [9] nozzles and cryogenic shock

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 161

DOI: 10.1007/978-90-481-3643-8_10, © Springer Science+Business Media Dordrecht 2013
162 10 Argon Nucleation

tubes [10–12]. The data obtained in these experiments showed significant scatter
and results of various groups turned out to be inconsistent with each other.
The experimental situation was largely improved in 2006 due to the construction of
the cryogenic Nucleation Pulse Chamber (NPC) [13], and its further development
[14, 15]. This chamber uses a deep adiabatic expansion of the argon–helium mixture
which causes argon nucleation at temperatures below the triple point.
The onset nucleation data obtained in NPC for the temperature range 42–58 K are
reproducible and refer to the estimated nucleation rates 107±2 cm−3 s−1 . An impor-
tant breakthrough in nucleation measurements was achieved by construction of Laval
Supersonic Nozzle (SSN) [16], making it possible to accurately determine the onset
conditions corresponding to significantly higher nucleation rates. Argon nucleation
experiments in SSN [17], carried out in the temperature range 35–53 K which partly
overlaps the NPC range, correspond to higher supersaturations yielding the esti-
mated nucleation rates as high as 1017±1 cm−3 s−1 . In what follows we refer to the
experimental data obtained by these two techniques [14, 15] and [17].
Nucleation is an example of a rare-event process, that is why molecular dynamic
simulations at low temperatures are usually performed for very high supersaturations
in order to get a good statistics of nucleation events. Kraska [18] carried out MD
simulations in the microcanonical ensemble (MD/NVE) in the temperature range
30 K < T < 85 K with the nucleation rates JMD/NVE ∼ 1025 − 1029 cm−3 s−1 .
MD simulations of Wedekind et al. [19] in the canonical ensemble (MD/NVT) are
performed in the temperature range 45 K < T < 70 K with the nucleation rates in the
range of JMD/NVT = 1023 − 1025 cm−3 s−1 . All these simulations yield nucleation
rates which are far beyond experimental values of both NPC and SSN data.
In Chap. 5 we discussed DFT of nucleation and demonstrated its predictions for
Lennard-Jones fluids—see Fig. 5.5. In terms of argon properties these calculations
correspond to relatively high temperatures 83 K < T < 130 K. At this temperature
range the detectable nucleation rates require relatively low supersaturations. The
DFT nucleation rates lie in the range JDFT = 10−1 − 105 cm−3 s−1 . Figure 10.1
illustrates experimental, simulation and DFT studies in the T − S plane indicating
the corresponding typical values of J.
According to Chap. 9, the experimentally achievable upper limit of supersaturation
for nucleation at a temperature T is the pseudospinodal corresponding to the nucle-
ation barrier ΔG∗ ≈ kB T . The MKNT pseudospinodal given by Eq. (9.44) is shown
in Fig. 10.1 by the line labeled “psp”. As it is seen from Fig. 10.1 the SSN experi-
ments at low temperatures 37 K < T < 40 K are carried out in the pseudospinodal
region which implies that one can expect critical nuclei to be nano-sized objects
with the number of molecules close to the coordination number in the liquid phase.
In Fig. 10.2 the pseudospinodal is compared to the estimates of the thermodynamic
spinodal, corresponding to the limit of thermodynamic stability of the fluid. One way
to estimate the spinodal is to use a suitable equation of state (EoS) [24]. Dashed line
in Fig. 10.2 shows the spinodal calculated from the LJ EoS of Kolafa and Nezbeda
10.1 Temperature-Supersaturation Domain 163

MD/NVE
15 25-27
J MD/NVE ~ 10 exp. Iland (NPC)
MD/NVT

psp
exp. Sinha (SSN)
DFT
10 pseudosp. MKNT

J SSN
ln S

SS 16-1
~ 10
N
23-26
J MD/NVT ~10
8
5

J NP JD
NP
C FT ~ 10 2 +_ 3
C ~1
0 7-9

0
40 60 80 100
T (K)

Fig. 10.1 T -S domain of experiments and simulations. Nucleation Pulse Chamber (NPC) experi-
ments [14, 15] (blue squares), Supersonic Nozzle (SSN) experiments [16] (green squares), MD/NVE
simulations [18] (filled squares), MD/NVT simulations [19] (open rhombs) and DFT simulations
[20] (filled triangles). The line labelled “psp” is the MKNT pseudospinodal Eq. (9.44)

15
Argon

pseudospinodal
LJ spinodal
EoS spinodal
10 Spinodal, equilibr. MD
ln S

0
40 60 80 100

T (K)

Fig. 10.2 Pseudospinodal and estimates of the thermodynamic spinodal for argon. Solid line:
pseudospinodal Eq. (9.44); dashed line: the spinodal from the Lennard-Jones equation of state of
Ref. [21]; upper half-filled circles: the spinodal from the simulations of the supersaturated Lennard-
Jones vapor of Ref. [22]; lower half-filled circles: the spinodal from equilibrium simulations of Ref.
[23]. The vertical dashed-dotted line corresponds to the criterion (9.47)
164 10 Argon Nucleation

[21]. Extrapolations below the argon triple point Ttr = 83.8 K are limited because
EoS are usually fitted to experimental data only in the stable region.
Spinodal can be also found in computer simulations of Lennard-Jones fluids. Linhart
et al. [22] performed MD simulations of the supersaturated vapor of a LJ fluid and
obtained the spinodal pressure for the LJ temperature range 0.7 ≤ kB T/εLJ ≤ 1.2. For
argon this range corresponds to 84 K < T < 143 K. Spinodal was estimated by the
appearance of an instantaneous phase separation in the supersaturated vapor increas-
ing the argon density in a series of simulations. Simulations were performed for a
large cut-off radius 10 σLJ and the non-shifted LJ potential. A large cut-off radius
makes it feasible to apply the LJ simulation results to real argon.1 Unfortunately,
these simulations cover only partly the temperature domain of MD [18, 19] and
DFT [20]. Imre et al. [23] estimated the spinodal from the extremes of the tangential
component of the pressure tensor obtained from the simulations of the vapor-liquid
interface. This approach is based on a single equilibrium simulation without any
constraints and is applicable also below the triple point. Figure 10.2 indicates that
theoretical predictions of the pseudospinodal are consistent with the calculations of
thermodynamic spinodal performed by various methods: within the common tem-
perature range the MKNT pseudospinodal lies slightly below the instability points
of simulations and equation of state.
Let us discuss the predictions of various nucleation theories discussed in this book—
CNT (Chap. 3), EMLD-DNT (Chap. 6), MKNT (Chap. 7),—experiment (NPC and
SSN), MD simulations and DFT within the T − S domain bounded from above by
the pseudospinodal and for the temperatures corresponding to the range of validity
of MKNT (Eq. (7.53)): T < 92 K. Thermodynamic properties of argon are presented
in Appendix A.
The behavior of various model parameters is shown in Fig. 10.3. The bulk (macro-
scopic) surface tension θ∞ determines the surface part of the nucleation barrier in
the CNT and EMLD-DNT. It decreases with the temperature as well as the micro-
scopic surface tension θmicro used in MKNT. For all temperatures θ∞ > θmicro . The
difference between them can be substantial: e.g. at T = 70 K: θ∞ (T = 70 K) =
10.68, θmicro (T = 70 K) = 4.96; at higher temperatures this difference decreases.
The dashed line in Fig. 10.3 labeled “LJ” corresponds to the universal form of
γmicro /kB Tc for Lennard-Jones fluids (see Sect. 7.9.2). In view of methodological
reasons the experimental temperature-supersaturation domain does not overlap with
that of MD and DFT as clearly seen from Fig. 10.1. Therefore we perform separate
comparisons: theory versus experiment and theory versus MD and DFT [27].

1 It has been shown that the usually used cut-off radii of 5 σLJ and 6.5 σLJ are sufficient [25], while
2.5 σLJ gives significant deviation in the thermophysical properties [26].
10.2 Simulations and DFT Versus Theory 165

Fig. 10.3 Equilibrium 25

properties of argon:
θ∞ , θmicro , γmicro/kB Tc . Argon
The dashed line labeled 20
“LJ” shows γmicro /kB Tc for
Lennard-Jones fluids accord-
ing to Eq. (7.89) with the 15
universal parameters given by
(7.93)–(7.93)
10 micro

5
LJ micro /kB Tc

0
0.2 0.3 0.4 0.5 0.6 0.7 0.8
T/Tc

10.2 Simulations and DFT Versus Theory

Figure 10.4 shows the ratio of nucleation rate log10 (Jsimul/Jtheor ), where Jsimul is the
nucleation rate found in MD simulations [18] and [19], and Jtheor refers to one of
the theoretical models. Open symbols correspond to Jtheor given by the CNT, and
filled symbols refer to the nonclassical models: MKNT and EMLD-DNT. The dashed
curve is the “ideal line” Jsimul = Jtheory . The agreement between simulations and
the nonclassical models in the whole temperature range is for most cases within 1–2
orders of magnitude, while the CNT rates are on average 3–5 orders of magnitude
lower than the simulation results. Figure 10.4 demonstrates that MKNT predicts a
better temperature dependence of the nucleation rates compared to EMLD-DNT.
An examination of MD results at T = 70 K shows that results obtained in NVE and
NVT simulations show a difference of one order of magnitude. There are two possible
reasons for this discrepancy. Firstly, in the NVT simulations the nucleation rate is
calculated from a mean first passage time analysis (MFPT) [28] while in the NVE
simulations the threshold method is employed. These two methods yield approxi-
mately one order of magnitude difference in the nucleation rate at given conditions
[29]. Since the nucleation rate obtained by the threshold method is larger, it is located
above the MFPT data. Secondly, in the NVE ensemble the latent heat heats up the
system allowing for the natural temperature fluctuations, while in the NVT simula-
tions velocity scaling is applied, which forces the system to stay at a fixed tempera-
ture thereby not allowing temperature fluctuations. Figure 10.5 compares theoretical
predictions with DFT of Ref. [20]. MKNT demonstrates a perfect agreement with the
DFT while both the CNT and EMLD-DNT underestimate DFT data by 3–5 orders
of magnitude. Recalling that nucleation rate is very sensitive to the intermolecular
interaction potential the agreement between MKNT and DFT is quite remarkable
166 10 Argon Nucleation

(a) 8
MD/NVE vs. MKNT (b) 8
S < S psp S < S psp MD/NVT vs. MKNT
MD/NVE vs. CNT
MD/NVT vs. CNT
6 MD/NVE vs. EMLD 6 MD/NVT vs. EMLD

log10 (Jsimul /Jtheor)

log10 (Jsimul /Jtheor)

4 4

2 2

0 0

-2 -2

-4 -4
70 75 80 85 60 65 70 75
T (K) T (K)

Fig. 10.4 MD simulations versus theory. a MD/NVE simulations of Ref. [18] versus theory. Open
circles: CNT, closed circles: MKNT, semi-filled squares: EMLD-DNT; b MD/NVT simulations
of Ref. [19] versus theory. Open triangles: CNT, filled upward triangles: MKNT, filled downward
triangles: EMLD-DNT

Fig. 10.5 DFT calculations 8

of Ref. [20] versus theory. S < Spsp DFT vs. CNT
Open rhombs: CNT, filled DFT vs. MKNT
6
DFT vs. EMLD
rhombs: MKNT, filled stars:
log10 (JDFT /Jtheor )

EMLD-DNT 4

-2

-4
85 90
T (K)

since DFT explicitly uses the interatomic interaction potential, while the MKNT is a
semi-phenomenological model using as an input the macroscopic empirical EoS, the
second virial coefficient, the plain layer surface tension and the coordination number
in the bulk liquid. Note, however, that the amount of available DFT data is insufficient
to formulate firm conclusions about the performance of different theoretical models.

10.3 Experiment Versus Theory

Consider first experiments in the nucleation pulse chamber [14, 15]. The relative
nucleation rates together with the error bars of the experimental accuracy are shown
in Fig. 10.6. It turns out that for argon (being a simple fluid), predictions of the
10.3 Experiment Versus Theory 167

30 MKNT
CNT
EMLD

Log10(Jexp /J theor )
20

10

40 45 50 55 60
T (K)

Fig. 10.6 Argon nucleation experiments in Nucleation Pulse Chamber [14, 15] versus theory: CNT
(open circles), EMLD-DNT (filled squares) [19], MKNT (filled circles)

Fig. 10.7 Volmer plot for argon nucleation in nucleation pulse chamber and supersonic nozzle.
Hexagons: NPC data; diamonds: SSN data. Open symbols (hexagons and diamonds) refer CNT,
closed symbols (hexagons and diamonds) refer MKNT. The solid lines in the CNT graphs are shown
to guide the eye. (Reprinted from Ref. [17] copyright (2010), American Institute of Physics.)
168 10 Argon Nucleation

CNT fail dramatically: the discrepancy with experiment reaches 26–28 orders of
magnitude! This result looks even more surprising taking into account that experi-
mental nucleation points (see Fig. 10.1) are located far from the pseudospinodal. For
other models the results are somewhat better but remain poor: the disagreement with
experiment is 12–14 orders for EMLD-DNT and 4–8 orders for MKNT.
Nucleation experiments in the supersonic nozzle provide a possibility to reach the
vicinity of pseudospinodal thereby entering into the regime with extremely small
critical clusters. Comparison of SSN experiment to theories shows both quantitative
and qualitative differences with respect to the NPC results. Figure 10.7, taken from
Ref. [16], depicts the relative nucleation rate as a function of the inverse temperature
(the so-called Volmer plot). In this form the logarithm of the saturation pressure,
given by the Clapeyron equation (2.14), is approximately the straight line as well as
the lines of constant nucleation rate (for not too high rates). The upper curve (open
hexagons) reproduces the NPC data with respect to CNT (similar to the upper curve
of Fig. 10.6) as a function of inverse temperature. The qualitative difference between
NPC and SSN data is apparent: while the NPC data is a strongly decreasing function
of temperature, the SSN data (open diamonds) is a weakly increasing function of
temperature. Quantitative comparison of SSN data with theories reveals that all SSN
experiments are in perfect agreement with MKNT: the relative nucleation rates lie
within the “ideality domain”
 
Jexp
−1 < log10 <1
JMKNT

References

1. A. Michels et al., Physica 15, 627 (1949)

2. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids (McGraw-Hill,
New York, 1987)
3. J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity (Clarendon Press, Oxford, 1982)
4. J.K. Johnson, J.A. Zollweg, K.E. Gubbins, Mol. Phys. 78, 591 (1993)
5. D. Frenkel, B. Smit, Understanding Molecular Simulaton (Academic Press, London, 1996)
6. R. Evans, Adv. Phys. 28, 143 (1979)
7. R. Evans, Density functionals in the theory of nonuniform fluids. in Fundamentals of Inhomo-
geneous Fluids, ed. by D. Henderson (Marcel Dekker, New York 1992), p. 85
8. B. Wu, P.P. Wegener, G.D. Stein, J. Chem. Phys. 69, 1776 (1978)
9. T. Pierce, P.M. Sherman, D.D. McBride, Astronaut. Acta 16, 1 (1971)
10. M.W. Matthew, J. Steinwandel, J. Aerosol. Sci 14, 755 (1983)
11. R.A. Zahoransky, J. Höschele, J. Steinwandel, J. Chem. Phys. 103, 9038 (1995)
12. R.A. Zahoransky, J. Höschele, J. Steinwandel, J. Chem. Phys. 110, 8842 (1999)
13. A. Fladerer, R. Strey, J. Chem. Phys. 124, 164710 (2006)
14. K. Iland, Ph.D. Thesis, University of Cologne, 2004
15. K. Iland, J. Wölk, R. Strey, D. Kashchiev, J. Chem. Phys. 127, 154506 (2007)
16. S. Sinha, H. Laksmono, B. Wyslouzil, Rev. Sci. Instrum. 79, 114101 (2008)
17. S. Sinha, A. Bhabbe, H. Laksmono, J. Wölk, R. Strey, B. Wyslouzil, J. Chem. Phys. 132,
064304 (2010)
References 169

18. T. Kraska, J. Chem. Phys. 124, 054507 (2006)

19. J. Wedekind, J. Wölk, D. Reguera, R. Strey, J. Chem. Phys. 127, 154516 (2007)
20. X.C. Zeng, D.W. Oxtoby, J. Chem. Phys. 94, 4472 (1991)
21. J. Kolafa, I. Nezbeda, Fluid Phase Equilib. 100, 1 (1994)
22. A. Linhart, C.-C. Chen, J. Vrabec, H. Hasse, J. Chem. Phys. 122, 144506 (2005)
23. A.R. Imre, G. Meyer, G. Hazi, R. Rozas, T. Kraska, J. Chem. Phys. 128, 114708 (2008)
24. T. Kraska, Ind. Eng. Chem. Res. 43, 6213 (2004)
25. M. Mecke, J. Winkelmann, J. Fischer, J. Chem. Phys. 107, 9264 (1997)
26. B. Smit, J. Chem. Phys. 96, 8639 (1992)
27. V.I. Kalikmanov, J. Wölk, T. Kraska, J. Chem. Phys. 128, 124506 (2008)
28. J. Wedekind, R Strey, D. Reguera. J. Chem. Phys. 126, 134103 (2007)
29. R. Römer, T. Kraska, J. Chem. Phys. 127, 234509 (2007)
Chapter 11
Binary Nucleation: Classical Theory

11.1 Introduction

An increase of dimensionality of a problem usually brings about a new physics.

Speaking about nucleation, a step from a single-component to a binary system intro-
duces an additional thermodynamic degree of freedom: the phase equilibrium of a
binary system is characterized by two thermodynamic variables—and not one as
in the single-component case. Due to this feature a binary cluster of an arbitrary
composition in the surrounding binary vapor at the pressure p v and temperature T
has the properties which are different from the p v − T equilibrium properties of
the environment. This consideration shows a crucial role of cluster composition in
nucleation behavior.
Even in equilibrium building of a binary cluster in the vapor, besides the creation
of the gas-liquid interface, is accompanied by the free energy change associated
with the difference in the chemical potential of a molecule inside and outside the
cluster; the latter difference can be both positive and negative depending on cluster
composition. This does not happen in a single-component case, where in equilibrium
a molecule inside and outside cluster has the same chemical potential depending only
on temperature.
In the binary case the free energy of cluster formation forms a surface in the space
of cluster compositions. Similarly, kinetics of binary nucleation is characterized by
an infinite number of nucleation paths. In this chapter we consider the binary clas-
sical nucleation theory (BCNT). Its history dates back to the works of Flood [1],
Volmer [2], Neumann and Döring [3]. The BCNT, as it is known now, is associ-
ated with the classical work of Reiss [4]. Generalizing the CNT of Becker-Döring-
Zeldovich to the binary mixtures, Reiss put forward the kinetic and thermodynamic
arguments to show that the nucleation rate in the binary problem is associated with
the passage over the saddle point of the free energy surface in the space of droplet
compositions.

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 171

DOI: 10.1007/978-90-481-3643-8_11, © Springer Science+Business Media Dordrecht 2013
172 11 Binary Nucleation: Classical Theory

11.2 Kinetics

Kinetics of binary nucleation describes formation of binary clusters at given external

conditions. A cluster with n a particles of component a and n b particles of component
b is denoted as a point in the two-dimensional (n a , n b ) composition space (see
Fig. 11.1). As in the single-component nucleation we assume that
• the elementary process which changes the size of a nucleus is the attachment to it
or loss by it of one molecule of either component a or b; thus, kinetics is governed
by the following reactions:

(n a , n b ) + (1, 0) ↔ (n a + 1, n b )
(n a , n b ) + (0, 1) ↔ (n a , n b + 1)

• if a monomer collides a cluster it sticks to it with probability unity

• there is no correlation between successive events that change the number of par-
ticles in a cluster
The last assumption means that binary nucleation is a Markov process.
The nucleation flux at the point (n a , n b ) is a vector

J = (Ja (n a , n b ), Jb (n a , n b ))

with coordinates Ja and Jb , where Ja (n a , n b ) is a net rate at which (n a , n b )-clusters

become (n a + 1, n b )-clusters, and Jb (n a , n b ) is a net rate at which (n a , n b )-clusters
become (n a , n b + 1)-clusters. The vectorial nature of the flux implies the existence
of a large (in fact, infinite) number of nucleation paths resulting from variety of the
possible directions of J. This makes an important difference with a single-component

Fig. 11.1 Schematic repre-

sentation of binary kinetics on
(n a , n b )-plane
nb
Jb(na,nb)

(na,nb)
nb
Ja(na,nb)

na na
11.2 Kinetics 173

case, in which J is a scalar. From the assumptions made the kinetic equation describ-
ing the evolution of the cluster distribution function ρ(n a , n b , t) becomes

∂ρ(n a , n b , t)
= Ja (n a − 1, n b , t) − Ja (n a , n b , t) + Jb (n a , n b − 1, t) − Jb (n a , n b , t)
∂t
(11.1)
or in differential notations
 
∂ρ(n a , n b , t) ∂ Ja ∂ Jb
=− + (11.2)
∂t ∂n a ∂n b

The last expression manifests the conservation law for the number of particles and
can be rewritten as
∂ρ(n)
= − div J(n) (11.3)
∂t
In the steady state
div J = 0 (11.4)

As usual in the rate theories we write the fluxes along n a and n b axis in terms of the
forward (condensation) and backward (evaporation) rates:

Ja (n a , n b ) = νa A(n a , n b ) ρ(n a , n b , t) − βa A(n a + 1, n b ) ρ(n a + 1, n b , t)

(11.5)
Jb (n a , n b ) = νb A(n a , n b ) ρ(n a , n b , t) − βb A(n a , n b + 1) ρ(n a , n b + 1, t)
(11.6)

Here νi , i = a, b is the impingement rate (per unit surface) of component i, i.e. the
rate of collisions of i-monomers with a unit surface of the cluster; βi , i = a, b is the
evaporation rate per unit surface of the component i, A(n a , n b ) is the surface area
of the (n a , n b )-cluster. The impingement rates for gas-liquid nucleation follow the
ideal gas kinetics (cf. (3.38))

yi p v
νi = √ (11.7)
2π m i kB T

where p v is the total vapor pressure, yi is the molar fraction of component i in

the vapor. As in the single-component theory the evaporation rates are found from
the detailed balance condition at the constrained equilibrium assuming that βi is
independent of the actual vapor pressure of the component i. By definition of con-
strained equilibrium (which throughout this chapter we denote by the subscript “eq”):
νi,eq = νi (cf. Chap. 3). Then from (11.5)–(11.6) the detailed balance condition
Ja = Jb = 0 reads:
174 11 Binary Nucleation: Classical Theory

νa ρeq (n a , n b )A(n a , n b )
βa = (11.8)
ρeq (n a + 1, n b )A(n a + 1, n b )
νb ρeq (n a , n b )A(n a , n b )
βb = (11.9)
ρeq (n a , n b + 1)A(n a , n b + 1)

where ρeq (n a , n b ) is the cluster distribution function in constrained equilibrium.

From the thermodynamic fluctuation theory it can be written as

ρeq (n) = C exp[−βΔG(n)] (11.10)

where ΔG(n) is a minimal (reversible) work required to form the (n a , n b )-cluster

and C is the normalizing factor. Given an appropriate thermodynamic model for
ΔG(n), the most comprehensive way to evaluate the nucleation rate is by summing
all fluxes Ja and Jb in Eqs. (11.5)–(11.9) that cross any arbitrary line joining the n a
and n b axis [5, 6]. At the steady state the resulting nucleation rate must be constant.

11.3 “Direction of Principal Growth” Approximation

Carrying out the kinetic procedure outlined in Sect. 11.2 requires considerable com-
putational effort due to the existence of a large number of nucleation paths. To proceed
with an analytical approach it is then necessary to identify the domain in the (n a , n b )
space bringing the major contribution to the nucleation rate. In its simplest form this
approach requires
1. identification of the “critical point” in the cluster space (corresponding to the
critical cluster),
2. determination of the direction of the flow in the critical point, and
3. making an assumption about the flow in the vicinity of the critical point
To cope with the problem of large number of paths contributing to the overall nucle-
ation rate, Reiss [4] showed that the nucleation rate is primarily determined by the
passage over the saddle point of the free energy surface ΔG(n a , n b ). This approx-
imation is based on the exponential dependence of ρeq on ΔG (recall that in the
single-component case the main contribution to the nucleation rate comes from the
vicinity of the maximum of ΔG(n)). This statement addresses the first question
raised above. Addressing the second one, Reiss suggested that the direction of the
flow at the saddle point is determined by the direction of the steepest descent of the
energy surface at this point, in other words this direction is determined solely by
energetic factors. The latter issue was later revisited by Stauffer [7] who showed that
the direction of the flow in the saddle point is also influenced by kinetics. Discussion
below follows Stauffer’s representation of binary nucleation kinetics [7].
11.3 “Direction of Principal Growth” Approximation 175

Let us start with presenting the kinetic equation in vector notations. Equations (11.5)–
(11.6) with the evaporation coefficients given by (11.8) and (11.9), can be written as


ρ(n)
J = −ρeq (n) F(n) ∇ , n = (n a , n b ) (11.11)
ρeq (n)

where the diagonal matrix F contains the forward collision rates:

 
νa A(n) 0
F(n) =
0 νb A(n)

The steady-state nucleation rate is obtained by integration of Eq. (11.11) along all
possible nucleation paths subject to the boundary conditions
ρ ρ
lim = 1, lim =0 (11.12)
n a ,n b →0 ρeq n a ,n b →∞ ρeq

which are similar to the single-component case (see discussion in Sect. 3.3). Multi-
plying Eq. (11.11) from the left by (1/ρeq )F−1 and taking curl we obtain
 
1
curl (F−1 J) = 0 (11.13)
ρeq

Applying the general identity

curl (a x) = a curl x − x × (∇a)

for
1
a≡ , x ≡ F−1 J
ρeq

we derive using (11.10)

curl (F−1 J) = (F−1 J) × ∇(βΔG) (11.14)

This equation determines the direction of the nucleation flux in any point (n a , n b )
of the cluster space. It implies that the direction of the nucleation flux depends not
only on the geometry of the energy surface ΔG(n a , n b ) but also on the impingement
rates of the components (through the matrix F).
The saddle point n∗ = (n a∗ , n ∗b ) of the free energy surface satisfies
 
∂ΔG  ∂ΔG 
= =0 (11.15)
∂n a n∗ ∂n b n∗
176 11 Binary Nucleation: Classical Theory

nb
y x

nb *
Jb
Ja
*

na * na

Fig. 11.2 Schematic illustration of the direction of principal growth approximation. Dashed lines:
curves of constant Gibbs free energy ΔG(n a , n b ). The origin of the x − y coordinate system corre-
sponds to the saddle point of ΔG. The angle ϕ gives the direction of principal growth determined
by Eq. (11.25); the angle ϕ ∗ is the approximation to ϕ given by Eq. (11.46)

In its vicinity ΔG can be expanded as:

ΔG(n a , n b ) = ΔG(n∗ ) + m a2 Daa + m 2b Dbb + 2m a m b Dab (11.16)

where 
1 ∂ 2 ΔG 
m i = n i − n i∗ , Di j = , i, j = a, b
2 ∂n i ∂n j n∗

At the saddle point two eigenvalues of the symmetric Hessian matrix D have different
signs implying that
det D < 0

It is convenient to introduce a new, rotated, coordinate system x(m a , m b ), y(m a , m b )

with the origin at n∗ and the x axis pointing along the direction of the flow at n∗ :

x = m a cos ϕ + m b sin ϕ, y = −m a sin ϕ + m b cos ϕ (11.17)

where ϕ is the (yet unknown) angle between the x and n a axis (see Fig. 11.2). The
rate components in the new coordinates are:

Jx = Ja cos ϕ + Jb sin ϕ, Jy = −Ja sin ϕ + Jb cos ϕ (11.18)

At the saddle point itself by definition of the rotated system

Jx = Jx∗ (n∗ ), Jy (n∗ ) = 0

11.3 “Direction of Principal Growth” Approximation 177

Fig. 11.3 Schematic illus-

tration of the flux around the
saddle-point in the rotated
coordinate system (x, y). The
flux is along the x-direction x
having a Gaussian form given Jx *
by Eq. (11.19)

Jx
(y
y Jy )
=0

Further BCNT invokes a rather strong Ansatz—the direction of principal growth

approximation: it is assumed that the direction of the flow (not the absolute value!)
remains constant in the entire saddle point region:

Jy = 0 in the saddle point region

Then, the continuity equation (11.4) written in the rotated system yields

∂ Jx (x, y)
=0
∂x
implying that in the saddle point region the absolute value of the flux depends only
on y: Jx = Jx (y). Taking into account that in the same region ΔG is approximately
parabolic, Eq. (11.14) suggests that Jx can be cast in the form:

Jx (y) = Jx∗ e−βW y

2
(11.19)

where Jx∗ is the flux at the saddle point and the dimensionless factor βW describes
the width of the saddle point region (Fig. 11.3). Together with the direction of the
flow, ϕ, it is found by substituting (11.19) into Eq. (11.14) which takes the form of
a linear combination
Qa ma + Qb mb = 0 (11.20)

with
Q a ≡ −w sin3 ϕ − wr sin ϕ cos2 ϕ + r cos ϕ + da sin ϕ (11.21)

Q b ≡ −wr cos3 ϕ − w sin2 ϕ cos ϕ + r db cos ϕ + sin ϕ (11.22)

178 11 Binary Nucleation: Classical Theory

Here
νb Daa Dbb
r= , da = − , db = − (11.23)
νa Dab Dab

and
W
w=− (11.24)
Dab

(We neglected variations of F in the saddle point region). Expression (11.20) is

supposed to be valid in the entire saddle point region with m a and m b varying
independently. This implies
Qa = Qb = 0

The solution of these equations for the two unknowns ϕ and w is:
 1
tan ϕ = s + s2 + r , with s = (da − r db ) (11.25)
2
 
1 tan ϕ + r db
w= (11.26)
sin ϕ cos ϕ tan ϕ + tanr ϕ

Equation (11.25) states that the direction of the flux in the saddle point region is
determined from a combination of energetic and kinetic factors. The steepest descent
approximation of Ref. [4] would give cot(2ϕ) = (db − da )/2. This would agree with
Eq. (11.25) only when r = 1, i.e. when the impingement rates of the two components
are equal. Let us look at the limiting cases of large and small r . From (11.25)
1
tan ϕ = , for νb νa (11.27)
db
tan ϕ = da , for νb νa (11.28)

The remaining unknown quantity in (11.19) is the flux at the saddle point. To find it
let us write the vector equation (11.11) in the rotated system. Since Jy = 0 we are
interested only in the x-component of this equation which reads
 
1 ∂ ρ(x, y)
(F−1 J)x =− (11.29)
ρeq (x, y) ∂x ρeq (x, y)

The boundary conditions (11.12) in the rotated system are:

ρ ρ
lim = 1, lim =0
x→−∞ ρeq x→∞ ρeq

implying that integration of Eq. (11.29) over x gives:

∞ 1
dx (F−1 J)x =1 (11.30)
−∞ ρeq (x, y)
11.3 “Direction of Principal Growth” Approximation 179

The x-coordinate of the vector F−1 J reads (cf. (11.18)):

(F−1 J)x = (F−1 J)a cos ϕ + (F−1 J)b sin ϕ (11.31)

where
Ja Jb
(F−1 J)a = , (F−1 J)b = (11.32)
νa A νb A

Using the standard linear algebra we express from (11.18) the “old” flux coordinates
in terms of the “new” ones, taking into account that Jy = 0:

Ja = Jx cos ϕ, Jb = Jx sin ϕ (11.33)

Substituting (11.33) into (11.31)–(11.32) we find

 
−1 1 νa sin2 ϕ + νb cos2 ϕ
(F J)x = Jx (11.34)
A νa νb

Now it is convenient to introduce the average impingement rate

νa νb
νav = (11.35)
νa sin2 ϕ + νb cos2 ϕ

so that (11.34) takes the form of expressions (11.32)

1
(F−1 J)x = Jx (y) , A = A(x, y) (11.36)
νav A

Let us substitute (11.36) into (11.30); in view of the exponential dependence of ρeq
on x and y we can replace A = A(x, y) by its value A∗ at the saddle point and take
it out from the integral:
 +∞ −1
∗ 1
Jx (y) = νav A dx (11.37)
−∞ ρeq (x, y)

The equilibrium distribution in (x, y) coordinates is:

ρeq (x, y) = C e−βΔG(x,y)

In the saddle-point region ΔG(x, y) has the parabolic form:

βΔG(x, y) = g ∗ + p11 x 2 + 2 p12 x y + p22 y 2

180 11 Binary Nucleation: Classical Theory

where g ∗ = βΔG(0, 0) is the Gibbs free energy at the saddle point,

  
1 ∂ 2 βΔG  1 ∂ 2 βΔG  1 ∂ 2 βΔG 
p11 = , p12 = , p22 =
2 ∂ x 2 (0,0) 2 ∂ x ∂ y (0,0) 2 ∂ y 2 (0,0)

Here p11 < 0, p22 > 0. The integral on the right-hand side of (11.37) reads

+∞ ∗ +∞
1 eg
dx = exp( p22 y 2 ) dx exp[ p11 x 2 + 2 p12 x y]
−∞ ρeq (x, y) C −∞

Gaussian integration gives

+∞ ∗
1 eg π 2 y2
p12
dx = exp + p22 y 2
−∞ ρeq (x, y) C (− p11 ) 4 (− p11 )

Then Eq. (11.37) takes the form

 
∗ −g ∗ (− p11 ) 2
p12
Jx (y) = C νav A e exp − + p22 y 2
(11.38)
π 4 (− p11 )

Comparing it with (11.19) we identify

(− p11 ) −g∗
Jx∗ = C νav A∗ e (11.39)
π
2
p12
βW = + p22 (11.40)
4 (− p11 )

Finally, the total steady-state nucleation rate is given by Gaussian integration of Jx (y)
over y
+∞ π
J= dy Jx (y) = Jx∗ (11.41)
−∞ βW

resulting in
∗
J = K e−g (11.42)
∗
K = Z νav A(n ) C (11.43)

The prefactor K has the form analogous to the prefactor J0 in the single-component
case (cf. (3.54)) in which the impingement rate ν is replaced by νav . The Zeldovich
factor Z determines the shape of the Gibbs free energy surface in the saddle point
region:
1 (∂ 2 ΔG/∂ x 2 )n ∗
Z =− √ (11.44)
2 − det D
11.3 “Direction of Principal Growth” Approximation 181

In the original (n a , n b )-coordinates:

 
1 Daa + 2Dab tan ϕ + Dbb tan2 ϕ 1
Z =− √ (11.45)
2 1 + tan2 ϕ − det D

In simplified approaches the angle ϕ is approximated by the angle ϕ ∗ characterizing

the critical cluster [8], as shown in Fig. 11.2:
 
∗ n ∗b
ϕ = arctan (11.46)
n a∗

Note, that although Eqs. (11.42)–(11.43) are similar to the single-component case,
it is not possible to recover the single-component nucleation rate from it by setting
one of the impingement rates to zero: this would lead to νav = 0. This result is a
manifestation of the general statement concerning the reduction of the dimensionality
of the physical problem. Such a reduction implies the abrupt change of symmetry
which can not be derived by smooth vanishing of one of the parameters of the system.

