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FUNDAMENTALS OFLIQUID CRYSTALDEVICES

Wiley-SID Series in Display Technology

Series EditorsAnthony C Lowe and Ian Sage

Display Systems Design and ApplicationsLindsay W MacDonald and Anthony C Lowe (Eds)

Electronic Display Measurement Concepts Techniques and InstrumentationPeter A Keller

Reflective Liquid Crystal DisplaysShin-Tson Wu and Deng-Ke Yang

Colour Engineering Achieving Device Independent ColourPhil Green and Lindsay MacDonald (Eds)

Display Interfaces Fundamentals and StandardsRobert L Myers

Digital Image Display Algorithms and ImplementationGheorghe Berbecel

Flexible Flat Panel DisplaysGregory Crawford (Ed)

Polarization Engineering for LCD ProjectionMichael G Robinson Jianmin Chen and Gary D Sharp

Fundamentals of Liquid Crystal DevicesDeng-Ke Yang and Shin-Tson Wu

Introduction to MicrodisplaysDavid Armitage Ian Underwood and Shin-Tson Wu

Mobile Displays Technology and ApplicationsAchintya K Bhowmik Zili Li and Philip Bos (Eds)

Photoalignment of Liquid Crystalline Materials Physics and ApplicationsVladimir G Chigrinov Vladimir M Kozenkov and Hoi-Sing Kwok

Projection Displays Second EditionMatthew S Brennesholtz and Edward H Stupp

Introduction to Flat Panel DisplaysJiun-Haw Lee David N Liu and Shin-Tson Wu

LCD BacklightsShunsuke Kobayashi Shigeo Mikoshiba and Sungkyoo Lim (Eds)

Liquid Crystal Displays Addressing Schemes and Electro-Optical Effects Second EditionErnst Lueder

Transflective Liquid Crystal DisplaysZhibing Ge and Shin-Tson Wu

Liquid Crystal Displays Fundamental Physics and TechnologyRobert H Chen

3D DisplaysErnst Lueder

OLED Display Fundamentals and ApplicationsTakatoshi Tsujimura

Illumination Colour and Imaging Evaluation and Optimization of Visual DisplaysTran Quoc Khanh and Peter Bodrogi

Interactive Displays Natural Human-Interface TechnologiesAchintya K Bhowmik (Ed)

Modeling and Optimization of LCD Optical PerformanceDmitry A Yakovlev Vladimir G Chigrinov Hoi-Sing Kwok

Addressing Techniques of Liquid Crystal DisplaysTemkar N Ruckmongathan


Second Edition

Deng-Ke YangLiquid Crystal Institute Kent State University USA

Shin-Tson WuCollege of Optics and Photonics University of Central Florida USA

This edition first published 2015copy 2015 John Wiley amp Sons Ltd

First Edition published in 2006copy 2006 John Wiley amp Sons Ltd

Registered OfficeJohn Wiley amp Sons Ltd The Atrium Southern Gate Chichester West Sussex PO19 8SQ United Kingdom

For details of our global editorial offices for customer services and for information about how to apply forpermission to reuse the copyright material in this book please see our website at wwwwileycom

The right of the author to be identified as the author of this work has been asserted in accordance with theCopyright Designs and Patents Act 1988

All rights reserved No part of this publication may be reproduced stored in a retrieval system or transmittedin any form or by any means electronic mechanical photocopying recording or otherwise except as permittedby the UK Copyright Designs and Patents Act 1988 without the prior permission of the publisher

Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not beavailable in electronic books

Designations used by companies to distinguish their products are often claimed as trademarks All brand namesand product names used in this book are trade names service marks trademarks or registered trademarks of theirrespective owners The publisher is not associated with any product or vendor mentioned in this book

Limit of LiabilityDisclaimer of Warranty While the publisher and author have used their best efforts in preparingthis book they make no representations or warranties with respect to the accuracy or completeness of the contentsof this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purposeIt is sold on the understanding that the publisher is not engaged in rendering professional services and neither thepublisher nor the author shall be liable for damages arising herefrom If professional advice or other expert assistanceis required the services of a competent professional should be sought

Library of Congress Cataloging-in-Publication Data

Yang Deng-KeFundamentals of liquid crystal devices Deng-Ke Yang and Shin-Tson Wu ndash Second edition

pages cm ndash (Wiley series in display technology)Includes bibliographical references and indexISBN 978-1-118-75200-5 (hardback)

1 Liquid crystal displays 2 Liquid crystal devices 3 Liquid crystals I Wu Shin-Tson II TitleTK7872L56Y36 201462138150422ndashdc23

2014027707

A catalogue record for this book is available from the British Library

Set in 1012pt Times by SPi Publisher Services Pondicherry India

1 2015

Contents

Series Editorrsquos Foreword xiii

Preface to the First Edition xv

Preface to the Second Edition xvii

1 Liquid Crystal Physics 111 Introduction 112 Thermodynamics and Statistical Physics 5

121 Thermodynamic laws 5122 Boltzmann Distribution 6123 Thermodynamic quantities 7124 Criteria for thermodynamical equilibrium 9

13 Orientational Order 10131 Orientational order parameter 11132 Landaundashde Gennes theory of orientational order in nematic phase 13133 MaierndashSaupe theory 18

14 Elastic Properties of Liquid Crystals 21141 Elastic properties of nematic liquid crystals 21142 Elastic properties of cholesteric liquid crystals 24143 Elastic properties of smectic liquid crystals 26

15 Response of Liquid Crystals to Electromagnetic Fields 27151 Magnetic susceptibility 27152 Dielectric permittivity and refractive index 29

