Sven T. Lagerwall
Ferroelectric and Antiferroelectric Liquid Crystals
Further titles of .interest
D. Demus, S. Goodby, G. W. Gray, H.-W. Spiess, V. Vill (Eds.): Handbook of Liquid Crystals, Four Vols. ISBN 3-527-29502-X; 1998 D. Demus, S. Goodby, G. W. Gray, H.-W. Spiess, V. Vill (Eds.): Physical Properties of Liquid Crystals ISBN 3-527-29747-2; 1999
J. L. Serrano (Ed.): Metallomesogens ISBN 3-527-29296-9; 1995
Sven T. Lagerwall
Ferroelectric and Antiferroelectric Liquid Crystals
@WILEY-VCH Weinheim * New York * Chichester Brisbane - Singapore * Toronto
Prof. Sven T. Lagenvall Physics Department Chalmers University of Technology S-412 96 Goteborg Sweden
This book was carefully produced. Nevertheless, author and publisher do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Cover picture: Zigzag defects in a smectic C*. Courtesy of Noel Clark and Tom Rieker. Library of Congress Card No. applied for A catalogue record for this book is available from the British Library Deutsche Bibliothek Cataloguing-in-Publication Data: Lagerwall, Sven T.: Ferroelectric and antifemoelectric liquid crystals I Sven T. Lagerwall. - Weinheim ; New York ;Chichester ; Brisbane ; Singapore ; Toronto : Wiley-VCH, 1999 ISBN 3-527-29831-2
0WILEY-VCH Verlag GmbH. D-69469 Weinheim (Federal Republic of Germany), 1999 Printed on acid-free and chlorine-free paper. All rights reserved (including those of translation in other languages). No part of this book may be reproduced in any form - by photoprinting, microfilm, or any other means - nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Composition and Printing: Konrad Triltsch, Druck- und Verlagsanstalt GmbH, D-97070 Wurzburg. Bookbinding: J. Schaffer GmbH & Co. KG, D-67269 Griinstadt Printed in the Federal Republic of Germany.
Preface This text has grown out of the chapter Ferroelectric Liquid Crystals which I wrote for the Handbook of Liquid Crystals, published in 1998. Although this was an unusually large chapter in the Handbook, the available space and time could not possibly permit a full coverage of the subject. The editors therefore proposed to extend the material into a monograph, which then also ought to cover antiferroelectric materials. In fact, not to treat the different kinds of polar order from a unified point of view would be to set artificial borders between indissolubly connected phenomena. The new text is substantially extended in several ways. This does not only concern the discussion of antiferroelectric materials, but foremost the inclusion of two chapters (11 and 12) on the elastic continuum theory of smectics, in particular smectic C and smectic C*. This description, which is indispensible in order to understand the peculiar intrinsic smectic C* properties, of great importance also for the applications of the material, has so far been absent in the literature. (It is not even treated in the Handbook.) Much of this discussion concerns the spontaneous bend deformations present in smectic C* and is written in a language - this has been the aim of the whole book - accessible to both experimentalists and theoreticians. Corresponding to its character of monograph I have tried to have a fair balance in the text between basic physics and applications. I have further attempted the treatment to be self-contained as far as possible, in order to give it likewise the character of textbook. Therefore, priority has also been given to conceptual clarity. Essentially all important equations have been derived from basic principles. In the same spirit, the text contains a quite detailed and in may opinion, necessary, introduction to the Landau formalism in the description of phase transitions. In comparison with the Handbook article this part has been further extended by an introductory discussion of order parameters, including many examples, in sections 2.5 and 2.6. The inclusion has necessitated a numbering of equations in those sections which deviates from that of the rest of the book, but otherwise should not affect the reading. The most important events in the subject-matter of polar liquid crystals can be traced back to 1969, 1975 and 1980. Serious industrial involvement began in 1985 (Canon Inc., Tokyo) after some year of initial testing. The first international conference on ferroelectric liquid crystals (FLCs) was held, on French initiative, in Arcachon (Bordeaux) in 1987 and six more such conferences have since been held: 1989 Goteborg (Sweden), 1991 Boulder/Colorado (USA), 1993 Tokyo, 1995 Cambridge (UK) and 1997 Brest (France). However, outside of these special conferences, the topic has been a dominating one also at the general liquid crystal conferences since the mid go’s, and it still is. The reason lies, no doubt, in its rich physics and chemistry, which continually pours out new surprises. Many of these have been very detrimental for applications and have required extraordinary efforts to cope with. As the
VI
Preface
industrial pioneer, Canon not only discovered most of those but had to overcome them, which they did in an outstanding combination of academic research and applied display work. However, neither is, by far, the physics and chemistry of these materials exhausted, nor are they sufficiently understood and mastered, theoretically and experimentally. Liquid crystals are a delight to the condensed matter physicist. Concepts developed to understand magnetism, superfluid helium and superconductivity have shown their unifying power when applied to liquid crystals, as they have when applied to nuclear matter, particle physics and weak and electromagnetic interactions. But liquid crystals are also competing with other technologies for large-area high-resolution displays, which are considered to be the real bottleneck and therefore halting the otherwise very rapid development in information technology. Liquid crystals hold the first position in this area, but the high definition LCD-TV does not seem to be around the comer as has usually been claimed during the last fifteen years. For personal computer screens though, both lap-top and desk-top, the combination nematic-TFT has few rivals. Within liquid crystals a number of different technologies compete. With the exception of antiferroelectric (AFLC) displays it does not seem likely that passively addressed smectic screens will be able to compete in the area of PCs, due to the substantial decrease in cost for transistor arrays. Competing liquid crystal technologies also profit from each other. For instance, the FLC technology has shown that it is feasible to go down to much lower cell thickness (-2-4 pm) and other technologies have followed with a resulting increase in performance. The symmetrically bistable in-plane switching of FLCs has also inspired both the development of nematic in-plane switching (IPS) devices and the present development of fast-switching symmetrically bistable nematics. On the other hand, it is likely that FLC will now in turn profit from the rapid development in TFTs and silicon backplanes, because no nematic-TFT combination can compete in performance with FLC-TFT or “FLT”, in particular for the forthcoming market of TV, the biggest flat screen market beyond comparison. The FLT development is the newest aspect of liquid crystal displays. In this book the discussion of applications has been limited to displays. The main reason is that non-display FLC applications are extensively covered in the Handbook article by W. A. Crossland and T. D. Wilkinson. The book does also not discuss the chemistry of FLCs, neither in basic nor applied aspects. These things are covered in the Handbook article by S . M. Kelly and in those by J. W. Goodby. Finally, the polymer FLCs are treated very superficially, as illustrative applications and the discotic FLCs not at all. Again, the reader is referred to Handbook articles by R. Zentel and by J. C. Dubois, P. Le Bamy, M. Mausac and C. Noel, in the first case, in the second case by A. N. Cammidge and R. J. Bushby and by N. Boden and B. Movaghar. As always however, there are certainly topics I wish I would have discussed, had more space and time been available. To these belong, in particular, the dielectric properties of FLC and AFLC materials, an the non-linear optical properties, but al-