11.4 Energetics of Binary Cluster Formation

Energetics of cluster formation determines the minimum reversible work ΔG(n a , n b )

needed to form the (n a , n b )-cluster in the surrounding vapor at the constant temper-
ature T and the vapor pressure p v . As in the single-component case we introduce an
arbitrary located Gibbs dividing surface distinguishing between the bulk (superscript
“l”) and excess (superscript “exc”) molecules of each species in the cluster. The state
of the cluster is characterized by the total numbers of molecules n i :

n i = n il + n iexc , i = a, b (11.47)

Each of the quantities in the right-hand side depend on the location of the dividing
surface while their sum can be assumed independent of this location to the relative
accuracy of O(ρ v /ρ l ), where ρ v and ρ l are the number densities in the vapor and
liquid phases. Therefore only n i ≥ 0 are observable physical properties; in this sense
the model quantities n il and n iexc can be both positive or negative.
In a unary system (see Sect. 3.2) we chose the equimolar dividing surface char-
acterized by zero adsorption n exc = 0. This choice made it possible to deal only
with the bulk numbers of cluster molecules. For a mixture, however, it is impossible
to choose a dividing surface in such a way that all excess terms n iexc vanish [9].
This is the reason for occurrence of the surface enrichment—preferential adsorption
of one of the species relative to the other. As a result the composition inside the
droplet can be different from that near its surface. For binary (and in general, multi-
component) nucleation problem introduction of the Gibbs surface is a nontrivial issue.
182 11 Binary Nucleation: Classical Theory

Thermodynamic considerations [10, 11] analogous to those of Sect. 3 yield a binary

mixture analogue of Eq. (3.19):
     
ΔG = ( p v − p l )V l +γ A+ n il μil ( p l ) − μiv ( p v ) + n iexc μiexc − μiv ( p v )
i=a,b i=a,b
(11.48)
Here p l is the pressure inside the cluster,

Vl = n il vil
i

is the cluster volume, vil is the partial molecular volume of component i in the
liquid phase (see Appendix D), A is the surface area of the cluster calculated at
the location of the dividing surface; γ is the surface tension at the dividing surface.
Equation (11.48) presumes that formation of a cluster does not affect the surround-
ing vapor and therefore the chemical potential of a molecule in the vapor remains
unchanged. Within the capillarity approximation the bulk properties of the cluster
are those of the bulk liquid phase in which liquid is considered incompressible. This
yields:
μil ( p l ) = μil ( p v ) + vil ( p l − p v ) (11.49)

Then
     
n il μil ( p l ) − μiv ( p v ) = n il μil ( p v ) − μiv ( p v ) − ( p v − p l ) V l
i i

The last term in this expression cancels the first term in (11.48), leading to
   
ΔG = γ A − n il Δμi + n iexc μiexc − μiv ( p v ) (11.50)
i i

where
Δμi ≡ μiv ( p v ) − μil ( p v ) (11.51)

is the difference in the chemical potential of a molecule of component i between the

vapor and liquid phases taken at the vapor pressure p v . The chemical potential of a
molecule of species i in the liquid phase μil ( p v , xbl ) depends on the bulk composition
of the (n a , n b )-cluster
nl
xbl = l b l (11.52)
na + nb

The last term in (11.50) contains the excess quantities. The chemical potentials μiexc
refer to a hypothetical (non-physical) surface phase and therefore can not be measured
in experiment or predicted theoretically. Therefore, one has to introduce an Ansatz
for them which serves as a closure of the model [12, 13]. The diffusion coefficient
11.4 Energetics of Binary Cluster Formation 183

in liquids is much higher than in gases (see e.g. [14]). This implies that diffusion
between the surface and the interior of the cluster is much faster than diffusion
between the surface and the mother vapor phase surrounding it. Hence, it is plausible
to assume equilibrium between surface and the interior (liquid) phase of the cluster,
resulting in the equality of the chemical potentials

μiexc = μil ( p l , xbl ) (11.53)

Following [13] the Ansatz (11.53) can be termed “the equilibrium μ conjecture”.
Using (11.49) we may write
 
μiexc −μiv ( p v ) = μil ( p v ) − μiv ( p v ) +vil ( p l − p v ) ≡ −Δμi +vil ( p l − p v ) (11.54)

Substituting (11.54) into (11.50) and using Laplace equation we obtain an alternative
form of the Gibbs formation energy of the binary cluster

  2γ (xbl ) 
ΔG = γ (xbl ) A − n il + n iexc Δμi + n iexc vil (11.55)
r
i    i
ni

where γ (xbl ) is the surface tension of the binary solution of liquid composition xbl .
Here the second term contains the total numbers of molecules in the cluster, and not
the bulk liquid numbers n il . At the same time all the thermodynamic properties are
functions of the bulk composition xbl and not the total composition

xbtot = n b /(n a + n b ) (11.56)

Those two are not identical: xbl = xbtot . The same refers to the volume and the surface
area of the cluster
 2/3
4π 3  l l 
V =
l
r = n i vi , A = (36π ) 1/3
n il vil (11.57)
3
i i

since by definition any dividing surface has a zero thickness. An important feature
of the binary problem is the presence of the last term in the Gibbs energy (11.55).

11.5 Kelvin Equations for the Mixture

In the previous section we derived the Gibbs energy of formation for an arbitrary
binary cluster. Consider now the critical cluster corresponding to the saddle point
of ΔG:
184 11 Binary Nucleation: Classical Theory

∂ΔG 
 = 0, j = a, b (11.58)
∂n lj 
n∗

∂ΔG 
 = 0, j = a, b (11.59)
∂n exc 
j n∗

Since the formation of a cluster does not affect the vapor properties, we have

dμiv = dp v = 0 (11.60)

Equations (11.50) and (11.58)–(11.59) then result in:

 ∂μl ( p v ) 
∂A ∂γ exc ∂μi
exc
i
γ + A −Δμ j + n l
i + n i = 0, j = a, b (11.61)
∂n lj ∂n lj i
∂n lj i
∂n lj

∂γ  ∂μiexc  l ∂μil ( p v )
A −(μvj −μexc
j )+ n iexc exc + ni = 0, j = a, b (11.62)
∂n j
exc ∂n j ∂n exc
j
i i

Gibbs adsorption equation (2.29) for the mixture at constant T reads


Adγ + n iexc dμiexc = 0 (11.63)
i

We rewrite it as

∂γ exc ∂μi
exc
A + n = 0, α = l, exc (11.64)
∂n αj i
∂n αj
i

Gibbs-Duhem equation (2.5) for the bulk liquid phase of the mixture at constant T is

−V l d p l + n il dμil ( p l ) = 0
i

Using (11.49) and (11.60) it can be presented as


n il dμil ( p v ) = 0 (11.65)
i

resulting in
 ∂μil ( p v )
n il = 0, α = l, exc (11.66)
∂n αj
i
11.5 Kelvin Equations for the Mixture 185

Substituting (11.64) and (11.66) into Eqs. (11.61)–(11.62) we obtain for the saddle
point:

∂A
γ − Δμ j = 0 (11.67)
∂n lj
μexc
j (p ) = μj(p )
l v v
(11.68)

The meaning of the second equality is transparent. As we discussed, for any cluster
the molecules belonging to the dividing surface are in equilibrium with the interior of
j ( p ) = μ j ( p ). For the critical cluster this condition is supplemented
the cluster μexc l l l

by the condition of unstable equilibrium with surrounding vapor yielding

j (p ) = μj(p ) = μj(p )
μexc l l l v v

which is exactly the equation (11.68).

The first equality (Eq. (11.67)) is nontrivial. In view of (11.57) it reads

2γ val
− Δμa + =0 (11.69)
r∗
2γ v l
−Δμb + ∗ b = 0 (11.70)
r
This set of equations is known as the Kelvin equations for a mixture; they determine
the composition and the size of the critical cluster. In particular, the critical cluster
composition satisfies
Δμa Δμb
= l (11.71)
va
l vb
∗
Once the composition xbl is determined, the critical radius is given by

2γ v lj
r∗ = (11.72)
Δμ j

Here the surface tension γ refers to the dividing surface of the radius r ∗ .
Substituting the Kelvin equations into (11.55) we find the Gibbs free energy at the
saddle point (or, equivalently, the nucleation barrier):
  
 2γ v l 2
∗ l∗
ΔG = γ A + ni − ∗i =γ A − Vl
r r
i

resulting in:
1
ΔG ∗ = γA (11.73)
3
186 11 Binary Nucleation: Classical Theory

where both quantities on the right-hand side depend on the critical cluster composi-
tion. Within the phenomenological approach droplets are considered to be relatively
∗
large, so that one can replace γ by γ∞ (xbl )—the surface tension of the plain layer
of the binary vapor- binary liquid system when the composition of the bulk liquid is
that of the critical cluster.
Note that if within the capillarity approximation we would set n aexc = n exc
b = 0,
then the terms with the excess quantities in the free energy would disappear and
the Gibbs adsorption equation (11.63) can not be invoked. The resulting equations
for the critical cluster would contain then the uncompensated term with the surface
tension derivative:

2γ∞ v lj ∂γ∞
− Δμ j + +A = 0, j = a, b (11.74)
r∗ ∂ x tot
j

The inconsistency of this result becomes obvious if we recall that in equilibrium

μil ( p l ) = μiv ( p v ) leading to Eqs. (11.69)–(11.70). That is why the phenomenological
binary nucleation model with the critical cluster given by Eqs. (11.69)–(11.70) is
called the internally consistent form of the BCNT ([10, 11]).

11.6 K-Surface

In the general expression for the Gibbs energy (11.50) (or its equivalent form (11.55))
the bulk and excess numbers of molecules are not specified and treated as independent
variables. Their specification is related to a choice of the dividing surface for a cluster.
This is not a unique procedure. One of the appropriate options is the equimolar surface
for the mixture, termed also the K -surface [13, 15], defined through the requirement

n iexc vil = 0 (11.75)
i=a,b

This choice ensures that the macroscopic surface tension is independent of the
curvature of the drop; however it does depend on the composition of the cluster.
This can be easily seen if we present Eq. (11.55) in the form
 
ΔG = − n il + n iexc Δμi + γ (xbl ; r ) A
i   
ni

where we introduced the curvature dependent surface tension

 
2 
γ (xbl ; r ) = γ (xbl ) 1+ n iexc vil
rA
i
11.6 K-Surface 187

For the K -surface the second term in the curl brackets vanishes implying that
γ (xbl ; r ) = γ (xbl ). Laaksonen et al. [15] showed that the K -surface brings together
various derivations of the free energy of cluster formation in the classical theory—
due to Wilemski [10], Debenedetti [16] and Oxtoby and Kashchiev [17]. In what
follows we adopt the K-surface formalism for binary clusters. Usually, the partial
molecular volumes of both components in the liquid phase are positive, implying
from (11.75) that the excess quantities n iexc have different signs. As we mentioned
already, a negative value of one of n iexc is not unphysical as soon as the total num-
ber n i (the quantity which does not depend on the choice of dividing surface) is
nonnegative.
Within the K -surface formalism, the volume of the cluster and its surface area can
be expressed either in terms of n il or in terms of the total numbers of molecules
n itot ≡ n i . Equation (11.57) reads:
 
Vl = n il vil = n itot vil (11.76)
i i
 2/3  2/3
 
A = (36π ) 1/3
n il vil = (36π ) 1/3
n itot vil (11.77)
i i

It is important to stress that consistent evaluation of thermodynamic properties

requires that μil , vil , γ are functions of the bulk composition of the cluster xbl —
and not the total composition xbtot ) [10]—and the difference matters. Combination of
(11.75) with Gibbs adsorption equation (11.63) results in the set of linear equations
for the excess numbers

n aexc val + n exc

b vb = 0
l
(11.78)
n aexc dμal ( p l , xbl ) + n exc
b dμb ( p , x b ) +
l l l
A dγ = 0 (11.79)

From the incompressibility of the liquid phase and Laplace equation

2γ vil
μil ( p l , xbl ) = μil ( p v , xbl ) + (11.80)
r

Excluding n aexc from (11.78), we obtain from (11.79) and (11.80)

  
vl 2γ l vbl
n exc
b − bl dμa + dμb + v d ln l + A dγ = 0 (11.81)
va r b va

where for brevity we used the notation μi ≡ μil ( p v , xbl ). Partial molecular volumes
can be written as (see Appendix D):
188 11 Binary Nucleation: Classical Theory

1 l
vil = η (11.82)
ρl i

with ηil given by (D.7)–(D.8). Combining (11.81) and (11.82) with the Gibbs-Duhem
equation 
xil dμi = 0 (11.83)
i

we find
−1
∂γ 1 ∂μa 2γ ηal ∂ ln(ηal /ηbl )
n aexc = −A l + (11.84)
∂ xb xbl ηbl ∂ xbl r ρl ∂ xbl
−1
∂γ 1 ∂μb 2γ ηbl ∂ ln(ηbl /ηal )
n exc
b = −A l + (11.85)
∂ xb xal ηal ∂ xbl r ρl ∂ xbl

To perform calculations of n iexc according to (11.84)–(11.85) we need to specify

∂μil /∂ xbl . This can be done using a correlation for activity coefficients [14]. To a
good approximation the activity coefficients in the liquid can be set equal to unity,
resulting in

∂μal 1
= −kB T l (11.86)
∂ xb
l x a
∂μlb 1
= kB T (11.87)
∂ xbl xbl

It is easy to see that these expressions satisfy the Gibbs-Duhem relation (11.83).
A more accurate approximation can be formulated using one of the more sophisticated
models for activity coefficients—e.g. van Laar model discussed in Sect. 11.9.1.
Finally, the terms ∂ ln(ηil /ηlj )/∂ xbl in (11.84)–(11.85) are found from (D.7)–(D.9):

∂ ln(ηal /ηbl ) 1
= (τ2 − τ12 ) (11.88)
∂ xbl ηal ηbl
∂ ln(ηbl /ηal ) ∂ ln(ηal /ηbl ) 1
=− =− (τ2 − τ12 ) (11.89)
∂ xbl ∂ xbl ηalηbl

where
∂ ln ρ l ∂τ1
τ1 = , τ2 = l
∂ xbl ∂ xb

If the excess numbers, derived using this procedure, turn out to be not small compared
to the bulk numbers then the classical theory, probably, falls apart [8]. This happens in
11.6 K-Surface 189

the mixtures with strongly surface active components exhibiting pronounced adsorp-
tion on the K -surface. An example of such a system is the water/ethanol mixture
(discussed in Sect. 11.9.3).

11.7 Gibbs Free Energy of Cluster Formation

Within K -Surface Formalism

If we choose the K -dividing surface, the Gibbs energy of cluster formation (11.55)
becomes
  
ΔG = γ (xbl , T ) A − n il + n iexc μiv ( p v , T ) − μil ( p v , xbl , T ) (11.90)
i   
ni

Within the K -surface formalism for each pair of bulk cluster molecules (n al , n lb ) the
excess quantities are constructed

n aexc (n al , n lb ), n exc
b (n a , n b )
l l

implying that n il and n iexc are not any more the independent quantities. Therefore, the
saddle point of ΔG has to be determined in the space of independent variables—the
total number of molecules:
∂ΔG
= 0, j = a, b (11.91)
∂n j

leading again to the Kelvin equations (11.69)–(11.70). From the first sight it may
seem that to construct ΔG(n a , n b ) according to Eq. (11.90) one needs to know
only the total numbers of molecules n a and n b ; however this is not true, since the
thermodynamic properties—the surface tension γ , partial molecular volumes vil and
chemical potentials μil —depend on the bulk composition xil rather than on xitot . So,
in order to calculate ΔG(n a , n b ), one has to know also the bulk numbers n al and n lb
(and therefore the bulk composition xbl ), giving rise to these n a and n b .
Equation (11.90) formally coincides with the BCNT expression
  
ΔG BCNT = γ (xbtot , T ) A − n i μiv ( p v , T ) − μil ( p v , xbtot , T ) (11.92)
i

except for the argument of μil and γ . This means that the standard BCNT does not
discriminate between the bulk and excess molecules in the cluster and thus does not
account for adsorption effects: a cluster in this model is a homogeneous object.
190 11 Binary Nucleation: Classical Theory

The Gibbs free energy contains the chemical potentials of the species which are not
directly measurable quantities. Therefore, it is necessary to cast ΔG in an approxi-
mate form containing the quantities which are either measurable or can be calculated
from a suitable equation of state. Let us first recall that μiv is imposed by external
conditions and does not depend on the composition of the cluster; at the same time
μil is essentially determined by the cluster composition. For this quantity using the
incompressibility of the liquid phase we may write

μil ( p v , xbl ) = μil ( p0 , xbl ) + vil ( p v − p0 ) (11.93)

where p0 is an arbitrary chosen reference pressure. Let us choose it from the condition
of bulk (xbl , T )-equilibrium. The latter is the equilibrium between the bulk binary
liquid at temperature T having the composition xbl and the binary vapor. Fixing
xbl and T , we can calculate from the EoS the corresponding coexistence pressure
p coex (xbl , T ) and coexistence vapor fractions of the components yicoex (xbl , T ). Now
we choose p0 as
p0 = p coex (xbl , T )

which transforms (11.93) into

   
μil ( p v , xbl ) = μil p coex (xbl ) + vil p v − p coex (11.94)

For the gaseous phase we assume the ideal mixture behavior, implying that each
component i behaves as if it were alone at the pressure piv = yi p v . Then the chemical
potential of component i in the binary vapor is approximately equal to its value for
the pure i-vapor at the pressure piv :

μiv ( p v , yi ) ≈ μi,pure
v
( piv ) (11.95)

The latter can be written as

piv
μi,pure
v
( piv ) = μi,pure
v
( pi ) + viv ( p) d p (11.96)
pi

where viv ( p) is the molecular volume of the pure vapor i and pi is another arbitrary
reference pressure. Let us choose it equal to the partial vapor pressure of component
i at (xbl , T )-equilibrium:

pi = yicoex (xbl ) p coex (xbl ) ≡ picoex (xbl ) (11.97)

Applying the ideal gas law to (11.96), we obtain

11.7 Gibbs Free Energy of Cluster Formation Within K -Surface Formalism 191

yi p v
μiv ( p v , yi ) = μi,pure
v
( picoex ) + kB T ln (11.98)
yicoex (xbl ) p coex (xbl )

In (xbl , T )-equilibrium μi,pure

v ( picoex ) = μil ( p coex ). Subtracting (11.94) from (11.98),
we find
  
yi p v p coex vil pv
βΔμi = ln coex l coex l − −1
yi (xb ) p (xb ) kB T p coex

The quantity in the curl brackets is proportional to the liquid compressibility factor,
which is a small number (∼10−6 − 10−2 ), implying that the second term can be
neglected in favor of the first one:

yi p v
βΔμi = ln (11.99)
yicoex (xbl ) p coex (xbl )

Substituting (11.99) into (11.90), we deduce the desired approximation for the free
energy containing now only the measurable quantities

 yi p v
βΔG(n a , n b ) = − n i ln + β γ (xbl ) A (11.100)
i
yicoex (xbl ) p coex (xbl )

This result coincides with the BCNT expression [4] except for the argument xbl of
the coexistence properties:

  
yi p v
βΔG BCNT (n a , n b ) = − n i ln + β γ (xbtot ) A
yicoex (xbtot ) p coex (xbtot )
i
(11.101)
Equations (11.100)–(11.101) imply that for an arbitrary (n a , n b )-cluster the bulk
(logarithmic) terms can be both positive and negative depending on the cluster com-
position. Consequently, even at equilibrium conditions (say, at a fixed p v and T ) the
free energy of formation of a cluster with a composition, different from the equi-
librium bulk liquid composition at given p v and T , will contain a non-zero bulk
contribution. This situation is distinctly different from the single-component case,
for which formation of an arbitrary cluster in equilibrium (saturated) vapor is asso-
ciated only with the energy cost to build its surface and the bulk contribution to ΔG
vanishes.
192 11 Binary Nucleation: Classical Theory

11.8 Normalization Factor of the Equilibrium Cluster

Distribution Function

To accomplish the formulation of the theory it is necessary to determine the “nor-

malization factor” C of the equilibrium cluster distribution function entering the
prefactor K of the nucleation rate (see Eq. (11.43)). Its form is not “dictated” by the
model presented in this chapter and remains a matter of controversy. In his seminal
paper [4] Reiss proposed the following expression:

CReiss = ρav + ρbv (11.102)

where ρiv is the number density of monomers of species i in the vapor. Such a choice,
however, violates the law of mass action (recall the similar feature of the single-
component CNT discussed in Sect. 3.6). Another difficulty associated with (11.102),
is that the number density of pure a-clusters, ρeq (n a , 0), becomes proportional to
the number density of b-monomers and vice versa. Wilemski and Wyslouzil [18]
proposed an alternative form of C which is free from these inconsistencies. It was
suggested that C should depend on the cluster composition:
 x tot  x tot
CWW = ρav,coex (xatot ) a ρbv,coex (xatot ) b , xatot + xbtot = 1 (11.103)

where ρiv, coex (xitot ) is the equilibrium number density of monomers of species i in
the binary vapor at coexistence with the binary liquid whose composition is xitot .
Another possibility discussed by the same authors, is the self-consistent classical
(SCC) form of C based on the Girshick-Chiu ICCT model (3.103):
  x tot  x tot
CWW,SSC = exp xatot θ∞,a + xbtot θ∞,b ρav, coex (xatot ) a ρbv, coex (xatot ) b
(11.104)
where θ∞,i (T ) is the reduced macroscopic surface tension of pure component i.
Mention, that Eqs. (11.103), (11.104) are just two of possible choices of the
prefactor C.

11.9 Illustrative Results

11.9.1 Mixture Characterization: Gas-Phase-and

Liquid-Phase Activities

Experimental results in binary nucleation are frequently expressed in terms of the

gas-phase- and liquid-phase activities of the mixture components. The gas-phase
activity of component i measures the deviation of the vapor of component i from
equilibrium at a certain reference state. If we characterize the state of component i
11.9 Illustrative Results 193

in the vapor by its chemical potential μiv , the gas-phase activity of component i is
defined through  
Aiv = exp β(μiv − μi,0 v
) (11.105)

v is the value of μv at the reference state which has yet to be specified.

where μi,0 i
Let us assume that the binary vapor at the total pressure p v is a mixture of ideal
gases with the partial vapor pressures piv = yi p v . Then, the chemical potential of
component i in the binary vapor is given by Eqs. (11.95)–(11.96):
 
yi p v
μiv ( piv ) = μi,pure
v
( pi ) + kB T ln (11.106)
pi

The most frequent choice of the reference state is the vapor-liquid equilibrium of
pure component i at temperature T (assuming, of course, that such a state exists!).
Then
pi = psat,i (T ), μi,pure
v
( pi ) = μi,0
v
= μsat,i (11.107)

where μsat,i (T ) and psat,i (T ) are the saturation values of the chemical potential and
pressure of component i, respectively. With this choice (11.105) yields

piv
Aiv = (11.108)
psat,i (T )

This form contains the experimentally controllable parameters (in contrast to

(11.105)) which makes it a convenient tool for representation of experimental results.
In the single-component case the gas-phase activity coincides with the usual defini-
tion of the supersaturation A v = S = p v / psat .
Now, let us consider a binary mixture with a bulk liquid composition xi at the temper-
ature T in equilibrium with the binary vapor. The partial vapor pressure of component
i over the bulk binary liquid at (xi , T )-equilibrium can be written as

picoex (xi , T ) = Γi (xi , T ) xi psat,i (T ) (11.109)

Here, Γi (xi , T ) is called the activity coefficient of component i. If a mixture is ideal,

which means that the compositions of the bulk liquid and bulk vapor are identical, then
Γi = 1; this choice was employed in Eqs. (11.86)–(11.87). For non-ideal mixtures
Γi (xi , T ) = 1. Hence, the activity coefficients describe the degree of non-ideality of
the system. A microscopic origin of the non-ideality is interaction between molecules
of different species.
A number of empirical correlations for activity coefficients is known in the literature
[14]. One of the widely used correlations is given by van Laar model (see e.g. [19])
194 11 Binary Nucleation: Classical Theory

AL
ln Γa =  2
A L xb
1+ B L xa

BL
ln Γb =  2
B L xa
1+ A L xb

where the van Laar constants A L and B L are determined from the equilibrium vapor
pressure measurements.
The liquid-phase activity of component i is defined as

picoex (xi , T )
Ail = = Γi xi (11.110)
psat,i (T )

This quantity describes the influence of the bulk liquid composition on the equi-
librium vapor pressure of components. For an ideal mixture Γi = 1, yielding
Ai,l ideal = xi , and for a single-component case (which is equivalent to the ideal
mixture with xi = 1): Ail = 1. In terms of activities Eqs. (11.100)–(11.101) read

 Aiv
βΔG(n a , n b ) = − n i ln + β γ (xbl ) A (11.111)
i
Ail (xil )
 Aiv
βΔG BCNT (n a , n b ) = − n i ln + β γ (xbtot ) A (11.112)
i
Ail (xitot )

It is important to mention that if component i is supercritical at temperature T the

choice of the reference state according to (11.107) becomes inappropriate and one
has to resort to Eq. (11.97).

11.9.2 Ethanol/Hexanol System

Ethanol–hexanol system is a natural candidate to test predictions of the BCNT against

experiment. An important feature of this system is that to a high degree of accuracy
it represents the ideal liquid mixture, which makes it possible to set Γi = 1. Further-
more, the surface tensions of pure ethanol and hexanol are nearly identical which
implies that the adsorption effects (surface enrichment) can be neglected. These
observations justify the use of the BCNT with the free energy of cluster formation
given by

  
Aiv
βΔG BCNT (n a , n b ) = − n i ln + β γ (xbtot ) A (11.113)
xitot psat,i (T )
i
11.9 Illustrative Results 195

Fig. 11.4 Nucleation rates for

ethanol–hexanol mixture at
T = 260 K as a function of the
mean vapor phase activity a
defined through Eq. (11.114).
Squares: experiment of Strey
and Viisanen [19]; solid
lines—BCNT with Stauffer’s
expression for K . Labels are
relative activities (11.115)
(Reprinted with permission
from Ref. [18], copyright
(1995), American Institute of
Physics.)

The nucleation rate is given by Eqs. (11.42)–(11.43) with the Stauffer form of the
prefactor K , and C = CReiss . In Fig. 11.4 we compare BCNT with the experimental
results of Strey and Viisanen [19] for nucleation of ethanol–hexanol mixture in argon
as a carrier gas at T = 260 K. Thermodynamic parameters used in calculations are
taken from Table I of Ref. [19]. The rates are plotted against the mean vapor phase
activity 
a= (AEv )2 + (AHv )2 (11.114)

where AEv and AHv are the ethanol and hexanol vapor phase activities, respectively.
The labels in Fig. 11.4 indicate the activity fractions

AHv
y= (11.115)
AEv + AHv

The line y = 0 corresponds to the pure ethanol nucleation and y = 1—to the pure
hexanol nucleation. As one can see, BCNT is in good agreement with experiment for
y < 0.9 but substantially underpredicts the experimental data for y → 1, i.e. at the
pure hexanol limit. It is instructive to present the same data as an activity plot. The
latter is the locus of gas phase activities of components required to produce a fixed
nucleation rate at a given temperature. Figure 11.5 is the activity plot corresponding
to the nucleation rate J = 107 cm−3 s−1 . The difference between the Stauffer from
of K and that of Reiss is quite small indicating that for this system the direction
of cluster growth at the saddle point is to a good approximation determined by the
steepest descend of the Gibbs free energy.
196 11 Binary Nucleation: Classical Theory

Fig. 11.5 Activity plot for ethanol–hexanol nucleation at T = 260 K. Activities a E = AEv and
a H = AHv correspond to the nucleation rate J = 107 cm−3 s−1 . Squares: experiment of Strey and
Viisanen [19]; solid lines—BCNT with Stauffer’s expression for K ; short dashed line—BCNT with
Reiss expression for K ; long dashed line—BCNT with SCC form of the prefactor C (11.104) and
Stauffer’s expression for K (Reprinted with permission from Ref. [18], copyright (1995), American
Institute of Physics.)

11.9.3 Water/Alcohol Systems

The classical theory is quite successful in predictions of nucleation in fairly ideal

mixtures. However, for mixtures showing non-ideal behavior with strong segregation
effects predictions of the BCNT may lead to quantitatively or even qualitatively
wrong results. This is in contrast to the single-component case where the CNT can
be quantitatively in error, but remain qualitatively correct.
To illustrate the situation, let us consider water/alcohol mixtures. It was found that
for these systems BCNT predicts the decrease of the nucleation rate when the vapor
density is increased [11, 20–22]. The fact that such a behavior is unphysical can be
most clearly seen by studying the activity plot. According to the nucleation theorem
for a binary system (4.22)
 
∂ ln J
Δn i,c = , i = a, b (11.116)
∂(βμiv ) T

where n a,c and n b,c are the (total) numbers of molecules of components in the critical
cluster (we neglected the contribution of the prefactor K which is between 0 and 1).
Recalling the definition of gas phase activities, this expression can be written as
11.9 Illustrative Results 197
 
∂ ln J
Δn a,c = (11.117)
∂(ln Aav ) A v
  b
∂ ln J
Δn b,c = (11.118)
∂(ln Abv ) A v
a

Using the general identity

     
∂A ∂B ∂C
= −1
∂B C ∂C A ∂A B

in which A ≡ ln J, B ≡ ln Aav , C ≡ ln Abv , we have

     
∂ ln J ∂ ln Aav ∂ ln Aav
= −1 (11.119)
∂ ln Aav A v ∂ ln Abv ln J ∂ ln J A v
b b

The first and the third term on the left-hand side of this expression can be written
using Eq. (11.116):
 
∂ ln Aav 1
Δn a,c = −1 (11.120)
∂ ln Abv ln J,T Δn b,c

resulting in  
∂ ln Aav Δn b,c
=− (11.121)
∂ ln Abv ln J,T Δn a,c

The left-hand side of (11.121) gives the slope of the activity plot ln Aav = f (ln Abv ).
Since Δn a,c , Δn b,c > 0 this slope should be negative:
 
∂ ln Aav
<0 (11.122)
∂ ln Abv ln J,T

Nucleation theorem is a general statement independent of the model, implying that

(11.122) must be true for all binary mixtures. For the ethanol–hexanol system this
requirement is satisfied—as clearly seen from Fig. 11.5.
Figure 11.6 shows the activity plot—experimental and theoretical—for nucleation
in the water/ethanol mixture at T = 260 K corresponding to the nucleation rate
J = 107 cm−3 s−1 . Experimental data of Viisanen et al. [21] (shown by points) are
in agreement with the requirement (11.122). Meanwhile, the BCNT curve (solid
line) shows the increasing part— a “hump”—(featured also by other water/alcohol
systems (see [11] and reference therein)) which violates the requirement (11.122).
This unphysical “hump” corresponds to one of the Δn i,c (let it be Δn a,c ) being
negative, while the other one is positive. Then, from (11.117) we would get
198 11 Binary Nucleation: Classical Theory

Fig. 11.6 Activity plot for the water/ethanol mixture corresponding to the nucleation rate J =
107 cm−3 s−1 and T = 260 K. Aw,g and Ae,g are the gas-phase activities of water and ethanol,
respectively. Points: experiment of Viisanen et al. [21], full line: BCNT predictions (Reprinted with
permission from Ref. [8], copyright (2006), Springer-Verlag.)

 
∂ ln J
<0
∂ ln Aiv A v ,T
j

Since Aiv is proportional to the vapor pressure (see Eq. (11.108)), the last inequality
would mean the decrease of the nucleation rate when the vapor density is increased.
The origin of this unphysical behavior lies in the prediction of the critical cluster
composition: BCNT predicts very water-rich critical clusters. Since the surface ten-
sion of pure water is much higher than that of the pure ethanol, the resulting surface
tension of the mixture becomes very high yielding low nucleation rates. These results
show that BCNT fails in describing the behavior of surface enriched nuclei. For this
system the adsorption effects, not taken into account by the BCNT, play an important
role: surface excess numbers n iexc turn out to be large and fluctuating.