16 Anchoring Effects of Nematic Liquid Crystal at Surfaces 38161 Anchoring energy 38162 Alignment layers 39

17 Liquid crystal director elastic deformation 40171 Elastic deformation and disclination 40172 Escape of liquid crystal director in disclinations 42

Homework Problems 48References 49

2 Propagation of Light in Anisotropic Optical Media 5121 Electromagnetic Wave 5122 Polarization 54

221 Monochromatic plane waves and theirpolarization states 54

222 Linear polarization state 55223 Circular polarization states 55224 Elliptical polarization state 56

23 Propagation of Light in Uniform Anisotropic Optical Media 59231 Eigenmodes 60232 Orthogonality of eigenmodes 65233 Energy flux 66234 Special cases 67235 Polarizers 69

24 Propagation of Light in Cholesteric Liquid Crystals 72241 Eigenmodes 72242 Reflection of cholesteric liquid crystals 81243 Lasing in cholesteric liquid crystals 84


3 Optical Modeling Methods 8731 Jones Matrix Method 87

311 Jones vector 87312 Jones matrix 88313 Jones matrix of non-uniform birefringent film 91314 Optical properties of twisted nematic 92

32 Mueller Matrix Method 98321 Partially polarized and unpolarized light 98322 Measurement of the Stokes parameters 100323 The Mueller matrix 102324 Poincareacute sphere 104325 Evolution of the polarization states on the Poincareacute sphere 106326 Mueller matrix of twisted nematic liquid crystals 110327 Mueller matrix of non-uniform birefringence film 112

33 Berreman 4 times 4 Method 113Homework Problems 124References 125

vi Contents

4 Effects of Electric Field on Liquid Crystals 12741 Dielectric Interaction 127

411 Reorientation under dielectric interaction 128412 Field-induced orientational order 129

42 Flexoelectric Effect 132421 Flexoelectric effect in nematic liquid crystals 132422 Flexoelectric effect in cholesteric liquid crystals 136

43 Ferroelectric Liquid Crystal 138431 Symmetry and polarization 138432 Tilt angle and polarization 140433 Surface stabilized ferroelectric liquid crystals 141434 Electroclinic effect in chiral smectic liquid crystal 144


5 Freacuteedericksz Transition 14951 Calculus of Variation 149

511 One dimension and one variable 150512 One dimension and multiple variables 153513 Three dimensions 153

52 Freacuteedericksz Transition Statics 153521 Splay geometry 154522 Bend geometry 158523 Twist geometry 160524 Twisted nematic cell 161525 Splay geometry with weak anchoring 164526 Splay geometry with pretilt angle 165

53 Measurement of Anchoring Strength 166531 Polar anchoring strength 167532 Azimuthal anchoring strength 169

54 Measurement of Pretilt Angle 17155 Freacuteedericksz Transition Dynamics 175

551 Dynamics of Freacuteedericksz transition in twist geometry 175552 Hydrodynamics 176553 Backflow 182


6 Liquid Crystal Materials 19161 Introduction 19162 Refractive Indices 192

621 Extended Cauchy equations 192622 Three-band model 193623 Temperature effect 195

viiContents

624 Temperature gradient 198625 Molecular polarizabilities 199

63 Dielectric Constants 201631 Positive Δε liquid crystals for AMLCD 202632 Negative Δε liquid crystals 202633 Dual-frequency liquid crystals 203

64 Rotational Viscosity 20465 Elastic Constants 20466 Figure-of-Merit (FoM) 20567 Index Matching between Liquid Crystals and Polymers 206

671 Refractive index of polymers 206672 Matching refractive index 208

Homework problems 210References 210

7 Modeling Liquid Crystal Director Configuration 21371 Electric Energy of Liquid Crystals 213

711 Constant charge 214712 Constant voltage 215713 Constant electric field 218

72 Modeling Electric Field 21873 Simulation of Liquid Crystal Director Configuration 221

731 Angle representation 221732 Vector representation 225733 Tensor representation 228


8 Transmissive Liquid Crystal Displays 23581 Introduction 23582 Twisted Nematic (TN) Cells 236

821 Voltage-dependent transmittance 237822 Film-compensated TN cells 238823 Viewing angle 241

83 In-Plane Switching Mode 241831 Voltage-dependent transmittance 242832 Response time 243833 Viewing angle 246834 Classification of compensation films 246835 Phase retardation of uniaxial media

at oblique angles 246836 Poincareacute sphere representation 249837 Light leakage of crossed polarizers

at oblique view 250838 IPS with a positive a film and a positive c film 254

viii Contents

839 IPS with positive and negative a films 2598310 Color shift 263

84 Vertical Alignment Mode 263841 Voltage-dependent transmittance 263842 Optical response time 264843 Overdrive and undershoot voltage method 265

85 Multi-Domain Vertical Alignment Cells 266851 MVA with a positive a film and a negative c film 269852 MVA with a positive a a negative a and a negative c film 273

86 Optically Compensated Bend Cell 277861 Voltage-dependent transmittance 278862 Compensation films for OCB 279


9 Reflective and Transflective Liquid Crystal Displays 28591 Introduction 28592 Reflective Liquid Crystal Displays 286

921 Film-compensated homogeneous cell 287922 Mixed-mode twisted nematic (MTN) cells 289

93 Transflector 290931 Openings-on-metal transflector 290932 Half-mirror metal transflector 291933 Multilayer dielectric film transflector 292934 Orthogonal polarization transflectors 292

94 Classification of Transflective LCDs 293941 Absorption-type transflective LCDs 294942 Scattering-type transflective LCDs 296943 Scattering and absorption type transflective LCDs 298944 Reflection-type transflective LCDs 300945 Phase retardation type 302