Preface
VII
so the very important and physically intriguing phenomena of electro-mechanical properties (pioneered by the Budapest group), in field-induced rotational instabilities (Goteborg) and the newest experimental results with relevance to the Landau description (Clausthal, Minneapolis). I due a lot of what I have written to my long-time colleagues and collaborators Noel Clark and David Walba at the University of Colorado in Boulder, with whom I always have had the most inspiring and profitable discussions. I likewise due a lot to my colleague Bengt Stebler in Goteborg and to my younger collaborators Joe Maclennan (Boulder), David Hermann and Per Rudquist (Goteborg). With Marek Maruszczyk I have had long discussions about device structures and, in particular, chevron geometries, and he has transferred my handmade sketches of not only those corresponding figures but the majority of the figures in the book, to a shape suitable for publication. The others have been prepared by Tomasz Mastuszczyk and by Jan Lagerwall. My wife has typed this manuscript (as so many others) without complaining, in spite of her own professional activity as an artist. I am also indebted to Dr. Jorn Ritterbusch at Wiley-VCH, who with much patience and encouragement has guided me through this project. Finally, my special thanks go to Dr. H.-R. Dubal of Hoechst, now Clariant, who during many years has been a constant source of support and inspiration. Sven Torbjorn Lagerwall
This Page Intentionally Left Blank
Contents List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi11
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Polar Materials and Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polar and Nonpolar Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nonpolarity of Liquid Crystals in General . . . . . . . . . . . . . . . . . . . Behavior of Dielectrics in Electric Fields: Classification of Polar Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Developments in the Understanding of Polar Effects . . . . . . . . . . . . . . The van der Waals Attraction and Born’s Mean Field Theory . . . . . . . . Landau Preliminaries . The Concept of Order Parameter . . . . . . . . . . . . The Simplest Description of a Ferroelectric . . . . . . . . . . . . . . . . . . . . . Improper Ferroelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Piezoelectric Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 16 22 29 40 48 51
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10
The Necessary Conditions for Macroscopic Polarization . . . . . . . . . The Neumann and Curie Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . Franz Neumann. Konigsberg. and the Rise of Theoretical Physics . . . . Neumann’s Principle Applied to Liquid Crystals . . . . . . . . . . . . . . . . . The Surface-Stabilized State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chirality and its Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Curie Principle and Piezoelectricity ........................ Hermann’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Importance of Additional Symmetries . . . . . . . . . . . . . . . . . . . . . . Optical Activity and Enantiomorphism . . . . . . . . . . . . . . . . . . . . . . . . . Non-Chiral Polar and NLO-Active Liquid Crystals . . . . . . . . . . . . . . .
57 57 58 61 63 72 76 79 81 83 88
4
The Flexoelectric Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Deformations from the Ground State of a Nematic . . . . . . . . . . . . . . . . 93 The Flexoelectric Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 The Molecular Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Analogies and Contrasts to the Piezoelectric Effect . . . . . . . . . . . . . . . 97 The Importance of Rational Sign Conventions . . . . . . . . . . . . . . . . . . . 97 99 Singularities are Charged in Liquid Crystals . . . . . . . . . . . . . . . . . . . . . The Flexoelectrooptic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Why Can a Cholesteric Phase not be Biaxial? . . . . . . . . . . . . . . . . . . . . 105
4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
1
7 7 10
X
4.9 4.10
Contents
Flexoelectric Effects in the Smectic A Phase ..................... Flexoelectric Effects in the Smectic C Phase . . . . . . . . . . . . . . . . . . . . .
106 107
The SmA*-SmC* Transition and the Helical C* State . . . . . . . . . . The Smectic C Order Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The SmA*-SmC* Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Smectic C* Order Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Helical Smectic C* State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Flexoelectric Contribution in the Helical State . . . . . . . . . . . . . . . . 5.5 Nonchiral Helielectrics and Antiferroelectrks .................... 5.6 Mesomorphic States without Director Symmetry . . . . . . . . . . . . . . . . . 5.7 Simple Landau Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 The Electroclinic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 5.10 The Deformed Helix Mode in Short Pitch Materials . . . . . . . . . . . . . . . 5.11 The Landau Expansion for the Helical C* State . . . . . . . . . . . . . . . . . . 5.12 The Pikin-Indenbom Order Parameter . . . . . . . . . . . . . . . . . . . . . . . . .
115 115 118 129 131 134 135 138 140 147 154 156 161
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9
Electrooptics in the Surface-StabilizedState . . . . . . . . . . . . . . . . . . . The Linear Electrooptic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Quadratic Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Switching Dynamics ....................................... The Scaling Law for the Cone Mode Viscosity . . . . . . . . . . . . . . . . . . . Simple Solutions of the Director Equation of Motion . . . . . . . . . . . . . . Electrooptic Measurements .................................. Optical Anisotropy and Biaxiality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Effects of Dielectric Biaxiality . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Viscosity of the Rotational Modes in the Smectic C Phase ......
169 169 172 175 178 179 180 185 187 191
7
Dielectric Spectroscopy:
5 5.1
7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10
.................... .......................
Components . . . . . . . . . . . . . . . . . . . . . . . . . Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modes . . . . . . . . . . . . . . . . . . . . . . . . . Measurements ...................
