The Atomistic Nature of Crystal Growth

369 Pages·2001·12.153 MB·English

by  Prof. Boyan Mutaftschiev (auth.)

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Boyan Mutaftschiev
The Atomistic Nature
of Crystal Growth
With 98 Figures
' Springer
Prof. Boyan Mutaftschiev
Laboratoire de Mineralogie-Cristallographie
Universites de Paris VI et Paris VII
4, Place Jussieu
752521'ans
France
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Mutaftschiev, Boyan, 1932-
The atomistic nature of crystal growth / Boyan Mutaftschiev.
p. cm. -- (Springer series in materials science, ISSN 0933-033X ; v. 43)
Includes bibliographical references and index.
ISBN 978-3-642-08577-2 ISBN 978-3-662-04591-6 (eBook)
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l. Crystal growth. I. Title. If. Series.
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To my teacher Rostislav Kaischew
To Lea, for her constant encouragement
throughout the writing of this book
Preface
Throughout my long involvement in research on crystal growth and nucle
ation, I have witnessed the exponential expansion of this field. It was during
my first year of college in 1949 that F. C. Frank presented his, for the time,
surprising ideas about the role of screw dislocations in the growth kinetics
at a Discussion of the Faraday Society. They were immediately appreciated
and propagated by my later teacher, R. Kaischew, who has been, along with
W. Kossel and I. N. Stranski, one of the founders of the modern atomistic
theories of crystal growth. Only four years later, we were able to observe in
situ and film the spreading of spiral fronts during the electrocrystallization
of silver.
For several more years, crystal growth remained an academic branch of
science. At that time it was difficult to present papers on the topic as part of
the programs of conferences on solid state physics or crystallography. It was
the demand for semiconductor materials with specific qualities that boosted
research and increased interest from a wide, interdisciplinary community in
the problems of nucleation, bulk growth of large crystals and, more recently,
in the growth of objects with nanometric dimensions.
The almost explosive development of this field resulted, however, in the
establishment of schools with quite different backgrounds and theoretical ap
proaches, unlike in classical disciplines, the slow development of which pro
vided the basis for a large consensus on the major issues (a typical example is
the nucleation theory, which is periodically "revised" while its sister theories
treating chemical kinetics have been much less subject to controversies).
The title of this book already reveals my intention, not to present a com
pilation of all existing theoretical approaches to the considered phenomena
but to focus on their microscopic, atomistic interpretation. This intention has
both an ambition and a limitation. The ambition is to go as far as possible
in the understanding of nucleation, crystal growth and thin film formation
through the link with other phenomena, the atomistic approach to which is
evident; the limitation requires leaving aside numerous topics related to the
transport of heat or matter and treating crystals as continua. However, the
reader will encounter some of them (Wulff theorem, BCF theory, morpholog
ical stability, etc.), as they may be considered as milestones in the evolution
VIII Preface
of the field. Also, in these cases, an effort is made to suggest their microscopic
interpretation in parallel.
Many persons have contributed to the ripening of the ideas presented in
this book, beginning with the great I. N. Stranski and his close associate,
R. Kaischew. It is a pleasure to acknowledge fruitful discussions over many
years, with R. Kern, J. M. Cases, A. Bonissent, C. Chapon, J. J. Metois and
C. R. Henry. X. Duval and A. Thorny introduced me to the world of two
dimensional phases, A. Baronnet shared with me his great enthusiasm for
dislocations in crystals. My interest in the atomistic interpretation of equilib
rium shapes and their modification by foreign adsorption has been stimulated
by the contacts I had with B. Honigmann and R. Lacmann, while H. Reiss,
J. L. Katz, F. F. Abraham and the late K. Nishioka contributed to my under
standing of nucleation with their critiques and suggestions. I am indebted to
Robert F. Sekerka for the revision of parts of the manuscript, and to Terry
Joseph for her improvement of the English.
The work on the manuscript was completed during my time as visiting
professor at the Satellite Venture Business Laboratory of the Tokushima Uni
versity in Japan. I am most grateful to Professor Shiro Sakai for his hospitality
during my tenure of this position.
Years ago, my professor, Rostislav Kaischew, told me that writing a textbook
should be the swansong of a scientist. Perhaps it is because he never wrote
one that this year he celebrated his 93nd birthday. It is to him that this
book is dedicated. I feel that I have inherited from him the way of looking at
the phenomena of nature and his way of teaching. Thus, I hope that in this
book I have succeeded in bridging the gap between the ideas of the "classics",
such as Volmer, Stranski and Kaischew, whom I met in the time when few
were interested in crystal growth, and the ideas of the scientists of today's
generation.
B. Mutaftschiev
Contents
Part I Introduction and Background
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Principles of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Internal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Other Thermodynamic Functions; Maxwell's Relations . . . . . . 10
2.5 Work and Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 Integrated Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2. 7 Chemical Potentials and Supersaturation . . . . . . . . . . . . . . . . . . 17