95 Dual-Cell-Gap Transflective LCDs 31296 Single-Cell-Gap Transflective LCDs 31497 Performance of Transflective LCDs 314

971 Color balance 314972 Image brightness 315973 Viewing angle 315


10 Liquid Crystal Display Matrices Drive Schemes and Bistable Displays 321101 Segmented Displays 321102 Passive Matrix Displays and Drive Scheme 322103 Active Matrix Displays 326

1031 TFT structure 328

ixContents

1032 TFT operation principles 329104 Bistable Ferroelectric LCD and Drive Scheme 330105 Bistable Nematic Displays 332

1051 Introduction 3321052 Twisted-untwisted bistable nematic LCDs 3331053 Surface-stabilized nematic liquid crystals 339

106 Bistable Cholesteric Reflective Display 3421061 Introduction 3421062 Optical properties of bistable Ch reflective displays 3441063 Encapsulated cholesteric liquid crystal displays 3471064 Transition between cholesteric states 3471065 Drive schemes for bistable Ch displays 355


11 Liquid CrystalPolymer Composites 363111 Introduction 363112 Phase Separation 365

1121 Binary mixture 3651122 Phase diagram and thermal induced phase separation 3691123 Polymerization induced phase separation 3711124 Solvent-induced phase separation 3741125 Encapsulation 376

113 Scattering Properties of LCPCs 377114 Polymer Dispersed Liquid Crystals 383

1141 Liquid crystal droplet configurations in PDLCs 3831142 Switching PDLCs 3851143 Scattering PDLC devices 3871144 Dichroic dye-doped PDLC 3911145 Holographic PDLCs 393

115 PSLCs 3951151 Preparation of PSLCs 3951152 Working modes of scattering PSLCs 396

116 Scattering-Based Displays from LCPCs 4001161 Reflective displays 4001162 Projection displays 4021163 Transmissive direct-view displays 403

117 Polymer-Stabilized LCDs 403Homework Problems 407References 409

12 Tunable Liquid Crystal Photonic Devices 413121 Introduction 413122 Laser Beam Steering 414

1221 Optical phased array 4151222 Prism-based beam steering 417

123 Variable Optical Attenuators 419

x Contents

124 Tunable-Focus Lens 4231241 Tunable-focus spherical lens 4231242 Tunable-focus cylindrical lens 4261243 Switchable positive and negative microlens 4281244 Hermaphroditic LC microlens 434

125 Polarization-Independent LC Devices 4351251 Double-layered homogeneous LC cells 4361252 Double-layered LC gels 438


13 Blue Phases of Chiral Liquid Crystals 445131 Introduction 445132 Phase Diagram of Blue Phases 446133 Reflection of Blue Phases 447

1331 Basics of crystal structure and X-ray diffraction 4471332 Bragg reflection of blue phases 449

134 Structure of Blue Phase 4511341 Defect theory 4521342 Landau theory 459

135 Optical Properties of Blue Phase 4711351 Reflection 4711352 Transmission 472


14 Polymer-Stabilized Blue Phase Liquid Crystals 477141 Introduction 477142 Polymer-Stabilized Blue Phases 480

1421 Nematic LC host 4821422 Chiral dopants 4831423 Monomers 483

143 Kerr Effect 4841431 Extended Kerr effect 4861432 Wavelength effect 4891433 Frequency effect 4901434 Temperature effects 491

144 Device Configurations 4961441 In-plane-switching BPLCD 4971442 Protruded electrodes 5011443 Etched electrodes 5041444 Single gamma curve 504

145 Vertical Field Switching 5071451 Device structure 5071452 Experiments and simulations 508

146 Phase Modulation 510References 510

xiContents

15 Liquid Crystal Display Components 513151 Introduction 513152 Light Source 513153 Light-guide 516154 Diffuser 516155 Collimation Film 518156 Polarizer 519

1561 Dichroic absorbing polarizer 5201562 Dichroic reflective polarizer 521

157 Compensation Film 5301571 Form birefringence compensation film 5311572 Discotic liquid crystal compensation film 5311573 Compensation film from rigid polymer chains 5321574 Drawn polymer compensation film 533

158 Color Filter 535References 536

16 Three-Dimensional Displays 539161 Introduction 539162 Depth Cues 539

1621 Binocular disparity 5391622 Convergence 5401623 Motion parallax 5401624 Accommodation 541

163 Stereoscopic Displays 5411631 Head-mounted displays 5421632 Anaglyph 5421633 Time sequential stereoscopic displays with shutter glasses 5421634 Stereoscopic displays with polarizing glasses 544

164 Autostereoscopic Displays 5461641 Autostereoscopic displays based on parallax barriers 5461642 Autostereoscopic displays based on lenticular lens array 5501643 Directional backlight 552

165 Integral imaging 553166 Holography 554167 Volumetric displays 556

1671 Swept volumetric displays 5561672 Multi-planar volumetric displays 5571673 Points volumetric displays 560

References 560

Index 565

xii Contents

Series Editorrsquos Foreword

The first edition of this book marked a new departure for the Wiley-SID Series in Display Tech-nology because it had been written primarily as a text for postgraduate and senior undergraduatestudents It fulfilled that objective admirably but the continuing advances in liquid crystal displaytechnology over the intervening eight years havemade necessary some additions to keep the bookcurrentAccordingly the following new sections have been added elastic deformation of liquid

crystals in Chapter 1 polarisation conversion with narrow and broadband quarter-wave platesin Chapter 3 and measurement of anchoring strength and pretilt angle in Chapter 5With each chapter is designed to be self-contained the first chapters cover the basic physics