........................ .............................. .......................... ................ ............. The
from .............................................................. Smectic ..................................... Three ...................................................... ............................................................ Smectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Methods . . . . . . . . . . . . . . . . . . . . . . . . . . Limitations
Contents
XI
8 8.1 8.2 8.3 8.4 8.5 8.6
FLC Device Structures and Local-Layer Geometry . . . . . . . . . . . . . 215 215 The Application Potential of FLC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Surface-Stabilized States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FLC with Chevron Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 226 Analog Grey Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Thin Walls and Thick Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 C1 and C2 Chevrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 9.1 9.2 9.3 9.4 9.5 9.6
FLCDevices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The FLC Technology Developed by Canon . . . . . . . . . . . . . . . . . . . . . . The Microdisplays of Displaytech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Idemitsu's Polymer FLC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Stuttgart Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Problems in FLC Technology . . . . . . . . . . . . . . . . . . . . . . . . . Nonchevron Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
241 241 244 245 247 255 257
Digital Grey and Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10.1 Analog versus Digital Grey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Spatial and Temporal Dither . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
263 263 265
Elastic Properties of Smectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Continuum Description of the Smectic A Phase . . . . . . . . . . . . . . . . . . 275 Continuum Description of the Smectic C Phase . . . . . . . . . . . . . . . . . . 282 The Smectic C Continuum Theory in the Local Frame of Reference . . 286 1 1.4 The Case of Undistorted Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 11.5 The Elastic Energy Expression for Smectic C" . . . . . . . . . . . . . . . . . . . 295 I 1.6 The Energy Expression in an Electric Field . . . . . . . . . . . . . . . . . . . . . . 299
11 11.1 11.2 11.3
12 12.1 12.2 12.3 12.4 12.5
Smectic Elasticity Applied to SSFLC Cells . . . . . . . . . . . . . . . . . . . . The P (9) - c (p) Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helical States. Unwinding and Switching . . . . . . . . . . . . . . . . . . . . . . . Splayed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristic Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Electrostatic Self-Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
301 301 306 312 316 319
13 Antiferroelectric Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 13.I The Recognition of Antiferroelectricity in Liquid Crystals . . . . . . . . . . 326 1 3.2 Half-Integral Disclinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 335 13.3 Antiferroelectric and Ferrielectric Phases . . . . . . . . . . . . . . . . . . . . . . . 13.4 A Complicated Surface Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 13.5 Landau Descriptions of Antiferroelectric and Ferrielectric Phases . . . . 346
XI1
Contents
13.6 13.7 13.8 13.9 13.10 13.11
Ising Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helix Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Present Understanding of the Antiferroelectric Phases . . . . . . . . . . Freely Suspended Smectic Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antiferroelectric Liquid Crystal Displays . . . . . . . . . . . . . . . . . . . . . . . Thresholdless Smectic Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
349 358 369 376 383 390
14
Current Trends and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
401
15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
405
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
417
List of Symbols and Abbreviations general coefficient; Landau expansion coefficient amplification at zero feedback area surface area of pixel SmC layer torsional constants vector potential Burgers vector Landau expansion coefficient Compressional elastic constant for SmA SmC elastic constant in the one-constant approximation SmC in-layer elastic constants bend vector Landau expansion coefficient concentration c director components elastic constants of solids c director (local tilt direction along smectic layer) Curie constant SmC* elastic constants chiral coupling coefficient (between tilt and polarization) cell gap thickness, sample thickness, width cross diameter of molecule smectic layer thickness, stereospecific length SmA layer periodicity SmC layer periodicity piezocoefficients dielectric displacement first order SmC* elastic constants charge of electron electroclinic coefficient flexoelectric constants for bend and splay, respectively SmC flexoelectric coefficients flexoelectric coupling coefficient in Landau expansion structure coefficient applied electric field external influence symmetry group activation energy ac signal aligning field threshold field
List of Symbols and Abbreviations
probability relaxation frequency of Goldstone mode relaxation frequency of soft mode relaxation frequency for molecular rotation around long axis relaxation frequency for molecular rotation around short axis force elastic energy density, free energy density free energy density due to c director distortions free energy density due to coupling of in-layer and layer distortions nematic-like and polar surface interactions, respectively total surface energy from G , and Gp free energy density due to layer strains hamiltonian magnetic field strength electric current intensity of transmitted light Boltzmann constant cholesteric wave vector local smectic layer normal elastic constant in the one-constant approximation KI1,K2*,K 3 3 ,K24 Oseen elastic constants distance between singularities, penetration length 1 L Langevin function L Lifshitz invariant L sample thickness unit vector along chloesteric axis m M magnetization n refractive index n1, n2, n3 director components n director All birefringence n (r) director field molecular density, number of layers, number of spins N N Pilun-Indenbom order-parameter P dipole moment P secondary order parameter P cholesteric pitch local polarization direction (= k x c ) in a smectic layer P P polarization PI p2 polarization of sublattices p2 second Legendre polynomial pf flexoelectric polarization 3
List of Symbols and Abbreviations
Tlj
Tr U
U U
U U V V W
W WS X
XV
induced polarization mesoscopic polarization magnitude of spontaneous polarization per radian of tilt spontaneous polarization primary order parameter helical wave vector bend vector twist wave vector wave vector value of helical smectic C* wave vector at the A*-C* transition total charge saturation charge nematic tensor order parameter polar coordinate in three dimensions, rank of tensor radius vector (spatial variable) radial distance relative concentration of enantiomer ( R ) , resistance strain electrostrictive strain piezoelectric strain reduced (scalar) nematic order parameter relative concentration of enantiomer (S) splay vector transmission coefficient time temperature transmitted light intensity Curie temperature, critical temperature, transition temperature rotation matrices second rank tensor trace of matrix layer displacement along z direction total energy per unit area distortion field energy potential energy of dipole in electric field voltage volume wall thickness equilibrium width for thick wall surface anchoring energy distance across cell
XVI
List of Symbols and Abbreviations
axes in Cartesian reference frame layer normal SmC* helical periodicity
a a a
7P Y Y/?Yt
angular velocity general coefficient, Landau expansion coefficient polarizability thermal expansion coefficient feedback coefficient strength of spontaneous bend general viscosity longitudinal (long axis) and transverse (short axis) rotational viscosities, respectively twist viscosity viscosity components in molecular frame of reference surface coupling constants Goldstone mode viscosity electroclinic or soft mode viscosity viscosity tensor in diagonal form total torque dielectric torque torque acting on director tilt viscous torque dielectric constant or permittivity permittivity of free space permittivity principal values in molecular frame of reference dielectric anisotropy dielectric constant contribution from Goldstone mode permittivity of sample in quasi-homeotropic geometry permittivity of QBS sample in presence of helix dielectric constant for planar alignment relative dielectric constant dielectric constant for racemate dielectric constant contribution from soft mode permittivity of QBS sample for unwound helix permittivity tensor in diagonal form dielectric anisotropy ( E ~ - E ~ ) dielectric biaxiality ( E ~ - E , ) measure of dielectric nonlinearity order parameter Eulerian nutation angle angular part of polar coordinates
List of Symbols and Abbreviations
tilt angle with respect to layer normal enantiomeric excess coupling constant Weiss (molecular field) proportionality constant surface strength wavelength of light magnetic moment characteristic length tilt vectors dielectric coherence length magnetic coherence length polarization coherence length measure of dielectric nonlinearity radial part of polar coordinates polarization charge density applied stress surface charge density electrical conductivity characteristic time response time response time due to dielectric torque response time due to ferroelectric torque response time for electroclinic effect azimuthal angle tilt angle of optic axis Eulerian precession angle dimensionless parameter susceptibility anisotropy of magnetic susceptibility azimuthal angle indicating direction of tilt in layer plane complex order parameter Eulerian angle of eigen rotation biquadratic coupling coefficient angular rotation araound the i axis chevron angle
AC, ac AFLC AMLCD ANN1 ANNNI 8CB
alternating current antiferroelectric liquid crystal active matrix liquid crystal display axial nearest neighbor king axial next nearest neighbor Ising cyanobiphenyl compound
