The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction

by Iain W. Stewart

Overview

Given the widespread interest in macroscopic phenomena in liquid crystals, stemming from their applications in displays and devices. The need has arisen for a rigorous yet accessible text suitable for graduate students, whatever their scientific background. This book satisfies that need.
The approach taken in this text, is to introduce the basic continuum theory for nematic liquid crystals in equilibria, then it proceeds to simple application of this theory- in particular, there is a discussion of electrical and magnetic field effects which give rise to Freedericksz transitions, which are important in devices. This is followed by an account of dynamic theory and elementary viscometry of nemantics Discussions of backflow and flow-induced instabilities are also included. Smetic theory is also briefly introduced and summarised with some examples of equilibrium solutions as well as those with dynamic effects. A number of mathematical techniques, such as Cartesian tensors and some variational calculus, are presented in the appendices.

Product Details

ISBN-13:	9780203646335
Publisher:	CRC Press
Publication date:	06/29/2004
Series:	Liquid Crystals Book Series
Sold by:	Barnes & Noble
Format:	eBook
Pages:	351
File size:	13 MB
Note:	This product may take a few minutes to download.

About the Author

Iain W. Stewart

Table of Contents

	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