Structure and Properties of Liquid Crystals (eBook)

Epigraph 8
Foreword 10
Contents 14
Chapter 1: Introductory Notes 20
References 23
Part I: Structure of Liquid Crystals 24
Chapter 2: Symmetry 25
2.1 Point Group Symmetry 25
2.1.1 Symmetry Elements and Operations 25
2.1.2 Groups 28
2.1.3 Point Groups 30
2.1.4 Continuous Point Groups 32
2.2 Translational Symmetry 33
References 36
Chapter 3: Mesogenic Molecules and Orientational Order 37
3.1 Molecular Shape and Properties 37
3.1.1 Shape, Conformational Mobility and Isomerization 37
3.1.2 Symmetry and Chirality 39
3.1.3 Electric and Magnetic Properties 40
3.2 Intermolecular Interactions 42
3.3 Orientational Distribution Functions for Molecules 46
3.3.1 Molecules with Axial Symmetry 47
3.3.2 Lath-Like Molecules 50
3.4 Principal Orientational Order Parameter (Microscopic Approach) 51
3.5 Macroscopic Definition of the Orientational Order Parameter 53
3.5.1 Tensor Properties 53
3.5.2 Uniaxial Order 54
3.5.3 Macroscopic Biaxiality 56
3.6 Apparent Order Parameters for Flexible Chains 57
References 57
Chapter 4: Liquid Crystal Phases 59
4.1 Polymorphism Studies 59
4.1.1 Polarized Light Microscopy 59
4.1.2 Differential Scanning and Adiabatic Calorimetry (DSC and AC) 60
4.1.3 X-Ray Analysis 61
4.2 Main Calamitic Phases 62
4.2.1 Nematic Phase 62
4.2.2 Classical Smectic A Phase 63
4.2.3 Special SmA Phases 64
4.2.4 Smectic C Phase 66
4.2.5 Smectic B 67
4.3 Discotic, Bowl-Type and Polyphilic Phases 68
4.4 Role of Polymerization 69
4.5 Lyotropic Phases 71
4.6 General Remarks on the Role of Chirality 73
4.7 Cholesterics 75
4.7.1 Intermolecular Potential 75
4.7.2 Cholesteric Helix and Tensor of Orientational Order 76
4.7.3 Tensor of Dielectric Anisotropy 77
4.7.4 Grandjean Texture 79
4.7.5 Methods of the Pitch Measurements 80
4.8 Blue Phases 81
4.9 Smectic C* Phase 83
4.9.1 Symmetry, Polarization and Ferroelectricity 83
4.9.2 Helical Structure 84
4.10 Chiral Smectic A* 86
4.10.1 Uniform Smectic A* 86
4.10.2 TGB Phase 86
4.11 Spontaneous Break of Mirror Symmetry 87
References 90
Chapter 5: Structure Analysis and X-Ray Diffraction 92
5.1 Diffraction Studies and X-Ray Experiment 92
5.1.1 General Consideration 92
5.1.2 X-Ray Experiment 93
5.2 X-Ray Scattering 94
5.2.1 Scattering by a Single Electron 94
5.2.2 Scattering by Two Material Points 96
5.2.3 Scattering by a Stack of Planes (Bragg Diffraction) 97
5.2.4 Amplitude of Scattering for a System of Material Points 98
5.2.5 Scattering Amplitude for an Atom 100
5.3 Diffraction on a Periodic Structure 101
5.3.1 Reciprocal Lattice 101
5.3.2 Intensity of Scattering 103
5.3.3 Form Factor and Structure Factor 104
5.4 Fourier Transforms and Diffraction 105
5.4.1 Principle 105
5.4.2 Example: Form Factor of a Parallelepiped 106
5.4.3 Convolution of Two Functions 108
5.4.4 Self-Convolution 110
5.5 X-Ray Diffraction by Crystals 111
5.5.1 Density Function and Structure Factor for Crystals 112
5.5.1.1 Density Function 112
5.5.1.2 The Structure Factor 112
5.5.2 A Crystal of a Finite Size 113
5.6 Structure of the Isotropic and Nematic Phase 114
5.6.1 Isotropic Liquid 114
5.6.2 Nematic Phase 116
5.7 Diffraction by Smectic Phases 118
5.7.1 Smectic A 118