11.9.4 Nonane/Methane System

In a number of practically relevant situations one deals with gas–liquid nucleation in

a mixture, in which one of the components, say, component b, is supercritical, i.e.
its critical temperature Tc,b is lower than the nucleation temperature T . This implies
that should it be pure, it could not nucleate. In the absence of a carrier gas nucleation
takes place inside the vapor-liquid coexistence region of the binary system and can be
induced by decreasing the total pressure. This phenomenon, termed the retrograde
nucleation, occurs in a number of applications, e.g. during production and processing
of natural gas [23].
Schematically the process is depicted in Fig. 11.7. The gaseous a − b mixture is
initially outside the coexistence region at the state characterized by the total pressure
p0 , temperature T0 and composition y = (ya , yb ). After a fast, usually adiabatic,
expansion the mixture is brought inside the coexistence region to a state with the
11.9 Illustrative Results 199

Lf = 0 (p0,T0,y)
-
-
p -
-

coexistence -
-
- -
region (pv ,T,y) vapor
vapor + liquid

Fig. 11.7 Schematic representation of retrograde nucleation of a binary mixture; ( p0 , T0 , y) is the

initial gaseous state, ( p v , T, y) is the state after the expansion of the mixture, located inside the
coexistence region; T > Tc,b . The boundary of the coexistence region corresponds to the liquid
fraction L f = 0

total pressure p v < p0 and temperature T < T0 ; the latter is characterized by the
equilibrium values of the thermodynamic parameters: the chemical potentials of
the species and their vapor and liquid molar fractions. The actual vapor composition
differs from the equilibrium one at the same p v and T showing that the mixture finds
itself in a nonequilibrium state. If p v is sufficiently high, the supercritical component
not only removes the latent heat (acting as a carrier gas) but also takes part in the
nucleation process due to the unlike a − b interactions becoming highly pronounced
at high pressures. These strong real gas effects attract considerable experimental
[24–28] and theoretical [29–31] attention.
As an example of a system showing retrograde nucleation behavior we consider the
n-nonane/methane mixture. The choice of the system is motivated by the availability
of data obtained in expansion wave tube experiments [32–36] carried out at the
nucleation pressures ranging from 10 to 40 bar and temperatures—from 220 to 250 K.
In this range methane is supercritical: Tc,b = 190 K [14]. In the vapor phase methane
is in abundance, yb ≈ 1, while ya ∼ 10−4 ÷ 10−3 . Let us first study the pressure
dependence of equilibrium properties influencing the nucleation behavior. For these
calculations we need an EoS. The most appropriate one for mixtures of alkanes is
the Redlich-Kwong-Soave equation [37].
As follows from Fig. 11.8, the miscibility of methane xb,eq in the bulk liquid grows
with the pressure and can be as high as ≈50 % for p v = 100 bar. The process of
methane dissolution in liquid nonane is accompanied by the decrease of the reduced
macroscopic surface tension θ∞ of the mixture. As opposed to the previously dis-
cussed examples, θ∞ can not be expressed as a sum of the corresponding individual
properties, θ∞,a and θ∞,b , since methane is supercritical. The surface tension for the
mixture is found from the Parachor method [14].
200 11 Binary Nucleation: Classical Theory

Fig. 11.8 Nonane/methane 1 20

equilibrium properties at nonane/methane
T = 240 K as a function of T=240 K
0.8
total pressure p v : miscibility 10
of methane xb,eq (left y-axis),

, ln ya,eq
reduced macroscopic surface 0.6

xb,eq
tension θ∞ (right y-axis) and
0
the vapor molar fraction of 0.4
nonane ya,eq (right y-axis).
Calculations are carried out xb,eq ln ya,eq
using the Redlich-Kwong- 0.2 -10
Soave equation of state (the
binary interaction parameter 0
0 20 40 60 80 100
ki j = 0.0448 [37])
pv (bar)

11.9.4.1 Compensation Pressure Effect

The equilibrium vapor molar fraction of nonane ya,eq shows nonmonotonous behav-
ior: at low pressures it decreases with p v since the increase of pressure results in the
growth of nonane fraction in the bulk liquid at the expense of its fraction in the vapor;
however, at higher pressures this process is partially blocked by penetration of the
supercritical methane into the liquid phase; ya,eq reaches minimum at p v ≈ 18 bar.
Qualitatively the presence and location of this minimum can be understood in terms
of the “compensation pressure effect” [30]. Consider the partial molecular volume
of component a in the vapor phase. By definition (D.1), vav is the change of the total
volume V v of the binary vapor when one extra a molecule is inserted into the vapor
at the fixed total pressure p v and the number of b-molecules. One can identify two
competing factors related to this process. The first factor is the tendency to increase
the volume in order to preserve p v . The opposite factor, manifested by the second
term in (11.123), is the tendency to reduce V v . The latter becomes pronounced for
mixtures of (partially) miscible components at sufficiently high pressures when the
separation between b molecules becomes of the order of the range of unlike (a − b)
attractions. As a result, a certain number of b molecules move in the direction of the
a molecule (usually relatively big compared to the b-molecule) thereby decreasing
V v . According to Eq. (D.7):
 
1 ∂ ln ρ v
vav = v 1 − yb (11.123)
ρ ∂ ya

where we took into account that ya + yb = 1. Despite the extreme smallness of

ya (usually ya ∼ 10−4 ÷ 10−5 ), the term with the derivative in (11.123) can be
substantial. At a certain pressure pcomp , satisfying
 
∂ ln ρ v  1
 = (11.124)
∂ ya pcomp ,T yb
11.9 Illustrative Results 201

the two opposing trends—expansion and squeezing—compensate each other result-

ing in vav = 0. At p v > pcomp : vav becomes negative implying that the “squeezing
tendency” prevails and a number of b-molecules find themselves attached to an
a-molecule. Since component a is supersaturated, the a-molecules tend to form a
liquid-like cluster, “entraining” the attached b-molecules into it. The presence of the
more volatile component decreases the specific surface free energy of a cluster. We
term pcomp the compensation pressure. The simplest way to estimate pcomp (T ) is
the virial expansion [14]

ρ v = βp v (1 − b2 ), b2 ≡ β B2 p v (11.125)

valid when |b2 | 1. Here


B2 = yi y j B2,i j
i j

is the second virial coefficient of the gas mixture; B2,aa (T ), B2,bb (T ) are the second
virial coefficients of the pure substances, and the cross term B2,ab is constructed
according to the combination rules [14]. Taking the logarithmic derivative in (11.125)
and linearizing in b2 we obtain from (11.124):

1 kB T
pcomp (T ) = (11.126)
2 [B2,bb (T ) − B2,ab (T )]

(after taking the derivative we can set yb ≈ 1 since methane is in abundance). This
result demonstrates the leading role in the compensation pressure effect played by a−
b interactions giving rise to B2,ab , which usually satisfies |B2,ab | > |B2,bb |. For the
nonane/methane mixture pcomp (T = 240 K) = 17.8 bar. This value approximately
coincides with the pressure, at which ya,eq is at minimum (see Fig. 11.8).

11.9.4.2 Experiment Versus BCNT

As usual we characterize the state of the system by the vapor-phase activities of the
components. Since methane is supercritical, the reference state should be different
from the pure component vapor–liquid coexistence at temperature T discussed in
Sect. 11.9.1. To this end we choose as a reference the ( p v , T )-equilibrium of the
v = μv ( p v , T ). Within the ideal gas approximation
mixture, so that in (11.105) μi,0 i,eq
Eq. (11.105) becomes

yi p v yi
Aiv ≈ = ≡ Si (11.127)
yi,eq ( p v , T ) p v yi,eq

The quantity Si will be termed the metastability parameter of component i. It is a

directly measurable quantity which takes into account the presence of the second
202 11 Binary Nucleation: Classical Theory

Fig. 11.9 Nonane/methane 25

nucleation. Nucleation rate nonane/methane
versus metastability parameter 20 T=240 K
of nonane Snonane at various BCNT
pressures and T = 240 K. 15

log10J (cm-3s-1)
Closed circles: experiments of 40 33 25
expt 10
Luijten [33, 34]; open circles: 10
experiments of Peeters [35];
half-filled squares: exper- 5
iments of Labetski [36].
Dashed lines: BCNT. Labels: 0
total pressure in bar
-5

40
33
25

10
-10
0.5 1 1.5
log10Snonane

component via yi,eq = yi,eq ( p v , T ) calculated through an appropriate equation of

state. The BCNT expression for the Gibbs free formation energy (11.101) reads

  
yi,eq p v
βΔG BCNT (n a , n b ) = − n i ln Si + β γ (xbtot ) A
yicoex (xbtot ) p coex (xbtot )
i
(11.128)
Figure 11.9 shows the BCNT predictions of nucleation rate as a function of
Sa = Snonane for temperature T = 240 K and pressures 10, 25, 33, 40 bar along
with the experimental results of [33–35] and [36]. Theoretical predictions are fairly
close to experiment for 10 and 25 bar. For higher pressures, however, BCNT largely
underestimates the experimental data. In particular, the 40 bar data show extremely
large deviation: more than 30 orders of magnitude. The analysis of the experimen-
tal data using nucleation theorem [34] reveals that the critical cluster is a small
object, containing 10–20 molecules. With this in mind it comes as no surprise that
application of a purely phenomenological BCNT approach to such clusters becomes
conceptually in error. Moreover, as opposed to the ethanol/hexanol mixture studied
in Sect. 11.9.2, surface enrichment in the nonane/methane mixture is expected to be
highly pronounced while the standard BCNT scheme does not take it into account.

References

1. H. Flood, Z. Phys, Chem. A 170, 286 (1934)

2. M. Volmer, Kinetik der Phasenbildung (Steinkopf, Dresden, 1939)
3. K. Neumann, W. Döring, Z. Phys, Chem. A 186, 203 (1940)
4. H. Reiss, J. Chem. Phys. 18, 840 (1950)
5. D.E. Temkin, V.V. Shevelev, J. Cryst. Growth 66, 380 (1984)
6. B. Wyslouzil, G. Wilemski, J. Chem. Phys. 103, 1137 (1995)
References 203

7. D. Stauffer, J. Aerosol Sci. 7, 319 (1976)

8. H. Vehkamäki, Classical Nucleation Theory in Multicomponent Systems (Springer, Berlin,
2006)
9. J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity (Clarendon Press, Oxford, 1982)
10. G. Wilemski, J. Chem. Phys. 80, 1370 (1984)
11. G. Wilemski, J. Phys. Chem. 91, 2492 (1987)
12. K. Nishioka, I. Kusaka, J. Chem. Phys. 96, 5370 (1992)
13. Y.S. Djikaev, I. Napari, A. Laaksonen, J. Chem. Phys. 120, 9752 (2004)
14. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids (McGraw-Hill,
New York, 1987)
15. A. Laaksonen, R. McGraw, H. Vehkamaki, J. Chem. Phys. 111, 2019 (1999)
16. P.G. Debenedetti, Metastable Liquids (Princeton University Press, Princeton, 1996)
17. D.W. Oxtoby, D. Kashchiev, J. Chem. Phys. 100, 7665 (1994)
18. G. Wilemski, B. Wyslouzil, J. Chem. Phys. 103, 1127 (1995)
19. R. Strey, Y. Viisanen, J. Chem. Phys. 99, 4693 (1993)
20. J.L. Schmitt, J. Witten, G.W. Adams, R.A. Zalabsky, J. Chem. Phys. 92, 3693 (1990)
21. Y. Viisanen, R. Strey, A. Laaksonen, M. Kulmala, J. Chem. Phys. 100, 6062 (1994)
22. R. Strey, P.E. Wagner, Y. Viisanen, in Nucleation and Atmospheric Aerosols, ed. by N. Fukuta,
P.E. Wagner (Deepak, Hampton, 1992), p. 111
23. M.J. Muitjens, V.I. Kalikmanov, M.E.H. van Dongen, A. Hirschberg, P. Derks, Revue de
l’Institut Français du Pétrole 49, 63 (1994)
24. R.H. Heist, H. He, J. Phys. Chem. Ref. Data 23, 781 (1994)
25. R.H. Heist, M. Janjua, J. Ahmed, J. Phys. Chem. 98, 4443 (1994)
26. J.L. Fisk, J.L. Katz, J. Chem. Phys. 104, 8649 (1996)
27. D. Kane, M. El-Shall, J. Chem. Phys. 105, 7617 (1996)
28. J.L. Katz, J.L. Fisk, V. Chakarov, in Nucleation and Atmospheric Aerosols, ed. by N. Fukuta,
P.E. Wagner (Deepak, Hampton, 1992), p. 11
29. D.W. Oxtoby, A. Laaksonen, J. Chem. Phys. 102, 6846 (1995)
30. V.I. Kalikmanov, D.G. Labetski, Phys. Rev. Lett. 98, 085701 (2007)
31. J. Wedekind, R. Strey, D. Reguera, J. Chem. Phys. 126, 134103 (2007)
32. K.N.H. Looijmans, C.C.M. Luijten, M.E.H. van Dongen, J. Chem. Phys. 103, 1714 (1995)
33. C.C.M. Luijten, Ph.D. Thesis, Eindhoven University, 1999
34. C.C.M. Luijten, P. Peeters, M.E.H. van Dongen, J. Chem. Phys. 111, 8535 (1999)
35. P. Peeters, Ph.D. Thesis, Eindhoven University, 2002
36. D.G. Labetski, Ph.D. Thesis, Eindhoven University, 2007
37. K.N.H. Looijmans, C.C.M. Luijten, G.C.J. Hofmans, M.E.H. van Dongen, J. Chem. Phys. 102,
4531 (1995)
Chapter 12
Binary Nucleation: Density Functional Theory

12.1 DFT Formalism for Binary Systems.

General Considerations

Classical theory of binary nucleation can be drastically in error and even lead to
unphysical behavior when applied to strongly non-ideal systems with substantial
surface enrichment—a vivid example is the water/alcohol system, for which BCNT
predicts the decrease of nucleation rate with increasing partial pressures. An alter-
native to the classical treatment, based on purely phenomenological considerations,
is the density functional theory based on microscopical considerations. The basic
feature of DFT, discussed in Chap. 5, is the existence of the unique Helmholtz free
energy functional of the nonhomogeneous one-particle density ρ(r). The liquid-
vapor equilibrium corresponds to the minimum of this functional in the space of
admissible density profiles under the constraint of fixed particle number N. In a
nonequilibrium state (like supersaturated vapor) one has to search for the saddle
point of the free energy functional which corresponds to the critical nucleus in the
surrounding supersaturated vapor.
DFT of Chap. 5 can be extended to the case of nonhomogeneous binary mixtures.
Let us first discuss the two-phase equilibrium of the binary liquid and binary vapor.
The Helmholtz free energy functional F and the grand potential functional Ω for
a mixture is now written in terms of the one-body density profiles of components
1 and 2, ρ1 (r) and ρ2 (r), normalized as

ρi (ri ) dri = Ni , i = 1, 2

where Ni is the total number of molecules of component i in the two-phase sys-

tem. Intermolecular interactions in the mixture are described by the potentials
u11 (r), u22 (r) and u12 (r). The first two of them refer to interactions of the molecules
of the same type, while u12 (r) describes the unlike interactions between differ-
ent species and is defined through an appropriate mixing rule. As in the single-

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 205

DOI: 10.1007/978-90-481-3643-8_12, © Springer Science+Business Media Dordrecht 2013
206 12 Binary Nucleation: Density Functional Theory

component DFT, we decompose each intermolecular potential uij (r) (i, j = 1, 2)

into the reference (repulsive) part, uij(1) (r), and perturbation, uij(2) (r), following the
Weeks-Chandler-Andersen scheme (5.20)–(5.21)


(1) uij (r) + εij for r < rm,ij
uij (r) = (12.1)
0 for r ≥ rm,ij


(2) −εij for r < rm,ij
uij (r) = (12.2)
uij (r) for r ≥ rm,ij

where εij is the depth of the potential uij (r) and rm,ij is the corresponding value of r:
uij (rm,ij ) = −εij . We assume that all interactions are pairwise additive.
The reference model is approximated by the hard-sphere mixture with appropriately
chosen effective diameters dij . Within the local density approximation the reference
part of the free energy is

Fd [ρ1 , ρ2 ] ≈ dr ψd (ρ1 (r), ρ2 (r)) (12.3)

where ψd (ρ1 (r), ρ2 (r)) is the free energy density of the uniform hard-sphere mixture
with the densities of components ρ1 and ρ2 . Using the standard thermodynamic
relationship (cf. (5.24)) it can be written as


2
ψd (r) = ρi μd,i (ρ1 (r), ρ2 (r)) − pd (ρ1 (r), ρ2 (r))
i=1

where μd,i is the chemical potential of component i in the uniform hard-sphere

fluid and pd is the pressure of the hard-sphere mixture. The quantities μd,i and pd
are obtained by means of the binary form of the Carnahan-Starling equation due to
Mansoori et al. [1] described in Appendix E.
For the pair distribution function we use the random phase approximation

(2)
ρij (r, r ) ≈ ρi (r) ρj (r ) (12.4)

From (12.3) and (12.4) the Helmholtz free energy functional for the mixture takes
the form
 2 
1  (2)
F [ρ1 , ρ2 ] = dr ψd (ρ1 (r), ρ2 (r)) + dr dr ρi (r) ρj (r ) uij (|r − r |)
2
i,j=1
(12.5)
12.1 DFT Formalism for Binary Systems. General Considerations 207

2
The grand potential functional for the mixture Ω[ρ1 , ρ2 ] = F [ρ1 , ρ2 ] − i=1
μi Ni is

 2 

Ω[ρ1 , ρ2 ] = − dr pd (ρ1 (r), ρ2 (r)) + dr ρi μd,i (ρ1 (r), ρ2 (r))
i=1
2  
1  (2)
+ dr ρi (r) dr ρj (r ) uij (|r − r |)
2
i,j=1


2 
− μi dr ρi (r) (12.6)
i=1

where μi is the chemical potential of component i. Two-phase binary equilibrium

corresponds to its variational minimization:

δΩ
= 0, i = 1, 2 (12.7)
δρi (r)

or, equivalently

2 
 (2)
μd,i (ρ1 (r), ρ2 (r)) = μi − dr ρj (r )uij (|r − r |), i = 1, 2 (12.8)
j=1

The integral equations (12.8) are solved iteratively. First it is necessary to determine
the bulk equilibrium properties of the system: μi , ρiv ρil . Fixing two degrees of free-
dom (temperature and the total pressure, or temperature and bulk composition in one
of the phases), the two-phase equilibrium of the mixture is given by

μl1 = μv1 ≡ μ1 (12.9)

μl2 = μv2 ≡ μ2 (12.10)
pl = pv ≡ p (12.11)

Equations (12.5)–(12.8) written for a homogeneous bulk mixture ρi (r) = ρi become


2
F (ρ1 , ρ2 ) = Fd (ρ1 , ρ2 ) − V ρi ρj aij (12.12)
i,j=1


2
μi = μd,i (ρ1 , ρ2 ) − 2 ρj aij (12.13)
j=1
208 12 Binary Nucleation: Density Functional Theory

where 
1 (2)
aij = − dr uij (r)
2

is the background interaction parameter. The virial equation for a mixture reads


2
p = pd (ρ1 , ρ2 ) − ρi ρj aij (12.14)
i,j=1

Consider the flat geometry with the inhomogeneity along the z-axis, directed towards
the bulk vapor. The bulk densities of the components in both phases provide asymp-
totic limits for the equilibrium density profiles in the inhomogeneous system:

ρi (z) → ρiv in the bulk vapor

ρi (z) → ρil in the bulk liquid

The density profiles are calculated iteratively from Eq. (12.8) starting with an ini-
tial guess for each ρi (z), which can be a step-function or a continuous function that
varies between the bulk limits. When the equilibrium profiles are found, they can
be substituted back into the thermodynamic functionals which then become the cor-
responding thermodynamic potentials of the two-phase system. In particular, from
(12.6) and (12.8) the grand potential of the two-phase system in equilibrium reads:

 2  
1  (2)
Ω[ρ1 , ρ2 ] = − dr pd (ρ1 (r), ρ2 (r)) − dr ρi (r) dr ρj (r ) uij (|r − r |)
2
i,j=1
(12.15)

The plain layer surface tension of the binary system can be determined from the
general thermodynamic relationship (5.35):

γ = (Ω[{ρi }] + pV )/A (12.16)

where A is the interfacial area. For inhomogeneity along the z direction dr = A dz,
and Eqs. (12.15) and (12.16) yield

   
1 
2
  (2) 
γ =− dz pd (z) + ρi (z) dr ρj (z ) uij (|r − r |) − p (12.17)
2
i=1
12.2 Non-ideal Mixtures and Surface Enrichment 209

12.2 Non-ideal Mixtures and Surface Enrichment

As we know from Chap. 11, talking about a mixture we can not avoid the dis-
cussion of adsorption effects. On the phenomenological level it means that for a
binary (or, more generally, a multi-component) mixture it is impossible to choose
the Gibbs dividing surface in such a way that the excess (adsorption) terms for all
species simultaneously vanish. This feature gives rise to the surface enrichment:
a preferential adsorption of one of the species in the interfacial region between the two
bulk phases.
On the microscopic level the issue of adsorption boils down to the strength and range
of unlike interactions. They determine the degree of non-ideality of the system. The
DFT yields the density profiles of components in the inhomogeneous system. These
profiles have no rigid boundaries, their form is based on the microscopic interactions
in the system, implying that adsorption (surface enrichment) is naturally built into
the DFT scheme.
It is instructive to study the effects of non-ideality on the behavior of the mixture
considering the simplest system: a binary mixture of Lennard-Jones fluids with the
interaction potentials

σij 12 σij 6
uij (r) = 4εij − (12.18)
r r

where σii and εii are the Lennard-Jones parameters of the individual components.
For illustrative reasons (in order to have realistic numbers) we choose their values
corresponding to the argon/krypton mixture:

σAr = σ11 = 3.405 Å, σKr = σ22 = 3.632 Å

and
εAr /kB = ε11 /kB = 119.8 K, εKr /kB = ε22 /kB = 163.1 K

The unlike interactions u12 are defined via the mixing rules. We assume that u12 has
also the Lennard-Jones form. For conformal potentials it is common to present the
mixing rules in the form [2]:

1
σ12 = (σ11 + σ22 ) (12.19)
2
and √
ε12 = ξ12 ε11 ε22 (12.20)

where ξ12 is called the binary interaction parameter and is found from the fit to
experiment. When ξ12 = 1 one speaks about a Lorentz-Berthelot mixture. For real
mixtures ξ12 is usually significantly less than unity. It is necessary to have in mind
210 12 Binary Nucleation: Density Functional Theory

(a) 0.6
Kr T= 115.77 K (b) 0.6
T = 115.77 K
xAr=0.3 Kr
xAr=0.3
0.5 0.5
=1 = 0.88
12 12

3
0.4 0.4
3

Kr
= 13.7 mN/m = 10.1 mN/m
Kr

0.3 0.3

i (z)
i (z)

Ar Ar
0.2 0.2

0.1 0.1

0 0
0 5 10 15 20 0 5 10 15 20
z/ Kr z/ Kr

Fig. 12.1 Density profiles of argon and krypton at the flat vapor-liquid interface with the bulk liquid
molar fraction of argon xAr = 0.3 at T = 115.77 K and different values of the binary interaction
parameter ξ12 ; (a) ξ12 = 1 (Lorentz-Berthelot mixture); (b) ξ12 = 0.88. Distances and densities are
scaled with respect to σKr = σ22 . The decrease of ξ12 leads to the increase of the surface activity of
argon (surface enrichment), accompanied by the decrease of the surface tension of the mixture γ

that the laws of the ideal mixture are obtained only if all the potentials are the same;
in this sense even for ξ12 = 1 the mixture should not necessarily be ideal. Note
that for the equation of state of the mixture the mixing rule (12.20) leads to the
corresponding form of the “energy parameter” a12 :
√
a12 = ξ12 a11 a22

Consider implications of the mixing rule (12.20). The decrease of ξ12 from unity
will decrease the depth of unlike interactions thereby enhancing separation in the
solution. Since in our example ε11 < ε22 , this separation leads to the increase of the
surface activity of component 1 (argon) which is manifested by its pronounced surface
enrichment. These features are illustrated in Fig. 12.1. Equilibrium calculations are
performed for the argon/krypton mixture at T = 115.77 K and the bulk liquid molar
fraction of argon xAr = 0.3 (hence, we discuss the (x, T )-equilibrium).
The left graph (a) refers to the Lorentz-Berthelot mixture: ξ12 = 1. The surface
enrichment of argon in this case is very weak. The surface tension calculated from
Eq. (12.17) gives γ (ξ12 = 1) = 13.7 mN/m. On the right graph (b) we show the
DFT calculations for ξ12 = 0.88. The density profile of argon shows considerable
surface enrichment, meaning that the vapor-liquid interface is argon-rich leading to
the decrease of the surface tension: γ (ξ12 = 0.88) = 10.1 mN/m.

12.3 Nucleation Barrier and Activity Plots: DFT Versus BCNT

Formulation of the density functional theory for the study of equilibrium properties
of inhomogeneous binary mixtures can be extended to the nonequilibrium case of
binary nucleation. The corresponding development was carried out by Oxtoby and
12.3 Nucleation Barrier and Activity Plots: DFT Versus BCNT 211

coworkers [3–5] and represents a generalization of the similar approach for a single-
component case. In DFT of binary nucleation one studies the system “droplet in a
nonequilibrium vapor”, where both the droplet and the vapor are binary mixtures.
A droplet is associated with a density fluctuation which has no rigid boundary. The
Helmholtz free energy and grand potential functionals for this system are given as
before by Eqs. (12.5)–(12.6). However, the chemical potentials of the components in
this expressions refer now to the actual nonequilibrium state of the system (and not
to the thermodynamic equilibrium as in Sect. 12.1). This state can be characterized,
e.g. by fixing the gas-phase activities of the components.
The critical nucleus is in unstable equilibrium with the environment and corresponds
to the saddle point of Ω[ρ1 , ρ2 ] (as opposed to the minimum of Ω[ρ1 , ρ2 ] in the case
of equilibrium conditions). The density profiles in the critical nucleus are found as
before from the solution of Eq. (12.8). If the profiles are determined, then the change
in the grand potential
ΔΩ ∗ = Ω[ρ1 , ρ2 ] − Ωu (12.21)

where Ωu is the grand potential of the uniform nonequilibrium vapor (i.e. the binary
vapor prior to the appearance of the droplet) is the free energy associated with the
formation of the critical cluster. As in the single-component case it is straightforward
to see that ΔΩ ∗ is equal to the Gibbs energy of the critical cluster formation

ΔΩ ∗ = ΔG∗

When we discussed Eq. (12.8) for the conditions of thermodynamic equilibrium,

the iterative procedure converged rapidly (within several iterations) to the desired
solution irrespective of the initial guess (as soon as it satisfies the boundary con-
ditions)—just due to the fact that equilibrium profiles minimize the functional. The
solution of the same equations for the critical cluster in nucleation are far from triv-
ial. Since we deal here with the unstable equilibrium, the convergent solution of
Eq. (12.8) does not exist. Meanwhile, if the initial guess for ρ1 (r), ρ2 (r)) is close to
the profiles of components in the critical cluster, the iteration process after several
steps reaches a plateau with ΔΩ remaining constant over several iterations. This
plateau corresponds to the critical cluster. Continuation of the iterative process will
after a certain amount of iterations lead to a deviation of ΔΩ from the plateau (no
convergency!). Clearly, the search for the critical cluster is very sensitive to the initial
guess. The details of iterative procedure are described in Refs. [4, 5].
After determination of the nucleation barrier ΔΩ ∗ the steady-state nucleation rate
follows from:
∗
J = K e−βΔΩ (12.22)

where the pre-exponential factor K can be taken from the classical binary nucleation
theory (see Eq. (11.43)).
In the previous section we studied the effects of non-ideality of the mixture (expressed
in the terms of the unlike interaction potential) on the equilibrium properties. Let
212 12 Binary Nucleation: Density Functional Theory

Fig. 12.2 Gas phase activities

for the mixture of argon
and krypton with ξ12 = 1.
Diamonds: DFT, full line:
BCNT. The results correspond
to the nucleation rate of
1 cm−3 s−1 and T = 115.77 K
(Reprinted with permission
from Ref. [4], copyright
(1995), American Institute of
Physics.)

Fig. 12.3 Gas phase activities

for the mixture of argon and
krypton with ξ12 = 0.88.
Diamonds: DFT; full line:
BCNT. The results correspond
to the nucleation rate of
1 cm−3 s−1 and T = 115.77 K
(Reprinted with permission
from Ref. [4], copyright
(1995), American Institute of
Physics.)

us study their impact on the nucleation behavior. As before we consider the binary
mixture argon/krypton with the mixing rule (12.20). The role of non-ideality effects
can be clearly demonstrated by means of activity plots. Figures 12.2 and 12.3 show
the BCNT and DFT activity plots for T = 115.77 K corresponding to the nucleation
rates

JBCNT ≈ JDFT ≈ 1 cm−3 s−1

12.3 Nucleation Barrier and Activity Plots: DFT Versus BCNT 213

Figure 12.2 refers to the Lorentz-Berthelot mixture: ξ12 = 1. As we know from the
equilibrium considerations of the previous section, surface enrichment of argon in
this case is very weak and the mixture is fairly ideal. It is therefore not surprising that
the difference in nucleation behavior between the BCNT and DFTs is of quantitative
nature; note that it increases with the krypton activity.
This situation is distinctly different from the case ξ12 = 0.88 shown in Fig. 12.3:
BCNT produces a “hump”—resembling the similar predictions for water/alcohol
systems (cf. Fig. 11.6). This hump, as discussed earlier, is unphysical since it violates
the nucleation theorem. Its occurrence is a consequence of the neglect within the
BCNT scheme of adsorption effects giving rise to surface enrichment. The DFT
approach does not have this drawback because adsorption is “built into it” on the
microscopic level. The DFT predictions therefore are in qualitative agreement with
the nucleation theorem. Figure 12.1b demonstrated the effect of surface enrichment
for the planar interface. DFT calculations reveal the same effect for the critical cluster
in binary nucleation [4].

References

1. G.A. Mansoori, N.F. Carnahan, K.E. Starling, T.W. Leland, J. Chem. Phys. 54, 1523 (1971)
2. J.S. Rowlinson, F.L. Swinton, Liquids and Liquid Mixtures (Butterworths, Boston, 1982)
3. X.C. Zeng, D.W. Oxtoby, J. Chem. Phys. 95, 5940 (1991)
4. A. Laaksonen, D.W. Oxtoby, J. Chem. Phys. 102, 5803 (1995)
5. I. Napari, A. Laaksonen, J. Chem. Phys. 111, 5485 (1999)
Chapter 13
Coarse-Grained Theory of Binary
Nucleation

13.1 Introduction

As we saw in Chap. 11, the classical theory proved to be successful for fairly ideal
mixtures. Meanwhile, for non-ideal mixtures BCNT can be sufficiently in error
[1–4] and even lead to unphysical results as in the case of water-alcohol systems.
The reasons for this failure are the neglect of adsorption effects and the inappropriate
treatment of small clusters. These issues are strongly coupled; they determine the
form of the Gibbs free energy of cluster formation and subsequently the composi-
tion of the critical cluster and the nucleation barrier. One can correct the classical
treatment by taken into account adsorption using the Gibbsian approximation (see
Sect. 11.6).
Taking into account adsorption within the phenomenological approach does not
resolve another deficiency associated with the capillarity approximation: the surface
energy of a cluster is described in terms of the planar surface tension. Obviously,
for small clusters the concept of macroscopic surface tension looses its meaning
and this assumption fails. This difficulty is not unique for the binary problem. In
the single-component mean-field kinetic nucleation theory (MKNT) of Chap. 7 this
problem was tackled by formulating an interpolative model between small clusters
treated using statistical mechanical considerations and big clusters described by the
capillarity approximation. In the present chapter we extend these considerations to
the binary case and incorporate them into a model which takes into account the
adsorption effects [5].
The statistical mechanical treatment of binary clusters, which we discuss in the
present chapter, originates from the analogy with the soft condensed matter theory,
where the description of complex fluids can be substantially simplified if one elim-
inates the degrees of freedom of small solvent molecules in the solution. By per-
forming such coarse-graining one is left with the pseudo-one-component system of
solute particles with some effective Hamiltonian. This approach opens the possibil-
ity to study the behavior of a complex fluid using the techniques developed in the

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 215

DOI: 10.1007/978-90-481-3643-8_13, © Springer Science+Business Media Dordrecht 2013
216 13 Coarse-Grained Theory of Binary Nucleation

theory of simple fluids [6]. Situation in nucleation theory is somewhat similar: the
complexity of binary nucleation problem can be substantially reduced by tracing out
the degrees of freedom of the molecules of the more volatile component in favor
of the less volatile one. This pseudo-one-component system can be studied using
the approach developed in Chap. 7 making it possible to adequately treat clusters of
arbitrary size and composition.