XVII
XVIII CMOS DC, dc DHM DOBAMBC ED FLC FLCD FLCP HDTV HF HOBACPC IC IT0 KDP LCD LED MHPOBC MHTAC NLO NMR NOBAPC 0.a. PAR PC PES PET PI QBS RGB SmA SmA*
SmC SmC* SHG SSFLC STN TFT TGB TN
uv
VGA VLSI XGA
List of Symbols and Abbreviations
complementary metal oxide semiconductor direct current deformed helix mode decyloxybenzylidene amino 2-methyl butyl cinnamate error diffusion ferroelectric liquid crystal ferroelectric liquid crystal display ferroelectric liquid crystal polymer high definition television high frequency hexy loxybenzylidene amino 2-chloro a-propyl-cinnamate integrated circuit indium tin oxyde KH2P04 liquid crystal display light emitting diode methyl heptyloxycarbonyl phenyl octyloxy biphenyl carboxylate methyl heptyl terephtalylidene-bis-4-aminocinnamate nonlinear optics nuclear magnetic resonance 4-nonyloxybenzylidene-4'-aminopentyl cinnamate optic axis pol yarylate polycarbonate poly estersulfonate poly(ethy1ene terephthalate) polyimide quasi-bookshelf red- green- blue smectic A chiral smectic A smectic C chiral smectic C second-harmonic generation surface-stabilized ferroelectric liquid crystal supertwisted nematic thin film transistor twist grain boundary twisted nematic ultraviolet virtual graphic adapter very large scale integration extended graphic adapter
1 Introduction Ferroelectric liquid crystals are a novel state of matter, a very recent addition to the science of ferroelectrics which, in itself, is of relatively recent date. The phenomenon which was later called ferroelectricity was discovered in the solid state (on Rochelle salt) in 1920 by Joseph Valasek, then a PhD student at the University of Minnesota. His first paper on the subject [ 11 had the title Piezo-Electric and Allied Phenomena in Rochelle Salt. This was at the time when solid state physics was not a fashionable subject and it took several decades until the importance of the discovery was recognized. Valasek had then left the field. Later, however, the development of this branch of physics contributed considerably to our understanding of the electrical properties of matter, of polar materials in particular and of phase transitions and solid state physics in general. In fact, the science of ferroelectrics is today an intensely active field of research. Even though its technical and commercial importance is substantial, many breakthrough applications may still lie ahead of us. The relative importance of liquid crystals within this broader area is also constantly growing. This is illustrated in Fig. 1, showing how the proportion of the new materials, which are liquid-crystalline, has steadily increased since the 1980s. The general level of knowledge of ferroelectricity, even among physicists, is far lower than in the older and more classical subjects like ferromagnetism. It might therefore be worthwhile to discuss briefly the most important and characteristic fea-
250
-
I
I
I
,
I
I
liquid crystals
'
5:
Figure 1. Number of known ferroelectrics. Solid line: solid state ferroelectrics, where each pure compound is counted as one. Dashed line: total number, including liquid crystal ferroelectrics, for which a group of hoinologs is counted as one. From about 1984, the proportion of liquid crystals has steadily grown which has been even more pronounced after 1990. (After Deguchi [2] as cited by Fousek [3].)
200
-
50
-
L
1920 1930 1940 1950 1960 1970 1980 1990 year
2
1
Introduction
tures of solid ferroelectrics and polar materials, before turning to liquid crystals. This will facilitate the understanding and allow us to appreciate the striking similarities as well as distinctive differences in how polar phenomena appear in solids and how they appear in liquid crystals. One of the aims, of course, is also to make a bridge to existing knowledge. Those not aware of this important knowledge are apt to coin new words and concepts, which are bound to be in contradiction to already established concepts or even contradictory to themselves. When dealing with ferroelectric liquid crystals, we use the same conceptual framework already developed for solid polar materials. An important part of this is the Landau formalism describing phase transitions (still not incorporated in any textbook on thermodynamics), based on symmetry considerations. It is important to gain some familiarity with the peculiarities of this formalism before applying it to ferroelectric liquid crystals. In this way it will be possible to recognize cause and effect more easily than if both subject matters were introduced simultaneously. Concepts like piezoelectric, pyroelectric, ferroelectric, ferrielectric, antiferroelectric, paraelectric, electrostrictive, and several more, relate to distinct phenomena and are themselves interrelated. They are bound to appear in the description of liquid crystals and liquid crystal polymers, as they do in normal polymers and crystalline solids. Presently, great confusion is created by the uncritical use of these terms. For example, in the latest edition of the Encyclopedia Britannica [4]it is stated that pyroelectricity was discovered in quartz in 1824. This is remarkable, because quartz is not pyroelectric at all and cannot be for symmetry reasons. To clarify such issues (and the confusion is no less in the area of liquid crystals), we will have to introduce some simple symmetry considerations that generally apply to all kinds of matter. In fact, symmetry considerations will be the basic guidelines and will probably play a more important role here than in any other area of liquid crystals. Chirality is a special property of dissymmetry with an equally special place in these considerations. It certainly plays a fundamental role for ferroelectric liquid crystals at least so far. Therefore we will have to check how exactly the appearance of polar properties in liquid crystals is related to chiral properties, and if chirality is dispensable, at least in principle. Finally, flexoelectricity is also a polar effect, and we will have to ask ourselves if this is included in the other polar effects or, if not, if there is an interrelation. Can liquids in which the constituents are dipoles be ferroelectric? For instance, if we could make a colloidal solution of small particles of the ferroelectric BaTiO,, would this liquid be ferroelectric? The answer is no, it would not. It is true that such a liquid would have a very high value of dielectric susceptibility and we might call it superparaelectric in analogy with the designation often used for a colloidal solution of ferromagnetic particles, which likewise does not show any collective behavior. An isotropic liquid cannot have polarization in any direction, because every possible rotation is a symmetry operation and this of course is independent of whether the liquid lacks a center of inversion, is chiral, or not. Hence we have at least to di-
1
Introduction
3
minish the symmetry and go to anisotropic liquids, that is, to liquid crystals, in order to examine an eventual appearance of pyroelectricity or ferroelectricity.To search for ferroelectricity in an isotropic liquid would be futile, because a ferroelectric liquid cannot be isotropic. In order to have a bulk polarization, a medium must have a direction, the polarity of which cannot be reversed by any symmetry operation of the medium. On the other hand, an isotropic liquid consisting of dipoles may show a polarization during flow, because a shear diminishes the symmetry and will partially order the dipoles, thus breaking the randomness. This order will be polar if the liquid is chiral. However, we would not consider such a liquid ferroelectric or pyroelectric - no more than we would consider a liquid showing flow birefringence to be a birefringent liquid. It is clear that there may be lots of interesting polar effects yet to be explored in flowing liquids, particularly in fluids of biological significance (which are very often chiral). Nevertheless, these effects should not be called “ferroelectric”. They should not even be called piezoelectric, even if setting up shear flow in a liquid certainly bears some resemblance to setting up shear strain in a crystal. Are there magnetic liquids? Yes, there are. We do not mean the just-mentioned “ferrofluids”, which are not true magnetic liquids, because the magnetic properties are due to the suspended solid particles (of about 10 nm size). As we know, ferromagnetic materials become paramagnetic at the Curie temperature and this is far below the melting point of the solid. However, in some cases it has been possible (with extreme precautions) to supercool the liquid phase below the Curie temperature. This liquid has magnetic properties, though it is not below its own Curie temperature (the liquid behaves as if there is now a different Curie temperature), but it would be wrong to call the liquid ferromagnetic. The second example is the equally extreme case of the quantum liquid He-3 in the A1 phase. Just like the electrons in a superconductor, the He-3 nuclei are fermions and have to create paired states to undergo Bose-Einstein condensation. However, unlike the electron case, the pairs are created locally, and the axis between two He-3 then corresponds to a local director. Thus He-3 A is a kind of nematic-like liquid crystal, and because of the associated magnetic moment, He-3 A is undeniably a magnetic liquid, but again, it is not calledferromagnetic. Thus, in the science of magnetism a little more care is normally taken with terminology, and a more sound and contradiction-free terminology has been developed: a material, solid or liquid, may be designated magnetic, and then what kind of magnetic order is present may be further specified. When it comes to polar phenomena, on the other hand, there is a tendency to call everything “ferroelectric”, a usage that leads to tremendous confusion and ambiguity. It would be very fortunate if in future we could reintroduce the more general concepts of “electric materials” and “electric liquids” in analogy with magnetic materials and magnetic liquids. Then, in every specific case, it would be necessary to specify which electric order (paraelectric, dielectric, etc.) is present, just as in the magnetic case (paramagnetic, diamagnetic, etc.).
