Preface ....................................................... x
Acknowledgements ............................................. xi
1 Introduction .................................................. 1
1.1 The Discovery of Liquid Crystals .......................... 1
1.2 Basic Descriptions of Liquid Crystals ..................... 2
1.3 The Development of the Continuum Theory of Liquid
Crystals .................................................. 7
1.4 Notation and Conventions .................................. 9
2 Static Theory of Nematics .................................... 13
2.1 Introduction ............................................. 13
2.2 The Frank-Oseen Elastic Energy ........................... 14
2.2.1 The Nematic Energy ................................. 20
2.2.2 The Cholesteric Energy ............................. 23
2.3 Electric and Magnetic Fields ............................. 26
2.3.1 Electric Fields and the Electric Energy ............ 26
2.3.2 Magnetic Fields and the Magnetic Energy ............ 28
2.3.3 Comments on Fields and Units ....................... 31
2.4 Equilibrium Equations .................................... 33
2.4.1 Preliminaries ...................................... 34
2.4.2 Derivation of the Equilibrium Equations ............ 35
2.5 General Equilibrium Solutions ............................ 42
2.6 Anchoring and Boundary Conditions ........................ 46
2.6.1 No Anchoring ....................................... 46
2.6.2 Strong Anchoring ................................... 47
2.6.3 Conical Anchoring .................................. 48
2.6.4 Weak Anchoring ..................................... 49
2.7 Reformulation of Equilibrium Equations ................... 50
2.7.1 Bulk Equilibrium Equations ......................... 51
2.7.2 Reformulation of Boundary Conditions ............... 53
3 Applications of Static Theory of Nematics .................... 57
3.1 Introduction ............................................. 57
3.2 Some Equilibrium Solutions ............................... 57
3.2.1 Elementary Equilibrium Solutions ................... 58
3.2.2 Tilt and Twist Equilibrium Solutions ............... 60
3.3 Magnetic Coherence Length ................................ 68
3.4 Freedericksz Transitions ................................. 72
3.4.1 The Classical Freedericksz Transitions in
Nematics ........................................... 72
3.4.2 Pretilt at the Boundaries .......................... 82
3.4.3 Tilted Fields ...................................... 85
3.5 Electric Field Effects ................................... 89
3.6 Weak Anchoring Effects ................................... 95
3.7 The Twisted Nematic Device .............................. 101
3.8 Defects ................................................. 109
3.8.1 Axial Line Disclinations .......................... 110
3.8.2 Perpendicular Disclinations ....................... 118
3.8.3 Boundary Line Disclinations ....................... 120
3.8.4 Point Defects at a Surface ........................ 127
4 Dynamic Theory of Nematics .................................. 133
4.1 Introduction ............................................ 133
4.2 The Ericksen-Leslie Dynamic Equations ................... 134
4.2.1 Kinematics and Material Frame-Indifference ........ 134
4.2.2 Balance Laws ...................................... 138
4.2.3 Constitutive Equations ............................ 142
4.2.4 The Dynamic Equations ............................. 147
4.2.5 Summary of the Ericksen-Leslie Dynamic
Equations ......................................... 150
4.3 Reformulation of the Dynamic Equations .................. 153
4.4 The Nematic Viscosities ................................. 154
5 Applications of Dynamic Theory of Nematics .................. 161
5.1 Introduction ........................................... 161
5.2 A Simple Flow Alignment ................................ 162
5.3 A Transverse Flow Effect ............................... 167
5.4 The Zwetkoff Experiment ................................ 170
5.5 Shear Flow ............................................. 176
5.5.1 Newtonian and Non-Newtonian Behaviour ............ 176
5.5.2 Governing Equations for Shear Flow ............... 177
5.5.3 Shear Flow Near a Boundary ....................... 180
5.5.4 Shear Flow between Parallel Plates ............... 182
5.5.5 Scaling Properties ............................... 185
5.6 Oscillatory Shear Flow ................................. 187
5.6.1 Oscillatory Shear Flow Solutions ................. 188
5.6.2 Stability and Instability ........................ 191
5.7 Couette Flow ........................................... 197
5.7.1 The Anisotropic Fluid Case ....................... 197
5.7.2 The Nematic Liquid Crystal Case .................. 202
5.8 Poiseuille Flow ........................................ 209
5.8.1 The Anisotropic Fluid Case ....................... 210
5.8.2 The Nematic Liquid Crystal Case .................. 213
5.8.3 Results from a Scaling Analysis .................. 217
5.9 Dynamics of the Freedericksz Transition ................ 218
5.9.1 Dynamics in the Twist Geometry ................... 218
5.9.2 Backflow and Kickback in the Splay Geometry ...... 223
5.9.3 Backflow in the Bend Geometry .................... 234
5.10 Light Scattering ....................................... 236
6 Theory of Smectic C Liquid Crystals ......................... 247
6.1 Introduction ............................................ 247
6.2 Static Theory of Smectic C .............................. 248
6.2.1 The Elastic Energy for Smectic C .................. 250
6.2.2 The Magnetic and Electric Energies ................ 258
6.2.3 Equilibrium Equations ............................. 259
6.2.4 Focal Conic Defects: Dupin and Parabolic
Cyclides .......................................... 267
6.2.5 A Freedericksz Transition in Bookshelf
Smectic C ......................................... 275
6.2.6 Smectic Layer Compression ......................... 281
6.3 Dynamic Theory of Smectic C ............................. 291
6.3.1 Dynamic Equations for SmC Liquid Crystals ......... 291
6.3.2 The Smectic C Viscosities ......................... 296
6.3.3 Simple Flow Alignment in Smectic C ................ 301
6.4 Theory of Smectic C* Liquid Crystals .................... 306
6.4.1 Energies for Smectic C* ........................... 309
6.4.2 Static and Dynamic Theory for Smectic C* .......... 312
6.4.3 Director Reorientation in Smectic C* .............. 313
6.5 Comments on Theories of Smectics ........................ 319
A Results Employing Variational Methods ....................... 321
B Identities .................................................. 323
C Physical Components in Cylindrical Polar Coordinates ........ 325
D Tables ...................................................... 329
Bibliography ................................................ 331
Index ....................................................... 351
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