Table Of ContentDefects in Liquid Crystals:
Computer Simulations, Theory and Experiments
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SeriesII:Mathematics,PhysicsandChemistry- Vol.43
Defects in Liquid Crystals:
Computer Simulations, Theory
and Experiments
edited by
Oleg D. Lavrentovich
Liquid Crystallnstitute and Chemical Physics Interdisciplinary Program,
Kent State University, Kent, Ohio, U.S.A.
Paolo Pasini
Istituto Nazionale di Fisica Nucleare,
Sezione di Bologna, Bologna, Italy
Claudio Zannoni
Dipartimento di Chimica Fisica ed Inorganica,
Universita di Bologna, Italy
and
Siobodan LUmer
Physics Department,
University of Ljubljana, Slovenia
~.
"
Springer-Science+Business Media, B.V.
Proceedings of the NATO Advanced Research Workshop on
Computer Simulations of Defects in Liquid Crystals Including their
Relation to Theory and Experiment
Erice, Sicily, ltaly
19-23 September 2000
A C.1. P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-0170-3 ISBN 978-94-010-0512-8 (eBook)
DOI 10.1007/978-94-010-0512-8
Printed on acid-free paper
AII Rights Reserved
© 2001 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2001
Softcover reprint of the hardcover 1s t edition 2001
No part of the material protected by this copyright notice may be reproduced or utilized
any form or by any means, electronic or mechanical, including photocopying, recognized
or by any information storage and retrieval system, without written permission fram
copyright owner.
CONTENTS
Preface xiii
1 Classification ofdefects in liquid crystals 1
H.-R. Trebin
1 Introduction........................... 1
2 Order and defects in a prototype model: the planar ferro-
magnet or the XY-model. . . . . . 2
2.1 Perfect order . . . . . . . . 2
2.2 The distorted ferromagnet . 3
2.3 Singularities......... 4
3 Classification ofdefects 4
3.1 Testloops and winding numbers . 4
3.2 Defect equivalence and stability. 5
3.3 Defect processes and group structure . 7
4 Furtherexamplesofreducedorderparameterspacesandfun-
damental groups . . . . . . . . . . . . . . . . . . . . 8
4.1 Two-dimensional nematics . . . . . . . . . . . 8
4.2 Two-dimensional smectics without rotations . 9
4.3 Two-dimensional periodic crystals .. . . 9
4.4 Two-dimensional smectics with rotations. 9
5 Singularities in three dimensions . . . . . . . . . 10
5.1 Line singularities . . . . . . . . . . . . . . 10
5.2 Thethree-dimensionalferromagnet (Heisenberg-model)
and point singularities . . . . . . . . . . . . . . 11
5.3 Defects ofthree-dimensional nematics . . . . . . .. 11
5.4 Biaxial nematics and cholesteric liquid crystals . .. 12
5.5 Homotopy classification ofdefects in three dimensions 13
6 Topological solitons. . . . . . . . . . . . . . . 14
6.1 Planar and linear topological solitons 14
6.2 Periodic boundary conditions 15
7 Summary and conclusions . . . . . . . . . . . 16
vi
2 Alignment tensor versus director description in nematic
liquid crystals 17
A.M. Sonnet and S. Hess
1 Introduction.................. 17
2 Description ofthe Alignment . . . . . . . . 18
2.1 Distribution Function and Averages 18
2.2 Alignment Tensor . 19
3 Dynamic equation for the alignment tensor 27
3.1 Director description 27
3.2 Tensor description . 28
4 Examples . 31
4.1 Visualization and s=1/2 disclination 31
4.2 Capillary 32
4.3 Droplet................ 33
3 Liquid crystal colloidal dispersions 37
H. Stark, A. Borstnik and S. turner
1 Introduction.................. 37
2 Colloidal Dispersions in a Nematic Solvent. 40
2.1 What is the Director Field? . . . . . 40