4
1
Introduction
Coherent and contradiction-free terminology is certainly important, because vagueness and ambiguity are obstacles for clear thinking and comprehension. In the area of liquid crystals, the domain of ferroelectric and antiferroelectric liquid crystals probably suffers from the greatest problems in this respect, because in the implementation of ideas, concepts, and general knowledge from solid state physics, which have been of such outstanding importance in the development of liquid crystal research, the part of ferroelectrics and other polar materials has generally not been very well represented. Presumed ferroelectric effects in liquid crystals were reported by Williams at RCA in Princeton, U. S. A., as early as 1963, and thus at the very beginning of the modern era of liquid crystal research [5]. By subjecting nematics to rather high dc fields, he provoked domain patterns that resembled those found in solid ferroelectrics. The ferroelectric interpretation seemed to be strengthened by subsequent observations of hysteresis loops by Kapustin and Vistin [6] and by Williams and Heilmeier [7]. However, these patterns turned out to be related to electrohydrodynamic instabilities, which are well understood today (see, for instance, [8], Sec. 4.3 or [9], Sec. 4.2), and it is also well known that certain loops (similar to ferroelectric hysteresis) may be obtained from a nonlinear lossy material (see [lo], Sec. 4.2). As we know today, nematics do not show ferroelectric or even polar properties. In order to find such properties we have to lower the symmetry until we come to the tilted smectics, and further lowering their symmetry by making them chiral. The prime example of such a liquid crystal phase is the smectic C*. In principle, the fascinating properties of the smectic C* phase could have been detected long before their discovery in 1974. Such materials were synthesized by Vorlander [ 111 and his group in Halle before the first World War. The first one seems to have been an amyloxy terephthal cinnamate with a smectic C* phase from 133"C to 247 "C and a smectic A* phase from 247 "C to 307 "C,made in 1909 [ 111 at a time far earlier than the first description of the smectic C phase as such [12] in 1933. At that time it was not, and could not possibly have been realized as ferroelectric. The concept did not even exist. In a classic review from 1969, Saupe (at Kent State University) discussed a hypothetical ferroelectric liquid crystal for the first time [ 131. While a nematic does not have polar order, such order, he pointed out, could possibly be found in the smectic state. The ferroelectric smectic, according to Saupe, is an orthogonal nonchiral smectic in which all molecular dipoles are pointing along the layer normal in one single direction (a longitudinal ferroelectric smectic). He also discussed a possible antiferroelectric arrangement. Among the other numerous topics discussed in this paper (suggesting even the first blue phase structure), Saupe investigated the similarities between nematics and smectics C and introduced the twisted smectic structure as the analog of a cholesteric. In the same year, Gray in Hull [ 141 synthesized such materials (actually the first members of the DOBAMBC series), but only reported on an orthogonal smectic phase; no attention was paid to smectic polymorphism in those
1
Introduction
5
days. Actually, in the year before, Leclerq et al. [ 151 (in Paris) had reported on a material having two distinct chiral and strongly optically active phases, which they interpreted as two distinct nematic phases. They were thus very close to discovering the helicoidal smectic. (Their material had a first order N*-C* transition.) A helicoidal smectic was then reported for the first time by Helfrich and Oh, at RCA in 1971, who described the first smectic liquid crystal (“spiraling” or “conical” smectic) identified as optically active [16]. Like the substances in the above-mentioned examples, this one belongs to the category that is the topic of this book, but who could have expected them to have special polar properties? While the smectic C* phase was gradually becoming recognized, the question of ferroelectricity was again brought up by McMillan at the University of Illinois, Urbana, U. S. A. In 1973 he presented a microscopic molecular theory of the smectic C phase [ 171 based on dipole-dipole interactions, which predicted three different polar phases. McMillan’s model molecules have a central dipole and two outboard dipoles perpendicular to the long axis. Either all three can line up or only the outboard ones with the central dipoles random, or the central dipoles can line up with the outboard dipoles random. The transition from the A phase to these polar phases is thought to take place through different rotational transitions where the rotational freedom is lost or frozen out due to the dipole-dipole interaction. The net polarization in the condensed phases lies in the smectic plane and gives rise to a two-dimensional ferroelectric. Actually, whether the order is ferroelectric or antiferroelectric depends on the sign of the interplanar interaction, which cannot be predicted. McMillan’s dipolar theory, which does not involve chirality at all, never really applied to liquid crystals and was rapidly superseded by the ideas of Meyer in the following year. In fact, the discovery and introduction of practically all polar effects in liquid crystals go back to the ideas of Meyer, at that time working at Harvard. In 1969 he published an epoch-making paper entitled Piezoelectric Effects in Liquid Crystals [ 181. It must be said that the new phenomena described in that paper are beautifully analogous to piezoelectric effects in solids. Nevertheless they are of a different nature. Therefore de Gennes instead proposed the name flexoelectric [19], in order to avoid misunderstanding. Seven years later, together with his student Garoff, Meyer presented a new, original effect which he called the piezoelectric effect in smectic A liquid crystals [20]. The analogies between this effect and the piezoelectric effect in solids are here perhaps even more striking, as we will see. However, it is not the same thing and, after much consideration by the authors, the new phenomenon was finally published under the name electroclinic effect [21], a term which has since been generally adopted. In the following twenty years, there were numerous publications, in which different workers reported measuring a piezoelectric effect in liquid crystals (normally without stating what it meant and why they used this term). Obviously, they meant neither the flexoelectric nor the electroclinic effect, because the meanings of these are by now well established. Therefore the question arises as to whether a third effect exists in liquid crystals, which would finally qualify for the name pi-
6
1 Introduction
ezoelectric. Obviously, this state of affairs is not very encouraging. A critical review of the terminology is therefore necessary in this area and should contribute to clarifying the concepts. Ferroelectric liquid crystals have been a field of research for about twenty years, and have certainly been in the forefront of liquid crystal research, with an increasing number of researchers involved. The first state-of-the-art applications have recently appeared. This account concentrates on the basic physics, but also treats in considerable detail the topics of highest relevance for applications. Literature references have been given, as far as possible, for topics that, for space reasons, could not be treated [8- 10,22-581. A big help for finding access to previous work is the bibliography of [52],which extends to 1989, as well as the series of conference proceedings published by Ferroelectrics [54-58a], covering a great deal of the work from 1987 to 1997. New ferroelectric liquid crystal materials are continually included in the Liqcryst-Database [59] set up by Vill at the University of Hamburg.