of liquid crystals their interaction with light and electric fields and the means by which theycan be modelled Next are described the majority of the ways in which liquid crystals can beused in displays and Chapter 12 the final chapter of the first edition deals with photonicdevices such as beam steerers tunable-focus lenses and polarisation-independent devicesIn this second edition four new chapters have been added two on blue phase and polymerstabilised blue phase liquid crystals which are emerging from the realm of academic researchto show promise for very fast response display and photonic devices a chapter which discussesLCD componentry and a final chapter on the use of LCDs in 3D display systemsAs with the first edition and following a standard textbook format each chapter concludes

with a set of problems the answers to which may be found on the Wiley web siteNew electro-optic technologies continue to be developed and some of them make inroads

into the LCD market Nevertheless liquid crystal technology ndash the first other than the CRT tomake a significant breakthrough into the mass market and which made possible flat displaysand transformed projection display technology ndash continues to hold a dominant position Thissecond edition of Fundamentals of Liquid Crystal Devices with its additions which includereferences to some very recent work will ensure that this volume will continue to provide stu-dents and other readers at the professional level with a most useful introduction to the subject

Anthony C LoweBraishfield UK 2014

Preface to the First Edition

Liquid crystal displays have become the leading technology in the information display industryThey are used in small-sized displays such as calculators cellular phones digital cameras andhead-mounted displays in medium-sized displays such as laptop and desktop computers and inlarge-sized displays such as direct-view TVs and projection TVs They have the advantages ofhigh resolution and high brightness and being flat paneled are lightweight energy saving andeven flexible in some cases They can be operated in transmissive and reflective modes Liquidcrystals have also been used in photonic devices such as laser beam steering variable opticalattenuators and tunable-focus lenses There is no doubt that liquid crystals will continue to playan important role in the era of information technologyThere are many books on the physics and chemistry of liquid crystals and on liquid crystal

devices There are however few books covering both the basics and applications of liquidcrystals Our main goal therefore is to provide a textbook for senior undergraduate and grad-uate students The book can be used for a one- or two-semester course The instructors canselectively choose the chapters and sections according to the length of the course and the inter-est of the students The book can also be used as a reference book by scientists and engineerswho are interested in liquid crystal displays and photonicsThe book is organized in such a way that the first few chapters cover the basics of liquid

crystals and the necessary techniques to study and design liquid crystal devices The later chap-ters cover the principles design operation and performance of liquid crystal devices Becauseof limited space we cannot cover every aspect of liquid crystal chemistry and physics andall liquid crystal devices but we hope this book will introduce readers to liquid crystals andprovide them with the basic knowledge and techniques for their careers in liquid crystalsWe are greatly indebted to Dr A Lowe for his encouragement We are also grateful to the

reviewers of our book proposal for their useful suggestions and comments Deng-Ke Yangwould like to thank Ms E Landry and Prof J Kelly for patiently proofreading his manuscriptHe would also like to thank Dr Q Li for providing drawings Shin-TsonWuwould like to thankhis research group members for generating the new knowledge included in this book especiallyDrs Xinyu Zhu Hongwen Ren Yun-Hsing Fan and Yi-Hsin Lin and Mr Zhibing Ge for kind

help during manuscript preparation He is also indebted to Dr Terry Dorschner of RaytheonDr Paul McManamon of the Air Force Research Lab and Dr Hiroyuki Mori of Fuji Photo Filmfor sharing their latest results We would like to thank our colleagues and friends for usefuldiscussions and drawings and our funding agencies (DARPA AFOSR AFRL and Toppoly)for providing financial support Finally we also would like to thank our families (Xiaojiang LiKevin Yang Steven Yang Cho-Yan Wu Janet Wu and Benjamin Wu) for their spiritualsupport understanding and constant encouragement

Deng-Ke YangShin-Tson Wu

xvi Preface to the First Edition

Preface to the Second Edition

Liquid crystal displays have become the leading technology in the information display indus-try They are used in small-sized displays such as calculators smart phones digital camerasand wearable displays medium-sized displays such as laptop and desktop computers andlarge-sized displays such as direct view TVs and data projectors They have the advantagesof having high resolution and high brightness and being flat paneled lightweight energy sav-ing and even flexible in some cases They can be operated in transmissive and reflectivemodes Liquid crystals have also been used in photonic devices such as switching windowslaser beam steering variable optical attenuators and tunable-focus lenses There is no doubtthat liquid crystals will continue to play an important role in information technologyThere are many books on the physics and chemistry of liquid crystals and on liquid crystal

devices There are however few books covering both the basics and the applications of liquidcrystals The main goal of this book is to provide a textbook for senior undergraduate and grad-uate students This book can be used for a one- or two-semester course The instructors canselectively choose the chapters and sections according to the length of the course and the inter-est of the students It can also be used as a reference book for scientists and engineers who areinterested in liquid crystal displays and photonicsThe book is organized in such a way that the first few chapters cover the basics of liquid

crystals and the necessary techniques to study and design liquid crystal devices The later chap-ters cover the principles design operation and performance of liquid crystal devices Becauseof limited space we cannot cover every aspect of liquid crystal chemistry and physics andall the different liquid crystal devices We hope that this book can introduce readers to liquidcrystals and provide them with the basic knowledge and techniques for their career in liquidcrystalsSince the publication of the first edition we have received a lot of feedback suggestions

corrections and encouragements We appreciate them very much and have put them intothe second edition Also there are many new advances in liquid crystal technologies We haveadded new chapters and sections to cover them