5.7.2 Landau-Peierls Instability 119
5.7.2.1 Displacement and Free Energy 119
5.7.2.2 Stability of Crystallographic Lattices of Different Dimensionality 121
5.7.3 ``Bond´´ Orientational Order in a Single Smectic Layer and Hexatic Phase 122
5.7.4 Three-Dimensional Smectic Phases 123
5.7.4.1 Uniaxial Orthogonal 123
5.7.4.2 Biaxial Orthogonal 125
5.7.4.3 Biaxial Tilted 125
References 126
Chapter 6: Phase Transitions 128
6.1 Landau Approach 128
6.2 Isotropic Liquid-Nematic Transition 132
6.2.1 Landau-De Gennes Equation 132
6.2.2 Temperature Dependence of the Nematic Order Parameter 133
6.2.3 Free Energy 135
6.2.4 Physical Properties in the Vicinity of the N-Iso Transition 136
6.3 Nematic-Smectic A Transition 138
6.3.1 Order Parameter 138
6.3.2 Free Energy Expansion 139
6.3.3 Weak First Order Transition 141
6.3.3.1 Role of Higher Order Fourier Components 142
6.3.3.2 Interaction of Two Order Parameters 143
6.3.4 Re-entrant Phases 144
6.4 Smectic A-Smectic C Transition 145
6.4.1 Landau Expansion 145
6.4.2 Influence of External Fields 146
6.5 Dynamics of Order Parameter 147
6.5.1 Landau-Khalatnikov Approach 147
6.5.2 Relaxation Rate 147
6.6 Molecular Statistic Approach to Phase Transitions 150
6.6.1 Entropy, Partition Function and Free Energy 150
6.6.1.1 Entropy 150
6.6.1.2 Partition Function and Free Energy 152
6.6.2 Equations of State for Gas and Liquid 153
6.6.2.1 Ideal Gas of Spherical Particles 153
6.6.2.2 Equation of State for a Dense Gas or a Liquid 155
6.7 Nematic-Isotropic Transition (Molecular Approach) 157
6.7.1 Interaction Potential and Partition Function 157
6.7.2 Onsager´s Results 158
6.7.3 Mean Field Approach for the Nematic Phase 160
6.7.3.1 Interaction Potential and Partition Function 160
6.7.3.2 Maier-Saupe Theory 163
References 164
Part II: Physical Properties 166
Chapter 7: Magnetic, Electric and Transport Properties 167
7.1 Magnetic Phenomena 167
7.1.1 Magnetic Anisotropy 167
7.1.2 Diamagnetism 168
7.1.2.1 Single Electron 168
7.1.2.2 Molecules 169
7.1.3 Paramagnetism and Ferromagnetism 171
7.1.3.1 Paramagnetism 171
7.1.3.2 Ferromagnetism 172
7.2 Dielectric Properties 173
7.2.1 Permittivity of Isotropic Liquids 173
7.2.1.1 Dielectric Spectrum 173
7.2.1.2 Local Field, Clausius-Mossotti and Onsager Equations 173
7.2.2 Static Dielectric Anisotropy of Nematics and Smectics 177
7.2.2.1 Maier-Meier Theory 177
7.2.2.2 SmA Phase and the Role of the Positional Order 180
7.2.2.3 Smectic C Case 181
7.2.3 Dipole Dynamics of an Isotropic Liquid 181
7.2.3.1 Dipole Relaxation 181
7.2.3.2 Debye and Cole-Cole Diagrams 184
7.2.4 Frequency Dispersion of epsi|| and epsi in Nematics 186
7.2.4.1 Relaxation Modes 186
7.2.4.2 Dual Frequency Addressing 188
7.3 Transport Properties 188
7.3.1 Thermal Conductivity 188
7.3.2 Diffusion 190
7.3.3 Electric Conductivity 192
7.3.3.1 Mobility of Ions 192
7.3.3.2 Ion Concentration 193
7.3.3.3 Current-Voltage Curve for Thin Cells 194
7.3.3.4 Frequency Dependence of Ionic Conductivity 197
7.3.3.5 Conductivity due to Dielectric Losses 198
7.3.3.6 Space Charge Relaxation 200
7.3.3.7 Measurements of Anisotropy epsia (omega) and sigmaa(omega) 202
7.3.3.8 Characteristic Times Related to the Discussed Phenomena (Resume) 202