2 Polar Materials and Effects 2.1 Polar and Nonpolar Dielectrics A molecule that has an electric dipole moment in the absence of an external electric field is called a polar (or dipolar) molecule. Such a molecule will tend to orient itself in an electric field. In a material consisting of polar molecules, the induced polarization P due to the average molecular reorientation is typically 10-100 times larger than the contribution from the electronic polarization present in all materials. In contrast, a nonpolar molecule has its distributions of positive and negative charges centered at the same point. A characteristic of materials consisting of nonpolar molecules is that the polarization P induced by a field E is small and independent of temperature, whereas in the first-mentioned case, P is a function P(T)with an easily observable temperature dependence. Hence the same goes for the dielectric constant E and the susceptibility If we write the relations between dielectric displacement D , induced polarization P , and applied electric field E , assuming that P is linear in E
x.
we get
with E~ as the relative, E as the total dielectric constant or permittivity, and % the perC V-' m-'. It will later be necessary mittivity of free space, where %=8.85x to consider that E and in reality are second rank tensors, and in that context we will write Eqs. (1) and (2) in the corresponding forms
x
and (2.2b)
x
The scalar dielectric susceptibility in Eq. (2) is, like E,, a dimensionless number, and lies in the range 0- 10 for most materials, although it may attain values higher than lo4 for certain ferroelectric substances. We will equally use this term, the susceptibility, for its dimensional form = aP/aE. As examples of nonpolar molecules we may take H,, O,, CO,, CS,, CH,, and CCl,, and as well-known polar molecules CO (0. lo), NH, (1.47), C,H,OH (1.70),
8
2 Polar Materials and Effects
H,O (1.85), C,H5NO2 (4.23),where we have stated the value of the dipole moment in parentheses, expressed in Debye (D), a unit commonly used for molecules. One Debye equals lo-'' cgs units and, expressed in SI units, 1 D = 3 . 3 x lop3' C m. For a comparison, let us consider a dipole consisting of two elementary charges + e (i.e., with e the charge of the electron, 1 . 6 ~ C) at a typical atomic distance of 1 %, or 0.1 nm from each other. This gives a dipole moment p = 4 d = 1 . 6 O-I9x ~ lo-'' = 1.6x 0-29C m equal to 4.85 D. Let us assume that we had a gas consisting of molecules with this dipole moment and that we had a field sufficiently strong to align the dipoles with the field. With a density N = 3 x lo2' molecules/m3 this would correspond to a polarization of P = 5 x lo4 C m-,= 50 nC cm-'. However, this is a completely unrealistic assumption because the orientational effect of the field is counteracted by the thermal motion. Therefore the distribution of dipolar orientation is given by a Boltzmann factor ePUlkT, where U=-p . E is the potential energy of the dipole in the electric field. Integrating over all angles forp relative to E gives the polarization P as a function of E according to the Langevin function L (to be discussed in Sec. 2.5) which expresses the average of cos(p,E), P = N~L(')
(3)
shown in Fig. 2. For small values of the argument, L [pEl(kT)]-pEl(3kT), and thus
p = - N P E~ 3kT
(4)
corresponding to the linear part around the origin. However, even at a field E = lo7 V m-', corresponding to dielectric breakdown, the value of pEl(kT) is only about 0.03 at room temperature, giving a resulting polarization of 1% of the saturation value. A similar result would be true for the liquid phase of the polar molecules. In liquid crystal phases, it will generally be even harder to polarize the medium in an external field. In the very special polar liquid crystals, on the other hand, the reverse is true: for quite moderate applied fields it is possible to align all dipoles, cor-
L(x)
I .o
0.8
-
1
2
3
I
I
I
I
4
5
6
7
x=pmT
Figure 2. The Langevin function L(~)=COthx-l/X.
2.1
9
Polar and Nonpolar Dielectrics
responding to polarization values in the range of 5 -500 nC cmP2(depending on the substance). According to Eq. (4), the polarization, at constant field, grows when we lower the temperature. By forming aP/aE, we may write for the susceptibility
x=-C
(5)
T
The fact that the susceptibility has a 1/T dependence, called the Curie law, is characteristic for gases and liquids, but may also be found in solids. Generally speaking, it indicates that the local dipoles are noninteracting. For another comparison, consider a crystal of rock salt, NaCl. It has a value of 4.8. If we apply the quite high but still realistic field of lo6 Vm-’ (1 V pm-’) we will, according to Eq. (2), induce a polarization of 4.25 nC cm-2 (or 42.5 pC m-2; the conversion between these units commonly used for liquid crystals is 1 nC = 10 pC mP2).The mechanism is now the separation of ionic charges and thus quite different from our previous case. It turns out that the displacement of the ions for this polarization is about nm (0.1 i.e., it represents only a small distortion of the lattice, less than 2%. Small displacements in a lattice may thus have quite strong polar effects. This may be illustrated by the solid ferroelectric barium titanate, which exhibits a spontaneous polarization of 0.2 C m-2 = 20 000 nC cm-*. Responsible for this are ionic displacements of about lop3nm, corresponding to less than half a percent of the length of the unit cell. Similar small lattice distortions caused by external pressure induce considerable polar effects in piezoelectric crystals. When atoms or molecules condense to liquids and solids the total charge is zero. In addition, in most cases the centers of gravity for positive and negative charges coincide. The matter itself is therefore nonpolar. For instance, the polar water molecules arrange themselves on freezing to a unit cell with zero dipole moment. Thus ice crystals are nonpolar. However, as already indicated, polar crystals exist. (In contrast, elementary particles have charge, but no dipole moment is permitted by symmetry.) These are then macroscopic dipoles and are said to be pyroelectric materials, of which ferroelectrics are a subclass. Pyroelectric materials thus have a macroscopic polarization in the absence of any applied electric field. Piezoelectric materials can also be polarized in the absence of an electric field (if under strain) and are therefore sometimes also considered as polar materials, though the usage is not general nor consistent. Clearly their polarization is not as “spontaneous” as in the pyroelectric case. Finally, there seems to be a consensus about the concepts of polar and nonpolar liquids. Water is a polar liquid and mixes readily with other polar liquids, i.e., liquids consisting of polar molecules, like alcohol, at least as long as the sizes of the molecules are not too different, whereas it is insoluble in nonpolar liquids like benzene. If in liquid form, constituent polar molecules interact strongly with other polar molecules and, in particular, are easily oriented in external fields. We will also
x
A),
10
2 Polar Materials and Effects
use this criterion for a liquid crystal. That is, we will call a liquid crystal polar if it contains local dipoles that are easily oriented in an applied electric field.