2.7.1 Supersaturated Vapor (Ideal Gas) . . . . . . . . . . . . . . . . . . 18
2.7.2 Supersaturated Solution... ....... ........ .... ..... 18
2.7.3 Undercooled Condensed Phase . . . . . . . . . . . . . . . . . . . . . 19
2.8 The "Surface Phase" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3. Statistical Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1 Partition Functions; General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Partition Function in a Continuum Phase Space . . . . . . . . . . . . 24
3.3 Examples of Simple Partition Functions . . . . . . . . . . . . . . . . . . . 26
3.3.1 Translational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.3 Free Rotation................ ..... ... ...... ...... 30
3.4 Canonical Partition Functions of Pure Phases;
Free Energy and Chemical Potential . . . . . . . . . . . . . . . . . . . . . . 32
3.4.1 The Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.2 The "Lattice Gas" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.3 The Monatomic Einstein Crystal . . . . . . . . . . . . . . . . . . . 33
3.4.4 The One-Dimensional Non-Einstein Crystal. . . . . . . . . . 34
3.5 Partition Function of Associated Vapor. . . . . . . . . . . . . . . . . . . . 38
X Contents
Part II Equilibria
4. Equilibrium Between Large Phases;
The Vapor Pressure of Solids........................ .... . 43
4.1 The Clausius-Clapeyron Equation . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Statistical-Thermodynamic Treatment . . . . . . . . . . . . . . . . . . . . 45
4.3 Repetitive-Step; Thermodynamic Frequency . . . . . . . . . . . . . . . 45
4.4 Kinetic Treatment of the Equilibrium Crystal-Vapor . . . . . . . . 49
4.5 Equilibrium in the Different Sites on the Crystal Surface . . . . 51
4.6 Adsorbed and Incorporated Molecules;
the Different Types of Crystal Face . . . . . . . . . . . . . . . . . . . . . . . 55
5. The Surface Tension of Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1 The Broken-Bond Approach to the Surface Energy of a Solid:
the Born-Stern Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Surface Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Stefan's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6. Equilibrium Between Large Three-
and Two-Dimensional Phases: Adsorption Phenomena . . . . 73
6.1 Partition Function and Chemical Potential
of an Adsorbed Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.1.1 The Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.1.2 The Mean Field Approximation . . . . . . . . . . . . . . . . . . . . 77
6.2 Desorption Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3 Frumkin-Fowler's Adsorption Isotherm.................... 79
6.4 Multilayer Adsorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.5 Two-Dimensional Phase Transitions; Spreading Pressure. . . . . 88
6.6 Two-Dimensional Versus Three Dimensional Phases;
a Link to Wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6. 7 Displacement and Mixing
of Condensed Two-Dimensional Phases . . . . . . . . . . . . . . . . . . . . 94
7. Thin Films, Surface Roughening, and Surface Alloys ...... 103
7.1 Chemisorbed Versus Physisorbed Layers; Thin Solid Films .. 103
7.2 Surface Roughness Considered as a Special Case
of Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3 Adsorption on a Thermally Rough Substrate:
Surface Alloying ................................... ..... 113
7.4 Surface Melting ....................................... . 119
Contents XI
8. Equilibrium Between a Small and a Large Phase .......... 123
8.1 The Gibbs Potential of a Small Phase;
the Capillarity Approximation ........................... 123
8.2 The Size-Dependent Chemical Potential .................. . 128
8.2.1 Droplets in a Vapor ... ............................ 129
8.2.2 Gas Bubbles in a Liquid ........................... 130
8.2.3 Solid Clusters in a Melt ........................... 132
8.3 Statistical Mechanical Treatment of the Free Energy
of a Solid Cluster ....... ................................ 133
8.4 Capillarity Approximation Versus Model Calculations .. .. ... 138
9. Equilibrium Shapes of Crystals ...................... .. ... 147
9.1 Curie-Wulff's Condition and Wulff's Theorem ............ . 148
9.2 Herring's 1-Plot ........................................ 151
9.3 Faceting of a K Face ................................... . 156
9.4 Equilibrium Shape on a Foreign Substrate ................ . 158
9.5 Equilibrium Shape of a Droplet on a Substrate:
Young's Equation ... .. ........... ..... ... ..... ......... 161
9.6 Entropy Effects on Surface Free Energy
and Equilibrium Shape .................................. 163
9.7 Equilibrium of Small Anisotropic Phases;
the Thermodynamic Approach .......................... . 166
9.8 Equilibrium of Small Anisotropic Phases;
the Atomistic Approach ... ...... ........................ 168
9.9 The Influence of Foreign Adsorption
on the Equilibrium Shape .. ............................. 174
9. 9.1 The Different Types of Crystal Face . . . . . . . . . . . . . . . . 177
9.9.2 The Equilibrium Shape ................ ........... 179
9.9.3 Faceting .. ........................ .. ............. 180
Part III Nucleation
10. Homogeneous Nucleation; the Phase Approach ...... ..... 183
10.1 The Classical Nucleation Work. Droplets in a Vapor ... .. .. . 184
10.2 Bubbles in Liquids: Boiling and Cavitation ................ 189
10.2.1 Boiling under Positive External Pressure .. .. ........ 190
10.2.2 Boiling under Negative External Pressure .... ....... 192
10.3 Anisotropic Embryos; the Thermodynamic Approach ....... 193
10.4 Anisotropic Embryos; the Atomistic Approach ............. 194
10.5 The Volmer-Weber Treatment of Nucleation Kinetics ....... 196
11. Homogeneous Nucleation; the Chemical Approach ........ 201
11.1 Equilibrium in Associated Vapor ......................... 201
11.2 Frenkel's Size Distribution .. ......... .... .... .... ........ 207