References 203
Chapter 8: Elasticity and Defects 204
8.1 Tensor of Elasticity 204
8.1.1 Hooke´s Law 204
8.1.2 Stress, Strain and Elasticity Tensors 205
8.1.2.1 Stress Tensor 205
8.1.2.2 Strain Tensor 206
8.1.2.3 Tensor of Elasticity 208
8.2 Elasticity of Nematics and Cholesterics 209
8.2.1 Elementary Distortions 209
8.2.1.1 Specific Features of Elasticity of Nematics 209
8.2.1.2 Elementary Distortions 210
8.2.1.3 Curvature Distortion Tensor 211
8.2.2 Frank Energy 212
8.2.2.1 Elasticity Tensors 212
8.2.2.2 Elastic Energy of the Conventional Nematic for n||z 213
8.2.2.3 Frank Formula 214
8.2.3 Cholesterics and Polar Nematics 215
8.2.3.1 Cholesterics 215
8.2.3.2 Polar Nematics 216
8.3 Variational Problem and Elastic Torques 216
8.3.1 Euler Equation 216
8.3.2 Application to a Twist Cell 218
8.3.3 ``Molecular Field´´ and Torques 220
8.3.4 Director Fluctuations 221
8.4 Defects in Nematics and Cholesterics 224
8.4.1 Nematic Texture and Volterra Process 224
8.4.1.1 Textures 224
8.4.1.2 Volterra Process 225
8.4.2 Linear Singularities in Nematics 226
8.4.2.1 Disclination Strength 226
8.4.2.2 The Director Field Around Disclination 227
8.4.2.3 Energy of a Disclination 229
8.4.3 Point Singularities and Walls 230
8.4.3.1 Point Singularities in the Bulk (Hedgehogs) 230
8.4.3.2 Point Singularities at the Surfaces (Boodjooms) 231
8.4.3.3 Walls 232
8.4.4 Defects in Cholesterics 233
8.4.4.1 Singular tau- and lambda-Lines in the Planar Texture 233
8.4.4.2 Defects in the Polygonal or Fingerprint Textures 234
8.5 Smectic Phases 235
8.5.1 Elasticity of Smectic A 235
8.5.1.1 Free Energy 235
8.5.1.2 Wave-Like Distortion 237
8.5.2 Peierls Instability of the SmA Structure 239
8.5.3 Defects in Smectic A 241
8.5.3.1 Steps and Dislocations 241
8.5.3.2 Cylinders, Tores, Hedgehogs 242
8.5.3.3 Focal-Conics 242
8.5.4 Smectic C Elasticity and Defects 243
8.5.4.1 Elastic Energy 243
8.5.4.2 Defects in Smectics C 244
References 246
Chapter 9: Elements of Hydrodynamics 247
9.1 Hydrodynamic Variables 247
9.2 Hydrodynamics of an Isotropic Liquid 248
9.2.1 Conservation of Mass Density 248
9.2.2 Conservation of Momentum Density 249
9.2.2.1 Ideal Liquid 249
9.2.2.2 Viscous Liquid 251
9.2.3 Navier-Stokes Equation 252
9.3 Viscosity of Nematics 253
9.3.1 Basic Equations 253
9.3.2 Measurements of Leslie coefficients 256
9.3.2.1 Laminar Shear Flow 256
Geometry (a), nz = 1 (Director Perpendicular to the Shear Plane) 257
Geometry (b), nx = 1 (Director in the Shear Plane Parallel to the Velocity of Upper Plate) 258
Geometry (c), ny = 1 (Director in the Shear Plane Perpendicular to the Upper Plate Velocity) 258
9.3.2.2 Poiseuille Flow in Magnetic Field 259
9.3.2.3 Capillary Flow and Determination of a2 and a3 260
9.3.2.4 Determination of gamma1 262
9.4 Flow in Cholesterics and Smectics 264
9.4.1 Cholesterics 264
9.4.1.1 Shear 264
Geometry I, h||x, s_yx = v_y/x||h and vh 264
Geometry II, h||z, s_yx = v_y/ x h and v h 265
Geometry III, h||y, s_yx = v_y/ x||h, and v||h 265
9.4.1.2 Permeation Effect 265
9.4.2 Smectic A Phase 267
9.4.2.1 Flow and Viscosity 267
9.4.2.2 Undulation Instability 269
References 269
Chapter 10: Liquid Crystal - Solid Interface 270
10.1 General Properties 270
