Table Of ContentPartially Ordered Systems
Editorial Board:
J. Charvolin • W. Helfrich • L. Lam
Advisory Board:
J.D. Litster • D.R. Nelson • M. Schadt
Partially Ordered Systems
Editorial Board: J. Charvolin . W. Helfrich' L. Lam
Solitons in Liquid Crystals
Lui Lam and Jacques Prost, Editors
Bond-Orientational Order in Condensed Matter Systems
Katherine J. Strandburg, Editor
Diffraction Optics of Complex-Structured Media
V.A. Belyakov
Katherine J. Strandburg
Editor
Bond -Orientational
Order in Condensed
Matter Systems
Foreword by David R. Nelson
With 163 Illustrations
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Katherine J. Strandburg
Argonne National Laboratory
Materials Science Division
Argonne, IL 60439
USA
Editorial Board:
Jean Charvolin Wolfgang Helfrich Lui Lam
Institut Max von Laue- Institut fiir Theorie der Department of Physics
Paul Langevin kondensierten Materie San Jose State University
Avenue des Martyrs Fachbereich Physik One Washington Square
38042 Grenoble Cedex Freie Universitiit Berlin San Jose, CA 95192
France Arnimallee 14 USA
1000 Berlin 33
Germany
Advisory Board:
John D. Litster David R. Nelson Martin Schadt
Francis Bitter National Department of Physics Department ZFE/RLC
Magnet Laboratory Harvard University F. Hoffman - La Roche
Massachusetts Institute of Cambridge, MA 02138 &Co.
Technology USA CH-4002 Basel
Cambridge, MA 02139 Switzerland
USA
Library of Congress Cataloging-in-Publication Data
Bond-orientational order in condensed matter systems 1 Katherine J.
Strandburg, editor.
p. cm. - (Partially ordered systems)
Includes bibliographical references and index.
ISBN -13:978-1-4612-7680-7
1. Condensed matter. 2. Crystals. 3. Glass. 4. Phase
transformations (Statistical physics) I. Strandburg, Katherine Jo,
1957- . II. Series.
QCI73.4.C65B66 1992
530.4'I-dc20 91-20237
Printed on acid-free paper.
© 1992 Springer-Verlag New York, Inc.
Softcover reprint of the hardcover 1st edition 1992
All rights reserved. This work may not be translated or copied in whole or in part without the
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ter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if
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understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by
anyone.
Photocomposed from a LaTex file.
987654321
ISBN -13 :978-1-4612-7680-7 e-ISBN -13 :978-1-4612-2812-7
DOl: 10.1007/978-1-4612-2812-7
Foreword
Most of us were taught in school that matter comes in three basic forms:
solid, liquid, and gas. This classification scheme goes back at least to the
Greeks, and has remained essentially unchanged for thousands of years. The
important embellishments have come during the last 100 years or so. We
now know, for example, that there is no fundamental distinction between a
liquid and a gas above the critical point. On the other hand, it is important
to distinguish between the repeating arrays of atoms and molecules in a
crystalline solid and the nonequilibrium disorderly arrangements in glassy
solids. We find additional phases in liquid crystals which, because of the
anisotropic shape of their constituent molecules, are in many ways interme
diate between liquids and solids. Despite such refinements, a strong basic
prejudice remains that simple pointlike atoms or molecules in equilibrium
should have just three basic phases.
This prejudice overlooks the fact that two broken symmetries separate
low-temperature crystalline solids from high-temperature liquids. The reg
ular rows of atoms in a crystal responsible for Bragg peaks reflect a break
down of the translational invariance characteristic of a liquid. The crys
tallographic axes of the solid represent a different, broken orientational
symmetry. A broken translational symmetry necessarily implies long-range
orientational order. It is possible to imagine phases of matter, however,
where the short-range translational order of a liquid coexists with a broken
rotational symmetry. The long-range rotational order of this liquid is asso
ciated with "bonds" (determined by, say, the Voronoi construction) joining
neighboring atoms or molecules. Thirteen years ago, Bertrand Halperin
and I discovered that such an intermediate phase exists in a model of
two-dimensional melting first proposed by Kosterlitz and Thouless and by
Berezinski. The usual latent heat of the crystal-to-liquid melting transition
was spread out over an intermediate "hexatic" phase, somewhat similar
to a nematic liquid crystal. Unlike a liquid crystal, though, the residual
sixfold orientational order of hexatics is not associated with the anisotropy
of the constituent particles. This fourth, hexatic phase of matter has now
been observed in free standing liquid crystal films, in colloidal crystals,
in magnetic bubble arrays, and even in the flux lines of high temperature
superconductors.
