Solitons in Liquid Crystals

350 Pages·1992·25.113 MB·English

by  L. Lam, J. Prost (auth.), Lui Lam, Jacques Prost (eds.)

Partially Ordered Systems
EditorialBoard:
1. Charvolin • W. Helfrich • L. Lam
AdvisoryBoard:
J.D. Litster • D.R. Nelson • M. Schadt
Partially Ordered Systems
EditorialBoard: 1. Charvolin . W. Helfrich' L. Lam
Solitolls ill Liquid Crystals
Lui Lam and Jacques Prost, Editors
BOlld-Orientatiollal Order in Condensed Matter Systems
Katherine J. Strandburg, Editor
Diffraction Optics ofComplex-SrrucltlredMedia
V. A. BelyakoY
Lui Lam Jacques Prost
Editors
Solitons in
Liquid Crystals
With 184 Illlustrations
Springer Science+Business Media, LLC
Lui Lam Jacques Prost
Department of Physics Ecole Superieure de Physique
San Jose State University el de Chimie lndustrielles
One Washington Square 10 Rue Vauquelin
San Jose, CA 95192 75231 Paris Cedex 05
USA France
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 Universităt 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
Solitons in liquid crystals I Lui Lam, Jacques Prost, editors.
p. cm. - (Partially ordered systems)
Includes bibliographical references (p. )
ISBN 978-1-4612-6946-5 ISBN 978-1-4612-0917-1 (eBook)
DOI 10.10071978-1-4612-0917-1
1. Liquid crystals. 1. Lam, Lui. II. Prost, Jacques.
IJI. Series.
Q0923.S64 1991
530.4'29-dc20 90-10244
Printed on acid-free paper.
© 19 92Springer Science+Business Media New York
Originally published by Springer-Verlag New York in 1992
Softcover reprint of the hardcover Ist edition 19 92
Ali rights reserved. This work may not be translated or copied in whole or in part without the
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Photocomposed from a laTex file.
987654321
ISBN 978-1-4612-6946-5
Preface
Solitons are localized traveling waves first discovered in shallow water by
John Scott Russell, an engineer, in 1834. They are ubiquitous and exist
in many kinds ofsystems from the sky to the laboratory, from physical to
biological systems. In general, their importance as nonlinear waves and as
nonlinear excitations in condensed matter was well recognized in the past
two decades.
Knowledge of liquid crystals also goes back more than a century. They
werediscovered in 1888,oneyearbeforethecompletionofthe EiffelTower,
by the botanist Frederich Reinitzer. Sincethe industrial application ofliq
uid crystals as displays was proposed in the late sixties, there has been a
resurrection of intense interest in these materials. Because of their com
bined properties of liquids and crystals, they occupy a unique position in
basic physics and materials science. The union ofthe "singular and beau
tiful" (in the words of John Scott Russell) phenomenon of solitons and
the "beautiful and mysterious" (according to Pierre Gilles de Gennes) liq
uid crystals occurred in 1968 with the publication ofa paper by Wolfgang
Helfrich. Thestorysincethis beautifulmarriageis thecontentofthisbook.
As editors we are very fortunate to be able to enlist the help of the
manyactivepioneersand experts in this field, whowrotedetailedaccounts
summarizing and explaining their own works and those of others. Many
chapters contain heretofore unpublished results. We are grateful to our
contributorsfor theknowledgeandskillfulpresentations,and to theeditors
at Springer-Verlag for their patience and professional assistance.