We are greatly indebted to Dr A Lowe for his encouragement We are also grateful to thereviewers of our book proposal for their useful suggestions and comments Deng-Ke Yangwould like to thank Ms E Landry Prof P Crooker his research group and coworkers forpatiently proofreading and preparing his sections of the book He would also like to thanksDr Q Li for providing drawings Shin-Tson Wu would like to thank his research group mem-bers for generating new knowledge included in this book especially Drs Xinyu Zhu HongwenRen Yun-Hsing Fan Yi-Hsin Lin Zhibing Ge Meizi Jiao Linghui Rao Hui-Chuan ChengYan Li and Jin Yan for their kind help during manuscript preparation He is also indebtedto Dr Terry Dorschner of Raytheon Dr Paul McManamon of Air Force Research Lab andDr Hiroyuki Mori of Fuji Photo Film for sharing their latest results We would like to thankour colleagues and friends for useful discussions and drawings and our funding agencies(DARPA AFOSR AFRL ITRI AUO and Innolux) for providing financial support Wewould also like to thank our family members (Xiaojiang Li Kevin Yang Steven YangCho-Yan Wu Janet Wu and Benjamin Wu) for their spiritual support understanding andconstant encouragement


xviii Preface to the Second Edition

1Liquid Crystal Physics

11 Introduction

Liquid crystals are mesophases between crystalline solid and isotropic liquid [1ndash3] The con-stituents are elongated rod-like (calamitic) or disk-like (discotic) organic molecules as shown inFigure 11 The size of the molecules is typically a few nanometers (nm) The ratio between thelength and the diameter of the rod-like molecules or the ratio between the diameter and thethickness of disk-like molecules is about 5 or larger Because the molecules are non-sphericalbesides positional order they may possess orientational orderFigure 11(a) shows a typical calamitic liquid crystal molecule Its chemical name is 40-n-

Pentyl-4-cyano-biphenyl and is abbreviated as 5CB [45] It consists of a biphenyl whichis the rigid core and a hydrocarbon chain which is the flexible tail The space-filling modelof the molecule is shown in Figure 11(c) Although the molecule itself is not cylindrical itcan be regarded as a cylinder as shown in Figure 11(e) in considering its physical behaviorbecause of the fast rotation (on the order of 10minus9 s) around the longmolecule axis due to thermalmotion The distance between two carbon atoms is about 15 Aring therefore the length and thediameter of the molecule are about 2 nm and 05 nm respectively The molecule shown hasa permanent dipole moment (from the CN head) but it can still be represented by the cylinderwhose head and tail are the same because in non-ferroelectric liquid crystal phases the dipolehas equal probability of pointing up or down It is necessary for a liquid crystal molecule tohave a rigid core(s) and flexible tail(s) If the molecule is completely flexible it will not haveorientational order If it is completely rigid it will transform directly from isotropic liquid phaseat high temperature to crystalline solid phase at low temperature The rigid part favors bothorientational and positional order while the flexible part disfavors them With balanced rigidand flexible parts the molecule exhibits liquid crystal phases

Fundamentals of Liquid Crystal Devices Second Edition Deng-Ke Yang and Shin-Tson Wucopy 2015 John Wiley amp Sons Ltd Published 2015 by John Wiley amp Sons Ltd

Figure 11(b) shows a typical discotic liquid crystal molecule [6] It also has a rigid coreand flexible tails The branches are approximately on one plane The space-filling model ofthe molecule is shown in Figure 11(d) If there is no permanent dipole moment perpendic-ular to the plane of the molecule it can be regarded as a disk in considering its physicalbehavior as shown in Figure 11(f ) because of the fast rotation around the axis which isat the center of the molecule and perpendicular to the plane of the molecule If there is apermanent dipole moment perpendicular to the plane of the molecule it is better to visualizethe molecule as a bowl because the reflection symmetry is broken and all the permanentdipoles may point in the same direction and spontaneous polarization occurs The flexibletails are also necessary otherwise the molecules form a crystal phase where there is posi-tional order

C N CH3 CH2 CH2 CH2CH2

(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)

Figure 11 Calamitic liquid crystal (a) chemical structure (c) space-filling model (e) physical modelDiscostic liquid crystal (b) chemical structure (d) space-filling mode (f ) physical model

2 Fundamentals of Liquid Crystal Devices

The variety of phases that may be exhibited by rod-like molecules are shown in Figure 12At high temperature the molecules are in the isotropic liquid state where they do not have eitherpositional or orientational order The molecules can easily move around and the material canflow like water The translational viscosity is comparable to that of water Both the long andshort axes of the molecules can point in any directionWhen the temperature is decreased the material transforms into the nematic phase which is

the most common and simplest liquid crystal phase where the molecules have orientationalorder but still no positional order The molecules can still diffuse around and the translationalviscosity does not change much from that of the isotropic liquid state The long axis of themolecules has a preferred direction Although the molecules still swivel due to thermal motionthe time-averaged direction of the long axis of a molecule is well defined and is the same for allthe molecules at macroscopic scale The average direction of the long molecular axis is denotedby n

which is a unit vector called the liquid crystal director The short axes of the moleculeshave no orientational order in a uniaxial nematic liquid crystalWhen the temperature is decreased further the material may transform into the Smectic-A

phase where besides the orientational order the molecules have partial positional orderie the molecules form a layered structure The liquid crystal director is perpendicular tothe layers Smectic-A is a one-dimensional crystal where the molecules have positional orderin the layer normal direction The cartoon shown in Figure 12 is schematic In reality the sep-aration between neighboring layers is not as well defined as that shown by the cartoon Themolecule number density exhibits an undulation with the wavelength about the molecularlength Within a layer it is a two-dimensional liquid crystal in which there is no positionalorder and the molecules can move around For a material in poly-domain smectic-A the trans-lational viscosity is significantly higher and it behaves like a grease When the temperature isdecreased further the material may transform into the smectic-C phase where the liquid crystaldirector is no longer perpendicular to the layer but tiltedAt low temperature the material is in the crystal solid phase where there are both positional

and orientational orders The translational viscosity becomes infinitely high and the molecules(almost) do not diffuse anymoreLiquid crystals get the lsquocrystalrsquo part of their name because they exhibit optical birefringence

as crystalline solids They get the lsquoliquidrsquo part of their name because they can flow and do notsupport shearing as regular liquids Liquid crystal molecules are elongated and have different