2.2 The Nonpolarity of Liquid Crystals in General The vast majority of molecules that build up liquid crystal phases are polar or even strongly polar. As an example we may take the cyanobiphenyl compound 8CB (Merck Ltd) with the formula
which has an isotropic -nematic transition at 40.1 "C and a nematic to smectic A transition at 33.3 "C. Whereas the molecules are strongly polar, the nematic and smectic A phases built up of these molecules are nonpolar. This means that the unit vector n, called director, describing the local direction of axial symmetry, does not represent a polar direction but rather one with the property of an optic axis. This means that n and -n describe the same state, such that all physical properties of the phase are invariant under the sign reversal of n
n
+ -n, symmetry operation
(6)
Repeatedly this invariance has, somewhat carelessly, been described as equivalent to the absence of ferroelectricity in the nematic phase, whereas it only expresses the much weaker condition that n is not a polar direction. How can we understand the nonpolarity expressed by Eq. (6), which up to now seems to be one of the most general and important features of liquid crystals? Let us simplify the 8CB molecules to cylindrical rods with a strong dipole parallel to the molecular long axis. In the isotropic phase these dipoles could not build up a macroscopic dipole moment, because such a moment would be incompatible with the spherical symmetry of the isotropic phase. But in the anisotropic nematic phase such a moment would be conceivable along n. However, this would set up a very strong external electrostatic field and such a polar nematic would tend to diminish the electrostatic energy by adjusting the director configuration n(r) to some complex configuration for which the total polarization and the external field would be cancelled. Such an adjustment could be done continuously because of the three-dimensional fluidity of the phase. It would lead to appreciably elastic deformations, but these deformations occur easily in liquid crystals and would carry small weight compared to the energy of the polar effects which, as we have seen, are very strong. Thus it is hard to imagine a polar nematic for energetic reasons. Moreover, the dipolar cancellation can, and therefore will, take place more efficiently on a local scale. This is illustrat-
2.2 The Nonpolarity of Liquid Crystals in General
11
ed in Fig. 3, where the local dipolar fields are shown for two molecules. Each dipole will want to be in the energetic minimum position in the total field of all its neighbors. If two molecules are end on, their dipoles tend to be parallel, but in a lateral position the dipoles will be antiparallel, as shown to the right. Now, on the average, there are many more neighbors in a lateral position, which is a result of the shape of the molecules, and hence the latter configuration will prevail. We will therefore have an antiparallel correlation on a local scale. This leads to the fundamental property of liquid crystals of being invariant under sign reversal of the director, i.e., to Eq. (6). We may note that the argument given above is even more conclusive for smectic phases, which are thus characterized by the same nonpolar, “nematic” order. The oblong (or more generally anisotropic) shape of the molecules, which both sterically and by the anisotropy of their van der Waals interactions leads to the liquid-crystalline order, at the same time makes this order nonpolar. It has nothing to do with the strength or weakness of the dipolar interaction. Whatever strength such an interaction may have, it cannot be assumed that it would favor a parallel situation giving a bulk polarization to a liquid, nor even to a solid. Undeniably there is also an entropic contribution to this order, because a polar orientation has a lower entropy than an apolar orientation for the same degree of parallel order. Thus, if we turn all dipoles into the parallel position, we diminish the entropy and increase the free energy. However, the electrostatic contribution seems to be the important one. This is confirmed by computer simulations [60], which show that the antiparallel correlation is quite pronounced, even in the isotropic phase of typical mesogenic molecules. Anyway, it is evident that the existence of a valid relation (6) really depends on the shape of the molecules. Different mesomorphic states are possible which do not permit such a relation. Their physics will be accordingly different and will be discussed in Sec. 5.7.
Figure 3. Why liquid crystals are in general nonpolar. Around any molecule a lateral neighbor molecule has the tendency to align with its dipole in an antiparallel fashion. This means that n +-n is a symmetry operation for liquid crystals.
12
2 Polar Materials and Effects
Although the nematic phase is nonpolar, there are very interesting and important polar effects in this phase, in a sense analogous to piezoelectric effects in solid crystals. This was recognized by Meyer [18] in 1969. These so-called flexoelectric effects are discussed in Sec. 4. Meyer also recognized [61] in 1974 that all chiral tilted smectics would be truly polar and the first example of this kind, the helielectric smectic C * , was presented [62] in 1975. Out of Meyer's discovery grew the whole research area of ferroelectric and antiferroelectric liquid crystals, which is today a major part of liquid crystal physics and chemistry.
2.3 Behavior of Dielectrics in Electric Fields: Classification of Polar Materials All dielectrics become polarized if we put them in an electric field. The polarization is linear in the field, P E, which means that it changes sign if we reverse the sign of the field. When the field is reduced to zero, the polarization vanishes. For very strong fields we will observe saturation effects (and eventually dielectric breakdown of the material). The typical behavior of a normal (nonpolar) dielectric is shown at the top of Fig. 4. In a piezoelectric, an appropriate strain s will have a similar influence to the electric field for a normal dielectric. The effect is likewise linear around the origin and shows saturation at high strains. Conversely, we can induce a strain
-
"normal" dielectrics
PI
non-polar piezoelectrics
Pi
converse piezoeffect
SI
piezo P- E
electrostriction s-E2
Y
E
Figure 4. Response of nonpolar dielectrics (which do not contain orientable local dipoles) to an applied electric field. Piezoelectric materials react both with polarization and distortion and, because of the way these are related, they are also polarized by a distortion in the absence of an electric field.
2.3 Behavior of Dielectrics in Electric Fields: Classification of Polar Materials
13
-
s E by applying an external field. This is the converse piezoelectric effect, which
is used to produce ultrasound, whereas the direct effect is used to transform a mechanical strain into an electric signal, for instance, in gramophone pick-ups. The linearity in P s and s - E means that there is linearity between P and E, that is, a nonpolar piezoelectric behaves dielectrically just like a normal dielectric. In addition, all materials show the electrostrictive effect. This is normally a very small field-induced strain, which originates from the fact that the equilibrium distance between atoms and the distribution of dipoles are to some degree affected by an applied field. It corresponds to the small induced polarization and thus small dielectric constant typical for ordinary materials. The electrostrictive effect therefore has an entirely different character; the strain itself is not related to any polarization so no converse effect exists (i.e., an electric field cannot be generated in ordinary materials simply by applying mechanical pressure). The electrostrictive strain is always superposed on, but can be distinguished from, a piezoelectric strain by the fact that it is quadratic and not linear in the field, thus s E2. The linear and quadratic dependence, respectively, of the piezoelectric and electrostrictive strain, of course applies near the origin for small fields. In the more general case, the effect is called piezoelectric if it is an odd function of E, and electrostrictive if it is an even function [63]. The piezoelectric effect can only be present in noncentrosymmetric materials. If we apply a field to a material with a center of symmetry, the resultant strain must be independent of the field direction. hence
-
-
se = a E 2
+ a’E4 + ...