13.2 Katz Kinetic Approach: Extension to Binary

Mixtures

Consider a binary mixture of component a and b in the gaseous state at the tempera-
ture T and the total pressure p v . The actual vapor mole fractions of components are
ya and yb . In the presence of a carrier gas with the mole fraction yc :

ya + yb = 1 − yc

In the present context the term “carrier gas” refers to a passive component, which
does not take part in cluster formation but serves to remove the latent heat. In some
cases, a passive carrier gas is absent (yc = 0), and one of the components of the
mixture (a or b) plays the double role: besides taking part in the nucleation process,
it removes the latent heat. In this case, this component should be in abundance in the
vapor phase.
Within the general formalism of Sect. 11.9.1 the state of component i is characterized
by the vapor phase activity
 
Aiv = exp β(μiv ( p v , T ; yi ) − μi,0
v
) , i = a, b (13.1)

where μiv ( p v , T ; yi ) is the chemical potential of component i in the vapor, μi,0

v is the

value of μi at some reference state. Usually one chooses as the reference the saturated
v

state of pure components at the temperature T . This choice implicitly assumes the
existence of such a state for both species. In the case when one of the components
is supercritical this choice becomes inappropriate. With this in mind we choose as a
reference the true equilibrium state of the mixture at p v , T ; yc , so that μi,0
v = μv .
i,eq
Throughout this chapter the subscript “eq” refers to the true equilibrium state of the
mixture, and not a constrained equilibrium as in Chap. 11; to avoid confusion the
latter will be denoted by the superscript “cons”. Within the ideal gas approximation

yi p v yi
Aiv ≈ = ≡ Si (13.2)
yi,eq ( p , T, yc ) p
v v yi,eq

where Si will be termed the metastability parameter of component i. Given the

equation of state for the mixture, Si is a directly measurable quantity. It is important
13.2 Katz Kinetic Approach: Extension to Binary Mixtures 217

to emphasize that its value takes into account the presence of all components in the
mixture through yi,eq = yi,eq ( p v , T ; yc ).
Let us consider a two-dimensional n a − n b space of cluster sizes. Here n i is the total
number of molecules of component i, which according to Gibbs thermodynamics is
the sum of the bulk and excess terms

n i = n il + n iexc

Kinetics of nucleation is governed by Eq. (11.1):

∂ρ(n a , n b , t)
= Ja (n a − 1, n b , t) − Ja (n a , n b , t) + Jb (n a , n b − 1, t) − Jb (n a , n b , t)
∂t
(13.3)
where the fluxes along the n a and n b directions are

Ja (n a , n b ) = va A(n a , n b ) ρ(n a , n b , t) − βa A(n a + 1, n b ) ρ(n a + 1, n b , t)

(13.4)
Jb (n a , n b ) = vb A(n a , n b ) ρ(n a , n b , t) − βb A(n a , n b + 1) ρ(n a , n b + 1, t)
(13.5)

Impingement rates (per unit surface) of component i, vi , are given by gas kinetics:

yi p v
vi = √ (13.6)
2π m i kB T

Evaporation rates βi are obtained from the detailed balance condition. Recall that in
BCNT it is applied to the constrained equilibrium state which would exist for the
vapor at the same temperature, pressure and vapor phase activities as the vapor in
question. Instead of using this artificial state, we apply the detailed balance to the
true (full) equilibrium of the system at ( p v , T, yc ):

0 = va,eq A(n a , n b ) ρeq (n a , n b ) − βa A(n a + 1, n b ) ρeq (n a + 1, n b ) (13.7)

0 = vb,eq A(n a , n b ) ρeq (n a , n b ) − βb A(n a , n b + 1) ρeq (n a , n b + 1) (13.8)

This procedure is a natural extension to binary mixtures of the Katz kinetic approach
discussed in Sect. 3.5. Assuming (following BCNT) that the evaporation rates do not
depend on the surrounding vapor, βi = βi,eq , we find from (13.7) and (13.8)

A(n a , n b ) ρeq (n a , n b )
βa = va,eq (13.9)
A(n a + 1, n b ) ρeq (n a + 1, n b )
A(n a , n b ) ρeq (n a , n b )
βb = vb,eq (13.10)
A(n a , n b + 1) ρeq (n a , n b + 1)
218 13 Coarse-Grained Theory of Binary Nucleation

Substituting (13.9)–(13.10) into (13.4) and (13.5) we write

 
ρ(n a , n b ) ρ(n a + 1, n b ) va,eq
Ja (n a , n b ) = ρeq (n a , n b ) A(n a , n b ) va −
ρeq (n a , n b ) ρeq (n a + 1, n b ) va
 
ρ(n a , n b ) ρ(n a , n b + 1) vb,eq
Jb (n a , n b ) = ρeq (n a , n b ) A(n a , n b ) vb −
ρeq (n a , n b ) ρeq (n a , n b + 1) vb

To make the expressions in the square brackets symmetric, it is convenient introduce

the function [7, 8]
 
ρ(n a , n b )  vi,eq n i ρ(n a , n b )  −n i
H (n a , n b ) = = Si (13.11)
ρeq (n a , n b ) vi ρeq (n a , n b )
i=a,b i=a,b

where the second equality results from (13.2) and (13.6). The fluxes along n a and
n b read:
⎡ ⎤

Ja = −ρeq A va ⎣ Sin i ⎦ [H (n a + 1, n b ) − H (n a , n b )] (13.12)
i=a,b
⎡ ⎤

Jb = −ρeq A vb ⎣ Sin i ⎦ [H (n a , n b + 1) − H (n a , n b )] (13.13)
i=a,b

We can write down the same fluxes in terms of the constrained equilibrium quantities.
cons = v . Repeating the previous steps, in which the equilibrium
By definition vi,eq i
properties are replaced by the corresponding constrained equilibrium ones, we find

Ja = −ρeq
cons
A va [H (n a + 1, n b ) − H (n a , n b )] (13.14)
Jb = −ρeq
cons
A vb [H (n a , n b + 1) − H (n a , n b )] (13.15)

where the function H has now a simple form:

ρ(n a , n b )
H (n a , n b ) = (13.16)
ρeq
cons (n , n )
a b

Comparing (13.14)–(13.16) with (13.12)–(13.13), we find the relationship between

the constrained and unconstrained distributions:
 n
ρeq
cons
= ρeq Si i (13.17)
i=a,b

Obviously, for Sa = Sb = 1 both equilibria become identical. A general form of

ρeq (n a , n b ) resulting from the thermodynamic fluctuation theory is
13.2 Katz Kinetic Approach: Extension to Binary Mixtures 219

ρeq (n a , n b ) = C e−βΔG eq (n a ,n b ) (13.18)

From (13.17) and (13.18)

(n a , n b ) = C e−βΔG eq
cons (n
a ,n b )
ρeq
cons
(13.19)

where
eq (n a , n b ) = −
βΔG cons n i ln Si + βΔG eq (n a , n b ) (13.20)
i=a,b

It is convenient to present Eqs. (13.12)–(13.13) in the vector notations:

⎡ ⎤

J = − ⎣ρeq (n) Sin i ⎦ F ∇ H, n = (n a , n b ) (13.21)
i=a,b

 
where ∇ ≡ ∂n∂ a , ∂n∂ b and the diagonal matrix F contains the rate of collisions of
a and b molecules with the surface of the cluster:
 
va A(n a , n b ) 0
F= (13.22)
0 vb A(n a , n b )

In these notations the kinetic equation (13.3) takes the form of the conservation law
for the “cluster fluid” (cf. Chap. 11)

∂ρ(n)
= − div J(n) (13.23)
∂t
with the steady state given by
div J = 0 (13.24)

In view of (13.17) we can discuss thermodynamics of nucleation focusing on the

model for the Gibbs free energy of cluster formation in vapor-liquid equilibrium,
ΔG eq (n a , n b ). An important feature of this construction is that the prefactor C in
(13.19) refers to the true equilibrium distribution ρeq (n a , n b ).
Repeating the considerations of Chap. 11, we find that the direction of the nucleation
flux in any point of the cluster space satisfies (cf. Eq. (11.14)):
⎛ ⎞

curl (F−1 J) = (F−1 J) × ∇ ⎝βΔG eq − n i ln Si ⎠ (13.25)

i=a,b

The saddle point n∗ = (n a∗ , n ∗b ) of the constrained equilibrium free energy

220 13 Coarse-Grained Theory of Binary Nucleation

β ΔG cons
eq (n a , n b ) = − n i ln Si + β ΔG eq (13.26)
i

corresponds to the critical cluster. The quadratic expansion of βΔG cons

eq near the
saddle point reads:
∗
eq (n a , n b ) = g + m a Daa + m b Dbb + 2 m a m b Dab
βΔG cons 2 2
(13.27)

Here g ∗ = βΔG cons ∗ ∗

eq (n a , n b ) is the reduced Gibbs free energy at the saddle point,

1 ∂ 2 βΔG eq 
mi = n i − n i∗ , Di j =  , i, j = a, b
2 ∂n i ∂n j 
n∗

Note, that in view of (13.20)

 
∂ 2 βΔG cons  ∂ 2 βΔG eq 
eq 
 = 
∂n i ∂n j  ∂n i ∂n j 
n∗ n∗

At the saddle point the eigenvalues of the symmetric matrix D have different signs
implying that det D < 0. Following the standard procedure, we introduce a rotated
coordinate system (x, y) in the (n a , n b )-space with the origin at n∗ and the x axis
pointing along the direction of the flow at n∗ (see Fig. 11.2):

x = m a cos ϕ + m b sin ϕ, y = −m a sin ϕ + m b cos ϕ

where ϕ is the yet unknown angle between the x and n a . At the saddle point Jb /Ja =
tan ϕ, which from (13.12)–(13.13) is written as:
∂H
vb ∂n b
∂H
= tan ϕ (13.28)
va ∂n a

Using the standard relationships between the derivatives in the original and the rotated
systems, we present after simple algebra Eq. (13.28) as:
 
∂H va tan2 ϕ + vb ∂ H
= (13.29)
∂x (va − vb ) tan ϕ ∂ y

The rate components in the new coordinates are:

Jx = Ja cos ϕ + Jb sin ϕ, Jy = −Ja sin ϕ + Jb cos ϕ

Right at the saddle point Jx = Jx∗ (n∗ ), Jy (n∗ ) = 0. As in the BCNT, we use the
direction of principal growth approximation assuming that
13.2 Katz Kinetic Approach: Extension to Binary Mixtures 221

Jy = 0 in the entire saddle point region (13.30)

Then, from the continuity condition (13.24)

∂ Jx (x, y)
=0
∂x

implying that in the saddle point region Jx = Jx (y), Jy = 0. In order to identify

the function Jx (y) we express the “old” coordinates Ja and Jb in terms of the “new”
ones, taking into account (13.30):

Ja = Jx cos ϕ, Jb = Jx sin ϕ

Then, using (13.14)–(13.15) we find

 
1 ∂H ∂H
Jx (y) = −ρeq (n) San a Sbn b A va = −ρeq San a Sbn b A vav (13.31)
cos ϕ ∂n a ∂x

where vav is the average impingement rate given by Eq. (11.35). Rewriting (13.31) as

1 ∂H
Jx (y) nb =−
ρeq San a Sb A vav ∂x

and integrating over x using the standard boundary conditions

lim H = 1, lim H = 0
x→−∞ x→+∞

we find  ∞
Jx (y) 1
dx =1 (13.32)
vav −∞ ρeq San a Sbn b A

Substituting the expansion (13.27) into Eq. (13.32) and performing Gaussian inte-
gration first over x and then over y, we find for the total nucleation rate
 
 n∗
∗
J = vav A Z Si i ρeq (n a∗ , n ∗b ) (13.33)
i

where A∗ = A(n a∗ , n ∗b ),
1 (∂ 2 βΔG/∂ x 2 )n∗
Z =− √ (13.34)
2 − det D

is the Zeldovich factor. Equation (13.33) contains the yet undetermined direction ϕ of
the flow in the saddle point. The latter is found by maximizing the angle-dependent
part of J .
222 13 Coarse-Grained Theory of Binary Nucleation

The quantities with the ϕ dependence are: vav and Z . It is convenient to present
vav as  
1 + t2 vb
vav = vb , t ≡ tan ϕ, r = (13.35)
r + t2 va

In the Zeldovich factor det D is invariant to rotation, implying that the only angle-
dependent part of Z is contained in
 
∂ 2 βΔG da − 2t + db t 2
= Daa cos2 ϕ +2Dab sin ϕ cos ϕ + Dbb sin2 ϕ = −Dab
∂x2 1 + t2
(13.36)
where we denoted
Daa Dbb
da = − , db = −
Dab Dab

Combining (13.35) and (13.36), the ϕ-dependent part of J is given by the function

da − 2t + db t 2
f (t) =
r + t2

Its extremum d f /dt = 0 yields

 1
tan ϕ = s + s 2 + r with s = (da − r db ) (13.37)
2
which coincides with Stauffer’s result (11.25) for BCNT. The advantage of
Eq. (13.33) is that it reduces the binary nucleation problem to the determination
of the equilibrium distribution of binary clusters ρeq (n a , n b ), which we discuss in
the next section.

13.3 Binary Cluster Statistics

13.3.1 Binary Vapor as a System of Noninteracting

Clusters

In line with the kinetic approach we discuss the full thermodynamic equilibrium of
the system at the total pressure p v , temperature T and carrier gas composition yc
(if present). The partition function of an arbitrary (n a , n b )-cluster is:

1
Z na nb ≡ Z n = qn a n b (13.38)
Λa3n a Λ3n
b
b
13.3 Binary Cluster Statistics 223

where Λi is the thermal de Broglie wavelength of a molecule of component i; qn a n b

is the configuration integral of (n a , n b )-cluster in a domain of volume V:

1
qn a n b (T ) = qn (T ) = dRn a drn b e−βUn (13.39)
na ! nb ! cl

where Rn a and rn b are locations of molecules a and b in the cluster, and

Un = Uaa (Rn a ) + Ubb (rn b ) + Uab (Rn a , rn b ) (13.40)

is the potential interaction energy of the cluster comprised of a −a, b−b and (unlike)
a − b interactions. The prefactor n1i ! takes into account the indistinguishability of

molecules of type i inside the cluster. The symbol cl indicates that integration is
only over those molecular configurations that belong to the cluster. The cluster as
a whole can move through the entire volume V, while the molecules inside it are
restricted to the configurations about cluster’s center of mass that are consistent with
a chosen cluster definition.
We represent the equilibrium gaseous state of the a − b mixture as a system of
noninteracting (n a , n b ) = n clusters. Since the clusters do not interact, the partition
function Z (n) of the gas of Nn of such n-clusters is factorized:

1
Z (n) = Z Nn
Nn ! n

where the prefactor 1/Nn ! takes into account the indistinguishability of (n a , n b )-

clusters viewed as independent entities. The Helmholtz free energy of this gas is:
F (n) = −kB T ln Z (n) which using Stirling’s formula becomes:
 
Nn
F (n) = Nn kB T ln
Zn e

The chemical potential of an n-cluster in this gas is

 
∂F (n) Nn
μn = = kB T ln
∂ Nn Zn

which using (13.38) reads

 
V Λa3n a Λ3n
b
b
μn = kB T ln ρeq (n a , n b ) (13.41)
qn

Here ρeq (n a , n b ) = Nn /V is the equilibrium distribution function of binary

clusters—the quantity we are aiming to determine. Equilibrium between the cluster
and the surrounding vapor requires
224 13 Coarse-Grained Theory of Binary Nucleation

μn = n a μa,eq
v
+ n b μvb,eq (13.42)

v ( p v , T ) is the chemical potential of a molecule of the component i in the

where μi,eq
equilibrium vapor. Combining (13.41) and (13.42), we find
q 
n
ρeq (n a , n b ) = [z a,eq ]n a [z b,eq ]n b (13.43)
V
where
eβμi ,eq
v

z i,eq = , i = a, b (13.44)
Λi3

is the fugacity of component i in the equilibrium vapor. Thus, we reduced the prob-
lem of finding ρeq (n a , n b ) to the determination of the cluster configuration integral.
Even though we discuss the clusters at vapor-liquid equilibrium, it is important to
realize that the chemical potential of a molecule inside an arbitrary binary cluster
depends on the cluster composition and therefore is not the same as in the bulk vapor
surrounding it. Equation (13.43) shows that the quantity qn/V plays the key role in
determination of the cluster distribution function. From the definition of qn it is clear
that qn/V involves only the degrees of freedom relative to the center of mass of the
cluster. Note also, that qn contains the normalization constant C of the distribution
function.

13.4 Configuration Integral of a Cluster: A Coarse-Grained

Description

Rewriting qn in the form

   
1 1
qn = dRn a e−βUaa drn b e−β(Ubb +Uab ) (13.45)
na ! cl nb ! cl

one can easily see that the expression in the curl brackets

1
qb/a ({Ran a }) ≡ drn b e−β(Ubb +Uab ) (13.46)
nb ! cl

is the configuration integral of b-molecules in the external field of a-molecules

located at fixed positions {Ran a }. The configurational part of the Helmholtz free energy
of this system is
Fb/a ({Ran a }, n a , n b , T ) = −kB T ln qb/a (13.47)

Substituting (13.47) into (13.45), we present qn in the coarse-grained form

13.4 Configuration Integral of a Cluster: A Coarse-Grained Description 225

1
dRn a e−β H
CG
qn = (13.48)
na ! cl

where the positions of b-particles are integrated out. By doing so we replaced the
binary cluster by the equivalent single-component one with the effective Hamiltonian

H CG = Uaa ({Ran a }) + Fb/a ({Ran a }; n a , n b , T ) (13.49)

which is the sum of the Hamiltonian of the pure a-system, Uaa , and the free energy
of b-molecules in the instantaneous environment of a molecules. Equation (13.48)
is formally exact.
In order to derive a tractable representation of the free energy Fb/a we perform
the diagrammatic expansion of ln qb/a in the Mayer functions of a − b and b − b
interactions:

f ab (|Ri − r j |) = exp[−β u ab (|Ri − r j |)] − 1

f bb (|rk − rl |) = exp[−β u bb (|rk − rl |)] − 1

As a result Fb/a is represented as the sum of m-body effective interactions between

a-molecules [6, 9]:

Fb/a ({Ran a }; n a , n b , T ) = F0 (n a , n b , T ) + U2 ({Ran a }; xbtot , T ) + · · · (13.50)

The zeroth order contribution F0 (n a , n b , T ), called the volume term, does not depend
on positions of molecules, but is important for thermodynamics since it depends on
cluster composition and therefore by no means can be neglected. The first-order term
U1 in (13.50) vanishes in view of translational symmetry [6]. Combining (13.49)–
(13.50) we write

H CG = F0 (n a , n b , T ) + U CG ({Ran a }; xbtot , T ) (13.51)

with the total coarse-grained interaction energy

U CG ({Ran a }; xb , T ) = Uaa ({Ran a }) + U2 ({Ran a }; xbtot , T ) + · · ·

Substituting (13.51) into (13.48), we obtain

qn = e−β F0 qnCG
a
(13.52)

where 
1
dRn a e−βU
CG
qnCG (xbtot , T ) = (13.53)
a
na ! cl

Interpretation of Eqs. (13.52)–(13.53) is straightforward: by tracing out the degrees

of freedom of b-molecules, we are left with the single-component cluster of
226 13 Coarse-Grained Theory of Binary Nucleation

pseudo—a molecules with the interaction energy U CG . The latter implicitly depends
on the fraction of b-molecules in the original binary cluster. The configuration
integral of this single-component cluster is qnCG
a
. Equation (13.52) is a key result
of the model.
Speaking about a binary cluster, we characterized it by the total numbers of molecules
n a and n b , not discriminating between the bulk and excess numbers of molecules of
each component
n i = n il + n iexc

Meanwhile, as we know, this distinction is important for capturing the adsorption

effects resulting in nonhomogeneous distribution of molecules within the cluster.
Description of adsorption requires introduction of the Gibbs dividing surface, for
which we will use the K -surface of Sect. 11.6. This means, that for an arbitrary bulk
cluster content (n al , n lb ) we find the excess numbers from Eqs. (11.84)–(11.85):

n aexc (n al , n lb ), n exc
b (n a , n b )
l l

Thus, the point (n al , n lb ) in the space of bulk numbers yields the point (n a , n b ) in
the space of total numbers. As a result the dependence of various quantities on the
total composition xbtot can be also viewed as a dependence (though a different one)
on the bulk composition xbl .
Since the carrier gas is assumed to be passive, the cluster composition satisfies the
normalization:
xal + xbl = xatot + xbtot = 1 (13.54)

13.4.1 Volume Term

Let us discuss in more detail the volume term in the effective Hamiltonian H CG . It
can be written as a sum of the free energy of ideal gas of pure b-molecules in the
cluster, Fb,id , and the excess (over ideal) contribution, ΔF0 , due to b − b and a − b
interactions
F0 = Fb,id + ΔF0 (13.55)

The ideal gas contribution reads (see e.g. [10]):

 
n b Λ3b
βFb,id = n b ln (13.56)
Vcl e

where Vcl = n al val + n lb vbl is the volume of the cluster. Within the K -surface for-
malism we can equivalently write it in terms of total numbers:
13.4 Configuration Integral of a Cluster: A Coarse-Grained Description 227

Vcl = n a val + n b vbl (13.57)

The specific feature of Eq. (13.56) is that b-molecules are contained in the cluster
volume which itself depends on their number n b . Substituting (13.57) into (13.56),
we write
βFb,id = n b f b,id (13.58)

where ⎧ ⎡ ⎤⎫
⎪
⎨  ⎪
⎬
xbtot ⎢ Λ3b ⎥
f b,id = ln ⎣  ⎦ ≡ f b,id (xbl , T )
⎪
⎩ xatot x tot
val + xbtot vbl e ⎪⎭
a

depends only on intensive quantities.

Calculation of ΔF0 in terms of interaction potentials is a challenging task; it has
been done for a limited number of model potentials: mixtures of hard spheres [9, 11]
and charged-stabilized colloidal suspensions [12, 13]. Fortunately, for our purposes
we do not need to know its exact form. Instead, we make use of the general statement
that F0,exc is a homogeneous function of the first order in n a and n b [14]:

na nb
βΔF0 = f 1 (xbl , T ) (13.59)
Vcl

where f 1 is some unknown function of xbl and T . Using (13.57), we present Eq.
(13.59) as
βΔF0 = n b f 0 (13.60)

where
f1
f0 = ≡ f 0 (xbl , T )
xbtot
val + xatot
vbl

depends only on intensive quantities. Combining (13.55), (13.58) and (13.60), we

present the volume term as
e−β F0 = Φbn b (13.61)

where Φb = exp[−( f b,id + f 0 )] is another unknown function of xbl and T .

13.4.2 Coarse-Grained Configuration Integral qnCG

a

The coarse-grained configuration integral qnCG

a
describes the cluster with n a identi-
cal particles (pseudo-a molecules), characterized by unknown complex interactions.
We will analyze it using the formalism of the mean-field kinetic nucleation the-
ory (MKNT) of Chap 7. Within MKNT the cluster configuration integral is given
228 13 Coarse-Grained Theory of Binary Nucleation

by Eq. (7.38):
qnCG
= C Φan a e−θmicro n a
a s
(13.62)
V
Here
1
Φa = (13.63)
z a,sat

z a,sat is the fugacity at saturation, n as (n a ) is the average number of surface particles in

the cluster, θmicro is the reduced microscopic surface tension. For the coarse-grained
cluster those are the functions of the cluster composition. It is important to stress,
that the division of the cluster molecules into the core- and surface particles, adopted
in MKNT, is different from the Gibbs construction (11.47).1
The properties of the pseudo-a fluid are functionals of the unknown interaction
potentials. It is practically impossible to restore these potentials from the microscopic
considerations. An alternative to the microscopic approach is the use of the known
asymptotic features of the distribution function.

13.5 Equilibrium Distribution of Binary Clusters

Using the basic result of the coarse-graining procedure Eq. (13.52), we express the
equilibrium distribution function (13.43) of binary clusters as
 
−β F0
qnCG
ρeq (n a , n b ) = e a
[z a,eq ]n a [z b,eq ]n b
V

Substitution of (13.61) and (13.62) into this expression yields

ρeq (n a , n b ) pv ,T = C [Φa (xbl ) z a,eq ]n a [Φb (xbl ) z b,eq ]n b e−g

surf (n
a ;x b ,T )
l
(13.64)

where
g surf (n a ; xbl , T ) = θmicro (xbl ) n as (n a ; xbl ) (13.65)

The right-hand side of (13.64) contains the unknown intensive quantities Φa , Φb and
θmicro which depend on the bulk composition of the cluster and the temperature. To
determine them we consider appropriate limiting cases, for which the behavior of
ρeq (n a , n b ) can be deduced from thermodynamic considerations.
The ( p v , T )-equilibrium corresponds to the bulk liquid composition xb,eq ( p v , T ).
An arbitrary (n a , n b )-cluster at ( p v , T )-equilibrium has the bulk composition xbl

1Note in this respect, that n as (n a ) is always positive, while the Gibbs excess numbers n iexc can be
both positive and negative.
13.5 Equilibrium Distribution of Binary Clusters 229

different from xb,eq . Let us now fix xbl and consider the two-phase equilibrium at the
pressure p coex (xbl , T ), representing the total pressure above the bulk binary solution
with the composition xbl . Obviously, p coex (xbl , T ) = p v (the equality occurs only for
xbl = xb,eq ). At this “xbl -equilibrium” state the fugacities are: z i,coex = eβμi,coex /Λi3 ,
where μi,coex (xbl ) is the chemical potential at xbl -equilibrium. Thus, in the distribution
function for this state z i,eq in (13.64) should be replaced by z i,coex .
Now, from the entire cluster size space let us consider the clusters falling on the
xbl -equilibrium line, i.e. those whose bulk numbers of molecules satisfy:

n lb = n al (xbl /xal )

For them the chemical potential of the molecule inside the cluster is equal to its value
in the surrounding vapor at the pressure p coex . The Gibbs formation energy of such
a cluster will contain only the (positive) surface term:

ρ(n a , n b ) pcoex ,T = C e−g

surf
(13.66)

This consideration leads to the determination of the functions Φi (xbl ) in (13.64):

1
Φi (xbl ) = (13.67)
z i,coex (xbl )

Considering the vapor to be a mixture of ideal gases, we write

 
z i,eq   yi,eq p v
= exp βμi,eq − βμi,coex ≈ , i = a, b
z i,coex (xbl ) yicoex (xbl ) p coex (xbl )

leading to
 n a  n b
ya,eq pv yb,eq pv
e−g
surf (n
a ;x b ,T )
l
ρeq (n a , n b ) pv ,T =C
yacoex (xbl ) p coex (xbl ) ybcoex (xbl ) p coex (xbl )
(13.68)

where yicoex (xbl ) is the equilibrium vapor fraction of component i at xbl -

equilibrium.
Now let us identify the parameters θmicro and C for the pseudo-a fluid. Within the
(single-component) MKNT they are expressed in terms of the equilibrium properties
of the substance:
 
B2 psat
θmicro = − ln − , (13.69)
kB T
psat θmicro
C= e (13.70)
kB T
230 13 Coarse-Grained Theory of Binary Nucleation

Here B2 (T ) is the second virial coefficient, psat (T ) is the saturation pressure. The
number of surface molecules n as depends parametrically on the coordination number
in the liquid phase N1 and the reduced plain layer surface tension θ∞ (T ).
Comparing (13.67) and (13.63), we can identify the “saturation state” of the pseudo-a
fluid: the latter is characterized by the chemical potential μa,coex (xbl ) and the pressure

psat = yacoex (xbl ) p coex (xbl ) (13.71)

Since the intermolecular potential u(r ; xbl , T ) of the pseudo—a fluid is not known,
we have to introduce an approximation for the second virial coefficient, which will
now depend on xbl . The simplest form satisfying the pure components limit, is given
by the mixing rule:

B2 = B2,aa (xal )2 + 2B2,ab xal xbl + B2,bb (xbl )2 (13.72)

where B2,ii (T ) is the second virial coefficient of the pure component i; the cross virial
term B2,ab (T ) can be estimated using the standard methods [15]. Having identified
psat and B2 for the pseudo-a system, we find the reduced microscopic surface tension
θmicro ≡ θmicro,a and the normalization factor C from Eqs. (13.69)–(13.70):
 
B2 yacoex p coex
θmicro,a (xbl ) = − ln − , (13.73)
kB T
 coex coex 
ya p
C(xbl ) = eθmicro,a (13.74)
kB T

The cluster distribution (13.68) with g surf given by (13.65) will be fully determined
if we complete it by the model for n as (n a ; xbl ). Calculation of this quantity requires
the knowledge of the reduced planar surface tension of the pseudo-a fluid, θ∞,a and
the coordination number in the liquid phase, N1,a . The latter can be estimated from
(7.68) in which the packing fraction of pseudo-a molecules is approximated as
π l l 3
η= ρ (xb ) σa (13.75)
6

Here ρ l (xbl ) is the binary liquid number density at xbl -equilibrium and σa is the
molecular diameter of component a.
To determine θ∞,a let us consider the distribution function ρeq (n a , n b ) for big
(n a , n b )-clusters. These clusters satisfy the capillarity approximation in which the
surface part of the Gibbs energy takes the form

g surf (n a , n b ; xbl ) = β γ (xbl ) A = β γ (xbl ) (36π )1/3 (n al val + n lb vbl )2/3 (13.76)

where γ (xbl ) is the planar surface tension of the binary system at xbl -equilibrium.
13.5 Equilibrium Distribution of Binary Clusters 231

Speaking about the single-component cluster with pseudo-a particles, we character-

ize it by the total number n a molecules (not n al ). Therefore, it is convenient to rewrite
(13.76) in terms of the total numbers, which we can do using the properties of the
K -surface:

g surf (n a , n b ; xbl ) = β γ (xbl ) (36π )1/3 (n a val + n b vbl )2/3 (13.77)

The partial molecular volume of component i can be expressed as (see Appendix

D.1):
1
vil = l ηil (13.78)
ρ

where 
∂ ln ρ l 
ηil = 1 + x j
∂ x j  pcoex ,T, j =i

Substituting (13.78) into (13.77) we find

  2/3 
 2/3 x tot
2/3
g surf (n a , n b ; xbl ) = θ∞ n a ηal + n b ηbl = θ∞ ηal + btot ηbl na
xa
(13.79)
where
θ∞ (xbl ) = β γ (36π )1/3 (ρ l )−2/3 (13.80)

is the reduced planar surface tension of the original binary system at

xbl -equilibrium.
At the same time, for a big single-component cluster with n a pseudo-a molecules
the surface free energy is
2/3
g surf = θ∞,a n a (13.81)

We require that the binary (n a , n b )-cluster has the same surface energy as the unary
n a -cluster of pseudo-a molecules for all sufficiently large n a . This implies that
 2/3
xbtot l
θ∞,a = θ∞ (xbl ) ηa + tot ηb
l
(13.82)
xa

The construction of the model thus ensures that for sufficiently large clusters, sat-
isfying the capillarity approximation, the distribution function recovers the clas-
sical Reiss expression [16] (see also [17]) and is symmetric in both components.
This will not be true for small clusters, for which the capillarity approximation
fails and the Gibbs energy of cluster formation differs from its phenomenological
counterpart.
232 13 Coarse-Grained Theory of Binary Nucleation

Having determined all parameters, entering Eq. (13.64), we are in a position to write
down the equilibrium distribution function:
( )
ρeq (n a , n b ) pv ,T = β yacoex (xbl ) p coex (xbl ) e−geq (n a ,n b ;xb )
l
(13.83)
* +, -
C(xbl )

where

geq (n a , n b ; xbl ) = g bulk (n a , n b ; xbl ) + g surf (n a , n b ; xbl ) (13.84)

 
yi,eq p v
g (n a , n b ; xb ) = −
bulk l
n i ln coex l coex l (13.85)
i
yi (xb ) p (xb )
g surf (n a , n b ; xbl ) = θmicro,a (xbl ) [ n as (n a ; xbl ) − 1] (13.86)

We have included the exponential factor with the microscopic surface tension in
(13.70) into the surface part of the free energy thereby redefining the prefactor C of
the distribution function:

C(xbl ) = β yacoex (xbl ) p coex (xbl ) (13.87)

The binary distribution function (13.83) recovers the single-component MKNT limit
when n b → 0.
An important feature of CGNT is that it eliminates ambiguity in the normalization
constant C in the nucleation rate inherent to BCNT and its modifications. This is
the direct consequence of replacing the constrained equilibrium concept by the full
thermodynamic equilibrium. Since all rapidly (exponentially) changing terms in the
distribution function are included into the free energy, we can safely set the value of
C in (13.90) to the one corresponding to the critical cluster:
∗ ∗ ∗
C(xbl ) = β yacoex (xbl ) p coex (xbl ) (13.88)

Our choice to trace out the b-molecules in the cluster in favor of a-molecules could
have been reversed: we could trace out a-molecules to be left with the effective
Hamiltonian for the b-molecules resulting in Eq. (13.52) with the single-component
cluster containing pseudo-b particles. This equation is formally exact. However,
calculation of the coarse-grained configuration integral in (13.52) invokes approxi-
mations inherent to MKNT. Its domain of validity is given by Eq. (7.53) which for
the system of pseudo-i particles (i = a or b) reads
 
 B2 yicoex p coex 
  1 (13.89)
 kB T 
13.5 Equilibrium Distribution of Binary Clusters 233

It is clear that to obtain accurate predictions within the present approach one has
to trace out the more volatile component, i.e. the one with the largest bulk vapor
fraction. Throughout this chapter we assume this to be component b: ybcoex > yacoex ;
thus, the coarse-grained cluster contains pseudo-a particles.