Crystal solid Smectic-A Nematic Isotropic liquid

Temperature

Smectic-C

Figure 12 Schematic representation of the phases of rod-like molecules


molecular polarizabilities along their long and short axes Once the long axes of the moleculesorient along a common direction the refractive indices along and perpendicular to the commondirection are different It should be noted that not all rod-like molecules exhibit all the liquidcrystal phases They may exhibit some of the liquid crystal phasesSome of the liquid crystal phases of disk-like molecules are shown in Figure 13 At high

temperature they are in the isotropic liquid state where there are no positional and orientationalorders The material behaves in the same way as a regular liquid When the temperature isdecreased the material transforms into the nematic phase which has orientational order butnot positional order The average direction of the short axis perpendicular to the disk is orientedalong a preferred direction which is also called the liquid crystal director and denoted by a unitvector n The molecules have different polarizabilities along a direction in the plane of the diskand along the short axis Thus the discotic nematic phase also exhibits birefringence as crystalsWhen the temperature is decreased further the material transforms into the columnar phase

where besides orientational order there is partial positional order The molecules stack up toform columnsWithin a column it is a liquid where the molecules have no positional order Thecolumns however are arranged periodically in the plane perpendicular to the columns Henceit is a two-dimensional crystal At low temperature the material transforms into the crystallinesolid phase where the positional order along the columns is developedThe liquid crystal phases discussed so far are called thermotropic liquid crystals and the

transitions from one phase to another are driven by varying temperature There is another typeof liquid crystals called lyotropic liquid crystals exhibited by molecules when they are mixedwith a solvent of some kind The phase transitions from one phase to another phase are drivenby varying the solvent concentration Lyotropic liquid crystals usually consist of amphiphilicmolecules that have a hydrophobic group at one end and a hydrophilic group at the other endand the water is the solvent The common lyotropic liquid crystal phases are micelle phase andlamellar phase Lyotropic liquid crystals are important in biology They will not be discussed inthis book because the scope of this book is on displays and photonic devicesLiquid crystals have a history of more than 100 years It is believed that the person who dis-

covered liquid crystals is Friedrich Reinitzer an Austrian botanist [7] The liquid crystal phaseobserved by him in 1888 was a cholesteric phase Since then liquid crystals have come a longway and become a major branch of interdisciplinary sciences Scientifically liquid crystals areimportant because of the richness of structures and transitions Technologically they have wontremendous success in display and photonic applications [8ndash10]

Crystal solid NematicColumnar Isotropic liquid

Temperature

Figure 13 Schematic representation of the phases of disk-like molecules


12 Thermodynamics and Statistical Physics

Liquid crystal physics is an interdisciplinary science thermodynamics statistical physicselectrodynamics and optics are involved Here we give a brief introduction to thermodynamicsand statistical physics

121 Thermodynamic laws

One of the important quantities in thermodynamics is entropy From the microscopic point ofview entropy is a measurement of the number of quantum states accessible to a system In orderto define entropy quantitatively we first consider the fundamental logical assumption that fora closed system (no energy and particles exchange with other systems) quantum states areeither accessible or inaccessible to the system and the system is equally likely to be in anyone of the accessible states as in any other accessible state [11] For a macroscopic systemthe number of accessible quantum states g is a huge number (~1023) It is easier to deal withln g which is defined as the entropy σ

σ = lng eth11THORN

If a closed system consists of subsystem 1 and subsystem 2 the numbers of accessible statesof the subsystems are g1 and g2 respectively The number of accessible quantum states of thewhole system is g = g1g2 and the entropy is σ = ln g = ln(g1g2) = ln g1 + ln g2 = σ1 + σ2Entropy is a function of the energy u of the system σ = σ(u) The second law of thermody-

namics states that for a closed system the equilibrium state has the maximum entropy Letus consider a closed system which contains two subsystems When two subsystems are broughtinto thermal contact (energy exchange between them is allowed) the energy is allocated to max-imize the number of accessible states that is the entropy is maximized Subsystem 1 has theenergy u1 and entropy σ1 subsystem 2 has the energy u2 and entropy σ2 For the whole systemu = u1 + u2 and σ = σ1 + σ2 The first law of thermodynamics states that energy is conserved thatis u = u1 + u2 = constant For any process inside the closed system δu = δu1 + δu2 = 0 From thesecond law of thermodynamics for any process we have δσ = δσ1 + δσ2 ge 0 When the two sub-systems are brought into thermal contact at the beginning energy flows For example anamount of energy |δu1| flows from subsystem 1 to subsystem 2 δu1 lt 0 and δu2 = minus δu1 gt 0

andpartσ

partu2=partσ1partu2

+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2

= minuspartσ1partu1

+partσ2partu2

ge 0 When equilibrium is reached the

entropy is maximized andpartσ1partu1

minuspartσ2partu2

= 0 that ispartσ1partu1

=partσ2partu2

We know that when two systems

reach equilibrium they have the same temperature Accordingly the fundamental temperatureτ is defined by

1=τ =partσ

partu

NV

eth12THORN

Energy flows from a high temperature system to a low temperature system The conventionaltemperature (Kelvin temperature) is defined by

T = τ=kB eth13THORN


where kB = 1381 times 10minus23 JouleKelvin is the Boltzmann constant Conventional entropy S isdefined by