10.1.1 Symmetry 270
10.1.2 Surface Properties of a Liquid 271
10.1.2.1 Surface Tension 271
10.1.2.2 Adsorption 272
10.1.2.3 Wetting 273
10.1.3 Structure of Surface Layers 273
10.1.3.1 Surface Induced Change in the Orientational Order Parameter 274
10.1.3.2 Surface-Induced Smectic Ordering 277
10.1.3.3 Polar Surface Order and Surface Polarization 279
Ionic Polarization 280
Dipolar Polarization 280
Ordoelectric (Quadrupolar) Polarization 281
10.2 Surface Energy and Anchoring of Nematics 284
10.2.1 Easy Axis 284
10.2.2 Variational Problem 285
10.2.3 Surface Energy Forms 287
10.2.4 Extrapolation Length 288
10.3 Liquid Crystal Alignment 290
10.3.1 Cells 290
10.3.2 Alignment 291
10.3.2.1 Planar Homogeneous and Tilted Alignment 291
10.3.2.2 Homeotropic Alignment 292
10.3.2.3 Multistable Alignment 292
10.3.3 Berreman Model 293
References 295
Part III: Electro-Optics 296
Chapter 11: Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals 297
11.1 Optical Properties of Uniaxial Phases 297
11.1.1 Dielectric Ellipsoid, Birefringence and Light Transmission 297
11.1.1.1 Dielectric Ellipsoid 297
11.1.1.2 Extraordinary Index of a Birefringent Layer 298
11.1.1.3 Light Ellipticity 300
Case 1 Corresponds to the So-Called lambda/4 Plate 301
Case 2 Corresponds to the lambda Plate 302
11.1.1.4 Light Transmission (Cell Between Polarizers) 302
11.1.1.5 Measurements of Birefringence of Nematics 303
11.1.1.6 Twist Structure 305
11.1.2 Light Absorption and Linear Dichroism 306
11.1.2.1 Extinction Index, Absorption Coefficient, Optical Density 306
11.1.2.2 Linear Dichroism 308
11.1.2.3 Kramers - Kronig Relations 309
11.1.3 Light Scattering in Nematics and Smectic A 311
11.1.3.1 Isotropic Phase 312
11.1.3.2 Nematic Phase 313
Case A: f || no||z and f deltan 314
Case B: f n0 314
11.1.3.3 Smectic A Phase 315
11.2 Frederiks Transition and Related Phenomena 316
11.2.1 Field Free Energy and Torques 316
11.2.2 Experiments on Field Alignment of a Nematic 318
11.2.3 Theory of Frederiks Transition 319
11.2.3.1 Simplest Model 320
11.2.3.2 Threshold Condition 321
11.2.4 Generalizations of the Simplest Model 324
11.2.4.1 Electric Field Case 324
11.2.4.2 Anisotropy of Elastic Properties 325
11.2.4.3 Oblique Field or Tilted Alignment 325
11.2.4.4 Weak Anchoring 325
11.2.4.5 Break of Anchoring 326
11.2.5 Dynamics of Frederiks Transition 327
11.2.6 Backflow Effect 328
11.2.7 Electrooptical Response 330
11.2.7.1 Splay-Bend Distortions 330
11.2.7.2 Twist and Supertwist Effects 333
11.2.7.3 Guest-Host Effect 333
11.3 Flexoelectricity 334
11.3.1 Flexoelectric Polarization 334
11.3.1.1 Dipolar and Quadrupolar Flexoelectricity 334
11.3.1.2 A Hybrid Cell 337
11.3.1.3 Measurements of Pf 338
11.3.2 Converse Flexoelectric Effect 339
11.3.2.1 Uniform Distortion 339
11.3.2.2 Electrooptical Properties 342
11.3.2.3 Dynamics of the Flexoelectric Effect 343
11.3.3 Flexoelectric Domains 344
11.4 Electrohydrodynamic Instability 346
11.4.1 The Reasons for Instabilities 346
11.4.2 Carr-Helfrich Mode 348
11.4.2.1 The Instability Threshold in the Simplest Model 349
11.4.2.2 Behaviour Above the Threshold 351
Reference 352
Chapter 12: Electro-Optical Effects in Cholesteric Phase 354