13.6 Steady State Nucleation Rate

Combining Eqs. (13.33) and (13.83)–(13.86), we obtain the total steady-state nucle-
ation rate for the binary mixture at the total pressure p v , temperature T and vapor
mole fractions yi :
∗ ∗ ∗ ∗
J = vav A∗ Z C(xbl ) e−g(n a ,n b ) (13.90)
* +, -
K

where star refers to the critical cluster being the saddle-point of the free energy
surface
∗ ∗
g(n a , n b ; xbl ) = − n i ln Si + geq (n a , n b ; xbl ) (13.91)
i

in the space of total numbers n i . Note, that search for the saddle point in the space of
bulk numbers can lead to an erroneous critical cluster composition and unphysical
results. Technical details of the saddle-point calculations are given in Appendix G.
The proposed model is termed the Coarse-Grained Nucleation Theory (CGNT).
It is instructive to summarize the steps leading to Eq. (13.90).
• Step 0. Determine the metastability parameters of components using as a reference
the ( p v , T )-equilibrium state of the mixture
yi
Si = , i = a, b
yi,eq ( p v , T ; yc )

• Step 1. Choose an arbitrary bulk composition of a cluster n al , n lb ; the bulk fraction

of component b is then
xbl = n lb /(n al + n lb )

• Step 2. Calculate the two-phase (xbl , T )-equilibrium properties from an appropri-

ate equation of state for the mixture:

yicoex (xbl , T ), p coex (xbl , T ), ρ l (xbl , T ), ρ v (xbl , T )

Find the planar surface tension γ∞ (xbl , T ) for the x(lb , T )-equilibrium, using e.g.
the Parachor method, or some other available (semi-)empirical correlation.
234 13 Coarse-Grained Theory of Binary Nucleation

• Step 3. Calculate the properties of the pseudo-a fluid: the second virial coefficient
from Eq. (13.72) and the reduced microscopic surface tension from Eq. (13.73)
• Step 4. Determine the excess numbers of molecules n aexc , n exc
b from the K -surface
equations (11.84)–(11.85). The total numbers of molecules in the cluster are

n il = n il + n iexc , i = a, b

• Step 5. Calculate the free energy of cluster formation g(n a , n b ; xbl ) from Eq.
(13.91).
• Step 6. Repeat Steps. 1–5 for a “reasonably chosen” domain of bulk compositions,
accumulating the data:

n al , n lb , n a , n b , g(n a , n b ; xbl )

• Step 7. Determine the saddle-point (n a∗ , n ∗b ) of g in the (n a , n b )-space (using e.g.

the method outlined in Appendix G)
• Step 8. Calculate the prefactor for the nucleation rate

∗ ∗
K = vav A∗ Z C(xbl )

• Step 9. Calculate the steady-state nucleation rate from Eq. (13.90)

For practical purpose it is usually sufficient to approximate the flow direction at
the saddle point by the angle satisfying tan ϕ ≈ n ∗b /n a∗ (see Fig. 11.2). Another
simplification refers to the Zeldovich factor. Recall that in CNT the Zeldovich factor
takes the form of Eq. (3.51), where 1/ρ l = vl is the molecular volume in the liquid
phase. Replacing in (3.51) γ∞ by the planar surface tension of the binary mixture
∗
taken at the saddle point concentration γ∞ (xbl ), and vl —by the average molecular
volume of a (virtual) monomer in the liquid phase
∗ ∗
l
vav = xal val + xbl vbl

we end up with the virtual monomer approximation for Z proposed by Kulmala and
Viisanen [18]: .
∗ ∗ ∗
γ∞ (xbl ) xal val + xbl vbl
Z = (13.92)
kB T 2π (r ∗ )2

where r ∗ is the radius of the critical cluster.

13.7 Results: Nonane/Methane Nucleation 235

13.7 Results: Nonane/Methane Nucleation

In Sect. 11.9.4 we considered nucleation in the binary mixture n-nonane/methane in

the absence of a carrier gas. Methane, being in abundance in the vapor phase, is a
natural candidate for component b in CGNT, whose degrees of freedom are traced
out within the coarse-graining procedure.
In Fig. 11.9 we showed the BCNT predictions for this system for T = 240 K
and pressures 10, 25, 33, 40 bar comparing them with the experimental data of
Refs. [4, 19–21]. Figure 13.1 completes this plot with the CGNT data. The agreement
between CGNT and experiment lies within the range of experimental accuracy for
most of the conditions except for extremely low Sa < 5 at the highest pressure 40 bar.
An important insight into the binary nucleation process can be obtained from the
analysis of the structure of the critical cluster. The latter determines the height of
the nucleation barrier as well as the average impingement rate. Figure 13.2 shows
the CGNT critical cluster content—bulk, excess, and total numbers of molecules
of each species—as a function of the total pressure at a fixed nucleation rate
J = 1010 cm−3 s−1 and temperature T = 240 K. At p v < 18 bar there are no
methane molecules in the critical cluster: n lb = n exc
b = 0. This feature indicates that
at low pressures nucleation can be viewed as an effective single-component process in
which the macroscopic thermodynamic properties—liquid density, surface tension—
are those of the mixture at the given p v and T . Even with this simplification one
has to bear in mind that since the critical cluster at these conditions is quite small—
n a ≈ 19 ÷ 21—the appropriate single-component treatment requires nonclassical
considerations. Beyond p v ≈ 18 bar methane starts penetrating into the critical clus-
ter, the nucleation process demonstrates binary features, becoming more and more
pronounced as the pressure grows. In Sect. 11.9.4.1 we discussed the compensation
pressure effect. For the nonane/methane system compensation pressure found from

Fig. 13.1 Nonane/methane 25

nucleation. Nucleation rate nonane/methane
versus metastability parameter 20 T=240 K
of nonane Snonane at various CGNT
pressures and T = 240 K. 15 BCNT
log 10 J (cm s )
-1

Solid lines: CGNT; dashed

expt
-3

lines: BCNT. Symbols: exper- 10

iment of Luijten [4, 19]
r

40

(closed circles), Peeters [20]

ba

5
40

(open circles) and Labetski

33
25

[21] (half-filled squares) 0

10
ar

r
b

ba

r
33

ba

-5
25

10

-10
0.5 1 1.5

log 10 Snonane
236 13 Coarse-Grained Theory of Binary Nucleation

Fig. 13.2 Critical cluster

10 -3 -1
at a fixed nucleation rate 50 J CGNT = 1 0 cm s
J = 1010 cm−3 s−1 at T = tot
240 K as a function of the tot +n b
40
total pressure; n itot ≡ n i = na
n il + n iexc . The vertical arrow

na , n b
indicates the compensation 30 nla
pressure
tot
na
20
tot
nb
10
l
nb

0
0 10 20 30 40 50 60
v
p (bar)
pcomp

Eq. (11.126) is
pcomp (T = 240 K) ≈ 17.8 bar

It is remarkable that the change in the nucleation behavior predicted by CGNT occurs
exactly at pcomp .
As the pressure is increased the total number of nonane molecules grows very slowly
while the total number of methane molecules increases rapidly, accumulating pre-
dominantly at the dividing surface; at the highest pressure of 60 bar shown in Fig. 13.2
there are only about 4 methane molecules in the interior of the cluster, while their
total number is ≈ 22 being close to the total number of nonane molecules n a ≈ 26.
Hence, at high pressures the critical cluster is a nano-sized object with a core-shell
structure: its interior is rich in nonane while methane is predominantly adsorbed on
the dividing surface.

References

1. C. Flageollet, M. Dihn Cao, P. Mirabel, J. Chem. Phys. 72, 544 (1980)

2. B. Wyslouzil, J.H. Seinfeld, R.C. Flagan, K. Okuyama, J. Chem. Phys. 94, 6827 (1991)
3. R. Strey, Y. Viisanen, J. Chem. Phys. 99, 4693 (1993)
4. C.C.M. Luijten, P. Peeters, M.E.H. van Dongen, J. Chem. Phys. 111, 8535 (1999)
5. V.I. Kalikmanov, Phys. Rev. E 81, 050601(R) (2010)
6. C.N. Likos, Phys. Rep. 348, 267 (2001)
7. G.C.J. Hofmans, M.Sc. Thesis, Eindhoven University of Technology, 1993
8. R. Flagan, J. Chem. Phys. 127, 214503 (2007)
9. M. Dijkstra, R. van Rooij, R. Evans, Phys. Rev. Lett. 81, 2268 (1998)
10. V.I. Kalikmanov, Statistical Physics of Fluids. Basic Concepts and Applications (Springer,
Berlin, 2001)
References 237

11. M. Dijkstra, R. van Rooij, R. Evans, Phys. Rev. E 59, 5744 (1999)
12. R. van Rooij, J.-P. Hansen, Phys. Rev. Lett. 79, 3082 (1997)
13. R. van Rooij, M. Dijkstra, J.-P. Hansen, Phys. Rev. E 59, 2010 (1999)
14. V.I. Kalikmanov, Phys. Rev. E 68, 010101 (2003)
15. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids (McGraw-Hill,
New York, 1987)
16. H. Reiss, J. Chem. Phys. 18, 840 (1950)
17. G. Wilemski, B. Wyslouzil, J. Chem. Phys. 103, 1127 (1995)
18. M. Kulmala, Y. Viisanen, J. Aerosol Sci. 22, S97 (1991)
19. C.C.M. Luijten, Ph.D. Thesis, Eindhoven University, 1999
20. P. Peeters, Ph.D. Thesis, Eindhoven University, 2002
21. D.G. Labetski, Ph.D. Thesis, Eindhoven University, 2007
Chapter 14
Multi-Component Nucleation

Understanding multi-component nucleation is of great importance for atmospheric

and environmental sciences. Vivid examples are: polar stratospheric clouds, acid
rains and air pollution. All these phenomena occur because Earth’s atmosphere is a
multi-component gaseous system in which nucleation leads to formation of droplets
of complex composition. A rich field of applications of multi-component nucleation
is associated with the natural gas industry since nucleation is the primary mechanism
responsible for formation of mist during the expansion of natural gas [1]. This is the
key process of the non-equilibrium gas-liquid separation technology [2].
Theoretical description of multi-component nucleation pioneered by Hirschfelder [3]
and Trinkaus [4] represents the extension of the phenomenological binary nucleation
theory, discussed in Chap. 11, to the N -component mixture.

14.1 Energetics of N-Component Cluster Formation

Consider an arbitrary cluster of the new phase (e.g. liquid) containing n 1 , . . . , n N

molecules of components 1, . . . , N , respectively. We will call it the n-cluster, where
the vector n = (n 1 , . . . , n N ) is represented by the point in the N -dimensional space
of the cluster sizes of components. The n-cluster is immersed in the “mother phase”
(e.g. supersaturated gas) characterized by the total pressure p v and temperature T .
Energetics of cluster formation determines the minimum reversible work ΔG(n)
needed to form the n-cluster in the surrounding vapor.
To take into account adsorption (leading to inhomogeneous density distribution of
species inside the cluster) we introduce an arbitrary located Gibbs dividing sur-
face distinguishing between the bulk (superscript “l”) and excess (superscript “exc”)
molecules of each species. Then, the total numbers of molecules n i are

n i = n il + n iexc , i = 1, . . . , N (14.1)

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 239

DOI: 10.1007/978-90-481-3643-8_14, © Springer Science+Business Media Dordrecht 2013
240 14 Multi-Component Nucleation

As discussed in Chap. 11, n il and n iexc separately depend on the location of the
dividing surface while their sum is independent of this location to the relative accu-
racy of O(ρ v /ρ l ), where ρ v and ρ l are the number densities in the vapor and liq-
uid phases. Straightforward generalization of Eq. (11.48) to N -component mixture
yields:


N   N
 
ΔG = ( p − p )V + γ A +
v l l
n il μi ( p ) − μi ( p ) +
l l v v
n iexc μiexc − μiv ( p v )
i=1 i=1
(14.2)
Here p l is the pressure inside the cluster,


N
Vl = n il vil (14.3)
i=1

 N 2/3

A = (36π ) 1/3
n il vil (14.4)
i=1

are, respectively, the cluster volume and surface area calculated at the location of
the dividing surface; vil is the partial molecular volume of component i in the liq-
uid phase, γ is the surface tension at the dividing surface. Within the capillarity
approximation
μil ( p l ) = μil ( p v ) + vil ( p l − p v ) (14.5)

Substituting (14.3) and (14.5) into (14.2) results in


N 
N
 
ΔG = γ A − n il Δμi + n iexc μiexc − μiv ( p v ) (14.6)
i=1 i=1

where
Δμi ≡ μiv ( p v ) − μil ( p v ) (14.7)

is the driving force for nucleation. As in the binary case, the chemical potentials in
both phases are taken at the vapor pressure p v .
Recalling that diffusion between the surface and the interior of the cluster is much
faster than diffusion between the surface and the mother vapor phase, we assume that
there is always equilibrium between the cluster (dividing) surface and the interior,
resulting in
μiexc = μil ( p l ) (14.8)
14.1 Energetics of N -Component Cluster Formation 241

Using incompressibility of the liquid this implies

μiexc − μiv ( p v ) = −Δμi + vil ( p l − p v ) (14.9)

Substituting (14.9) into (14.6) and using Laplace equation, we obtain a generalization
of Eq. (11.55):

N 

  2γ (xl ) 
N
ΔG = γ (xl ) A − n il + n iexc Δμi + n iexc vil (14.10)
r
i=1 i=1
ni

where r is the radius of the cluster (assumed to be spherical), γ (xl ) is the surface
tension of the N -component liquid solution with composition

xl = (x1l , . . . , x Nl ),
nl 
N
xil =  N i , xil = 1
l
k=1 n k i=1

Note, that the second term in (14.10) contains the total numbers of molecules, while
the thermodynamic properties depend on the bulk cluster composition xl .
If we choose the dividing surface according to the K -surface recipe (cf.
Sect. 11.6):
N
n iexc vil = 0 (14.11)
i=1

then the last term in ΔG disappears, leading to


N
ΔG = − n i Δμi + γ (xl ) A (14.12)
i=1

The volume and surface area of the cluster can now be written in terms of total
numbers of molecules


N
Vl = n i vil (14.13)
i=1
 N 2/3

A = (36π ) 1/3
n i vil (14.14)
i=1

The chemical potential of a molecule inside the cluster can be written as

242 14 Multi-Component Nucleation

μil (xl ) = μiv ({y coex

j (xl )})

where y coexj (xl ) is the fraction of component j in the N -component vapor which
coexists with the N -component liquid having the composition xl of the cluster;
the corresponding coexistence pressure is p coex (xl ). The superscript “coex” empha-
sizes that the corresponding quantity refers to (x, T )-equilibrium, rather than the
( p v , T )-equilibrium. The quantities p coex (xl ) and ycoex = (y1coex (xl ), y2coex (xl ), . . .)
are found from the (x, T )-equilibrium equations:

p l (T, ρ l , xl ) = p coex
p v (T, ρ v , ycoex ) = p coex (14.15)
μil (T, ρ l , xl ) = μiv (T, ρ v , ycoex ), i = 1, . . . , N

This is the system of N + 2 equations for the N + 2 unknowns:

ρ l , ρ v ; p coex ; y1coex , . . . , y Ncoex

−1
3 N −1

Using the ideal gas approximation we write


N  
yi p v
ΔG(n 1 , . . . , n N ) = − kB T n i ln + γ (xl ) A (14.16)
yicoex (xl ) p coex (xl )
i=1

If we replace the bulk fractions xl by the total fractions

xtot = (x1tot , . . . , x Ntot )

where ni
xitot =  N
k=1 n k

then Eq. (14.16) becomes a straightforward generalization of Reiss result (11.100) to

the multi-component case. The equilibrium distribution of multi-component clusters
has the general form:

ρeq (n 1 , . . . , n N ) = C e−βΔG(n 1 ,...,n N ) (14.17)

N
The prefactor C in Reiss’s approximation reads C = i=1 ρi , where ρi is the number
density of monomers of component i in the mother phase.
Let us the rotate the coordinate system (n 1 , . . . , n N ) of the N -dimensional space of
cluster sizes. The general form of rotation transformation is
14.1 Energetics of N -Component Cluster Formation 243


N
wi = U ji n j (14.18)
j=1

where the new set of coordinates is denoted as wi , and U is a unitary matrix with real
coefficients [5]. The latter by definition satisfies

UT = U−1 (14.19)

where UT is the transposed matrix and U−1 is the inverse matrix. From (14.19) it
follows that multiplication by U has no effect on inner products of the vectors, angles
or lengths. In particular, lengths of the vectors ||n|| are preserved

(U n)T (U n) = nT (UT U) n = nT n

which stands for

||U n||2 = ||n||2

The latter property actually provides rotation. Since UT U = I , where I is the unit
matrix, we have
det (UT U) = det UT det U = 1

resulting in
det U = 1

where we used the identity det UT = det U.

For example, for the binary case, rotation at the angle φ results in the matrix (cf. Eq.
(11.17)):  
cos φ − sin φ
U= (14.20)
sin φ cos φ

It is straightforward to see that its determinant is equal to unity.

Using the direction of principal growth approximation, we define w1 as the “reaction
path” (direction of principal growth). This means that the fluxes along other directions
w2 , . . . , w N are set to zero. The rotation angle, or equivalently, the form of the
rotation matrix U should be determined separately. The critical cluster (denoted by
the subscript “c”) is defined as the one corresponding to the saddle point of the free
energy surface. The latter satisfies the standard relationships
   
∂ΔG ∂ΔG
= 0, or = 0, i = 1, . . . , N (14.21)
∂wi c ∂n i c

Now we are in the position to impose a certain form of the rotation matrix. Since the
critical cluster is associated with the saddle point of the free energy, we define U so
244 14 Multi-Component Nucleation

that second derivatives of ΔG at the saddle point satisfy

 
∂ 2 ΔG
= Q i δi j , i, j = 1, . . . , N (14.22)
∂wi ∂w j c

where δi j is the Kronecker delta; Q 1 < 0 (maximum of ΔG along w1 ), while the rest
eigenvalues are all positive: Q 2 , . . . , Q N > 0 (minimum of ΔG along w2 , . . . , w N );
the values of Q i are yet to be defined. Then, the matrix U should satisfy

    
∂ 2 ΔG ∂ 2 ΔG
0= Uui Uv j = , i = j (14.23)
u,v
∂n u ∂n v c ∂wi ∂w j c

The unitary conditions for U read:

 
Ui j Uv j = δiv = U ji U jv (14.24)
j j

implying U −1
ji = Ui j . Equation (14.23) determines the rotation angle of the original
coordinate system.
Combining Eqs. (14.22) and (14.23) we find

  
−1 ∂ 2 ΔG
Uiu Uv j = Q i δi j (14.25)
u,v
∂n u ∂n v c

We multiply both sides of (14.25) by Uki and sum over k. Since U is the unitary
matrix, we obtain using Eq. (14.24):

  ∂ 2 ΔG  
− Q j δkv Uv j = 0 (14.26)
v
∂n k ∂n v c

This expression shows that Q j are the eigenvalues and Uv j are the eigenvectors of
the Hessian matrix  2 
∂ ΔG
G2 =
∂n k ∂n v c

The eigenvalues are the roots of the secular equation

det (G2 − Q I ) = 0

We choose the rotational system such that the critical cluster corresponds to

w1 = w1c , w2c = . . . = w N c = 0
14.1 Energetics of N -Component Cluster Formation 245

Expansion of the Gibbs energy in the vicinity of the saddle point of ΔG reads:

1 
N
1
ΔG = ΔG c + Q 1 (w1 − w1c )2 + Q i wi2 + . . . (14.27)
2 2
i=2

where ΔG c is the value of ΔG at the saddle point. Substituting (14.27) into (14.17),
we obtain the equilibrium cluster distribution in the vicinity of the saddle point:
⎧ ⎡ ⎤⎫
⎨ 1 
N ⎬
ρeq (w1 , . . . , w N ) = ρeq,c exp − β ⎣ Q 1 (w1 − w1c )2 + Q i wi2 + . . .⎦ (14.28)
⎩ 2 ⎭
i=2

where
ρeq,c = C e−βΔG c (14.29)

is the equilibrium number density of the critical clusters.

14.2 Kinetics

We assume that formation of the multi-component cluster results from attachment

or loss of a single monomer E j of one of the species, i.e. we consider the reactions
of the following type

k
E(n 1 , . . . , n j , . . . , n N ) + E j k j E(n 1 , . . . , n j + 1, . . . , n N ) (14.30)
j

Here k j (n 1 , . . . , n N ) and k j (n 1 , . . . , n N ) are the reaction rates. In the chemical

kinetics the forward reaction rate is proportional to the impingement rate v j of the
component j and the surface area of the cluster:

ρ j k j = v j A(n 1 , . . . , n j , . . . , n N ) (14.31)

The form of v j depends on the physical nature of the nucleation process; for the
gas-liquid transition it is given by the gas kinetics expression
pj
vj =  (14.32)
2π m j kB T

where p j is the partial pressure of component j in the mother phase, and m j is the
mass of the molecule of component j. Equation (14.30) describes a single act of the
cluster evolution. The net rate at which the clusters (n 1 , . . . , n j + 1, . . . , n N ) are
created in the unit volume of the system is
246 14 Multi-Component Nucleation

I j (n 1 , . . . , n j , . . . , n N ) = ρ j k j ρ(n 1 , . . . , n j , . . . , n N )
− k j ρ(n 1 , . . . , n j + 1, . . . , n N ) (14.33)

In equilibrium all net rates vanish (here, as in the BCNT, we consider the constrained
equilibrium) which yields the determination of the backward reaction rate

ρ j k j ρeq (n 1 , . . . , n j , . . . , n N )
k j (n 1 , . . . , n N ) = (14.34)
ρeq (n 1 , . . . , n j + 1, . . . , n N )

The evolution of the cluster distribution function is given by the kinetic equation
which includes all possible reactions with single molecules of species 1, . . . , N

∂ρ(n 1 , . . . , n j , . . . , n N )  
N
= I j (n 1 , . . . , n j − 1, . . . , n N )
∂t
j=1

−I j (n 1 , . . . , n j , . . . , n N ) (14.35)

Let us introduce the ratios of the actual (nonequilibrium) to equilibrium distribution

functions
ρ(n 1 , . . . , n j , . . . , n N )
f (n 1 , . . . , n j , . . . , n N ) = (14.36)
ρeq (n 1 , . . . , n j , . . . , n N )

Then using (14.34) I j can be written as

I j = ρ j k j ρeq (n 1 , . . . , n j , . . . , n N )
× [ f (n 1 , . . . , n j , . . . , n N ) − f (n 1 , . . . , n j + 1, . . . , n N )]

As usual in the phenomenological theories, we consider sufficiently large clusters,

so that derivatives can be replaced by finite differences resulting in

∂ f (n 1 , . . . , n j , . . . , n N )
I j = − ρ j k j ρeq (n 1 , . . . , n j , . . . , n N ) (14.37)
∂n j

Then, the kinetic equation (14.35) becomes

N 
 
∂ρ(n 1 , . . . , n j , . . . , n N ) ∂Ij
=− = −div I (14.38)
∂t ∂n j
j=1

where the vector I is defined as I = (I1 , . . . , I N ). The same equation in the rotated
system reads
⎡ ⎤
    ∂ ⎣ −1 ⎦
N N N N
∂ρ ∂Ij
=− Ui−1 =− Ui j I j ≡ −div J (14.39)
∂t j
∂wi ∂wi
i=1 j=1 i=1 j=1
14.2 Kinetics 247

where J = (J1 , . . . , J N ),


N
Ji = Ui−1
j I j , i = 1, . . . , N (14.40)
j=1

is the component of the nucleation flux along the wi -axis. Using (14.31) and (14.37)
Ji can be presented as


N  
∂f
Ji = −A ρeq Biu , i = 1, . . . , N (14.41)
∂wu
u=1

in which

N
Biu = Ui−1
j v j U ju (14.42)
j=1

We search for the steady state solution of Eq. (14.39) ignoring the short time-lag
stage. Then, Eq. (14.39) becomes

div J = 0

Recall that we have chosen the rotated system in such a way that the flux J is directed
along the w1 axis, implying
J2 = . . . = J N = 0 (14.43)

Then Eq. (14.41) for i = 2, . . . , N become

  
N  
∂f ∂f
Bi1 + Biu = 0, i = 2, . . . , N
∂w1 ∂wu
u=2

∂f
By separating the term with ∂w1 from the rest of the sum, we get a linear set of N − 1
∂f
equations for N − 1 unknown variables ∂wu , i = 2, . . . , N . Its determinant is
 
 B22 . . . B2N 
 
 · · 

D2 =  · · 
 · · 

 BN 2 . . . BN N 

and the solution reads:

 
∂f (−1)u+1 ∂f
= L u u = 2, . . . , N (14.44)
∂wu D2 ∂w1
248 14 Multi-Component Nucleation

where  
 B21 B22 . . . B2,u−1 B2,u+1 . . . B2N 

 · · · · · · · 
 
Lu =  · · · · · · · 
 · · · · · · · 

 BN 1 BN 2 . . . B N ,u−1 B N ,u+1 . . . BN N 

∂f ∂f
Having expressed N − 1 quantities ∂w u
in terms of ∂w 1
, we substitute the solution
(14.44) into the only remained equation from the set (14.41), namely the equation
for J1 :
  
N  
∂f ∂f
J1 = −A ρeq B11 + B1u
∂w1 ∂wu
u=2
  
∂f 1 
N
= −A ρeq B11 D2 + (−1) u+1
B1u L u
∂w1 D2
u=2

As can be easily seen, the expression in the square brackets is the determinant of the
N × N matrix of all Bi j ’s:
 
 B11 . . . B1N 
 
 · · 
 
D1 =  · ·  (14.45)
 · · 

 BN 1 . . . BN N 

Thus, J1 can be written in the compact form

 
∂f D1
J1 (w1 , . . . , w N ) = −A ρeq (14.46)
∂w1 D2

Integrating Eq. (14.46) along the reaction path w1 , we obtain

w1 J1 D2
f (w1 ) = − dw + f 1
1 D1 ρeq A 1

where f 1 is an unknown integration constant. In view of the exponential form of

ρeq (w1 , . . . , w N ) we may assign the slowly varying functions A, D1 , D2 , J1 their
values calculated at w1 = w1c and take them outside of the integral:
  w1
J1 D2 1
f (w1 ) = − dw1 + f 1 (14.47)
D1 A w1 =w1c 1 ρeq

Now we apply the (standard) boundary conditions for the cluster distribution function
(cf. (3.42)–(3.43)): the concentration of small clusters is nearly equal to equilibrium
14.2 Kinetics 249

one; for large clusters ρeq diverges, while the actual distribution function ρ remains
finite. These requirements yield:

f (w1 ) → 1 for w1 → 1, and f (w1 ) → 0 for w1 → ∞ (14.48)

The first condition results in f 1 = 1, then from the second one we obtain
   ∞  −1
D1 A 1
J1 (w2 , . . . , w N ) =  dw1
D2 w1 =w1c 1 ρeq (w1 , w2 , . . . , w N )
(14.49)
We can simplify this result by using the expansion (14.28) and extending the inte-
gration limits in (14.49) to ±∞:
  ! "
D1 A 1 N
J1 (w2 , . . . , w N ) = ρeq,c exp − β Q i wi2
D2 w1 =w1c 2
i=2
Q 1 (w1 −w1c ) 2  −1
∞
× dw1 e 2kB T
−∞

Taking into account that Q 1 < 0 and performing Gaussian integration we obtain

  ! " #
D1 A 1 
N
(−Q 1 )
J1 (w2 , . . . , w N ) = ρeq,c exp − β Q i wi2
D2
w1 =w1c 22π kB T
i=2
(14.50)
The total nucleation rate is found by integrating J1 over all possible values of
w2 , . . . , w N . In view of the previously presented considerations we set the limits
of integration to ±∞:
∞ ∞
J= ... dw2 . . . dwN J1 (w2 , . . . , wN ) (14.51)
−∞ −∞

Assuming further that D1 A/D2 varies slowly with wi ’s compared to the exponential
terms exp[−Q i wi2 /kB T ], we perform N − 1 Gaussian integrations

∞ Q i wi2  1/2
− 2π kB T
dwi e 2kB T =
−∞ Qi

resulting in
  #
D1 A (−Q 1 )
J = C e−βΔG c (2π kB T )(N −2)/2 (14.52)
D2 c Q2 . . . Q N

This result represents the extension of Reiss’s BCNT to multicomponent systems.

250 14 Multi-Component Nucleation

14.3 Example: Binary Nucleation

Let us consider application of the general formalism to the binary nucleation. Rotation
matrix U in this case is given by Eq. (14.20). The rotation angle φ is found from Eq.
(14.23) which reads
$ %
∂ 2 ΔG
2 ∂n 1 ∂n 2 c
tan(2φ) =      (14.53)
∂ 2 ΔG ∂ 2 ΔG
∂n 21
− ∂n 22
c c

The eigenvalues Q j of the matrix G2 are

     
∂ 2 ΔG ∂ 2 ΔG ∂ 2 ΔG
Q1 = cos2 φ + sin(2φ) + sin2 φ
∂n 21 ∂n 1 ∂n 2 c ∂n 22
c c
     
∂ 2 ΔG ∂ 2 ΔG ∂ 2 ΔG
Q2 = sin2 φ − sin(2φ) + cos2 φ
∂n 21 ∂n 1 ∂n 2 c ∂n 22
c c

Equation (14.53) has two solutions for the angle φ. We choose the solution which
gives Q 1 < 0 and Q 2 > 0 as required by construction of the model. Having deter-
mined the rotation angle, we write down the coefficients Bi j from Eq. (14.42):

B11 = v1 cos2 φ + v2 sin2 φ

B22 = v1 sin2 φ + v2 cos2 φ
B12 = B21 = (−v1 + v2 ) sin φ cos φ
(14.54)

The determinants D1 and D2 become

D1 = B11 B22 − B12

2
= v1 v2
D2 = B22

Then, the nucleation rate (14.52) is

#
−βΔG c (−Q 1 )
J =Ce νav Ac (14.55)
Q2

where  
v1 v2
vav = (14.56)
v1 sin φ + v2 cos2 φ
2
c
14.3 Example: Binary Nucleation 251

is the average impingement rate. This result agrees with Eqs. (11.35), (11.42)–(11.44)
derived in Chap. 11.

14.4 Concluding Remarks

Using similar assumptions as in the BCNT, the present model focuses on nucle-
ation in the vicinity of the saddle point of the free energy surface and completely
ignores nucleation along all paths other than the principal nucleation path (direction
of principal growth). This local approach identifies the predominant composition
of the observable nuclei—the one that corresponds to the saddle point. Although
this approach obviously has its merits, it, however, is unable to predict the rate of
formation of nuclei with arbitrary composition. The latter issue requires a global
approach which treats all cluster compositions on an equal footing. This problem
was addressed among others by Wu [6] who studied various nucleation paths (not
only the saddle-point nucleation). In [6] conditions were identified under which a
multi-component system behaves “as if it were simple” which means that the system
can be modelled by a one-dimensional Fokker-Planck equation (3.79)–(3.80).
Direction of principal growth approximation may not always be the best choice.
Depending on the form of the Gibbs free energy surface and values of the impinge-
ment rates of the components it is possible that the main nucleation flux bypasses
the saddle-point. In particular, as pointed out by Trinkaus [4], if one reaction rate is
essentially smaller than the other ones, the flux line can turn into the directions of the
fast-reacting component and pass a ridge before the saddle-point coordinate of the
slowly reacting component is reached. For binary systems these issues were studied
numerically by Wyslouzil and Wilemski [7].

References

1. M.J. Muitjens, V.I. Kalikmanov, M.E.H. van Dongen, A. Hirschberg, P. Derks, Revue de l’Institut
Français du Pétrole 49, 63 (1994)
2. V. Kalikmanov, J. Bruining, M. Betting, D. Smeulders, in 2007 SPE Annual Technical Conference
and Exhibition (Anaheim, California, USA, 2007), pp. 11–14. Paper No: SPE 110736
3. O. Hirschfelder, J. Chem. Phys. 61, 2690 (1974)
4. H. Trinkaus, Phys. Rev. B 27, 7372 (1983)
5. K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering
(Cambridge University Press, Cambridge, 2007)
6. D. Wu, J. Chem. Phys. 99, 1990 (1993)
7. B. Wyslouzil, G. Wilemski, J. Chem. Phys. 103, 1137 (1995)
Chapter 15
Heterogeneous Nucleation

15.1 Introduction

Heterogeneous nucleation is a first order phase transition in which molecules of the

parent phase nucleate onto surfaces forming embryos of the new phase. These preex-
isting foreign particles are usually called condensation nuclei (CN). To discriminate
between the cluster and condensation nuclei, we call CN “solid”, the cluster phase
“liquid” and the parent phase “vapor”. These notations are purely terminological:
clusters as well as CN can be liquid and solid. CN can be a planar macroscopic
surface, a spherical particle (liquid or solid); it can be also an ion—the latter case is
termed ion-induced nucleation [1–4]. In this chapter we do not discuss ion-induced
nucleation and consider CN to be electrically neutral and insoluble to the cluster
formed on its surface, i.e. there is no mass transfer between the CN and the liquid
or vapor phases. A vivid example of heterogeneous nucleation, one experiences in
the everyday life, is vapor-liquid nucleation on the surface of aerosol particles in
atmosphere. The process of condensational growth of aerosol particles got consid-
erable experimental and theoretical attention due to its role in the environmental
effects [5].
Classical theory of single-component heterogeneous nucleation was developed by
Fletcher in 1958 [6]. Later on Lazaridis et al. [7] extended Fletcher’s theory to
binary systems. The general approach to kinetics of nucleation processes discussed
in Sect. 3.3 remains valid for the heterogeneous case. The steady-state nucleation
rate has the general form of Eq. (3.52)
 
ΔG ∗
J = K exp − (15.1)
kB T

where K is a kinetic prefactor and ΔG ∗ is the free energy of formation of the critical
embryo on the foreign particle (CN). As in the case of homogeneous nucleation, the
heterogeneous nucleation rate is determined largely by the energy barrier ΔG ∗ . That
is why it is sufficient to know the prefactor K (its various forms are discussed in

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 253

DOI: 10.1007/978-90-481-3643-8_15, © Springer Science+Business Media Dordrecht 2013
254 15 Heterogeneous Nucleation

Sect. 15.5) to one or two orders of magnitude. We will show in the next section that
the presence of a foreign particle reduces the energy cost to build the cluster surface.