1=T = partS=partu eth14THORNHence

S= kBσ eth15THORN

122 Boltzmann Distribution

Now we consider the thermodynamics of a system at a constant temperature that is in thermalcontact with a thermal reservoir The temperature of the thermal reservoir (named B) is τ Thesystem under consideration (named A) has two states with energy 0 and ε respectively A andB form a closed system and its total energy u = uA + uB = uo = constant When A is in the statewith energy 0 B has the energy uo the number of accessible states g1 = gA times gB = 1 times gB(uo) =gB(uo) When A has the energy ε B has the energy uo minus ε the number of accessible states is g2 =gA times gB = 1 times gB(uo minus ε) = gB(uo minus ε) For the whole system the total number of accessiblestates is

G= g1 + g2 = gB uoeth THORN + gB uominusεeth THORN eth16THORN

(A + B) is a closed system and the probability in any of theG states is the sameWhen the wholesystem is in one of the g1 states A has the energy 0 When the whole system is in one of the g2states A has the energy ε Therefore the probability for A in the state with energy 0 is

P 0eth THORN= g1g1 + g2

=gB uoeth THORN

gB uoeth THORN+ gB uominusεeth THORN The probability for A in the state with energy ε is

P εeth THORN= g2g1 + g2

=gB uominusεeth THORN

gB uoeth THORN+ gB uominusεeth THORN From the definition of entropy we have gB uo

= eσB uoeth THORN

and gB uominus ε

= eσB uo minusεeth THORN Because ε uo σB uominusεeth THORNasympσB uoeth THORNminus partσBpartuB

ε = σB uoeth THORNminus 1τ ε Therefore

we have

P 0eth THORN= eσB uoeth THORN

eσB uoeth THORN + eσB uoeth THORNminusε=τ =1

1 + eminusε=τ=

1

1 + eminusε=kBTeth17THORN

P εeth THORN = eσB uoeth THORNminusε=τ

eσB uoeth THORN + eσB uoeth THORNminusε=τ =eminusε=τ

1 + eminusε=τ=

eminusε=kBT


p εeth THORNP 0eth THORN = e

minusε=kBT eth19THORN

For a system having N states with energies ε1 ε2 εi εi + 1 εN the probability for thesystem in the state with energy εi is

P εieth THORN= eminusεi=τ=XNj= 1

eminusεj=kBT eth110THORN


The partition function of the system is defined as

Z =XNi= 1

eminusεi=kBT eth111THORN

The internal energy (average energy) of the system is given by

U = lt ε gt =Xi

εiP εieth THORN = 1Z

Xi

εieminusεi=kBT eth112THORN

Because partZpartT =

Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT

2 part lnZeth THORNpartT

eth113THORN

123 Thermodynamic quantities

As energy is conserved the change of the internal energy U of a system equals the heat dQabsorbed and the mechanical work dW done to the system dU = dQ + dW When the volumeof the system changes by dV under the pressure P the mechanical work done to the system isgiven by

dW = minusPdV eth114THORN

When there is no mechanical work the heat absorbed equals the change of internal energyFrom the definition of temperature 1=T = partS

partU

V the heat absorbed in a reversible process at

constant volume is

dU = dQ = TdS eth115THORN

When the volume is not constant then

dU = TdS minus PdV eth116THORN

The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN


The internal energy U entropy S and volume V are extensive quantities while temperatureT and pressure P are intensive quantities The enthalpy H of the system is defined by

H =U +PV eth119THORN

Its variation in a reversible process is given by

dH = dU + d PVeth THORN= TdSminusPdVeth THORN+ PdV +VdPeth THORN= TdS+VdP eth120THORN

From this equation it can be seen that the physical meaning of enthalpy is that in a process atconstant pressure (dP = 0) the change of enthalpy dH is equal to the heat absorbed dQ (=TdS))The derivatives of the enthalpy are

T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN

The Helmholtz free energy F of the system is defined by

F =UminusTS eth123THORN


dF = dU minus d TSeth THORN= TdSminusPdVeth THORNminus TdS + SdTeth THORN= minusSdT minusPdV eth124THORN

The physical meaning of Helmholtz free energy is that in a process at constant temperature(dT = 0) the change of Helmholtz free energy is equal to the work done to the systemThe derivatives are

S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN

The Gibbs free energy G of the system is defined by

G=U minus TS + PV eth127THORN

The variation in a reversible process is given by

dG = dU minus d TSeth THORN minus d PVeth THORN= minusSdT +VdP eth128THORN


In a process at constant temperature and pressure Gibbs free energy does not change Thederivatives are

S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN

The Helmholtz free energy can be derived from the partition function Because ofEquations (113) and (125)

F =U minus TS = kBT2 part lnZeth THORN

partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V

= kBT2 part lnZeth THORN

partT

Hence

F = minuskBT lnZ = minuskBT lnXi

eminusεi=kBT

eth131THORN

From Equations (111) (125) and (131) the entropy of a system at a constant temperature canbe calculated

S = minuskB lt ln ρ gt = minuskBXi

ρi ln ρi eth132THORN

124 Criteria for thermodynamical equilibrium

Now we consider the criteria which can used to judge whether a system is in its equilibriumstate under given conditions We already know that for a closed system as it changes from anon-equilibrium state to the equilibrium state the entropy increases

δS ge 0 eth133THORNIt can be stated in a different way that for a closed system the entropy is maximized in theequilibrium stateIn considering the equilibrium state of a system at constant temperature and volume we con-

struct a closed system which consists of the system (subsystem 1) under consideration and athermal reservoir (subsystem 2) with the temperature T When the two systems are brought intothermal contact energy is exchanged between subsystem 1 and subsystem 2 Because the

whole system is a closed system δS = δS1 + δS2 ge 0 For system 2 1=T =partS2partU2