The idea of bond-orientational order has proven to be more general than
the model of two-dimensional dislocations we studied at that time. Bond
orientational order can exist in three dimensions, and in two and three
dimensional glasses, and need not be associated with a continuous dis-
vi Foreword
location unbinding transition. Although a "hexatic phase," with sixfold
bond-orientational order, is the most likely possibility in two dimensions,
"tetradic," "cubatic," and even icosahedral bond-oriented phases are pos
sible. In this book, Katherine J. Strandburg has brought together an out
standing collection of scientists who have studied bond-orientational order.
The focus is on hexatic bond-orientational order in two dimensions, and
icosahedral bond order in metallic glasses and three-dimensional quasicrys
tals.
The subject of bond-orientational order is introduced in the article by
Joel D. Brock. Intriguing computer simulations of bond order and qua
sicrystals are then discussed by Strandburg. Some of the important exper
iments on liquid crystal phases (where the hexatic was first discovered) are
reviewed by C.C. Huang. Cherry A. Murray then presents a very thorough
account of experiments on melting and hexatics in colloidal suspensions.
Prior to this work, many investigators thought the hexatic phase would not
occur in systems of pointlike particles in two dimensions.
The remainder of the book focuses on icosahedral bond order in glasses
and quasicrystals. Tin-Lun Ho discusses some fascinating issues which arise
in the faceting of quasicrystals. Icosahedral clusters are a natural packing
of (slightly distorted) tetrahedra in three dimensions, playing a role similar
to hexagonal clusters in two dimensions. Subir Sachdev discusses how local
orientational order in these clusters determines the structure of undercooled
liquids and metallic glasses. Landau expansions for long-range translational
and orientational order in quasicrystals are reviewed by Marko V. Jaric, and
the icosahedral glass model for quasicrystals is explained by A.I. Goldman.
Bond-oriented phases seem likely to be with us for quite some time.
This book provides an excellent introduction for both theorists and exper
imenters to the basic concepts in the field.
Cambridge, Massachusets David R. Nelson
May 1991
Contents
Foreword by David R. Nelson v
Contributors xii
1 Bond-Orientational Order 1
Joel D. Brock
1.1 Introduction 1
1.2 Elementary Ideas 4
1.2.1 Example: Two-Dimensional Harmonic Crystal 4
1.2.2 Mean Field Phase Diagram and Coupled
Order Parameters 10
1.2.3 Modern Theory of Phase Transitions 13
1.3 Experimental Results 17
1.3.1 Thermodynamic Measurements 17
1.3.2 Static Structural Studies 20
1.4 More Complicated Systems 25
1.4.1 Crystals 25
1.4.2 Glasses 26
1.4.3 Incommensurate Crystals 26
1.4.4 Quasicrystals 26
1.4.5 Quasilattice 27
1.4.6 Quasicrystalline Glass 28
1.5 Extensions to Three Dimensions 29
References 29
2 Computer Simulation Studies of Bond-Orientational
Order 32
Katherine J. Strandbury
2.1 Introduction 32
2.2 Numerical Simulation Techniques 35
2.2.1 Atomistic Simulations 35
2.2.2 Simulations of Abstract Objects 40
2.2.3 Basic Problems with Numerical Simulation Studies:
Short Times, Small Sizes, and the Importance of
Boundary Conditions 42
2.3 Examples of Computer Simulation Studies of
Bond-Orientational Order 48
viii Contents
2.3.1 Measurement of Bond-Orientational Order in a
Two-Dimensional Atomistic System 48
2.3.2 Behavior of a Two-Dimensional Atomistic System in
the Presence of a Hexatic Field 50
2.3.3 Study of the Hexatic Phase in a Defect-Based
Simulation 53
2.3.4 Effects of Frustration on Glass Formation in a
Two-Dimensional Lennard-Jones System 56
2.3.5 Square-Triangle Analysis 57
2.3.6 A Simple Atomistic Model Possessing an
Equilibrium Quasicrystal Phase 60
2.3.7 Consequences of Nontraditional Bond-Orientational
Order in a Random Tiling Model 66
2.3.8 Growth of a Perfect Quasicrystal 69
2.4 Conclusions 73
References 73