12.1 Cholesteric as One-Dimensional Photonic Crystal 354
12.1.1 Bragg Reflection 354
12.1.1.1 Experimental Data 354
12.1.1.2 The Simplest Model 356
12.1.2 Waves in Layered Medium and Photonic Crystals 358
12.1.2.1 Hill and Mathieu Equations 358
12.1.2.2 One Dimensional Photonic Band-Gap Structure (Modelling) 360
12.1.3 Simple Analytical Solution for Light Incident Parallel to the Helical Axis 362
12.1.3.1 Wave Equations 362
12.1.3.2 Dispersion Relation 363
12.1.3.3 Rotation of Linearly Polarised Light 366
12.1.3.4 Waveguide Regime 367
12.1.4 Other Important Cases 367
12.1.4.1 Cholesteric Slab of Finite Thickness 367
12.1.4.2 Oblique Incidence of Light 367
12.1.4.3 Diffraction and Scattering 368
12.2 Dielectric Instability of Cholesterics 369
12.2.1 Untwisting of the Cholesteric Helix 369
12.2.1.1 De Gennes-Meyer Model for Field Induced Cholesteric-Nematic Transition 369
12.2.1.2 Topological Limitation 372
12.2.2 Field Induced Anharmonicity and Dynamics of the Helix 375
12.2.3 Instability of the Planar Cholesteric Texture 377
12.3 Bistability and Memory 381
12.3.1 Naive Idea 381
12.3.2 Berreman-Heffner Model 382
12.3.2.1 A Cell and Free Energy 382
12.3.2.2 Backflow and Director Relaxation 384
12.3.2.3 Topological Problem and Trap States 384
12.3.3 Bistability and Field-Induced Break of Anchoring 386
12.4 Flexoelectricity in Cholesterics 387
References 389
Chapter 13: Ferroelectricity and Antiferroelectricity in Smectics 391
13.1 Ferroelectrics 391
13.1.1 Crystalline Pyro-, Piezo- and Ferroelectrics 391
13.1.1.1 Polarization Catastrophe in Liquids and Solids 391
13.1.1.2 Pyro-, Piezo- and Ferroelectrics 392
13.1.1.3 Simplest Description of a Proper Ferroelectric 394
13.1.2 Ferroelectric Cells with Non-ferroelectric Liquid Crystal 396
13.1.2.1 Meyer´s Discovery 396
13.1.2.2 Goldstone Mode and Helicity of the Structure 398
13.1.2.3 Smectic C* Phase and Criteria for Ferroelectricity 399
13.1.2.4 Surface Stabilised Ferroelectric Cells 400
13.1.3 Phase Transition SmA*-SmC* 402
13.1.3.1 Simplification 402
13.1.3.2 Soft Mode for Smectic A*-Smectic C* Transition 403
13.1.3.3 Goldstone and Soft Modes in Sm C* Phase 406
13.1.3.4 Measurements of Landau Expansion Coefficients 407
13.1.4 Electro-Optic Effects in Ferroelectric Cells 408
13.1.4.1 Electroclinic Effect in SmA 408
13.1.4.2 Helix Distortion and Deformed Helix Ferroelectric effect 410
13.1.4.3 Frederiks Transition and Clark-Lagerwall Bistability 413
13.1.5 Criteria for Bistability and Hysteresis-Free Switching 417
13.1.5.1 Cells with No Insulating Layers 417
13.1.5.2 Role of Aligning Layers in Bistability 418
13.1.5.3 V-Shape Effect 420
13.2 Introduction to Antiferroelectrics 420
13.2.1 Background: Crystalline Antiferroelectrics and Ferrielectrics 420
13.2.2 Chiral Liquid Crystalline Antiferroelectrics 423
13.2.2.1 Discovery and Polymorphism 423
13.2.2.2 Molecular Interactions 424
13.2.2.3 Models 426
13.2.2.4 Electric Field Switching 430
13.2.3 Polar Achiral Systems 433
13.2.3.1 The Problem 433
13.2.3.2 Achiral Ferroelectrics 434
13.2.3.3 Achiral Antiferroelectrics 435
13.2.3.4 Ferro- and Antiferroelectric Compounds Based on the Bent-Shape Molecules 437
References 439
Index 442