San Jose Lui Lam
Paris Jacques Prost
Contents
Preface v
Contributors xiii
1 Introduction 1
L. Lam and J. Prost
1.1 Liquid Crystal Phases 1
1.2 Solitons in Liquid Crystals 4
References 6
2 Solitons and Field Induced Solitons in Liquid Crystals 9
L. Lam
2.1 Introduction 9
2.2 Solitons 10
2.2.1 Origin and Definition 10
2.2.2 A BriefHistory 13
2.3 Soliton Equations 15
2.3.1 Korteweg-deVries Equation 15
2.3.2 Nonlinear Schrodinger Equation 15
2.3.3 Sine-Gordon Equation 16
2.3.4 Fisher Equation 19
2.3.5 The Damped (}4 Equation 20
2.3.6 Other Equations 22
2.4 Constructing Soliton Equations 23
2.5 Methods ofSolving Soliton Equations 24
2.5.1 Inverse Scattering Method 24
2.5.2 Backlund Transformation 25
2.5.3 Hirota Method 26
2.5.4 Perturbation 27
2.5.5 Numerical Method 28
2.5.6 Experimental Simulation 28
2.6 Formation ofSolitons 28
2.7 Magnetic Field Induced Solitons in Nematics 31
2.7.1 Early Works 31
2.7.2 Nematics Under Tilted Magnetic Fields 34
2.8 Electric Field Induced Solitons in Liquid Crystals 44
viii Contents
2.9 Conclusions 45
References 46
3 Solitons in Shearing Liquid Crystals 51
L. Lam and G.Q. Shu
3.1 Introduction 51
3.2 Steady Uniform ShearI: One-Dimensional Case 52
3.2.1 Equations ofMotion 52
3.2.2 Classification ofSolitons 53
3.2.3 Single Solitons 56
3.2.4 Multisolitons 58
3.2.5 Energy Dissipation 59
3.2.6 Transmitted Light Intensity 61
3.2.7 Flexoelectric Solitons 64
3.2.8 Spatiotemporal Distribution ofDirectors 67
3.2.9 Experimental Situation 69
3.3 Steady Uniform Shear II: Boundary Effects 72
3.3.1 Director Equation ofMotion 72
3.3.2 Steady States 73
3.3.3 Numerical Soliton Solutions 75
3.3.4 Analytic Soliton Solutions 78
3.3.5 Relaxation Processes 84
3.4 Unsteady Uniform Shear 84
3.4.1 Multiple Scale Analysis 85
3.4.2 Perturbed Solitons 85
3.5 Steady Nonuniform Shear I: Linear Cell 86
3.5.1 Theory ofPressure Gradient Induced Solitons 86
3.5.2 Experimental Results in Linear Cells 87
3.5.3 Perturbed Solitons Under Spatially Varying Shear 100
3.6 Steady Nonuniform Shear II: Radial Cell 101
3.6.1 Torsional Shear Flow 103
3.6.2 Radial Poiseuille Flow 103
3.6.3 Experiments in Radial Cells 105
3.7 Conclusions 106
References 107
4 Some Nonlinear Problems in Anisotropic Systems 110
P.E. Gladis and W. van Saarloos
4.1 Introduction 110
4.2 Nonlinear Aspects ofStatic Properties of
Liquid Crystals 112
4.2.1 Nonlinearities Associated with the
Freedericksz Transition 116
4.2.2 Escape into the Third Dimension 117
4.3 Nonlinear Macroscopic Dynamics ofLiquid Crystals 120
Contents ix
4.3.1 Dynamics ofLine Defects in Nematic
Liquid Crystals 122
4.3.2 Moving Wall Fronts in Helielectric
Liquid Crystals 127
4.3.3 An Exact Solitary Wave Solution for Ea < 0 133
4.3.4 Biological Significance ofChirality 134
4.3.5 Nonlinear Aspects ofLiquid Crystals
in Flow 135
4.4 Perspectives 144
4.5 Conclusions 145
References 146
5 Solitary Waves in Ferroelectric Liquid Crystals 151
J.E. Maclennan, N.A. Clark, and M.A. Handschy
5.1 Introduction 151
5.2 Equations ofMotion in One Dimension 153
5.2.1 Solving the Equation ofMotion 156
5.3 Wave Fronts in Infinite Systems 156
5.3.1 Computer Simulations 157
5.3.2 Marginal Stability 160
5.3.3 Analytic Solution for the Metastable-·