3 Nature of Phase Transitions Related to Stacked
Hexatic Phases in Liquid Crystals 78
C.C. Huang
3.1 Introduction 78
3.2 Fundmental Properties of the Hexatic Phase 80
3.2.1 Two-Dimensional Melting Theory 80
3.2.2 Existence of Hexatic Order in Other Physical Systems 84
3.2.3 Hexatic Phases in the Liquid Crystal 85
3.2.4 Pure Compounds and Mixtures Exhibiting
Hexatic Phases 92
3.3 Thermal Properties 100
3.3.1 Heat Capacity 102
3.3.2 Nature of the SmA-HexB-SmI Point 116
3.3.3 Thermal Transport Studies 120
3.4 Criticality of the Smectic-A-Hexatic-B Transition 124
3.5 Conclusions 129
References 130
4 Experimental Studies of Melting and Hexatic Order
in Two-Dimensional Colloidal Suspensions 137
Cherry A. Murray
4.1 Introduction 137
4.1.1 Theoretical Predictions of Two-Dimensional Melting 138
4.1.2 Advantages and Disadvantages of Colloid Direct
Imaging Experiments 147
4.2 Experimental Results-Melting of Two-Dimensional
Colloidal Systems 152
Contents ix
4.2.1 Charged, Confined between Flat Plates,
Wedge Geometry 152
4.2.2 Floating Monolayers on a Liquid Surface 175
4.2.3 Expansion between Rigid Plates 179
4.2.4 Dipole Holes in Ferrofiuid 188
4.2.5 Wedge Geometry Revisited: Comparison of Melting
in Two and Three Dimensions 196
4.3 Conclusions and Suggestions for Future Work 209
4.3.1 Equilibration and System Size 209
4.3.2 Possibility for Studying Driven Nonequilibrium
Phase Transitions 209
4.3.3 Relevant Energy Scales 210
4.3.4 Other Predictions of KTHNY 210
4.3.5 Substrates and Their Effects 210
4.3.6 Other Analog Molecular Dynamics Experiments 211
References 211
5 Faceting in Bond-Oriented Glasses and Quasicrystals 216
Tin-Lun Ho
5.1 Introduction 216
5.1.1 What Have Quasicrystals Brought Us? 216
5.1.2 The Problem of Quasicrystal Facets 217
5.1.3 Organization of This Chapter 218
5.2 Quasicrystal Facets 219
5.3 The Conventional View of Facet Formation 220
5.3.1 The Determination of the Equilibrium Shapes of
Solids: The Wulff Construction 223
5.3.2 The Origin of Facets: Cusps in the Surface Energy 225
5.4 Faceting in Perfect Bond-Oriented Systems 228
5.4.1 Sufficient Conditions for Facet Formation 229
5.4.2 The Herring Algorithm 231
5.4.3 The Equilibrium Shape of Simple Bond-Oriented
Systems with Icosahedral Symmetry 234
5.5 Equilibrium Shapes of Perfect and Random Quasicrystals 235
5.5.1 Tiling Model of Perfect and Disordered Quasicrystals 235
5.5.2 The Surface Energy of Perfect and Disordered
Quasicrystals 241
5.6 Final Remarks 248
References 253
6 Icosahedral Ordering in Supercooled Liquids
and Metallic Glasses 255
Subir Sachdev
6.1 Introduction 255
6.2 Three-Dimensional Sphere Packings and Frustration 256
x Contents
6.2.1 Frank-Kasper Phases 262
6.3 Structure Factor of Monoatomic Supercooled Liquids 262
6.3.1 Sphere Packings in Curved Three-Dimensional
Space 265
6.3.2 Order Parameter 266
6.3.3 Landau Free Energy 268
6.4 Application to Real Metallic Glasses 272
6.4.1 Metal-Metalloid Glasses 274
6.4.2 Metal-Metal Glasses 279
6.5 Conclusions 280
References 282
7 Orientational Order and Quasicrystals 284
Marko Vukobrat Jane
7.1 Introduction 284
7.2 Bond-Orientational Order Parameter 288
7.2.1 Definition 288
7.2.2 Symmetries 289
7.2.3 Measures 308
7.3 Bond-Orientational Phase Diagram 318
7.3.1 Summary 318
7.3.2 Free Energy 319
7.3.3 Minimization 320
7.3.4 Phase Diagram 323
7.4 Icosahedral Quasicrystals 327
7.4.1 Summary 327
7.4.2 Landau Theory of Freezing 328
7.4.3 Orientational-Translational Coupling 334
7.5 Conclusion 336
References 338
8 Icosahedral Glass Models for Quasicrystals 341
A.!, Goldman
8.1 Introduction 341
8.2 Icosahedral Crystals and Quasiperiodicity 342
8.3 Survey of Quasicrystalline Models for the
Icosahedral Phase 344
8.3.1 Perfect Quasicrystalline THings 348
8.3.2 Entropically Stabilized Structures 349
8.3.3 Quasicrystalline Tilings as Higher-Dimensional
Periodic Structures 349
8.4 Icosahedral Glass Structures 352
8.5 The Decagonal Glass in Two-Dimensions 355
8.5.1 The Basic Algorithm for Numerical Simulations 355
