GEOFFREY R. LUCKHURST
School of Chemistry, University of Southampton, UK

TIMOTHY J. SLUCKIN
School of Mathematics, University of Southampton, UK

About the Editors xiii

List of Contributors xv

Preface xvii

1 Introduction 1
Geoffrey R. Luckhurst and Timothy J. Sluckin

1.1 Historical Background 1

1.2 Freiser Theory 3

1.3 Nematic Order Parameters 4

1.4 Nematic Tensor Order Parameters 5

1.5 Theoretical Phase Diagrams 6

1.6 Landau–de Gennes Theory 9

1.7 Computer Simulation 10

1.8 Other Theoretical Issues 11

1.9 Applications 12

1.10 Characterisation 12

1.11 Lyotropic and Colloidal Systems 14

1.12 Molecular Design 15

References 19

2 Biaxial Nematics: Order Parameters and Distribution Functions 25
Geoffrey R. Luckhurst

2.1 Introduction 25

2.2 The Cartesian Language 26

2.2.1 Order Parameters 26

2.2.2 Molecular Symmetry 28

2.2.3 Measurement 30

2.3 The Spherical Tensor Language 31

2.3.1 The Order Parameters of Biaxial Molecules in a Uniaxial Phase 31

2.3.2 Molecular Symmetry 33

2.3.3 Measurement 33

2.4 Extension to Biaxial Nematics 35

2.4.1 Orientational Order Parameters 35

2.4.2 Systems with D2h Point Group Symmetry 36

2.4.3 Measurement of the Order Parameters 37

2.4.4 Systems with C2h Point Group Symmetry and Their Order Parameters 38

2.4.5 Systems with C2h Point Group Symmetry: The Cartesian Language 39

2.5 Fourth-Rank Order Parameters 42

2.6 The Singlet Orientational Distribution Function 44

2.7 Appendices 47

2.7.1 Point Group Symmetry and the Associated Symmetry Operations 47

2.7.2 Legendre Polynomials, Modified Spherical Harmonics and Wigner Rotation Matrices 48

Acknowledgements 51

References 51

3 Molecular Field Theory 55
Epifanio G. Virga

3.1 Introduction 55

3.2 General Mathematical Theory 57

3.2.1 Two-Particle Hamiltonian 57

3.2.2 Ensemble Potentials 62

3.2.3 Molecular Field Approximation 65

3.2.4 Variational Principles 69

3.2.5 Local Stability Criterion 71

3.3 Non-Polar Molecules 74

3.3.1 Quadrupolar Hamiltonians 74

3.3.2 Phase Transitions 80

3.3.3 Universal Phase Diagram 87

3.3.4 Steric Effects 91

3.4 Polar Molecules 99

3.4.1 Dipolar Fluids 100

3.4.2 Dipolar Hamiltonian 102

3.4.3 Condensed Polar Phases 107

References 112

4 Hard Particle Theories 117
Andrew J. Masters

4.1 Introduction 117

4.2 Theoretical Approaches 119

4.3 Board-Like Models 122

4.4 Bent-Core Models 124

4.5 Rod–Plate Mixtures 125

4.6 Conclusions and Speculations 128

Acknowledgements 129

References 129

5 Landau Theory of Nematic Phases 133
Lech Longa

5.1 Introduction 133

5.2 Symmetry of Biaxial Nematics and Primary Order Parameters 134

5.3 Landau Expansion 136

5.3.1 Generic NU –I Phase Transition 136

5.3.2 Generic NB –NU and NB –I Phase Transitions 138

5.3.3 Role of Coupling between Nematic Order Parameters 141

5.3.4 Landau–de Gennes Expansion in Terms of the Alignment Tensor 145

5.4 Conclusion 149

Acknowledgements 149

References 149

6 Computer Simulations of Biaxial Nematics 153
Roberto Berardi and Claudio Zannoni

6.1 Introduction 153

6.2 Order Parameters 156

6.3 Model Potentials and Applications 159

6.3.1 Lattice Models 159

6.3.2 Atomistic Models 162

6.3.3 Molecular Models 163

6.4 Conclusion 171

Acknowledgements 173

6.5 Appendices 173

6.5.1 Quaternions 173

6.5.2 Angular Momentum Operator 174

6.5.3 Kinematic and Dynamic Equations of Rotational Motion 175

6.5.4 Propagator/Integrator of Rotational Equations of Motion 176

6.5.5 Gradient of the Biaxial Gay–Berne Potential 176

6.5.6 Torques of the Biaxial Gay–Berne Potential 177

References 178

7 Continuum Theory of Biaxial Nematic Liquid Crystals 185
Iain W. Stewart

7.1 Introduction 185

7.2 Continuum Model and Energies 186

7.2.1 The Elastic Energy 187

7.2.2 The Magnetic and Electric Energies 187

7.2.3 The Total Energy 189

7.3 Dynamic Equations 189

7.3.1 Balance Laws 190

7.3.2 The Viscous Stress 192

7.3.3 The Dynamic Equations 194

7.3.4 Euler Angle Description 195

7.3.5 A Simple Shear Flow 196

7.4 Equilibrium Equations 198

7.4.1 The Equilibrium Equations 199

7.4.2 Alignment Induced by a Magnetic Field 200

7.5 Conclusion 202

References 202

8 The Alignment of Biaxial Nematics 205
Demetri J. Photinos

8.1 Introduction 205