15.2 Energetics of Embryo Formation

We consider a cluster (embryo) of phase 2 which has the form of a spherical liquid
cap of volume V2 and radius r and contains n molecules,

n = ρ l V2 (15.2)

where ρ l is the number density of the phase 2. The embryo is resting on the spherical
foreign particle (CN) 3 of a radius R p surrounded by the parent phase 1. The line
where all three phases meet, called a three-phase contact line, is characterized by
the contact angle θ . This configuration is schematically shown in Fig. 15.1. Within
the phenomenological approach embryos are considered to be the objects character-
ized by macroscopic properties of phase 2. Applying Gibbs thermodynamics to this
system, it is necessary to define two dividing surfaces: one for the gas-liquid (1–2)
interface and one for the solid-liquid (2–3) interface. Since we discuss the single-
component nucleation, we can choose for both of them the corresponding equimolar
surface so that the adsorption terms in the Gibbs energy vanish. Within the capillarity
approximation the Gibbs free energy of an embryo formation is

ΔG(n) = −n Δμ + ΔG surf (15.3)

Here Δμ = μ1 − μ2 > 0 is the difference in chemical potential of the phases 1

(supersaturated vapor) and phase 2 taken at the same pressure p v of the vapor phase.
The surface contribution contains two terms:

Rp r

Op O
d
2
3
1

Fig. 15.1 Embryo 2 (dashed area) on the foreign particle (condensation nucleus) 3 in the parent
phase 1. R p is the radius of the spherical CN, O p is its center; r is the radius of the sphere with the
center in O corresponding to the embryo, d = O p O, θ is the contact (wetting) angle
15.2 Energetics of Embryo Formation 255

ΔG surf = γ12 A12 + (γ23 − γ13 )A23 (15.4)

Here γi j is the interfacial tension between phases i and j, Ai j is the corresponding

surface area. The first term in (15.4) is the energy cost of the vapor-liquid interface,
the second one stems from the fact that when a cluster is built on the surface of CN,
the initially existed solid-vapor interface, characterized by the interfacial tension
γ13 , is replaced by the solid-liquid interface with the interfacial tension γ23 . From
geometrical considerations:

A12 = 2πr 2 (1 − cos α) (15.5)

A23 = 2π R 2p (1 − cos φ) (15.6)
4π 3
V2 = r fV (15.7)
3
where the angles α and φ are defined in Fig. 15.1:

Rp − rm r − R pm
cos φ = , cos α = − (15.8)
d d
Here

m = cos θ (15.9)

d = R 2p + r 2 − 2R p r m (15.10)

Geometrical factor f V relates the embryo volume (shaded area in Fig. 15.1) to its
homogeneous counterpart, being the full sphere of radius r . Standard algebra yields:

Rp
f V = q(cos α) − a 3 q(cos φ), a ≡ (15.11)
r
where the function q(y) is:

1
q(y) = (2 − 3 y + y 3 ) (15.12)
4
The angles φ and α depend not on r and R p individually, but on their ratio a; therefore
it is convenient to rewrite (15.8), (15.10) in dimensionless units:
 
a−m 1−ma
cos φ = , cos α = − (15.13)
w w

w = 1 + a2 − 2 m a (15.14)

Looking at (15.3)–(15.14), one can see that our model for ΔG is incomplete: the
surface areas A12 , A23 and the volume of the embryo depend on the yet undefined
256 15 Heterogeneous Nucleation

contact angle θ . Its value can be derived from the interfacial force balance at the
three phase boundary known as the Dupre-Young equation:

m ≡ cos θeq = (γ13 − γ23 )/γ12 (15.15)

(to emphasize the bulk equilibrium nature of the Dupre-Young equation we added
the subscript “eq” to the contact angle). An alternative way to derive this result is to
consider a variational problem: formation of an embryo with a given volume V2 with
an arbitrary contact angle. The equilibrium angle θeq will be the one that minimizes
ΔG at fixed V2 :

∂ΔG 
=0 (15.16)
∂m V2

Combining (15.15) with (15.3)–(15.4) we find

 
A23
ΔG = −n Δμ + γ12 A12 1−m (15.17)
A12

Let us compare the free energies necessary to form a cluster with the same number
of molecules n at the temperature T and the supersaturation S (or equivalently, Δμ)
for the homogeneous and heterogeneous nucleation. Clearly, for both cases the bulk
contribution to ΔG will be the same: −n Δμ. Consider now the surface term. In the
homogeneous case:
ΔG surf (n)hom = γ12 s1 n 2/3

where
 −2/3
s1 = (36π )1/3 ρ l (15.18)

In the heterogeneous case according to (15.17)

 
A23
ΔG surf (n)het = γ12 A12 1−m
A12

Using (15.5)–(15.12) and (15.18) we find

ΔG surf (n)het 21/3 [(1 − cos α) − m a 2 (1 − cos φ)]

=
ΔG surf (n)hom f
2/3
V

Straightforward inspection of this result shows that

ΔG surf (n)het
≤1 (15.19)
ΔG surf (n)hom
15.2 Energetics of Embryo Formation 257

implying that it is energetically more favorable to form an n-cluster on a foreign

surface than to form it directly in the mother phase 1. The equality sign in (15.19) is
realized for non-wetting conditions: θeq = π ⇔ m = −1. Mention several limiting
cases:
• R p →0 corresponds to the absence of foreign particles: f V (m, 0) = 1, thus leading
to homogeneous nucleation;
• the same (homogeneous) limit is obtained when the phase 2 does not wet the
phase 3, i.e. θeq = π , yielding from (15.11), (15.13)–(15.14): f V (−1, a) = 1.
An embryo in this case represents a sphere having a point contact with a foreign
particle.

15.3 Flat Geometry

The case of flat geometry deserves special attention. This is a limiting case when the
radius of an embryo is much smaller than the radius R p of the foreign particle, or
equivalently a→∞. Then, the embryo sees the particle as a flat wall while the particle
radius becomes irrelevant. An embryo becomes a spherical segment (cup) resting on
a plane as shown in Fig. 15.2. Taking the limit R p →∞ in Eqs. (15.5)–(15.8), we find

α = θeq
4
V2 = πr 3 q(m) (15.20)
3
A12 = 2πr 2 (1 − m) (15.21)
A23 = πr (1 − m )
2 2
(15.22)

where the function q(m) is given by (15.12)

1
q(m) = (2 − 3m + m 3 ) (15.23)
4

Fig. 15.2 Flat surface geom-

1
etry: a sphere-cup-shaped h 2
A B
embryo of radius r resting on
rt 3
a plane. The height of the cap
is h = r (1−cos θ), where θ is
r
the contact angle. Points A and
B belong to the three phase
contact line, which is a circle
of radius rt = r sin θ located
in the plane perpendicular to
the plane of the figure
258 15 Heterogeneous Nucleation

Fig. 15.3 Functions q(cos θ) 1

and q 1/3 (cos θ) given by
Eq. (15.23) q1/3(m)
0.8

0.6
q(m)

0.4

0.2

0
-1 -0.5 0 0.5 1
m=cos

The height of the cup is

h = r (1 − cos θeq )

and the 2 − 3 interface becomes a circle bounded by the three-phase contact line of
the radius rt = r sin θeq . From (15.2) and (15.20) we find the relation between the
number of molecules in the embryo and its radius:
 1/3  
3 −1/3
r= ρl q −1/3 n 1/3 (15.24)
4π

Using the general form (15.17) of the heterogenous free energy barrier together with
Eqs. (15.20)–(15.22) for the case of flat geometry we find:

ΔG(n)het = −nΔμ + (γ12 q 1/3 ) s1 n 2/3 (15.25)

For homogeneous nucleation of a cluster with the same number of molecules at the
same temperature T and supersaturation S one would have:

ΔG(n)hom = −nΔμ + γ12 s1 n 2/3 (15.26)

Comparing (15.25) and (15.26), one can see that the heterogeneous barrier has the
form of the homogeneous barrier in which the surface tension γ12 is replaced by the
effective surface free energy γeff defined by

γeff = γ12 q 1/3 (θeq ) (15.27)

Figure 15.3 shows the behavior of the functions q(m) and q 1/3 (m) for different values
of the contact angle. They increase monotonously from 0 at θ = 0 to 1 at θ = π .
15.3 Flat Geometry 259

Thus, γeff ≤ γ12 and therefore the heterogeneous barrier is always smaller or equal
to ΔG hom for the same values of T and S. Expressing ΔG in terms of the radius of
the embryo, we have using (15.24):

ΔG het (r ) = ΔG hom (r ) q(m) (15.28)

Note that Eq. (15.28) is solely based on macroscopic considerations. For nonwetting
conditions (m = −1) we recover the homogeneous limit: ΔG het = ΔG hom .

15.4 Critical Embryo: The Fletcher Factor

Let us return to the case of arbitrary geometry. The critical embryo rc satisfies

∂ΔG 
=0 (15.29)
∂r rc

Finding the critical radius using the general form (15.17) of the Gibbs formation
energy in the presence of a foreign particle requires a considerable amount of algebra.
Fortunately, calculation can be substantially simplified if we take into account that
irrespective of the presence or absence of the foreign particles, the critical cluster is
in metastable equilibrium with the surrounding vapor (phase 1) and thus the chemical
potentials of a molecule inside and outside the critical cluster are equal resulting in
the Kelvin equation (3.61):

2γ12
rc = (15.30)
ρ l Δμ

This expression manifests an important feature of the heterogeneous problem: the

radius of the critical embryo is determined solely by the temperature and the super-
saturation of the phase 2 and does not contain any information about the foreign
particles. At the same time, the number of molecules in the critical embryo depends
on the wetting properties of the phases through Eqs. (15.2) and (15.7):

rc = rc,hom (T, S)
Rp
n c = n c,hom (T, S) f V (m, ac ), ac = (15.31)
rc

assuming the same bulk liquid density of phase 2 in the homogeneous and heteroge-
neous cases. Substituting (15.30) into (15.17) and using (15.5)–(15.14), we derive
the nucleation barrier:

ΔG ∗ = ΔG ∗hom f G (m, ac ) (15.32)

260 15 Heterogeneous Nucleation

Fig. 15.4 Fletcher factor 10

f G (m, a) as a function of
a = R p /r . Labels are
the corresponding values m=-1
of m = cos θ 1
m=0

m=0.5

0.1

fG
m=0.8

0.01

m=1
0.001
0.1 1 10 100
a = Rp/r

where

1 16π γ123
ΔG ∗hom = γ12 (4π rc2 ) = (15.33)
3 3 (ρ l )2 (Δμ)2

is the homogeneous nucleation barrier given by the CNT; the function

 
1 1 1 − ma 3
f G (m, a) = +
2 2 w
     
1 3 a−m a−m 3 3 2 a−m
+ a 2−3 + + ma −1 (15.34)
2 w w 2 w

is called the Fletcher factor [6]. It varies between 0 and 1, depending on the contact
angle and the relative size of foreign particles with respect to the embryo. Thus,
heterogeneous nucleation reduces the nucleation barrier compared to the homoge-
neous case ΔG ∗hom due to the presence of foreign bodies; the Fletcher factor being
the measure of this reduction. It is important to bear in mind that the Fletcher fac-
tor f G refers exclusively to the critical embryo so that the expression (15.32) is
not true for an arbitrary embryo. The behavior of f G (m, a) as a function of a for
various values of the contact angle (parameter m) is shown in Fig. 15.4. The upper
line m = −1 corresponds to the non-wetting conditions recovering the homoge-
neous limit: f G = 1, ΔG ∗ = ΔG ∗hom for all R p . For the flat geometry the function
f G,∞ = lima→∞ f G (m, a) levels up yielding

f G,∞ = q(m) (15.35)

and ΔG ∗ = ΔG ∗hom q(m) in accordance with Eq. (15.28). As it follows from

Fig. 15.4, the flat geometry limit can be applied when R p /r > 10.
15.5 Kinetic Prefactor 261

15.5 Kinetic Prefactor

Recall that in homogeneous nucleation the kinetic prefactor takes the form (3.54)

J0 = Z (v A∗ ) ρ1 (15.36)

where v A∗ is the rate of addition of molecules to the critical cluster of the radius r ∗ ,
Z is the Zeldovich factor (3.50):

γ12 1
Z = (15.37)
kB T 2πρ l (r ∗ )2

and ρ1 is the number of monomers (per unit volume) of the mother phase. The
latter quantity coincides with the number of nucleation sites, since homogeneous
nucleation can occur with equal probability at any part of the physical volume.
Kinetic prefactor in the heterogeneous case has the same form

K = Z (v A∗ ) ρ1,s (15.38)

with the number of nucleation sites ρ1,s being the number of molecules in contact
with the substrate; clearly, ρ1,s is sufficiently reduced compared to ρ1 .
The value of the prefactor K depends on the particular mechanism of the cluster
formation. Two main scenarios are discussed in the literature. The first one assumes
that nucleation occurs by direct deposition of vapor monomers on the surface of the
cluster [6]. Another possibility is surface diffusion [8–10]: vapor monomers collide
with the CN surface and become adhered to it; the adsorbed molecules further migrate
to the cluster by two-dimensional diffusion. Fundamental aspects of surface diffusion
are discussed in the seminal monograph of Frenkel [11]. It was found theoretically
and experimentally [12, 13] that the surface diffusion mechanism is more effective
and leads to higher nucleation rates.
The dimensionality of K coincides with the dimensionality of the nucleation rate. The
number of critical embryos in the process of heterogeneous nucleation depends on
the amount of pre-existing foreign particles acting as CN. That is why the nucleation
rate can be expressed as:
• number of embryos per unit area of the foreign particle per unit time; or
• number of embryos per foreign particle per unit time; or
• number of embryos formed per unit volume of the system per unit time
Let us discuss the “adsorption—surface diffusion” mechanism of an embryo for-
mation. Adsorption can be visualized as a process in which molecules of phase 1
strike the surface of a foreign particle, remain on that surface for a certain adsorption
time τ and then are re-evaporated. The motion of adsorbed molecules on the surface
262 15 Heterogeneous Nucleation

of a particle can not be a free one but reminds the 2D random walk which can be
associated with the 2D diffusion process [8, 11].
Let Nads be the surface concentration of adsorbed molecules, i.e. the number of
molecules of phase 1 adsorbed on the unit surface of CN. It is difficult to determine
a realistic value of this quantity for a general case. Fortunately, Nads appears in the
pre-exponential factor K and errors in evaluating it will not influence the nucleation
rate as critically as errors in determination of ΔG ∗ (in particular, the uncertainty in
the contact angle is much more significant than the uncertainty in Nads ). In view of
these considerations we can use for Nads an “educated estimate”:

Nads = v τ (15.39)

where v is the impingement rate of the monomers of the phase 1 per unit surface of
CN; if phase 1 is the supersaturated vapor, v is given by the gas kinetics expression
(3.38). The time of adsorption can be written in the Arrhenius form [14]

τ = τ0 exp(E ads /kB T ) (15.40)

where E ads is the heat of adsorption (per molecule) and τ0 is the characteristic
time—the period of oscillations of a molecule on the surface of CN. The latter
can be estimated as the inverse of the characteristic absorption frequency f I of the
substance. For most of the substances the absorption frequencies lie in the ultraviolet
region [15]: f I = 3.3 × 1015 s−1 implying that

1
τ0 ≈ ≈ 3 × 10−16 s
fI

Obviously, E ads depends on the nature of the adsorbing surface and the adsorbed
molecules. One can find it from the data on diffusion coefficient D written in the
form of an Arrhenius plot:
E ads
ln D = const −
kB T

where we associate E ads with the activation energy for diffusion. Its value is given
by the slope of ln D as a function of the inverse temperature. Typical molar values
of E ads in liquids are found to be ≈10 − 30 kJ/mol.
If the heterogeneous nucleation rate is expressed as a number of embryos formed
per unit area of a foreign particle per unit time [16], the prefactor takes the form:

K S = Z (v A∗12 ) Nads = Z v2 A∗12 τ0 eβ E ads [cm−2 s−1 ] (15.41)

15.5 Kinetic Prefactor 263

If nucleation rate is expressed as the number of embryos per foreign particle per unit
time, Eq. (15.41) will be modified to

K p = v2 A∗12 Z τ0 eβ E ads 4π R 2p [s−1 ] (15.42)

(assuming that a foreign particle is a sphere of the radius R p ). The nucleation rate
 
ΔG ∗
J p = K p exp − [s−1 ] (15.43)
kB T

gives the rate at which a foreign particle is activated to growth.

Finally, if there are N p foreign particles in the volume V of the system, one can
define the nucleation rate as the total amount of embryos formed per unit volume of
the system per unit time. In this case the prefactor reads:

K V = v2 A∗12 Z τ0 eβ E ads 4π R 2p (N p /V )[cm−3 s−1 ] (15.44)

15.6 Line Tension Effect

15.6.1 General Considerations

The presence of two or more bulk phases in contact with each other gives rise to
the discontinuity of their thermodynamic properties and results in the corresponding
interfacial tensions. Formation of an embryo on the surface of a foreign particle
results in the occurrence of a line of three-phase contact. This line has an associated
with it tension τt , which is the excess free energy of the system due to three-phase
contact, per unit length of the contact line. A thermodynamic definition of the line
tension τt can be given in a way analogous to the definition of the surface tension in
Sect. 2.2 by introducing three dividing surfaces—gas-liquid, solid-liquid and solid-
vapor—and using the methodology of Gibbs thermodynamics of nonhomogeneous
systems. The excess Ωt of the free energy (grand potential) associated with the
three-phase contact line reads [17]:

Ωt = τt L (15.45)

where L is the length of the contact line. This relation defines τt and is the analogue of
Eq. (2.25) for the surface tension. The line tension does not depend on the location of
dividing surfaces like each of the 2D interfacial tensions γ12 , γ23 , γ13 [17]. However,
unlike the 2D interfacial tensions, τt can be of either sign. A two-phase interfacial
tension should be necessarily positive: if this would not be the case the increase of
the interfacial area would become energetically favorable leading to the situation
when two phases become mutually dispersed in each other on molecular scale, so
264 15 Heterogeneous Nucleation

that the interface between the phases disappears. Thus, the separation between any
two phases requires a positive interfacial tension. The situation with the three-phase
line is different. A negative line tension makes an increase of the length L of the triple
line energetically favorable (Ωt < 0), however this increase inevitably changes the
surface areas of the 2D interfaces—characterized by positive tensions—in a such a
way that the total free energy of the system increases.1
The classical (Fletcher) theory, described in the previous sections, does not take into
account the line tension effect. At the same time, several authors [19, 20] indicated
that for highly curved surfaces (i.e. small critical embryos) it can have a substantial
influence on the nucleation behavior. The presence of the line tension modifies the
Gibbs free energy of an embryo formation (15.3)–(15.4):

ΔG = −n Δμ + γ12 A12 + (γ23 − γ13 )A23 + τt 2πrt (15.46)

where rt is the radius of the contact line. Inclusion of the line tension changes
the force balance at the contact line implying that the contact angle correspond-
ing to the new situation, which we denote as θt and call the intrinsic (or micro-
scopic) contact angle, will be different from its bulk value θeq given by the
Dupre-Young equation.
For simplicity consider the flat geometry of Fig. 15.2. To derive the force balance at
the contact line of radius rt let us consider a small arc of this line seen from its center
under a small angle α as shown in Fig. 15.5. The length of this arc is l = rt α. The
line energy of the arc is E t = τt l. Consider a change of the contact line radius δrt .
The corresponding change of the arc length δl = α δrt induces the change of the line
energy
δ E t = τt δl.

Then, the line force acting on the arc of length l in the radial direction (with the unit
vector −
→
er pointing outwards from the center C of the contact line) is
   
−
→ δ Et −
→ δl −
→
Ft = − lim er = −τt lim er = −τt α −
→
er
δrt →0 δrt δrt →0 δrt

The line force per unit length of the contact line is

−
→
−
→ τt →
=− −
Ft
ft = er (15.47)
l rt

The balance of interfacial and line forces becomes:

τt
γ13 − γ23 = γ12 cos θt + (15.48)
rt

1Note, that a possibility of a negative tension of the three-phase contact line was already mentioned
by Gibbs [18].
15.6 Line Tension Effect 265

Fig. 15.5 Line tension effect.

A small arc of the contact line er
l+ l
with the center in point C is
characterized by the angle α. l
A change of the line radius rt
δrt causes the change in the
length of the arc δl = α δrt .
The line tension force is along rt
the r -axis which points away
from the center of curvature
C

This expression defines the contact angle in the presence of line tension. An alter-
native derivation of this result stems from the observation that θt minimizes the
Gibbs free energy (15.46) at a fixed embryo volume V2 . Comparing (15.48) with the
Dupre-Young equation (15.15) we find
τt 1
cos θt = cos θeq − (15.49)
γ12 r sin θt

This result is known as the modified Dupre-Young equation. A positive τt would mean
that the intrinsic angle of a small embryo is larger than the bulk value; a negative τt
leads to smaller contact angles compared to θeq . On a molecular level the line tension
stems from the intermolecular interactions between the three phases in the vicinity
of the contact line. This fact imposes the bulk correlation length ξ as a natural length
scale of the line tension effect. Equation (15.49) suggests that the relevant energy
scale (per unit area) is the gas-liquid surface tension γ12 . The order-of-magnitude
estimate of τt is then
τt ∼ γ12 ξ

In liquids far from the critical temperature ξ ∼ 5 ÷ 10 Å, typical gas-liquid sur-

face tensions are γ12 ∼ 50 ÷ 80 mN/m, yielding τt ∼ 10−11 − 10−10 N. Though
the actual measurements of τt meet serious difficulties, various theoretical studies
[17, 21–23] indeed reveal that its value should be of the order of 10−12 − 10−10 N in
agreement with our estimates. Hence, τt /γ12 ∼ 0.1 − 1 nm which implies that one
can expect the line tension to have a measurable influence on nucleation behavior if
embryos are nano-sized objects; for larger scales it becomes negligible compared to
the interfacial tensions.
Derivation of the modified Dupre-Young equation assumed that τt does not depend
on the radius of the contact line. At the same time it can not be independent on the
contact angle, since the relative inclination of the phases determines the net effect of
molecular interactions close to the contact line. Now we present (15.49) as
 
τt 1 1
cos θt = cos θeq − +O (15.50)
γ12 r sin θeq r2
266 15 Heterogeneous Nucleation

In this form the line tension can be interpreted as the first order correction for the
bulk contact angle in the inverse radius of the cap. Equation (15.50) can be viewed
as an alternative definition of τt (θeq ). If one can measure (or simulate) the intrinsic
angle as a function of the cluster radius r and the bulk angle θeq , than τt can be found
as a slope of cos θt versus 1/(r sin θeq ).

15.6.2 Gibbs Formation Energy in the Presence

of Line Tension

Consider implication of the line tension on the free energy of an embryo formation.
Combining Eqs. (15.46) and (15.48), we write
 
A23 τt A23
ΔG = −n Δμ + γ12 A12 1 − mt − + τt 2πrt
A12 rt

where m t = cos θt . Using (15.22) for the surface area A23 , this expression reduces to
 
A23
ΔG = −n Δμ + γ12 A12 1 − mt + τt π r sin θt (15.51)
A12

where we took into account that for flat geometry rt = r sin θt . The expression in the
square brackets represents the Fletcher model in which the bulk angle θeq is replaced
by the intrinsic one θt :

ΔG = ΔG hom (r ) q(m t ) + τt π r sin θt (15.52)

An important feature of this expression is that taking into account the line tension
has a two-fold effect on the energy barrier:
• an additional term in ΔG appears which is proportional to the length of the contact
line, and
• in the Fletcher factor q the bulk angle is replaced by the intrinsic contact angle
which depends on τt and the radius of the embryo through the modified Dupre-
Young equation
Denoting m eq = cos θeq and performing the first order perturbation analysis in (1/r),
we write

q(m t ) = q(m eq ) + Δq

dq  3
Δq =  Δm = − sin2 θeq Δm (15.53)
dm m eq 4

where from (15.50)

15.6 Line Tension Effect 267

τt 1
Δm ≡ m t − m eq ≈ − (15.54)
γ12 r sin θeq

In the same approximation the Gibbs energy reads:

 
ΔG = ΔG hom (r ) q(m eq ) + Δq + τt π r sin θeq (15.55)

The critical cluster corresponds to maximum of ΔG:

  d
ΔG hom (r ) q(m eq ) + Δq + ΔG hom (r ) Δq + τt π sin θeq = 0, where =
dr
(15.56)

Following the same thermodynamic arguments, as those used in the derivation of

the Fletcher theory, we can expect that the critical cluster size should not be affected
either by the foreign particles or the line tension and corresponds to the maximum
of the Gibbs energy for homogeneous case:

ΔG hom (r )r =rc = 0

Indeed, substituting the homogeneous nucleation barrier

4π
ΔG hom (rc ) = γ12 rc2
3
into Eq. (15.56) and taking into account (15.53) and (15.54), we find that Eq. (15.56)
becomes an identity. This means that the critical cluster size in the presence of line
tension is given by the classical Kelvin equation.2 Setting r = rc in (15.55), we
obtain the nucleation barrier in the presence of line tension:

4π
ΔG ∗ = γ12 rc2 q(m eq ) + 2 πrc τt sin θeq (15.57)
3
A positive τt increases the nucleation barrier given by the Fletcher theory (first term),
while a negative τt lowers it thus enhancing nucleation. Such an enhancement was
observed experimentally in Refs. [24, 25].

15.6.3 Analytical Solution of Modified Dupre-Young

Equation

The modified Dupre-Young equation (15.49) has an analytical solution θt (θeq , r ) for
all values of the bulk contact angle θeq . However, the general form of the solution

2 Note, that if in (15.52) the bulk Fletcher factor q(m eq ) is used instead of q(m t ), the critical cluster
size, maximizing ΔG, would violate the Kelvin equation.
268 15 Heterogeneous Nucleation

is rather complex except for the special case of θeq = π/2, for which Eq. (15.49) is
simplified to:

2p τt
sin(2θt ) = − , p≡ (15.58)
r γ12

It is instructive to study this equation in order to verify the perturbation approach of

the previous section. Obviously a solution of (15.58) exists if
 
 τt 
r ≥  
 (15.59)
2γ12

This constraint gives the range of validity of the modified Dupre-Young equation.
From the previous analysis (15.59) can be approximately expressed as

r > 0.5 nm (15.60)

When the radius of the embryo approaches the molecular size Eq. (15.58) fails.
At the same time at large r the classical expression should be recovered: limr →∞ θt =
θeq , which in our case results in

θt →r →∞ π/2 (15.61)

Solving (15.58) we find:

   
1 2p π 1 2p
θt,1 = arcsin − , and θt,2 = − arcsin −
2 r 2 2 r

At r →∞: θt,1 →0, θt,2 →π/2. Hence, the solution satisfying the asymptotic con-
dition (15.61) is
 
π 1 2p
θt = − arcsin − (15.62)
2 2 r
 
1 2p
m t = sin arcsin − (15.63)
2 r

Expansion in 1/r yields

 
p p3 1
mt = − − 3 −O (15.64)
r 2r r5

Since the second order term is absent, the approximate solution

τt 1
m t (r ) ≈ −
γ12 r
15.6 Line Tension Effect 269

coincides with the exact one up to the terms of order 1/r 2 . One can expect that for
angles close to π/2 the first order expansion in 1/r remains a good approximation.

15.6.4 Determination of Line Tension

To accomplish the model for the Gibbs formation energy (15.55) we need to have
information about the line tension for a given system. The extreme smallness of τt
makes its direct experimental measurement a very difficult task, that is why available
experimental data remains scarce. To obtain reliable estimates of τt it is important
that the droplets are of the same size as the deduced line tension values. Several
advanced techniques were used recently aiming to satisfy this requirement. Pompe
and Herminghaus [26] and Pompe [27] used Scanning Force Microscopy to study
the shape of the sessile droplets on solid substrates near the contact line. From the
droplet profile they deduced that τt lies in the range 10−12 −10−10 N and can be either
positive or negative depending on the system. In particular, it was found that line
tension increases with lowering contact angle; at large θeq it is negative and changes
sign at θeq ≈ 6◦ . Berg et al. [28] used Atomic Force Microscopy (AFM) to study
nanometer-size sessile fullerene (C60 ) droplets on the planar Si O2 interface and
observed the size-dependent variation of the contact angle which can be interpreted
as the line tension. From the modified Dupre-Young equation they found the negative
values

τt = −(0.7 ± 0.3) × 10−10 N (15.65)

and obtained a characteristic length scale of the effect τt /γ12 ≈ 1.4 nm.
In most of the heterogeneous nucleation studies, which take into account the line
tension effect, the value of τt is found from fitting to the experimental data on nucle-
ation rates [24, 29, 30]. However, such fitting can not be considered reliable in view
of a number of reasons. Homogeneous nucleation in the bulk has to be distinguished
from heterogeneous nucleation on impurities, and small changes of parameters (e.g.
substrate heterogeneities) can lead to considerable difference in measured nucleation
rates. These and other artifacts can then be erroneously interpreted as line tension
effects. In view of these reasons it is highly desirable to determine τt from an inde-
pendent source: model/simulations/experiment.
The advantage of computer simulations is that the properties of interest are con-
trollable parameters. The simplest model of a fluid is the lattice-gas with nearest
neighbor interactions (Ising model) on the simple cubic lattice. The Ising Hamil-
tonian (discussed in Sect. 8.9) reads:

H = −K si s j (15.66)
nn
270 15 Heterogeneous Nucleation

where the “spins” sk are equal to ±1, and K is the coupling parameter (interaction
strength); summation is over nearest neighbors. The presence of the foreign substrate
(a solid wall) is described by the corresponding boundary condition which is charac-
terized by a surface field H1 acting on the first layer of fluid molecules adjacent to the
surface. Monte Carlo simulations of this system performed by Winter et al. [31] at
temperatures far from Tc result in the appearance of a spherical cap-shaped (liquid)
droplet surrounded by the vapor and resting on the adsorbing solid wall (favoring
liquid). The surface free energy of the embryo formation ΔG surf MC is found in simula-
tions using thermodynamic integration. Simulation results reveal that the difference
between ΔG surf
MC and the Fletcher model

MC − γ12 4π r q(θeq )
ΔG surf 2

increases linearly with the droplet radius r . This difference can be attributed to the
line tension contribution. In Ref. [31] this linear dependence is presented in the form

MC − γ12 4π r q(θeq ) = τMC (2π r sin θeq )

ΔG surf 2
(15.67)

which is used as a definition of the line tension τMC . We introduced the notation τMC
to emphasize the difference between the latter and the previously defined quantity τt .
We require that the simulated surface free energy ΔG surfMC be equal to its theoretical
counterpart given by (15.55)
 
ΔG surf
th = γ 12 4π r 2
q(θeq ) + γ 12 4π r 2
Δq + τt π r sin θeq (15.68)

Comparing (15.67) and (15.68), we find

τMC
τt = (15.69)
2
Simulations, performed for the temperature

kB TMC /K = 3, (15.70)

reveal that for all contact angles studied τMC is negative and its absolute value
increases with θeq as shown in Fig. 15.6. The equilibrium contact angle is controlled
by varying the surface field H1 . It is important to note, that for a fixed tempera-
ture different contact angles (obtained by tuning the surface field H1 ) in Fig. 15.6
physically correspond to different solids.
Equation (15.66) being the simplest model of a magnet, can be also viewed as a model
of a fluid. Mapping of the Ising model to a model of a fluid is not a unique procedure,
therefore the results for τt for fluids can differ depending on the chosen procedure
but most probably will be qualitatively the same. One of the strategies, used e.g. in
the theory of polymers, is to equate the critical temperature of a substance to the
15.6 Line Tension Effect 271

Fig. 15.6 Line tension τMC as -0.05

a function of the equilibrium
contact angle θeq for the Ising
system at the temperature -0.1
kB TMC /J = 3; a0 is the

a0/(kBTMC )
lattice spacing. The solid line -0.15
is a fit to the Monte Carlo
results of Ref. [31]
-0.2

MC
-0.25

-0.3
20 30 40 50 60 70 80 90
eq (grad)

critical temperature of 3D Ising model [32]:

kB Tc,Ising/K ≈ 4.51

Comparing this expression with (15.70), we find

TMC = 0.665 Tc,Ising

Considering water as an example and setting Tc,Ising equal to the critical temperature
of water Tc,water = 647 K, we find that MC simulations of Ref. [31] correspond to
T = 430.3 K. Considering a substance which at this temperature has the bulk contact
angle θeq = 90◦ , the simulation results of Fig. 15.6, give

τMC σ
= −0.26
kB TMC

where we replaced the lattice spacing a0 in the Ising model by the water molecular
diameter σ = 2.64 Å [33]. Then, taking into account (15.69)

τt (T = 430.3 K, θeq = 90◦ ) = −0.41 × 10−11 N

One of the important issues that has to be considered is the temperature dependence
of the line tension. Experimentally the latter can be deduced from the measurements
of the microscopic contact angle using the modified Dupre-Young equation: τt is
found from the slope of cos θt as a function of 1/r at different temperatures. Such
study was performed by Wang et al. [34] for n-octane and 1-octene in the temperature
interval 301 < T < 316 K. Fig. 15.7 shows the plot of the line tension for n-octane
(solid circles) and 1-octene (open circles) as a function of reduced temperature

t = (Tw − T )/Tw
272 15 Heterogeneous Nucleation

Fig. 15.7 Line tension as a function of reduced temperature t = (Tw − T )/Tw for n-octane and
1-octene on coated silicon. (Reprinted with permission from Ref. [34], copyright (2001), American
Physical Society.)

where Tw is the wetting temperature (corresponding to cos θeq = 1). The solid
substrate in both cases is Si wafer.
For both liquids as temperature increases towards the wetting temperature Tw , the
line tension changes from a negative to a positive value with an increasing slope
|dτt /dT |. This behavior qualitatively agrees with theoretical predictions [35, 36].
The wetting temperatures of n-octane and 1-octene on Si wafer were found to be

Tw,octane = 318.5 K, Tw,octene = 324.3 K

Both of them lie well below the corresponding critical temperatures

Tc,octane = 568.7 K Tw,octene = 566.7 K

15.6.5 Example: Line Tension Effect in Heterogeneous

Water Nucleation

For illustration purposes let us analyze the implication of the line tension for water
nucleating on a large seed particle with the bulk contact angle θeq = 90◦ at T = 285 K
and supersaturation S = 2.93. At these conditions the critical cluster radius according
to CNT is rc = 1.03 nm. If the radius of a seed particle R p 1 nm, it can be
considered as a flat wall for the critical embryo and the Fletcher factor is given by
the function q(m). Figure 15.8 shows ΔG(r ) for 3 different models:
15.6 Line Tension Effect 273