V

and


therefore δS2 = δU2T (this is true when the volume of subsystem is fixed which also means thatthe volume of subsystem 1 is fixed) Because of energy conservation δU2 = minus δU1 Hence δS =δS1 + δS2 = δS1 + δU2T = δS1 minus δU1T ge 0 Because the temperature and volume are constantfor subsystem 1 δS1 minus δU1T = (1T)δ(TS1 minusU1) ge 0 and therefore

δ U1minusTS1eth THORN= δF1 le 0 eth134THORN

At constant temperature and volume the equilibrium state has the minimum Helmholtz freeenergyIn considering the equilibrium state of a system at constant temperature and pressure we

construct a closed system which consists of the system (subsystem 1) under considerationand a thermal reservoir (subsystem 2) with the temperature T When the two systems arebrought into thermal contact energy is exchanged between subsystem 1 and subsystem 2Because the whole system is a closed system δS = δS1 + δS2 ge 0 For system 2 because the vol-ume is not fixed and mechanical work is involved δU2 = TδS2 minus PδV2 that is δS2 = (δU2 +PδV2)T Because δU2 = minus δU1 and δV2 = minus δV1 δS = δS1 + (δU2 + PδV2)T = δS1 minus (δU1 +PδV1)T = (1T)δ(TS1 minusU1 minus PV1) ge 0 Therefore

δ U1 +PV1minusTS1eth THORN= δG1 le 0 eth135THORN

At constant temperature and pressure the equilibrium state has the minimum Gibbs freeenergy If electric energy is involved then we have to consider the electric work done tothe system by external sources such as a battery In a thermodynamic process if the electric

work done to the system is dWe δS gedQ

T=dUminusdWmminusdWe

T=dU +PdV minusdWe

T Therefore at

constant temperature and pressure

δ UminusWe +PV minusTSeth THORN= δ GminusWeeth THORN le 0 eth136THORN

In the equilibrium state G minusWe is minimized

13 Orientational Order

Orientational order is the most important feature of liquid crystals The average directions of thelong axes of the rod-like molecules are parallel to each other Because of the orientational orderliquid crystals possess anisotropic physical properties that is in different directions they havedifferent responses to external fields such as electric field magnetic field and shear In thissection we will discuss how to specify quantitatively orientational order and why rod-likemolecules tend to parallel each otherFor a rigid elongated liquid crystal molecule three axes can be attached to it to describe its

orientation One is the long molecular axis and the other two axes are perpendicular to the longmolecular axis Usually the molecule rotates fast around the long molecular axis Although themolecule is not cylindrical if there is no hindrance in the rotation in nematic phase the fastrotation around the long molecular axis makes it behave as a cylinder There is no preferreddirection for the short axes and thus the nematic liquid crystal is usually uniaxial If there is






























Second Edition
















2014027707



1 2015

Contents






















vi Contents















viiContents
















viii Contents















ixContents















x Contents















xiContents











References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at



































Second Edition
















2014027707



1 2015

Contents






















vi Contents















viiContents
















viii Contents















ixContents















x Contents















xiContents











References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at









Second Edition
















2014027707



1 2015

Contents






















vi Contents















viiContents
















viii Contents















ixContents















x Contents















xiContents











References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at





















2014027707



1 2015

Contents






















vi Contents















viiContents
















viii Contents















ixContents















x Contents















xiContents











References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at








Contents






















vi Contents















viiContents
















viii Contents















ixContents















x Contents















xiContents











References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at




















vi Contents















viiContents
















viii Contents















ixContents















x Contents















xiContents











References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at






















viiContents
















viii Contents















ixContents















x Contents















xiContents











References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at























viii Contents















ixContents















x Contents















xiContents











References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at






















ixContents















x Contents















xiContents











References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at






















x Contents















xiContents











References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at






















xiContents











References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at


















References 560

Index 565

xii Contents

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at
































11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at

























11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at




















11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at

















11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at












11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at









11 Introduction






(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at










(a)C7 H15

C7 H15

C7 H15H15 C7

H15 C7

H15 C7

(b)

(c) (d)

(e)L ~ 2 nm

D ~ 05 nm

(f)










Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at















Temperature

Smectic-C









Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at














Temperature







σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at












σ = lng eth11THORN



andpartσ


+partσ2partu2

=partσ1partu1

partu1partu2

+partσ2partu2


+partσ2partu2



minuspartσ2partu2


=partσ2partu2



1=τ =partσ

partu

NV

eth12THORN






S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at










S= kBσ eth15THORN






=gB uoeth THORN





= eσB uoeth THORN

and gB uominus ε



we have



1 + eminusε=τ=

1




1 + eminusε=τ=

eminusε=kBT









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at









Z =XNi= 1



U = lt ε gt =Xi


Xi



Xi

εikBT2

eminusεi=kBT =

1kBT2

Xi

εieminusεi=kBT

U =kBT2

Z

partZ

partT= kBT


eth113THORN





partU


constant volume is




The derivatives are

T =partU

partS

V

eth117THORN

P= minuspartU

partV

S

eth118THORN







T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at













T =partH

partS

P

eth121THORN

V =partH

partP

S

eth122THORN






S = minuspartF

partT

V

eth125THORN

P= minuspartF

partV

T

eth126THORN







S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at









S = minuspartG

partT

P

eth129THORN

V =partG

partP

T

eth130THORN



partT+T

partF

partT

V

F minus TpartF

partT

V

= minusT2 1T

partF

partT

V

+Fpart 1

T

partT

V

= minusT2 part F

T

partT

V


partT

Hence


eminusεi=kBT

eth131THORN









V

and










T=dU +PdV minusdWe

T Therefore at
















T=dU +PdV minusdWe

T Therefore at








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