2.2 One-Particle Properties . . . . . . . 44
2.3 Two-Particle Interactions in a Nematic Solvent 53
2.4 Nematic Colloidal Dispersions in Complex Geometries 56
3 ColloidalDispersionsabove theNematic-IsotropicPhaseTran-
sition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62
3.1 Stability criteria for colloidal dispersions . . . . . .. 62
3.2 Noncharged liquidcrystalcolloidaldispersionsat tem
peratures above TNl . . . . . . . . . . . . . . . . .. 64
3.3 Charged spherical particles immersed in an isotropic
liquid crystal 74
4 Conclusions........................... 80
4 Computersimulationsanddefects in confined liquid crystal
lattice models 87
C. Chiccoli, P. Pasini, 1. Feruli and C. Zannoni
1 Introduction............. 87
2 Liquid Crystal Lattice Models. . . 88
2.1 The Lebwohl-Lasher model 89
2.2 Monte Carlo simulations .. 90
vii
3 Polymer Dispersed Liquid Crystals . . . . . 91
3.1 Radial Boundary Conditions (RBC) 92
4 Nematic Films . 104
4.1 Hybrid aligned cell . 105
4.2 Schlieren textures in planar aligned cells. 109
5 Conclusions . . . . . . . . . . . . . . . . . . . . . 110
5 Molecular simulations and theory of planar interfaces
and defects in nematic liquid crystals 113
M.P. Allen
1 Computer simulations and theoretical approaches . 113
1.1 Simulation Models . . . . . 113
1.2 Simulation Methods . 114
1.3 Coarse-grained descriptions . . 114
2 Liquid crystals between parallel walls . 116
2.1 Surface anchoring coefficient 116
2.2 Elastic boundary condition .. 117
3 Nematic-isotropic interface . 119
3.1 Orientational order and density profiles 119
3.2 Results at high elongation. 120
3.3 Surface Tension. . . . . . 122
3.4 Capillary waves . . . . . . 126
4 Disclination in cylindrical pores . 128
5 Defects near a colloid particle 132
5.1 Saturn ring defect 133
5.2 Satellite defect 134
6 Off-center ring .. 137
7 Acknowledgements .. 138
6 Topological defect behavior in a quenched nematic liquid
crystal 141
R.A. Pelcovits, J.L. Billeter, A.M. Smondyrev and G.B. Loriot
1 Introduction........ 142
2 Numerical simulations . . 146
3 Defect-finding algorithms 148
4 Results........... 152
4.1 Coarsening sequence 152
4.2 Real-space correlation function 154
4.3 Structure factor. 156
5 Conclusions............... 160
viii
6 Acknowledgements....................... 163
7 Restoring forces on nematic disclinations 167
R. Rosso and E.G. Vi1:ga
1 Introduction ..... 167
2 Restoring force and torque . 169
3 Core's influence . . . 174
3.1 Elliptic core . 176
4 Director's moulding 177
5 Boundary's influence 179
8 Challenges in the dynamics of point defects 185
A.M. Sonnet and E.G. Virga
1 Introduction. 185
2 Elastic force . . . 186
3 Defect drift . . . 189
4 Discretization ofplanar director fields 193
5 Artefacts in numerical defect dynamics . 195
9 Numerical simulation of elastic anisotropy in nematic
liquid crystalline polymers 201
H. Tu, G. Goldbeck-Wood and A.H. Windle
1 Introduction.......... 202
2 Model.................. 203
3 The calibration ofthe model ..... 206
3.1 Reproducing the Freedericksz transitions. 206
3.2 Topological defects in thin films. 208
3.3 Coarsening ofthe defects 209
4 Disclinations in bulk samples . . . . 211
4.1 Twist type or wedge type . . 212
4.2 Escaped integer disclinations 214
5 Texture evolution under shear flow . 215
5.1 Out-of-plane......... 219
5.2 Interaction ofwedge disclination pairs under shear flow220
6 Summary............................ 224
ix
10 Computer Simulations and Fluorescence Confocal Pol