Stable Case 162
5.3.4 Summary 162
5.4 Director Reorientation in Finite Domains with
Fixed Boundaries 162
5.4.1 Thick Cells with a Helix 163
5.4.2 Thin Cells with Splay 163
5.4.3 Computer Simulations 164
5.5 Structures with Finite Interface Energies 179
5.5.1 Introduction 179
5.5.2 One-Dimensional Structures 182
5.5.3 Two-Dimensional Structures 182
5.6 Conclusions 188
References 189
6 Frustrated Smectics 191
P. Barois, J. Pommier, and J. Prost
6.1 Introduction 191
6.2 The Physics ofPolar Smectics 192
6.2.1 Intrinsic Incommensurability of
Coexisting Modulations 192
6.2.2 Phenomenological Theory ofFrustrated
Smectics 194
x Contents
6.2.3 Solitons in the Model ofFrustrated
Smectics 199
6.2.4 Connection with Experiments 205
6.3 Electric Properties ofthe Incommensurate Smectics 211
6.3.1 Intrinsic Ferroelectricity of
Incommensurate Smectics 211
6.3.2 Distortion ofthe Modulated Smectic
Structure by an Electric Field 217
6.3.3 Longitudinal Ferroelectricity 219
6.3.4 Conclusions 224
6.4 Escape from Incommensurability 225
6.4.1 Two-Dimensional Lockin ofthe
Wavevectors 225
6.4.2 Theoretical Phase Diagrams with 2D
Antiphases 227
6.4.3 Including Higher Order Harmonics in k 229
x
6.5 Conclusions 230
References 232
7 Soft Walls and Orientational Singularities in
Two-Dimensional Liquid Crystal Films 235
R. Pindak
7.1 Background 235
7.2 Experimental Techniques 239
7.3 Soft Tilt Director Walls in Ferroelectric
Smectic C' Films 242
7.4 Characteristic Orientational Singularities in Tilted
Hexatic Films 246
7.5 Concluding Remarks 249
References 250
8 Charged Twist Walls in Nematic Liquid Crystals 253
N.V. Madhusudana, J.P. Palierne, Ph. Mariinot-Lagarde,
and G. Durand
8.1 Introduction 253
8.2 Experiment 253
8.3 Model 257
8.4 Conclusions 263
References 263
9 Localized Instabilities in the Convection
ofNematic Liquid Crystals 264
R. Ribotta
9.1 Introduction 264
Contents xi
9.2 Localized Instabilities in the Evolution to
the Chaotic State 265
9.3 Theoretical Model: The Amplitude Equation 267
9.4 Convective Instabilities in Nematics Under A.C.
Electric Fields 269
9.5 Sequence ofHomogeneous Stationary States 271
9.6 Topology ofDislocations 274
9.7 Experimental Techniques 275
9.8 Nucleation ofDislocations in the Convective Rolls 276
9.9 Phase Propagation and Localization ofa
Convective Structure 284
9.10 Propagation ofSolitary Rolls 289
References 290
10 Solitons and Commensurate-Incommensurate Phase
Transitions in Ferroelectric Smectics 293
M. Yamashita
10.1 Introduction 293
10.2 Condensation ofSolitons 294
10.2.1 Soliton and Soliton Lattices ofthe
Sine-Gordon Equation 294
10.2.2 Cholesteric-Nematic Phase Transition 295
10.2.3 Flexoelectric Instability ofthe Nematic
Phase 297
10.3 The Chiral Smectic C-Smectic C Phase Transition 300
10.3.1 Chiral Smectic C Phase Under an
Electric Field 300
10.3.2 Interaction Between Solitons 302
10.3.3 Multisolitons and Soliton Lattices 303
10.304 Phase Transition ofInstability Type 309
10.3.5 Effect ofMagnetic Field 312
lOA Incommensurate and Rippled Phases Without Lifshitz
Invariant 314
1004.1 Rippled Phase 314
1004.2 Phase Transitions Between Smectic A
Phases 319
10.5 Summary 322
References 322
Author Index 326
Subject Index 334