8.2 Alignment by an External Electric or Magnetic Field 206

8.3 Surface Alignment 208

8.3.1 Macroscopic Description 208

8.3.2 Molecular Scale Description 210

8.4 Flow Alignment 210

8.5 Lower Symmetry Biaxial Nematics and Hierarchical Domain Structures 211

Acknowledgements 212

References 212

9 Applications 215
Paul D. Brimicombe

9.1 Introduction 215

9.1.1 Materials Considerations 215

9.1.2 Surface Alignment 216

9.2 Thin-Film Electro-Optic Devices 217

9.2.1 Minor-Director In-Plane Switching Devices 218

9.2.2 Electric Field-Induced Biaxiality Effects 220

9.2.3 Planar Biaxial Nematic Devices 221

9.2.4 Twist Effects in Biaxial Nematics and Biaxial Pi-Cells 222

9.2.5 Bistable Biaxial Nematic Devices 223

9.2.6 Spontaneous Chirality Effects 224

9.3 Non-Device Applications of Biaxial Nematic Liquid Crystals 225

9.3.1 Optical Compensation Films 225

9.4 Conclusion 225

References 226

10 Characterisation 229

10.1 Textures of Nematic Liquid Crystals 230
Ingo Dierking

10.1.1 Polarising Microscopy 230

10.1.2 Simple Liquid Crystal Optics 230

10.1.3 Optical Biaxiality 232

10.1.4 Textures 234

References 240

10.2 Refractive Index Studies 242
Antonio J. Palangana

10.2.1 Introduction 242

10.2.2 Optical Indicatrix 242

10.2.3 Optical Conoscopy 244

10.2.4 Results 246

10.2.5 Acknowledgements 250

References 250

10.3 Orientational Order Parameters of Nematic Liquid Crystals Determined by Infrared and Raman Spectroscopy 251
Jagdish K. Vij and Antoni Kocot

10.3.1 Introduction 252

10.3.2 Polarised IR Spectroscopy 252

10.3.3 Scalar Order Parameters of a Second-Rank Tensor 252

10.3.4 IR Absorbance Components 254

10.3.5 Experimental Method 256

10.3.6 Results for the Order Parameters for the Tetrapodes 256

10.3.7 Discussion of the Order Parameters 258

10.3.8 Raman Spectroscopy 259

10.3.9 Comparisons of IR and Raman Spectroscopy for Determining Order Parameters 263

References 264

10.4 NMR Spectroscopy 265
Louis A. Madsen

10.4.1 Introduction: NMR Basics, Advantages and Limitations 265

10.4.2 Probing Orientational Order 266

10.4.3 Creating a Director Distribution to Observe Biaxiality 267

10.4.4 Spectral Analysis Considerations: Fitting and Rotational Modulations 268

10.4.5 Incorporating Deuterium: Direct Mesogen Labelling Versus Probe Solutes 270

10.4.6 Powder Spectra and Monodomain Spectra: Examples 271

10.4.7 Alternative and Emerging Methods 272

References 274

10.5 Structural Studies of Biaxial Nematics: X-Ray and Neutron Scattering 276
Patrick Davidson

10.5.1 Introduction 276

10.5.2 Theoretical Considerations 276

10.5.3 Experimental Details 279

10.5.4 Specificities of the Scattering by Different Kinds of Biaxial Nematics 280

References 283

11 Lyotropic Systems 285
Antonio M. Figueiredo Neto and Yves Galerne

11.1 Introduction 285

11.2 Phase Diagrams 286

11.3 The Potassium Laurate–Decanol–Water Mixture: A Working Example 287

11.4 The Intrinsically Biaxial Micelles Model 294

11.5 Theoretical Reconstruction of the Lyotropic Nematic Phase Diagram: a Landau-Like Approach 298

11.6 Conclusions 302

Acknowledgements 302

References 302

12 Colloidal Systems 305
Gert Jan Vroege

12.1 Introduction 305

12.2 Onsager Theory and Extensions 306

12.3 Special Features of Colloids and Colloidal Liquid Crystals 307

12.4 Biaxiality in Mixtures of Rods and Plates 308

12.5 Particles with Inherent Biaxial Shape 311

12.6 Concluding remarks 315

References 316

13 Thermotropic Systems: Biaxial Nematic Polymers 319
Anke Hoffmann, Felicitas Brömmel, and Heino Finkelmann

13.1 Introduction 319

13.2 Main-Chain Liquid Crystal Polymers 321

13.3 Side-Chain Liquid Crystal Polymers 321

13.4 Comparison of Attachment Geometries – Influence of Molecular Dynamics and Molecular Shape 327

13.5 Conclusion 330

References 330

14 Low Molar Mass Thermotropic Systems 333
Matthias Lehmann

14.1 Preamble 333

14.2 Introduction and General Considerations 333

14.3 Single Component 336

14.3.1 Biaxial Board-Shaped Mesogens 336

14.3.2 V-Shaped Nematogens 338

14.3.3 Multipodes 350

14.4 Mixtures 354

14.5 Concluding Remarks 360

References 360

15 Final Remarks 369
Geoffrey R. Luckhurst and Timothy J. Sluckin

References 373

Index 375

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