arizing Microscopy of Structures in Cholesteric Liquid
Crystals 229
S. V. Shiyanovskii, I.I. Smalyukh and D.D. Lavrentovich
1 Introduction........................... 229
2 Mathematical basis for computer simulations ofequilibrium
LC structures . . . . . . . . . . . . . . . . . 231
2.1 LC structure and elastic properties. 231
2.2 Electric field 233
2.3 Surface anchoring 233
2.4 Domain and scheme ofsimulations . 235
3 Confocal Microscopy and Fluorescence Confocal Microscopy 235
3.1 Confocal Microscopy. . . . . . . . . . . . . . . 236
3.2 Fluorescence Confocal Microscopy. . . . . . . . . .. 236
3.3 Fluorescence Confocal Polarizing Microscopy. . . .. 238
3.4 Theoreticalfoundations ofconfocalmicroscopyofori-
entation patterns . . . . . . 241
4 Experimental set up and materials . . . . . . . . . . . . 243
4.1 Experimental set up . . . . . . . . . . . . . . . . 243
4.2 Liquid crystal materials and fluorescent probes. . 244
5 Results and Discussion. . . . . . . . . 246
5.1 FCPM of Fredericks transition .. . . . . . . . . 246
5.2 FCPM oftwisted nematic cell. . . . . . . . . . . 247
5.3 The FCPM imaging ofbasic cholesteric textures. 248
5.4 Field-induced cholesteric stripes in thin cells with
planar boundary conditions . . . . . . . . . . . . .. 252
5.5 Cholestericstripesinthincellswithhomeotropic bound-
ary conditions. 257
6 Conclusions.... 260
7 Acknowledgements.. 263
11 Defects and Undulation in Layered Liquid Crystals 271
T. Ishikawa and D. D. Lavrentovich
1 Introduction........................ 272
2 Bulk Elastic and Surface Properties ofLamellar Systems.. 273
2.1 Weakly distorted Smectic A phase in two dimensions 274
2.2 Lubensky-deGennescoarse-grainedmodelofthechole-
steric phase . . . . 275
2.3 Surface Anchoring 277
3 Edge Dislocations. . . 278
3.1 Linear Theory .. 279
x
3.2 ExperimentalDislocationProfileandNon-linearThe-
ory. 281
4 Layers undulations (Helfrich-Hurault instability) . .. 286
4.1 Helfrich-Hurault model 287
4.2 Undulations Profile near the Threshold: Experiment 289
4.3 Undulations in a Cell with a Finite Anchoring 292
4.4 Undulations profile well above the Threshold. 294
5 Conclusion........................ 298
12 Liquid crystals under shear: role ofdefects 301
M. Kleman and C. Meyer
1 Introduction.... 301
2 Disclinations.... 303
2.1 Generalities. 303
2.2 Instabilities and defects in the LE regime. 303
2.3 Instabilities and defects in the Doi regime (De» 0.1).304
2.4 Isolated disclination. . . . . . . . 305
3 Dislocations vs disclinations.. . . . . . . 306
4 Core structure and physical properties. . 309
5 Smectics . . . . . . . . . . . . . . . . . . 311
5.1 General relationships. . . . . . . 311
5.2 Movement ofisolated dislocations in a SmA phase. 314
5.3 Collective behavior ofdislocations: climb ofedges. 317
5.4 Collective behavior ofdislocations: glide ofscrews. 319
5.5 Aparticular case ofreorganization insmectic phases:
the onion texture. 320
13 Numerical simulation of defects in quasicrystals 323
H.-R. Trebin
1 Crystals and quasicrystals . . . . . . . . . . . . . . . . . .. 323
1.1 Characteristic features ofcrystals and quasicrystals. 323
1.2 Structure models for quasicrystals . . . . 325
1.3 Model quasicrystals 326
2 The cut formalism and phason-degree offreedom 327
2.1 The cut formalism . . . . . . . . . . . . 327
2.2 Phonon and phason degrees offreedom 328
3 Dislocations in crystals and quasicrystals . 329
3.1 Geometric basis. . . . . . . . . . . . . . 329
3.2 Experimental observations. . . . . . . . 330
3.3 Atomic interaction and numerical algorithm. 331