100
homogeneous Water
T=285 K
80
S=2.93

60

G/kBT
Fletcher
cont. angle
40 90
Fletcher+line tension
20 t = -1.1*10-11 N

0.5 1 1.5 2 2.5

r [nm]

Fig. 15.8 Gibbs free energy of a cluster formation for water nucleating on a large seed particle
at T = 285 K and supersaturation S = 2.93. Solid line: classical homogeneous nucleation theory
(CNT). Dashed line: the classical heterogeneous (Fletcher) theory with the bulk the contact angle
θeq = 90◦ . Dashed-dotted line: the Fletcher theory corrected with the line tension effect according
to Eq. (15.55); the value of line tension is τt = −1.1 × 10−11 N

• classical homogeneous nucleation theory (CNT),

• classical heterogeneous (Fletcher) theory, and
• Fletcher theory corrected with the line tension effect according to Eq. (15.55).
Using previous considerations we choose a typical value of the line tension

τt = −1.1 × 10−11 N

The homogeneous nucleation barrier is ≈86 kB T , the kinetic prefactor

J0 = 4 × 1025 cm−3 s−1

so that homogeneous nucleation is suppressed:

Jhom ∼ 10−12 cm−3 s−1

The Fletcher correction reduces the barrier to ΔG ∗Fletcher ≈ 43 kB T . The line tension
leads to further reduction ΔG ∗ ≈ 25 kB T resulting in considerable enhancement of
nucleation. Setting R p = 10 nm, τ0 = 2.55 × 10−13 s [7], E ads = 10.640 kcal/mol
[37] we find for the heterogenous nucleation prefactor (15.42)

K p ≈ 5.4 × 1013 s−1

Using a typical concentration of aerosol particles in experiments [38]

274 15 Heterogeneous Nucleation

Np
= 2 × 104 cm−3
V

the prefactor in the units of cm−3 s−1 is

Np
KV = K p ≈ 1018 cm−3 s−1
V
which is 7.5 orders of magnitude lower than the corresponding homogeneous
quantity J0 .
The Fletcher theory gives

JFletcher = K V exp(−βΔG ∗Fletcher ) ≈ 3.6 × 10−1 cm−3 s−1

while the incorporation of the line tension into the model yields a considerable
increase of the nucleation rate

JFletcher+line tension = K V exp(−βΔG ∗Fletcher+line tension ) ≈ 1.5 × 107 cm−3 s−1

15.7 Nucleation Probability

Activation of a foreign particle occurs when the first critical embryo is formed on its
surface. This is a random event and as such can be studied using the methodology
of the theory of random processes. This approach to heterogeneous nucleation can
be particularly useful when analyzing the experimental data.
Let us choose some characteristic time t, during which heterogeneous nucleation is
observed. Typical value of t in experiments is ∼1 ms. Let Pk (t) be the probability
that exactly k activation events occurred during time t. The average number of such
events per unit time is given by the nucleation rate J p . A probability of activation
during an infinitesimally small interval Δt is J p Δt (assuming that two simultaneous
activation events during Δt are highly unlikely). Then the probability that no events
happened during the same interval is 1 − J p Δt. Consider the quantity Pk (t + Δt)
which is the probability that exactly k activation events occurred during time t + Δt.
Straightforward probabilistic considerations yield:

Pk (t + Δt) = Pk (t) (1 − J p Δt) + Pk−1 (t) J p Δt (15.71)

The first term on the right-hand side refers to the situation when all k events happened
during time t and no events occurred during time Δt. The second term gives the
probability that exactly k − 1 events took place during time t and one event happened
during time Δt. Dividing both sides by Δt and taking the limit at Δt→0 we obtain
15.7 Nucleation Probability 275

d Pk (t)
= −Pk (t) J p + Pk−1 (t) J p , k = 0, 1, 2, . . . (15.72)
dt

Consider the first equation of this set, corresponding to k = 0; P0 (t) describes the
probability that no events happened during time t. Obviously, we must set P−1 (t) = 0
which yields

d P0 (t)
= −J p P0 (t) (15.73)
dt
Integration of Eq. (15.73) gives

P0 (t) = P0 (0) e−J p t

where P0 (0) is the probability that no events happened in zero time. Obviously,
P0 (0) = 1 resulting in
P0 (t) = e−J p t

Then, the quantity

Phet (t) = 1 − e−J p t (15.74)

is the probability that at least one foreign particle was activated to growth dur-
ing time t. For large number of events Phet (t), termed the nucleation probability
[37, 39, 40], describes the fraction of foreign particles activated to growth during
time t. The latter quantity is measured in heterogeneous nucleation experiments. Set-
ting Phet to 0.5 we refer to the situation when half of the foreign particles are activated
to growth. This can be viewed as the onset conditions for heterogeneous nucleation.
Since the nucleation rate is a very steep function of the supersaturation (activity), the
nucleation probability is expected to be close to the step-function centered around
the onset activity.
For illustration we use the example of water nucleation considered in Sect. 15.6.5.
Setting the characteristic experimental time to t = 1 ms [37] we find:

Phet,Fletcher = 0, Phet,Fletcher+line tension = 0.52

This result implies that S = 2.93 is the onset condition if the line tension effect is
taken into account; at the same S the Fletcher theory predicts no nucleation. From
experimental data (see e.g. [37]) it follows that the onset conditions are not much
sensitive to the choice of t.
276 15 Heterogeneous Nucleation

References

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2. J.L. Katz, J. Fisk, M. Chakarov, J. Chem. Phys. 101, 2309 (1994)
3. A.B. Nadykto, F. Yu, Atmosd. Chem. Phys. 4, 385 (2004)
4. H. Rabeony, P. Mirabel, J. Phys. Chem. 91, 1815 (1987)
5. R.J. Charlson, T. Wigley, Sci. Am. 270, 48 (1994)
6. N.N. Fletcher, J. Chem. Phys. 29, 572 (1958)
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8. M. Lazaridis, J. Coll. Interface Sci. 155, 386 (1993)
9. G.M. Pound, M.T. Simnad, L. Yang, J. Chem. Phys. 22, 1215 (1954)
10. H.R. Pruppacher, J.C. Pflaum, J. Coll. Int. Sci. 52, 543 (1975)
11. J. Frenkel, Kinetic Theory of Liquids (Clarendon, Oxford, 1946)
12. J.S. Sheu, J.R. Maa, J.L. Katz, J. Stat. Phys. 52, 1143 (1988)
13. H.R. Pruppacher, J.D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht,
1978)
14. P. Hamill et al., J. Aerosol Sci. 13, 561 (1982)
15. J. Israelashvili, Intermolecular and Surface Forces (Cambridge University Press, Cambridge,
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16. M. Lazaridis, I. Ford, J. Chem. Phys. 99, 5426 (1993)
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18. J.W. Gibbs, The Scientific Papers (Ox Bow, Woodbridge, NJ, 1993)
19. R.D. Gretz, J. Chem. Phys. 45, 3160 (1966)
20. L.F. Evans, J.E. Lane, J. Atmos. Sci. 30, 326 (1973)
21. J. Indekeu, Physica A 183, 439 (1992)
22. R. Lipowsky, J. Phys. II (France) 2, 1825 (1992)
23. L. Schimmele, M. Napiorkowski, S. Dietrich, J. Chem. Phys. 127, 164715 (2007)
24. A. Sheludko, V. Chakarov, B. Toshev, J. Coll. Int. Sci. 82, 83 (1981)
25. B. Lefevre, A. Saugey, J.L. Barrat, J. Chem. Phys. 120, 4927 (2004)
26. T. Pompe, S. Herminghaus, Phys. Rev. Lett. 85, 1930 (2000)
27. T. Pompe, Phys. Rev. Lett. 89, 076102 (2002)
28. J.K. Berg, C.M. Weber, H. Riegler, Phys. Rev. Lett. 105, 076103 (2010)
29. A.I. Hienola, P.M. Winkler, P.E. Wagner et al., J. Chem. Phys. 126, 094705 (2007)
30. P. Winkler, Ph.D. Thesis, University of Vienna, 2004
31. D. Winter, P. Virnau, K. Binder, Phys. Rev. Lett. 103, 225703 (2009)
32. R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academics Press, London, 1982)
33. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids (McGraw-Hill,
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35. J. Indekeu, Int. J. Mod. Phys. B 8, 309 (1994)
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37. P.E. Wagner, D. Kaller, A. Vrtala et al., Phys. Rev. E67, 021605 (2003)
38. M. Kulmala, A. Lauri, H. Vehkamaki et al., J. Phys. Chem. B 105, 11800 (2001)
39. M. Lazaridis, M. Kulmala, B.Z. Gorbuniv, J. Aerosol Sci. 23, 457 (1992)
40. H. Vehkamäki, Classical Nucleation Theory in Multicomponent Systems (Springer, Berlin,
2006)
Chapter 16
Experimental Methods

Throughout this book we compared the predictions of theoretical models with

available experimental data. This chapter is aimed at providing a reader with a flavor
of experimental methods used in nucleation research. As in the previous chapters,
we focus on vapor to liquid nucleation as most of the experimental studies refer to
this type of transition.
Prior to 1960–1970s most experiments dealt with the critical supersaturation mea-
surements, or more generally, the conditions accompanying the onset of nucleation
at various temperatures (for review see [1]). This research was pioneered by Wil-
son in the end of the nineteenth century [2] who studied the behavior of water
vapor in expansion chamber and observed the onset of the condensation process and
the associated with it light scattering. The main conclusion drawn from Wilson’s
experiments is that if the vapor is sufficiently supersaturated, thermal density fluc-
tuations trigger droplet formation in the chamber in the absence of impurities—the
process we now refer to as homogeneous nucleation. Starting with 1970s a number
of newly developed techniques appeared which make it possible to measure not only
the onset conditions but the nucleation rates themselves at various temperatures
and supersaturations. This big step in experimental research opened the way for
quantitative tests of nucleation theories (within the accessible range of temperatures
and pressures). At present quantitative nucleation rate measurements, using various
experimental techniques, span the range of nucleation rates from 10−3 cm−3 s−1 to
1018 cm−3 s−1 . Also combination of these measurements with nucleation theorems,
studied in Chap. 4, provides direct information on the properties of critical cluster.
Among the variety of methods (for a review see e.g. [3]) we describe the four most
widely used techniques:
• thermal diffusion cloud chamber
• expansion cloud chamber
• shock tube
• supersonic nozzle

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 277

DOI: 10.1007/978-90-481-3643-8_16, © Springer Science+Business Media Dordrecht 2013
278 16 Experimental Methods

Fig. 16.1 Cutaway view of diffusion cloud chamber. (Reprinted with permission from Ref. [4],
copyright (1975), American Institute of Physics.)

16.1 Thermal Diffusion Cloud Chamber

The thermal diffusion cloud chamber consists of two metallic cylindrical plates sep-
arated by the optically transparent cylindrical ring. The region between the plates
forms the working volume of the chamber. The substance under study is placed as
a shallow liquid pool on the lower plate of the chamber and the working volume is
filled by the carrier gas (aiming at removal of the latent heat emerging in the process
of condensation). The lower plate is heated while the upper plate is cooled. Due to
the temperature difference ΔT between the plates, vapor1 evaporates from the top
surface of the liquid pool, diffuses through a noncondensable carrier gas (usually
helium, argon or nitrogen), and condenses on the lower surface of the top plate.
Construction of the diffusion cloud chamber is illustrated in Fig. 16.1.
Thermal diffusion gives rise to the profiles of temperature, density, pressure and
supersaturation inside the chamber. These profiles can be calculated from the one-
dimensional energy and mass transport equations using an appropriate equation of
state for the vapor/carrier gas mixture as shown in Fig. 16.2. At certain values of ΔT
the supersaturation in the chamber becomes sufficiently large to cause nucleation
of droplets which are subsequently detected by light scattering using a laser and a
photo-multiplier.

1 The term “vapor” in this chapter is used for the condensible component; while the term “gas”
refers to the carrier gas.
16.1 Thermal Diffusion Cloud Chamber 279

Fig. 16.2 Profiles of density, temperature, supersaturation and the nucleation rate inside the cham-
ber. (Reprinted with permission from Ref. [5], copyright (1989), American Institute of Physics.)

Diffusion cloud chamber can operate in the temperature range from near the triple
point (of the substance under study) up to the critical temperature and in the pressure
range from below the ambient to elevated pressures. A typical range of accessi-
ble nucleation rates is 10−3 − 103 cm−3 s−1 . The growing droplets are removed
by gravitational sedimentation or by convective flow which ensures the steady-state
self-cleaning operational conditions. Due to this feature and to the relatively low
nucleation rates, e.g. relatively small number of droplets to be counted, the quantita-
tive nucleation rate measurements are straightforward [4–8]. Measuring nucleation
rate as a function of supersaturation at a constant temperature, one can determine the
size of the critical cluster using the nucleation theorem (Chap. 4). The experimentally
determined critical cluster can then be compared to the nucleation models.
Note, however, that nonlinear temperature and pressure profiles inside the chamber
can lead to substantial nonuniformities of temperatures and supersaturations in the
working volume making it difficult to assign particular values to supersaturations
and temperatures corresponding to the observed nucleation rates.

16.2 Expansion Cloud Chamber

Compared to the diffusion chamber, functioning of the expansion cloud chamber

relies upon a different mechanism: rapid adiabatic expansion of the vapor/gas mix-
ture which produces supersaturation and subsequent nucleation. The device can be
generally described as a cylinder piston-like structure containing the vapor/gas mix-
ture in the region above the cylinder and bounded by the piston walls [9] (or valves
connecting additional volumes to the chamber, as in the nucleation pulse chamber
of Ref. [10]). Initially the mixture has the temperature of the piston wall and the
vapor may or may not be saturated. After rapid withdrawal of the piston adiabatic
cooling occurs: the pressure and temperature of the mixture decrease. As a result,
the supersaturation of the vapor

pv
S=
psat (T )
280 16 Experimental Methods

Fig. 16.3 Time dependence of the supersaturation in the expansion cloud chamber during a single
nucleation experiment. Experiment starts when vapor is supersaturated. As a result of adiabatic
expansion vapor becomes supersaturated and nucleation occurs during the time of nucleation pulse.
After that a slight recompression terminates nucleation process; condensational growth of droplets
results in further reduction of the supersaturation due to vapor depletion. (Reprinted with permission
from Ref. [10], copyright (1994), American Chemical Society.)

increases because the decrease of its partial pressure p v during the isentropic expan-
sion is slower than the exponential decrease of the saturation pressure psat with
temperature, which is given by the Clapeyron equation (2.14). As opposed to the
diffusion chamber, nucleation of the supersaturated vapor in the working volume of
the expansion chamber takes place at uniform conditions.
Among various modifications of the expansion camber—single-piston chamber [11,
12], piston-expansion tube [13]—we will describe in a somewhat more detail the
nucleation pulse chamber (NPC) [10, 14, 15]. In order to ensure the constant con-
ditions during the nucleation period a small recompression pulse is issued in NPC
after the completion of the adiabatic expansion, which terminates the nucleation
process after a short time, of the order of 1 ms, called the nucleation pulse. Conden-
sational growth of droplets results in further reduction of the supersaturation due to
vapor depletion. Schematically evolution of the supersaturation in the NPC during
the nucleation experiment is depicted in Fig. 16.3. The cooling rate in the NPC is of
the order 104 K/s.
The values of supersaturation and temperature corresponding to the measured nucle-
ation rate are calculated from the following considerations. If p0 is the initial total
pressure of the vapor/gas mixture, T0 is the initial temperature and y is the vapor
molar fraction, then
p0v = y p0

is the partial vapor pressure at the initial conditions. After adiabatic expansion the
total pressure drops by Δpexpt becoming equal to

p = p0 − Δpexpt
16.2 Expansion Cloud Chamber 281

The nucleation temperature follows the Poisson law:

 (κ−1)/κ
T p
= (16.1)
T0 p0

where κ = c p /cv is the ratio of specific heats for the vapor/gas mixture. Then, from
the saturation vapor pressure at temperature T , psat (T ), given by the Clapeyron
equation (2.14), the supersaturation is found to be

y ( p0 − Δpexpt )
Sexpt = (16.2)
psat (T )

During the short (∼1 ms) nucleation pulse only a negligible fraction of the vapor
is consumed, i.e. depletion effects are negligible which leaves the supersaturation
practically constant. After recompression supersaturation drops, nucleation is sup-
pressed so that only particle growth at constant number density occurs (while no new
droplets are formed). Thus, the nucleation pulse method realized in the expansion
tube makes it possible to decouple nucleation and growth processes.
The last step is to determine the number density ρd of droplets formed during the
nucleation pulse. In the NPC the nucleated droplets grow to the sizes ∼1 μm when
they are detected by the constant angle Mie scattering (CAMS) technique leading
to determination of ρd . CAMS, which uses a laser operating in the visual, is based
on the Mie theory of scattering of electromagnetic waves by dielectric spherical
particles [16].
The basic idea behind the technique is straightforward: (i) analyzing the time evolu-
tion of the intensity of light scattered by the droplets and comparing it with the Mie
theory, one finds the size rd of the droplet at time t; (ii) analyzing the evolution of
the intensity of the transmitted light one determines the number density of droplets
using the value of extinction coefficient corresponding to the droplet size rd .

16.2.1 Mie Theory

To clarify CAMS, we briefly formulate the main results of the Mie theory (for details
the reader is referred to Refs. [16, 17]) relevant for the analysis of nucleation exper-
iments. Consider a single dielectric spherical particle of radius rd emerged in the
vacuum and having the refractive index m. The particle is illuminated by the inci-
dent light with a wavelength λ. Let us introduce the dimensionless droplet radius2
 
2π
α= rd = k rd
λ

2If instead of vacuum the particle is emerged in a homogeneous medium with the refractive index
m medium , the wavelength should be replaced by λvacuum /m medium .
282 16 Experimental Methods

If I0 is the intensity of the incident light (watt/m2 ), the sphere will intercept
Q ext πrd2 I0 watt from the incident beam, independently of the state of polarization of
the latter. The dimensionless quantity Q ext (m, α) is called the extinction efficiency.
In the Mie theory it is given by
∞
2 
Q ext (m, α) = (2n + 1) (an + bn ) (16.3)
α2
n=1

Here the complex Mie coefficients an and bn are obtained from matching the bound-
ary conditions at the surface of the spherical droplet. They are expressed in terms of
spherical Bessel functions evaluated at α and y = mα:

ψn (y) ψn (α) − m ψn (y) ψn (α)

an =
ψn (y) ζn (α) − m ψn (y) ζn (α)
m ψn (y) ψn (α) − ψn (y) ψn (α)
bn =
m ψn (y) ζn (α) − ψn (y) ζn (α)

where

ψn (z) = (π z/2)1/2 Jn+1/2 (z)

(2)
ζn (z) = (π z/2)1/2 Hn+1/2 (z)

(2)
and Jn+1/2 (z) is the half-integer-order Bessel function of the first kind, Hn+1/2 (z)
is the half-integer-order Hankel function of the second kind [18].
The intensity of the incident beam decreases with the a distance L (called the optical
path) as it proceeds through the cloud of droplets. The transmitted light intensity is
given by the Lambert-Beer law [19]

Itrans = I0 e−βext L (16.4)

where βext is the extinction coefficient computed from

βext = ρd πrd2 Q ext (m, α) (16.5)

(here we assumed that all ρd dielectric spheres in the unit volume are identical).
The behavior of the extinction efficiency Q ext as a function of the size parameter α
is illustrated in Fig. 16.4 for the two substances with the values of refractive index
m = 1.33 (water) and m = 1.55 (silicone oil).
Consider now the light scattered by a single sphere. The direction of scattering
is given by the polar angle θ and the azimuth angle φ. The intensity Iscat,1 of the
scattered light in a point located at a large distance r from the center of the particle
has a form
I0
Iscat,1 = 2 2 F(θ, φ; m, α) (16.6)
k r
16.2 Expansion Cloud Chamber 283

Fig. 16.4 Mie extinction efficiency versus size parameter α for water (m = 1.33) and silicone oil
(m = 1.55). For small particles Q ext ∼ α 4 (Rayleigh limit); for big particles Q ext → 2 (limit of
geometrical optics α → ∞). The largest value of Q ext is achieved when the particle size is close
to the wavelength

where F is the dimensionless function of the direction (not of r ). For the linearly
polarized incident light
F = i 1 sin2 φ + i 2 cos2 φ (16.7)

Here i 1 and i 2 refer, respectively, to the intensity of light vibrating perpendicularly

and parallel to the plain through the directions of propagation of the incident and
scattered beams. The quantities i 1 and i 2 are expressed in terms of the amplitude
functions S1 (m, α; θ ) and S2 (m, α; θ ):

i 1 = |S1 (m, α; θ )|2 , i 2 = |S2 (m, α; θ )|2

The amplitude functions are given by

∞
 2n + 1
S1 (m, α; θ ) = [an πn (cos θ ) + bn τn (cos θ )] (16.8)
n(n + 1)
n=1
∞
2n + 1
S2 (m, α; θ ) = [bn πn (cos θ ) + an τn (cos θ )] (16.9)
n(n + 1)
n=1

where
1
πn (cos θ ) = P 1 (cos θ ) (16.10)
sin θ n
d 1
τn (cos θ ) = P (cos θ ) (16.11)
dθ n

and Pn1 (cos θ ) is the associated Legendre polynomial [18].

284 16 Experimental Methods

Fig. 16.5 Time dependence

of the normalized scattered
light intensity (left y-axis)
and the total pressure (right
y-axis) for CAMS (scattering
angle is 15◦ ). It is clearly
seen that scattering is detected
after the nucleation pulse.
(Reprinted with permission
from Ref. [10], copyright
(1994), American Chemical
Society.)

Averaging F over the azimuth angle using the identity

 2π  2π
1 1 1
sin2 φ dφ = cos2 φ dφ =
2π 0 2π 0 2

we have
i1 + i2 |S1 |2 + |S2 |2
F φ = = (16.12)
2 2
If multiple scattering can be avoided, the total intensity of light scattered in the
direction θ by all spheres in the volume V is

I0 ρd V
Iscat = Iscat,1 ρd V = (|S1 |2 + |S2 |2 ) (16.13)
2 k 2r 2
At fixed m and θ the quantity in the round brackets as a function of α has a distinct
pattern of maxima and minima [16].
During the single nucleation experiment one measures the time dependence of the
expt
scattering intensity Iscat (see Fig. 16.5) which has a form of a sequence of maxima
and minima. Assuming that on the time scale of experiment droplets, nucleated
during the pulse, grow without coagulation and Ostwald ripening, one can state that
their number density ρd remains constant. That is why in order to determine ρd it
expt
is sufficient to compare the first peak of the function Iscat with the first maximum
of the scattering intensity from the Mie theory. This comparison yields the droplet
radius rd at the first peak, which after substitution into (16.4) and (16.5) yields

1
ln(I0 /I )
ρd = L
(16.14)
π rd2 Q ext (m, α)
16.2 Expansion Cloud Chamber 285

Note, that this procedure does not provide information about the droplet growth
rd (t)—the latter can be obtained from the analysis of series of peaks in the scattering
intensity.

16.2.2 Nucleation Rate

The nucleation rate is calculated as

ρd
J= (16.15)
Δt
where Δt is the duration of the nucleation pulse. In various versions of expansion
chambers accessible range of nucleation rates is approximately 102 − 109 cm−3 s−1
which nicely complements the range achieved in diffusion cloud chambers.
For studies of homogeneous nucleation it is important to provide the particle-free
operational regime of the chamber excluding heterogeneous effects. It has to be noted
that expansion chamber is not a self-cleaning device (as the previously considered
static diffusion chamber) and care has to be taken to avoid significant contamination
prior to nucleation experiment.

16.3 Shock Tube

Shock tube realizes the same idea of a short nucleation pulse, providing the separa-
tion in time of the nucleation and growth processes, which we discussed in Sect. 16.2.
In the shock tube this is achieved by means of the shock waves. The tube consists
of two sections: the driver-, or High-Pressure Section (HPS), and the driven-, or the
Low Pressure Section (LPS). The two sections are separated by the diaphragm. A
small amount of condensable vapor is added to the driver section. During the nucle-
ation experiment the diaphragm is rapidly ruptured and the high-pressure vapor/gas
mixture from the driver section sets up a nearly one-dimensional, unsteady flow and
the shock wave traveling from the diaphragm into the driven section. At the same
time the expansion wave travels back—from the diaphragm into the driver section.
Cooling of the rapidly expanding gas in the driver section imposes nucleation.
The construction of the shock tube for nucleation studies was proposed by Peters and
Paikert [21, 22] and further developed by van Dongen and co-workers [20, 23–26].
The scheme of the experimental set-up [20] is shown in Fig. 16.6. The HPS has a
length of 1.25 m, the length of the LPS is 6.42 m. The local widening in the LPS
plays an important role in creating the desired profile of pressure and, accordingly,
the supersaturation: after the rupture of the polyester diaphragm between HPS and
LPS the initial expansion wave traveling from LPS to HPS is followed by a set of
reflections of the shock wave at the widening (see Fig. 16.7). These reflections travel
286 16 Experimental Methods

Fig. 16.6 Pulse expansion wave tube set-up. (Reprinted with permission from Ref. [20], copyright
(1999), American Institute of Physics.)

back into the HPS and create the pulse-shaped expansion at the end wall of the HPS.
After the short pulse and a small recompression the pressure remains constant for a
longer period of time during which no nucleation occurs but the already nucleated
droplets are growing to macroscopic sizes to be detected by means of the scattering
technique. The temperature profile follows the adiabatic Poisson law (16.1).
Similar to the nucleation-pulse chamber, discussed in Sect. 16.2, the number density
of droplets, ρd , is obtained by means of a combination of the constant-angle Mie
scattering and the measured intensity of transmitted light—the procedure described
in Sect. 16.2.1. In the experiments of Refs. [20, 23–25] the droplet cloud in the
HPS was illuminated by the Ar-ion laser with a wavelength λ = 514.2 nm. Since
the observation section in the shock tube is located near the endwall of the HPS,
an obvious choice of the scattering polar angle is θ = 90◦ . Figure 16.8 shows the
optical set-up of the device. The laser beam passes the tube through two conical
windows. The transmitted light is focused by lens L 2 onto photodiode D2 . The
scattered intensity is recorded by the photomultiplier P M.
Because of the nature of the nucleation pulse method, the value of ρd should
be approximately constant in time. The steady-state nucleation rate is given by
Eq. (16.15)
ρd
J=
Δt
where Δt is the duration of the pulse.
As pointed out in the previous section, besides the steady-state nucleation rate one
can obtain from the same experimental data the growth law of the droplets. At each
moment of time during the nucleation experiment for which the measured scattered
16.3 Shock Tube 287

Fig. 16.7 Profiles of pressure

and temperature and the
wave propagation in the pulse
expansion shock tube. (Copied
from Ref. [27])

Fig. 16.8 Optical set-up used

for measurements of droplet
size and number density of
droplets in the endwall of the
shock tube. (Copied from Ref.
[27])

signal is at maximum, one can find the value of the droplet radius by comparison
with the corresponding maximum of the theoretical scattering intensity given by the
Mie theory of Sect. 16.2.1 as illustrated in Fig. 16.9. This gives the droplet growth
curve rd (t).
The absence of moving parts in the shock-tube (as opposed to the expansion chamber)
opens a possibility to study nucleation at sufficiently high nucleation pressures—
up to 40 bar—and reach nucleation rates in the range of 108 − 1011 cm−3 s−1
[20, 28].
288 16 Experimental Methods

Fig. 16.9 Theoretical and experimental scattering patterns for n-nonane droplets. From mutual
correspondence of extrema the time-resolved droplet radius rd (t) is found. (Copied from Ref. [27])

16.4 Supersonic Nozzle

The supersonic nozzle (SSN) relies upon adiabatic expansion of the vapor/gas mix-
ture flowing through a nozzle of some sort. The most widely used type of these
devices contain the Laval (converging/diverging) nozzle [29–32]. The vapor/gas
mixture is undersaturated prior to and slightly after entering the nozzle region. Dur-
ing the flow in the nozzle the mixture becomes saturated and then supersaturated.
Nucleation and growth of the droplets takes place when the flow passes the throat
region of the nozzle. Rapid increase of the supersaturation results in spontaneous
onset of condensation which depletes the vapor and subsequently terminates the
supersaturation.
A typical nucleation pulse is very short ∼10 μs, and cooling rates are very high: ∼5×
105 K/s leading to characteristic nucleation rates as high as 1016 − 1018 cm−3 s−1 .
The droplets formed in SSN are extremely small ∼1 − 20 nm; critical embryos are
even smaller ∼0.1 nm, containing 10–30 molecules. Clearly, droplets of this size can
not be detected by optical devices operating in the visual—shorter wavelength is
required. Methods used for particle characterization in SSN are small-angle neutron
scattering (SANS ) and small-angle x-ray scattering (SAXS).
The schematic diagram of the experimental set-up with the supersonic nozzle and
SAXS unit is shown in Fig. 16.10. The experiment consists of the pressure trace mea-
surements during the expansion and the SAXS measurements. A movable pressure
probe measures the pressure profile of the gas p(x) along the axis of the nozzle. The
condensible vapor mole fraction y is determined from the mass flow measurements.
Using the stagnation conditions p0 , T0 of the vapor/gas mixture, one determines the
pressure profile of the vapor along the nozzle
  
p(x) g(x)
p v (x) = y p0 1−
p0 g∞
16.4 Supersonic Nozzle 289

Fig. 16.10 Schematic diagram of the experimental set-up with supersonic nozzle and SAXS unit.
(Copied from Ref. [33])

where g(x) is the condensate mass fraction at point x,

ṁ v
g∞ =
ṁ v + ṁ gas

where ṁ v , ṁ gas are the mass flow rates of the vapor and gas, respectively. Then, the
supersaturation profile is
p v (x)
S(x) =
psat (T (x))

where T (x) is the temperature at point x found from the Poisson equation.
Using SAXS technique, one studies elastic scattering of X-rays by a cloud of droplets.
Scattering leads to interference effects and results in a pattern, which can be analyzed
to provide information about the size of droplets and their number density. Let us
define a scattering vector (length) according to

4π
q= sin(θ/2)
λ
where θ is the scattering angle, λ is wavelength of the incident beam; for SAXS
λ ≈ 1Å. The scattering intensity Iscat is proportional to the number density of
290 16 Experimental Methods

Fig. 16.11 SAXS spectrum

of n-butanol at plenum tem-
perature T0 = 50◦ C and
pressure p0 = 30.2 kPa. The
solid line is a fit to Gaussian
distribution of spherical sizes
with parameters given by
(16.16)–(16.17). (Copied
from Ref. [33])

particles, ρd , and the form-factor P of a single particle. For a spherical particle of

radius rd the form-factor takes a simple form [31]
 2
4π (sin(qrd ) − qrd cos(qrd ))
P(q, rd ) = ρ S2 L D
q3

where ρ S L D is the contrast factor, being the difference in the scattering length density
between the liquid droplet and surrounding bulk gas. Assuming Gaussian distribution
of droplet sizes with the mean rd and the width σ , Iscat can be written as
  
1 (rd − rd )2
Iscat (q) = ρd √ exp − P(q, rd ) drd
σ 2π 2σ 2

Fitting the measured scattering intensity to this expression, one finds the desired
quantities ρd and rd .
Figure 16.11 from Ref. [33] illustrates this procedure for SSN experiments with
n-butanol at plenum temperature T0 = 50◦ C and pressure p0 = 30.2 kPa. The solid
line gives the fit to the Gaussian distribution with the following set of parameters

rd = 7 nm; σ = 2.19 nm (16.16)

and the number density

N ≡ ρd = 1.7 × 1012 cm−3 (16.17)

Taking into account the pulse duration Δt ≈ 10 μs, this leads to the nucleation rate

J ≈ 1.7 × 1017 cm−3 s−1

16.4 Supersonic Nozzle 291

Thus, quantitative nucleation rate measurements, using various techniques discussed

in this chapter, cover the range of more than 20 orders of magnitude. These mea-
surements, in combination with nucleation theorems of Chap. 4, also provide direct
information about the properties of the critical clusters which can be compared to
predictions of theoretical models.

References

1. G.M. Pound, J. Phys. Chem. Ref. Data 1, 119 (1972)

2. C.R.T. Wilson, Phil. Trans. R. Soc. London A 189, 265 (1897)
3. R.H. Heist, H. He, J. Phys. Chem. Ref. Data 23, 781 (1994)
4. J.L. Katz, C. Scoppa, N. Kumar, P. Mirabel, J. Chem. Phys. 62, 448 (1975)
5. C. Hung, M. Krasnopoler, J.L. Katz, J. Chem. Phys. 90, 1856 (1989)
6. J.L. Katz, M. Ostermeier, J. Chem. Phys. 47, 478 (1967)
7. J.L. Katz, J. Chem. Phys. 52, 4733 (1970)
8. R. Heist, H. Riess, J. Chem. Phys. 59, 665 (1973)
9. P.E. Wagner, R. Strey, J. Chem. Phys. 80, 5266 (1984)
10. R. Strey, P.E. Wagner, Y. Viisanen, J. Phys. Chem. 98, 7748 (1994)
11. G.W. Adams, J.L. Schmitt, R.A. Zalabsky, J. Chem. Phys. 81, 5074 (1984)
12. J.L. Schmitt, G.J. Doster, J. Chem. Phys. 116, 1976 (2002)
13. T. Rodemann, F. Peters, J. Chem. Phys. 105, 5168 (1996)
14. J. Wölk, R. Strey, J. Phys. Chem. B 105, 11683 (2001)
15. K. Iland, J. Wölk, R. Strey, D. Kashchiev, J. Chem. Phys. 127, 154506 (2007)
16. H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981)
17. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press,
New York, 1969)
18. K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering
(Cambridge University Press, Cambridge, 2007)
19. J.D.J. Ingle, S.R. Crouch, Spectrochemical Analysis (Prentice Hall, New Jersey, 1988)
20. C.C.M. Luijten, P. Peeters, M.E.H. van Dongen, J. Chem. Phys. 111, 8535 (1999)
21. F. Peters, B. Paikert, J. Chem. Phys. 91, 5672 (1989)
22. F. Peters, B. Paikert, Exp. Phys. 7, 521 (1989)
23. K.N.H. Looijmans, P.C. Kriesels, M.E.H. van Dongen, Exp. Fluids 15, 61 (1993)
24. K.N.H. Looijmans, C.C.M. Luijten, G.C.J. Hofmans, M.E.H. van Dongen, J. Chem. Phys. 102,
4531 (1995)
25. K.N.H. Looijmans, C.C.M. Luijten, M.E.H. van Dongen, J. Chem. Phys. 103, 1714 (1995)
26. D.G. Labetski, Ph.D. Thesis, Eindhoven University, 2007
27. C.C.M. Luijten, Ph.D. Thesis, Eindhoven University, 1999
28. C. Luijten, M.E.H. van Dongen, J. Chem. Phys. 111, 8524 (1999)
29. A. Khan, C.H. Heath, U.M. Dieregsweiler, B.E. Wyslouzil, R. Strey, J. Chem. Phys. 119, 3138
(2003)
30. C.H. Heath, K.A. Streletzky, B.E. Wyslouzil, J. Wölk, R. Strey, J. Chem. Phys. 118, 5465
(2003)
31. Y.J. Kim, B.E. Wyslouzil, G. Wilemski, J. Wölk, R. Strey, J. Phys. Chem. A 108, 4365 (2004)
32. S. Tanimura, Y. Zvinevich, B. Wyslouzil et al., J. Chem. Phys. 122, 194304 (2005)
33. D. Ghosh, Ph.D. Thesis, University of Cologne, 2007
Appendix A
Thermodynamic Properties

Water

Molecular mass: M = 39.948 g/mol

Critical state parameters: pc = 221.2 bar, Tc = 647.3 K, ρc = 17.54 × 10−3
mol/cm3 [1]
Equilibrium liquid mass density [1]:

ρmass
l
= 0.08 tanh y + 0.7415 x 0.33 + 0.32 g/cm3
T
x = 1 − , y = (T − 225)/46.2
Tc

Saturation vapor pressure [1]:

psat = exp [77.3491 − 7235.42465/T − 8.2 ln T + 0.0057113 T ] Pa

Surface tension [1]:

γ∞ = 93.6635 + 9.133 × 10−3 T − 0.275 × 10−3 T 2 mN/m

Lennard-Jones interaction parameters [2]:

σLJ = 2.641Å, εLJ /kB = 809.1 K

Second virial coefficient [3]:

B2 (T ) = 17.1−102.9/(1−x)2 −33.6×10−3 (1−x) exp [5.255/(1 − x)] , cm3 /mol

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DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013
294 Appendix A: Thermodynamic Properties

Nitrogen

Molecular mass: M = 28.0135 g/mol

Critical state parameters: pc = 33.958 bar, Tc = 126.192 K, ρc = 0.3133 g/cm3 [4]
Equilibrium liquid mass density [4]:

ρsat
l
ln = 1.48654237 x 0.3294 − 0.280476066 x 4/6
ρc
+ 0.0894143085 x 16/6 − 0.119879866 x 35/6 ,
T
x = 1−
Tc

Saturation vapor pressure [4]:

psat Tc  
ln = −6.12445284 x + 1.2632722 x 3/2 − 0.765910082 x 5/2 − 1.77570564 x 5
pc T

Surface tension [4]:

γ∞ = 29.324108 x 1.259 mN/m

Lennard-Jones interaction parameters [2]:

σLJ = 3.798 Å, εLJ /kB = 71.4 K

Pitzer’s acentric factor: ω P = 0.037 [2].

Mercury

Molecular mass: M = 200.61 g/mol

Critical state parameters: pc = 1510 bar, Tc = 1765 K, ρc = 23.41 × 10−3 mol/cm3
[5].
Saturation vapor pressure [5]
a
log10 psat [Torr] = − + b + c lg10 T
T
with
a = 3332.7, b = 10.5457, c = −0.848

Equilibrium liquid mass density [5]

ρmass
l
[g/cm3 ] = 13.595 [1 − 10−6 (181.456 TCels + 0.009205 TCels
2

+ 0.000006608 TCels
3
+ 0.000000067320 TCels
4
)]
Appendix A: Thermodynamic Properties 295

where TCels = T − 273.15 is the Celsius temperature.

Surface tension [6]
γ∞ [mN/m] = 479.4 − 0.22 TCels

Second virial coefficient [7]:

        
TB ε1 ε2
B2 N A = c1 exp − 1 + c2 exp − −1
T TB TB
ε   ε 
1 2
− c1 exp − 1 + c2 exp − −1 cm3 /mol (A.1)
T T

where TB = 4286 K is the Boyle temperature of mercury, c1 = 69.87 cm3 /mol,

c2 = 22.425 cm3 /mol, ε1 = 655.8 K, ε2 = 7563 K.
The value of coordination number N1 can be obtained from the measurements of
the static structure factor. For fluid mercury it was studied over the whole liquid-
vapor density range by Tamura and Hosokawa [8] and Hong et al. [9] using X-ray
diffraction measurements. Their data show that the first peak of the pair correlation
function g(r ) in the liquid phase is located at ≈ 3 Å and is relatively insensitive to the
mass density in the range 10-13 g/cm3 . The packing fraction in the liquid mercury
at this range of densities is η ≈ 0.581. Using (7.68) one finds N1 ≈ 6.7.

Argon

Molecular mass: M = 39.948 g/mol

Critical state parameters: pc = 48.6 bar, Tc = 150.633 K, ρc = 13.29 × 10−3
mol/cm3 [4]
Equilibrium liquid mass density [4]:
  T
ρmass
l
= M 13.290 + 24.49248 x 0.35 + 8.155083 x × 103 g/cm3 , x =1−
Tc

Saturation vapor pressure [4]:

psat Tc  
ln = −5.904188529 x + 1.125495907 x 1.5 − 0.7632579126 x 3 − 1.697334376 x 6
pc T

Surface tension [4]:

γ∞ = 37.78 x 1.277 mN/m

Lennard-Jones interaction parameters [10]:

σLJ = 3.405 Å, εLJ /kB = 119.8 K

296 Appendix A: Thermodynamic Properties

Pitzer’s acentric factor: ω P = −0.002 [2].

The second virial coefficient is given by the Tsanopoulos correlation for nonpolar
substances Eq. (F.2).

N-nonane

Molecular mass: M = 128.259 g/mol

Critical state parameters: pc = 22.90 bar, Tc = 594.6 K, ρc = 1.824 × 10−3
mol/cm3 [2]
Equilibrium liquid mass density [2]:

ρmass
l
= 0.733503 − 7.87562 × 10−4 TCels − 9.68937 × 10−8 TCels
2
− 1.29616 × 10−9 TCels
3
g/cm3

where TCels = T − 273.15.

Saturation vapor pressure [3]:
 
psat = exp −17.56832 ln T + 1.52556 10−2 T − 9467.4/T + 135.974 dyne/cm2

Surface tension [3]:

γ∞ = 24.72 − 0.09347 TCels mN/m

The second virial coefficient [3]:

B2 N A = 369.2 − 705.3/Tr + 17.9/Tr2 − 427.0/Tr3 − 8.9/Tr8 cm3 /mol

where Tr = T /Tc .
Appendix B
Size of a Chain-Like Molecule

As one of the input parameters MKNT and CGNT use the size of the molecule.
For a chain-like molecule, like nonane, it can be characterized by the radius of
gyration Rg —the quantity used in polymer physics representing the mean square
length between all pairs of segments in the chain [11]:

Nsegm
1 
Rg2 = 2
(Ri − R j )2 
2Nsegm
i, j=1

where Nsegm is the number of segments. Equivalently Rg can be rewritten as

Nsegm
1 
Rg2 = (Ri − R0 )2 
Nsegm
i=1

where R0 is the position of the center of mass of the chain. The latter expression
shows that the chain-like molecule can be appropriately represented as a sphere with
the radius Rg . The radius of gyration can be found using the Statistical Associating
Fluid Theory (SAFT) [12]. Within the SAFT a molecule of a pure n-alkane can be
modelled as a homonuclear chain with Nsegm segments of equal diameter σs and
the same dispersive energy ε, bonded tangentially to form the chain. The soft-SAFT
correlations for pure alkanes read [13]:

Nsegm = 0.0255 M + 0.628 (B.1)

Nsegm σsegm
3
= 1.73 M + 22.8 (B.2)

where M is the molecular weight (in g/mol). Thus, the number of segments and the
size of a single segment depend only on the molecular weight. Note, that within this
approach Nsegm is not necessarily an integer number. Having determined Nsegm , the
radius of gyration can be calculated using the Gaussian chain model in the theory of

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 297

DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013
298 Appendix B: Size of a Chain-Like Molecule

polymers [11]: 
Nsegm
Rg = σsegm (B.3)
6

Then, the effective diameter of the molecule can be estimated as

σ = 2 Rg (B.4)

For n-nonane Eqs. (B.1)–(B.2) give:

Rg = 3.202 Å, σ = 6.404 Å (B.5)

Appendix C
Spinodal Supersaturation for van der Waals
Fluid

In reduced units ρ ∗ = ρ v /ρc , T ∗ = T /Tc , p ∗ = p v / pc the van der Waals equation

of state reads [14]
8ρ ∗ T ∗
p ∗ = −3ρ ∗ +
2
(C.1)
3 − ρ∗

The spinodal equation ∂ p ∗ /∂ρ ∗ = 0 is:

ρ∗
T∗ = (3 − ρ ∗ )2 (C.2)
4
∗ we obtain using the standard
Solving Eq. (C.2) for the spinodal vapor density ρsp
v

methods [15]:
 
∗ 1
(T ∗ ) = 2 − 2 cos β = arccos(1 − 2T ∗ )
v
ρsp β , (C.3)
3

Substitution of (C.3) into the van der Waals equation (C.1) yields the vapor pressure
at the spinodal:
      
∗ v 8 4T ∗ − 3 cos 13 β + 3 cos 23 β sin2 16 β
psp = 1  (C.4)
1 + 2 cos 3 β

from which the supersaturation at spinodal is

∗ (T ∗ )
v  
psp 1
Ssp (T ∗ ) = =
p ∗ (T ∗ ) ∗ (T ∗ )
psat
 sat      
8 4T ∗ − 3 cos 13 β + 3 cos 23 β sin2 16 β
  (C.5)
1 + 2 cos 13 β

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 299

DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013
Appendix D
Partial Molecular Volumes

D.1 General Form

In this section we present a general framework for calculation of partial molecular

volumes of components in a mixture. These quantities for a liquid phase are involved
in the Kelvin equations. We consider here a general case of a two-phase m-component
mixture viα , i = 1, . . . , m; α = v, l. The partial molecular volume of component
i in the phase α is defined as

∂ V α 
viα = , i = 1, 2, . . . , α = v, l (D.1)
∂ Niα  pα ,T, N α
j, j =i

where the quantities with the superscript α refer to the phase α. Since

Nα
Vα =
ρα

the change of the total volume due to the change of Niα is

1 Nα
dV α = dN α
− dρ α
ρα i
(ρ α )2

where we took into account that dN α = dNiα . Then

⎡ ⎤

α
α 1 ⎣ α ∂ ln ρ  ⎦
vi = α 1 − N (D.2)
ρ ∂ Niα  pα ,T, N α
j, j =i

The density ρ α is an intensive property which can be expressed as a function of the

set of intensive quantities—molar fractions of components in the phase α

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 301

DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013
302 Appendix D: Partial Molecular Volumes

N αj
y αj =  α (D.3)
k Nk
 α α α
In view of normalization m k=1 yk = 1, ρ is a function of m − 1 variables yk
and one is free to choose a particular component to be excluded from the list of
independent variables. Discussing the partial molecular volume of component i,
it is convenient to exclude this component from the list, i.e. to set

ρ α = ρ α (y1α , . . . , yi−1
α α
, yi+1 , . . . , ym )

Then the right-hand side of (D.2) can be expressed using the chain rule:

∂ ln ρ α   ∂ ln ρ α ∂ y αj
= (D.4)
∂ Niα  pα ,T, N α ∂ y αj ∂ Niα
j, j =i j=i

From (D.3) we find 

∂ y αj  y αj
 =− (D.5)
∂ Niα  Nα
N vj, j=i

Substituting (D.5) into ((D.2) and D.4) we obtain the general result

 α
1 α α ∂ ln ρ 
viα α
= α ηi , ηi = 1 + yj  , α = v, l (D.6)
ρ ∂ y αj 
j=i p α ,T

In particular, for the binary a − b mixture (ya + yb = 1) we have:


∂ ln ρ α 
ηaα = 1 − ybα (D.7)
∂ yaα  pα ,T

∂ ln ρ α 
ηbα = 1 + yaα (D.8)
∂ yα  α
a p ,T

Mention a useful identity resulting from (D.7)–(D.8):

ya ηaα + yb ηbα = 1 (D.9)

D.2 Binary van der Waals Fluids

Here we present calculation of the partial molecular volumes of components in binary

mixtures (vapor or liquid) described by the van der Waals equation of state:
Appendix D: Partial Molecular Volumes 303

ρkB T
p= − am ρ 2 (D.10)
1 − bm ρ

where the van der Waals parameters am , bm for the mixture read [2]:
√ √
am = (ya aa + yb ab )2 (D.11)
bm = ya ba + yb bb (D.12)

where yi is the molar fraction of component i in vapor or liquid; ai and bi are the
van der Waals parameters for the pure fluid i. According to the definition of the
partial molecular volume consider a small perturbation in a number of molecules
of component a at a fixed pressure, temperature and the number of molecules of
component b. This perturbation results in the change of ρ, am and bm :

m = am + Δam
a (D.13)

bm = bm + Δbm (D.14)
ρ
 = ρ + Δρ (D.15)

Substituting (D.13)–(D.15) into the van der Waals Eq. (D.10) and linearizing in
Δam , Δbm and Δρ we find

ρkB T Δρ kB T ρ kB T
p− + am ρ 2 = + (Δbm ρ + Δρ bm ) − ρ 2 Δam − 2am ρΔρ
1 − bm ρ 1 − bm ρ (1 − bm ρ)2

The left-hand side vanishes in view of (D.10) resulting in

 
Δbm kB T ρ 2 (1 − bm ρ)2
Δρ = A0 Δam − , A0 ≡ (D.16)
(1 − bm ρ)2 kB T − 2am ρ (1 − bm ρ)2

Van der Waals parameters am and bm change due to the variation in molar fractions
satisfying Δya = −Δyb :
√ √ √
Δam = 2 Δya am ( aa − ab ) (D.17)
Δbm = Δya (ba − bb ) (D.18)

Substituting (D.18) into (D.16) we obtain:

  
√ √ √ (ba − bb ) kB T
Δρ = Δya A0 2 am ( aa − ab ) − (D.19)
(1 − bm ρ)2

Thus, for the binary van der Waals system

304 Appendix D: Partial Molecular Volumes

Table D.1 Reduced partial

p v (bar) ηal ηbl
molecular volumes in the
liquid phase, ηil , i = a, b, for 1 1.005 0.289
the mixture n-nonane 10 1.057 0.305
(a)/methane (b) at T = 240 K 25 1.142 0.289
and various total pressures 40 1.228 0.365

 vdW  
√ √ √ 
∂ ln ρ (1 − bm ρ)2 a m ( aa − a b )
2ρ ρ(ba − bb )
= −
∂ ya mρ (1 − bm ρ)2
p,T 1 − 2akB T (1 − bm ρ)
2 kB T
(D.20)
Substituting this result into (D.7)–(D.8) we obtain the expression for ηi and hence
for the partial molecular volumes
   
1 ∂ ln ρ vdW
va = 1 − yb (D.21)
ρ ∂ ya p,T
   
1 ∂ ln ρ vdW
vb = 1 + ya (D.22)
ρ ∂ ya p,T

The reduced partial molecular volumes of the components in the liquid phase,
ηil , i = a, b play an important role for binary nucleation, especially at high pressures.
Table D.1 shows the values of these parameters for the mixture n-nonane (a)/methane
(b) for the nucleation temperature T = 240 K and various total pressures.
Appendix E
Mixtures of Hard Spheres

This appendix summarizes the relations describing the thermodynamic properties of a

binary mixture of hard spheres [16–18]. The hard-sphere diameters of the components
are d1 and d2 ; their respective number densities are ρ1 and ρ2 .
We start with defining the first three “moments” of the hard-sphere diameters:

Ri = di /2
Ai = π di2
Vi = π di3 /6

Note that Ai can be regarded as a molecular surface area, whereas Vi denotes the
molecular volume of component i. Next, the parameters ξ (k) are defined as

ξ (0) = ρ1 + ρ2
ξ (1) = ρ1 R1 + ρ2 R2
ξ (2) = ρ1 A1 + ρ2 A2
ξ (3) = ρ1 V1 + ρ2 V2 .

Note that ξ (3) is the total volume fraction occupied by hard spheres. For notational
convenience, we also introduce

η = 1 − ξ (3) .

On the basis of ξ (k) , new parameters c(k) are calculated according to

c(0) = − ln η
ξ (2)
c(1) =
η

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 305

DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013
306 Appendix E: Mixtures of Hard Spheres

 (2) 2
(2) ξ (1) ξ
c = +
η 8π η2
 (2) 3
(3) ξ (0) ξ (1) ξ (2) ξ
c = + + .
η η 2 12π η3

The pressure pd of the hard-sphere mixture follows from

p3 = c(3) k B T (E.1)
 (2) 3 (3)
ξ ξ
p2 = p3 − kB T (E.2)
12π η3
2 p3 + p2
pd = . (E.3)
3
Let us introduce for brevity of notations two additional quantities:
 
(1) 1 − 23 ξ (3) ln η
Y =3 + (3)
η2 ξ
  2 
ξ (2) η + 1 − 2ξ (3) 2 ln η
Y (2) = (3) + (3) .
6ξ η3 ξ

The chemical potentials μd,i can be derived from the virial equation using standard
thermodynamic relationships:

(3)
μi = c(0) + c(1) Ri + c(2) Ai + c(3) Vi (E.4)
 2
(2) (3) Ri ξ (2)  (1) 
μi = μi + (3)
Y − 2Ri Y (2) (E.5)
3ξ
(3) (2)
+ μi
2μi
μiex = kB T (E.6)
3 
μd,i = k B T ln ρi Λi3 + μiex (E.7)

The last expression presents the chemical potential of a species as a sum of an ideal
and excess contributions.
Appendix F
Second Virial Coefficient for Pure Substances
and Mixtures

Calculation of the second virial coefficient

1
B2 (T ) = 1 − e−βu(r ) dr (F.1)
2

from first principles requires the knowledge of the intermolecular potential u(r)
which is in most cases is not available. With this limited ability second virial
coefficient is calculated from appropriate corresponding states correlations. For
nonpolar substances such correlation has the form due to Tsanopoulos [2, 19]:

B2 pc
= f0 + ω P f1 (F.2)
kB Tc

where

f 0 = 0.1445 − 0.330/Tr − 0.1385/Tr2 − 0.0121/Tr3 − 0.000607/Tr8 (F.3)

f 1 = 0.0637 + 0.331/Tr2 − 0.423/Tr3 − 0.008/Tr8 (F.4)

Tr = T /Tc is the reduced temperature and ω P is Pitzer’s acentric factor.

For normal fluids van Ness and Abbott [20] suggested simpler expressions for f 0
and f 1

f 0 = 0.083 − 0.422/Tr1.6 (F.5)

f1 = 0.139 − 0.172/Tr4.2 (F.6)

Expressions (F.5)–(F.6) agree with (F.3)–(F.4) to within 0.01 for Tr > 0.6 and
ω P < 0.4, while for lower Tr the difference grows rapidly.
For the mixtures the second virial coefficient is written using the mixing rule

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 307

DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013
308 Appendix F: Second Virial Coefficient for Pure Substances and Mixtures

B2 = yi y j B2,i j (F.7)
i j

where B2,ii are the second virial coefficient of the pure components. For the cross term
B2,i j the combining rules should be devised to obtain Tc,i j , pc,i j and ω P,i j which
then are substituted into the pure component expression (F.2 ), with the coefficients
f 0 and f 1 satisfying (F.3)–(F.4) or (F.5)–(F.6). For typical applications the following
combining rules are used [2]

Tc,i j = (Tc,i Tc, j )1/2 (F.8)

! 1/3 1/3 "3
Vc,i + Vc, j
Vc,i j = (F.9)
2
Z c,i + Z c, j
Z c,i j = (F.10)
2
ω P,i + ω P, j
ω P,i j = (F.11)
2
Z c,i j kB Tc,i j
pc,i j = (F.12)
Vc,i j

where Vc,i = 1/ρc,i .

Appendix G
Saddle Point Calculations

In Chap. 13 we search for the saddle-point of the free energy of cluster formation in
the space of total numbers of molecules of each species in the cluster. For calculation
of g(n a , n b ) we choose an arbitrary bulk composition n il and find the excess numbers
n iexc according to Eqs. (11.84)–(11.85). The total numbers of molecules are: n i =
n il + n iexc . Then, g(n a , n b ) is found from Eq. (13.91). Is easy to see that although
we span the entire space of (nonnegative) bulk numbers (n al , n lb ), the space of total
numbers (n a , n b ) contains “holes”, i.e. the points, to which no value of g(n a , n b ) is
assigned. This feature complicates the search of the saddle point of g(n a , n b ).
To overcome this difficulty we apply a smoothing procedure aimed at elimination
of the holes in (n a , n b )-space by an appropriate interpolation procedure between the
known values. The simplest procedure for the 2D space is the bilinear interpolation
which presents the function g at an arbitrary point (n a , n b ) as

g(n a , n b ) = a n a + b n b + c n a n b + d (G.1)

where coefficients a, b, c, d are defined by the known values of g around the point
(n a , n b ). However, due to randomness of the location of the “holes”, the straightfor-
ward application of bilinear interpolation is quite complicated. This difficulty can be
avoided if we notice that (G.1) is the solution of the 2D Laplace equation

∂2 ∂2
Δg(n a , n b ) = 0, Δ ≡ + (G.2)
∂n a2 ∂n 2b

Thus, filling the holes in (n a , n b ) space by bilinear interpolation is equivalent to

solving the Laplace equation (G.2), which turns out to be a quick and efficient
procedure. Discretizing (G.2) on the 2D grid with the grid-size δn a = δn b = 1,
we have

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 309

DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013
310 Appendix G: Saddle Point Calculations

∂2g
= g(n a − 1, n b ) − 2g(n a , n b ) + g(n a + 1, n b )
∂n a2
∂2g
= g(n a , n b − 1) − 2g(n a , n b ) + g(n a , n b + 1)
∂n 2b

The discrete version of Eq. (G.2) becomes

g(n a − 1, n b ) + g(n a , n b − 1) + g(n a + 1, n b ) + g(n a , n b + 1)

g(n a , n b ) =
4
(G.3)
An iterative procedure of finding g(n a , n b ) satisfying Eq. (G.3) is known as
“Laplacian smoothing” [21]. Among various possibilities of performing iterations
the Gauss-seidel relaxation scheme [22] seems most computationally efficient:

g i+1 (n a − 1, n b ) + g i+1 (n a , n b − 1) + g i (n a + 1, n b ) + g i (n a , n b + 1)
g i+1 (n a , n b ) =
4
(G.4)
where g i is the value of g at i-th iteration step. The procedure is repeated until
g i+1 (n a , n b ) ≈ g i (n a , n b ).
The saddle point of the smoothed Gibbs function satisfies
 
∂g  ∂g 
= =0
∂n a n b ∂n b n a

Note, that computationally it is preferable to search for the saddle point by solving
the equivalent variational problem:
!  "2 !  "2
∂g  ∂g 
+ → min (G.5)
∂n a n b ∂n b n a

References

1. J. Wölk, R. Strey, J. Phys. Chem. B 105, 11683 (2001)

2. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, 4th edn. (McGraw-
Hill, New York, 1987)
3. A. Dillmann, G.E.A. Meier, J. Chem. Phys. 94, 3872 (1991)
4. K. Iland, Ph.D. Thesis, University of Cologne, 2004
5. International critical tables of numerical data, vol. 2, p. 457 (McGraw-Hill, New York,1927)
6. L.E. Murr, Interfacial Phenomena in Metals and Alloys (Addison-Wisley, London, 1975)
7. A. Kaplun, A. Meshalkin, High Temp. High Press. 31, 253 (1999)
8. K. Tamura, S. Hosogawa, J. Phys. Condens. Matter 6, A241 (1994)
9. H. Hong et al., J. Non-Cryst. Solids 312—314, 284 (2002)
10. A. Michels et al., Physica 15, 627 (1949)
11. M. Sano, S.F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, Oxford,
1988)
Appendix G: Saddle Point Calculations 311

12. W.G. Chapman, K.E. Gubbins, G. Jackson, M. Radosz, Fluid Phase Equilib. 52, 31 (1989)
13. J. Pamies, Ph.D. Thesis, Universitat Rovira i Vrgili, Tarragona, 2003
14. V.I. Kalikmanov, Statistical Physics of Fluids. Basic Concepts and Applications (Springer,
Berlin, 2001)
15. G.A. Korn, T.M. Korn, Mathematical Handbook (McGraw-Hill, New York, 1968)
16. G.A. Mansoori, N.F. Carnahan, K.E. Starling, T.W. Leland, J. Chem. Phys. 54, 1523 (1971)
17. Y. Rosenfeld, J. Chem. Phys. 89, 4272 (1988)
18. C.C.M. Luijten, Ph.D. Thesis, Eindhoven University, 1999
19. C. Tsanopoulos, AICHE J. 20, 263 (1974)
20. H.C. van Ness, M.M. Abbott, Classical Thermodynamics of Non-electrolyte Solutions
(McGraw, New York, 1982)
21. F. O’Sullivan, J. Amer. Stat. Assoc. 85, 213 (1990)
22. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer, New York, 1993)
Index

i, v-cluster, 71 equilibrium, 33, 80, 110, 192, 243

nonequilibrium, 24
Coarse-grained
A configuration integral, 227, 232
Absorption nucleation theory(CGNT), 233
frequency, 260 Coarse-graining, 215, 228, 234
Activation Coexistence
barrier, 141 pressure, 190, 240
energy for diffusion, 260 Compensation pressure
Activity effect, 200, 201, 236
coefficients, 188, 193 Compressibility factor, 18, 30, 91, 104, 191
gas-phase, 192 critical, 91
liquid-phase, 194 vapor, 89
Adiabatic expansion, 277, 278, 286 Condensation nuclei
Adiabatic system, 120, 121 in heterogeneous nucleation, 251
Aerosol particles Configuration integral, 84, 224
concentration, 271 Constant Angle Mie Scattering (CAMS), 279,
284
Constrained equilibrium, 25, 26, 173, 174,
B 216–219, 232, 244
Barrier Contact
nucleation, 55, 56, 67, 145, 150, 151, 162, angle, 141, 252, 254–256, 258, 260,
164, 258 262–265, 267–271
Binary interaction parameter, 200, 209, 210 line, 252, 255, 256, 261–264, 267
Binding energy, 84 Continuity equation, 134, 177
Binodal, 146 Cooling rate, 278, 286
Coordination number, 92, 95, 97, 129, 154,
162, 166, 229, 230
C Core-shell structure, 236
Capillarity approximat, 21, 28, 30, 32, 55, 72, Courtney correction, 32
77, 154, 182, 215, 230, 231, 252 Critical cluster, 23, 27, 28, 30, 32, 36, 37,
Cluster 43–45, 47–51, 55, 73, 74, 77,
distribution function, 173 97–99, 103, 136, 137, 138, 145,
growth law, 34, 35, 37 150–155, 158, 174, 181, 183, 185,
Cluster definition 186, 196, 198, 202, 211, 215, 220,
live-time criterion, 128 233–235, 241, 257, 258, 265, 270
Cluster distribution function Cut-off radius, 164

V.I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics, 860, 313

DOI: 10.1007/978-90-481-3643-8, Ó Springer Science+Business Media Dordrecht 2013
314 Index

D I
Detailed balance, 25, 31, 76, 173, 217 Ideal gas, 7, 18, 28, 29, 72, 73, 75, 90, 173,
Diagrammatic expansion, 225 190, 202, 216, 226, 240
Diffusion cloud chamber, 102 Ideal mixture, 190, 194, 209
Direction of principal growth approximation, Impingement rate, 25, 173, 180, 235, 243, 260
177, 220, 241, 242 average, in binary nucleation, 221, 235,
Distribution function 249
one-particle, 57 Importance sampling
Dupre-Young equation, 254, 262 in MC simulations, 126
Intrinsic
chemical potential, 60
free energy, 57, 58, 61, 63, 65, 67
E Ising model, 268
Entropy Isothermal compressibility, 148
configurational, 84
bulk per molecule, 85
Exclusion volume, 130 K
Expansion cloud chamber, 275, 277, 278 Kelvin equation
classical, 30

F
Fisher droplet model, 79 L
Fletcher theory, 251 Lagrange equation, 115
Fluctuation theory, 19, 88, 174, 218 Landau
Fokker-Planck equation, 35 expansion, 146
Frenkel distribution, 32 Laplace equation
Fugacity, 82, 87, 88, 224, 228 generalized, 12
Functional standard, 12
grand potential, 65, 67, 205, 207 Latent heat, 7, 51, 121, 122, 141, 165, 199, 216
Helmholtz free energy, 55, 57–59, 65, 69, Lattice-gas model, 142
149, 205, 206 Laval Supersonic Nozzle, 162, 275, 286
Law of mass action, 32
Lennard-Jones potential, 141
Limiting consistency, 33
G Line tension, 261–265, 267–273
Gibbs Local density approximation (LDA), 62, 206
dividing surface, 9–13, 21, 44, 47, 51, 56,
76, 83, 92, 181–183, 185–187, 189,
209, 226, 236, 238, 239, 252 M
equimolar surface, 10, 12, 13, 21, Matrix
107, 186 inverse, 241
surface of tension, 12, 13 transposed, 241
free energy, 20 unit, 241
of droplet formation, 20 unitary, 241, 242
Mean-field
approximation, 83, 86, 96
H Mean-field Kinetic Nucleation Theory, 80, 97
Hard-spheres Metastability
Carnahan-Starling theory, 63, 64, 67, 96 parameter , 200, 202, 216, 235
cavity function, 96 Metastable state, 17
effective diameter, 63 Microscopic surface tension, 79, 88, 90, 106,
Heat capacity, 121 112, 157, 164, 230, 232, 233
Heat of adsorption, 260 reduced, 88, 228, 230, 233
Hill equation, 51 Mie theroy, 279, 280, 282, 285
Index 315

amplitude functions, 281 Refractive index, 279, 280

extinction coefficient, 279, 280 Retrograde nucleation, 198, 199
extinction efficiency, 280, 281 Rotation
Minimum image convention, 119 angle, 241, 242, 248
Mixing rule, 205, 210, 212 transformation, 240
Modified Drupe-Young equation, 263

S
N Saddle point, 55, 68, 149, 150, 171, 174,
Nucleation 176–180, 185, 189, 195, 205,
boundary conditions, 35, 36, 63, 119, 120, 211, 220, 221, 233, 234,
127, 136, 178, 211, 246 241–243, 249
Nucleation barrier, 22, 47, 50, 76, 77, 98, 99, Saturation
130, 132, 152–155, 158, 185, 211, pressure, 6, 168, 229
215, 235, 258, 265, 271 Scattering intensity, 282, 283, 285, 288
Nucleation pulse, 277–279, 282–284, 286 Shock tube, 275, 283–285
Nucleation pulse chamber, 278, 279 Small Angle Neutron Scattering (SANS), 286
Nucleation Theorem Small Angle X-ray Scattering
first, 44, 48 (SAXS), 286–288
pressure, 53 Spherical particle
second, 50 contrast factor of, 288
form-factor of, 288
Spinodal, 56, 69, 132, 145, 146, 150, 151–154,
O 162–163
Order parameter, 17, 146, 149, 152 decomposition, 145, 147, 153
kinetic, 153
thermodynamic, 145, 149, 162–164
P Supercritical solution
Packing fraction, 95, 97 rapid expansion of (RESS method), 124
Partial Supersaturation, 18, 151
molecular volume, 52, 182, 200, 231, 238 Surface diffusion, 259
vapor pressure, 190 Surface enrichment, 141, 181, 194, 202, 205,
Partition function 207, 209, 210, 213
canonical, 58 Surface tension
grand, 60 macroscopic, 69, 72, 100, 103, 104, 164,
Periodic boundary conditions, 120, 127 186, 192, 199, 200, 215
Phase transition
first order, 1, 7
Poisson law, 279, 284 T
Pseudospinodal, 98, 145, 152–158, 162, Thermal diffusion cloud chamber, 275, 276
163, 168 Thermodynamics
first law, 7
Threshold method
R in MD simulations, 134–136, 138,
Radius of gyration 141, 165
of a polymer molecule, 295 Time-lag, 39
Random number, 126 Tolman equation, 108
Random phase approximation (RPA), 63, 67 Tolman length, 13, 108
Random processes
theory of, 272
Rate U
forward, 24–26, 31, 80 Umbrella sampling, 127, 133
316 Index

V Virtual monomer approximation, 234

Van Laar Volume term, 225–227
constants, 194
model, 193
Vapor depletion, 278 W
Variational transition state theory, 76 Weeks-Chandler-Anderson theory
Velocity scaling decomposition scheme, 96
algorithm in MD simulations, 121
Velocity-Verlet algorithm, 118
Verlet algorithm, 118 Z
Virial coefficient Zeldovich factor, 27, 98
second, 90, 106, 157, 166, 201, 229, Zeldovich relation, 37
230, 233