(7)
whereas the field reversibility of the piezoeffect will only admit odd powers
s P = P E + p ’ E 3 + ...
(8)
Sometimes care has to be taken not to confuse the two effects. If, for some reason, the sample has been subjected to a static field E, and then a small ac signal E,, is applied, the electrostrictive strain will be
se = a(Eo + Eat)'= a E i
+ 2aEoEac
(9)
giving a linear response with the same frequency as the applied field. An aligning field E, may, for instance, be applied to a liquid crystal polymer, for which the electrostrictive coefficient is often particularly large and the signal coming from this socalled biased electrostriction may easily be mistaken for piezoelectricity. Therefore, in the search for piezoelectricity, which would indicate the lack of a center of symmetry in a material, the direct piezoeffect should be measured, whenever possible, and not the converse effect. In Fig. 5 we have in the same way illustrated how polar materials may behave in response to an external electric field E. The P- E trace is the fingerprint of the cate-
14
2
Polar Materials and Effects
,I---
two stable states
ferroelectric
t t t t t t t t t t t t t t t
t t t t t
t t t t t
antiferroelectric
helielectric
t t t t t
J
l l l h
t t t t t
1t
l l l l
t t t t
E+
two stable states
Figure 5. Response of polar dielectrics (containing local permanent dipoles) to an applied electric field; from top to bottom: paraelectric, ferroelectric, ferrielectric, antiferroelectric, and helielectric (helical antiferroelectric).A pyroelectric in the strict sense hardly responds to a field at all. A paraelectric, antiferroelectric, or helieletric phase shows normal, i.e., linear dielectric behavior and has only one stable, i.e., equilibrium, state for E=O. A ferroelectric as well as a ferrielectric (a subclass of ferroelectric) phase shows the peculiarity of two stable states. These states are polarized in opposite directions (*P) in the absence of an applied field (E=O). The property in a material of having two stable states is called bistability. A single substance may exhibit several of these phases, and temperature changes will provoke observable phase transitions between phases with different polar characteristics.
gory that we are dealing with and is also characteristic of the technological potential. At the top there is the normal dielectric response with P increasing linearly up to a saturation value at high fields. In principle this behavior is the same whether we have local dipoles or not, except that with local dipoles the saturation value will be high and strongly temperature-dependent. If the molecules lack permanent dipoles, the induced local polarization is always along the field and temperature-independent. Dipolar molecules may, on the other hand, align spontaneously at a certain temperature (Curie temperature) to a state of homogeneous polarization. On approaching such a temperature the susceptibility (aP/aE) takes on very high values and is strongly temperature-dependent. We will describe this state - which is unpolarized in the absence of a field, but with a high and strongly temperature-dependent value of the dielectric susceptibility - as paraelectric, independent of whether there actually is a transition to an ordered state or not. It should be noted from the previous figure that a piezoelectric material has the same shape as the P - E curve of a normal dielectric, but it often shows paraelectric behavior with a large and even diverging susceptibility. Next, the contrasting, very strongly nonlinear response of a ferroelectric is shown in Fig. 5. The two stable states (+ P , - P ) at zero field are the most characteristic feature of this hysteresis curve, which also illustrates the threshold (coercive force) that the external field has to overcome in order to flip over from one state to the other. In the solid state this behavior may be represented by BaTiO,.
2.3 Behavior of Dielectrics in Electric Fields: Classification of Polar Materials
15
The response of an antiferroelectric is shown two diagrams below. The initial macroscopic polarization is zero, just as in a normal dielectric and the P- E relation is linear at the beginning until, at a certain threshold, one lattice polarization flips over to the direction of the other (the external field is supposed to be applied along one of the sublattice polar directions). This is the field-induced transition to the so-called ferroelectric state of the antiferroelectric. (Not ferroelectric phase, as often written the phase is of course antiferroelectric. Transitions between different thermodynamic phases by definition only occur on a change of temperature, pressure, or composition, that is, the variables of a phase diagram.) The very characteristic doublehysteresis loop reveals the existence of two sublattices as opposed to a random distribution of dipoles in a paraelectric. From the solid state we may take NaNbO, as representative of this behavior. It is not without interest to note that the structure of NaNbO, is isomorphous with BaTiO,. The fact that the latter is ferroelectric whereas the former is antiferroelectric gives a hint of the subtleties that determine the character of polar order in a lattice. At the limit where the hysteresis loops shrink to thin lines, as in the diagram at the bottom, we get the response from a material where the dipoles are ordered in a helical fashion. Thus this state is ordered but has no macroscopic polarization and therefore belongs to the category of antiferroelectrics. It is called helical antiferroelectric or heliectric for short. If an electric field is applied perpendicular to the helical axis, the helix will be deformed as dipoles with a direction almost along the field start to line up, and the response P - E is linear. As in the normal antiferroelectric case, the induced P value will be relatively modest until we approach a certain value of E at which complete unwinding of the helix takes place rather rapidly. Although the helielectric is a very special case, it shares the two characteristics of normal antiferroelectrics: to have an ordered distribution of dipoles (in contrast to random) and a threshold where the linear response becomes strongly nonlinear (see Fig. 5). In the solid state this behavior is found in NaN02. In the middle diagram of Fig. 5 we have also traced the P-E response for the modification of an antiferroelectric, which we get in the case where the two sublattices have a different polarization size. This phase is designated ferrielectric. Because we have, in this case, a macroscopic polarization, ferrielectrics are a subclass of ferroelectrics. It also has two stable states as it should, although the spontaneous macroscopic polarization is only a fraction of that which can be induced. If the polarization values of the sublattices are P, and P,
16
2
Polar Materials and Effects
pyroelectrics and antiferroelectrics, we have no spontaneous polarization domains, i.e., polarization up and polarization down domains appearing in the field-free state. The fact that it has not yet been possible to develop solid state ferroelectric memory devices is related to the very involved interactions with charge carriers and to the fact that switching between the two states is normally coupled to a change in the lattice distortion, which may eventually cause a total breakdown of the lattice (crystal fatigue). If the same or analogous memory property could be found in a liquid, this might be an attractive way to rule out at least the second problem. This is one of the facts that make ferroelectric liquid crystals very promising. As a result of these inherent problems, solid state ferroelectrics have, strangely enough, not been used for their ferroelectric properties, but rather for their superior pyroelectric (heat detectors), piezoelectric, and dielectric (very high E) properties. In addition, they are the dominating electrooptic materials as Pockels modulators as well as for nonlinear optics (NLO), e.g., for second-harmonic generation.
2.4 Developments in the Understanding of Polar Effects The denominations dielectric, paralectric, and ferroelectric were of course created as analogs to the names of the previously recognized magnetic materials. In the first treatise on magnetism, De Magnete (1600), Gilbert (W. Gilbert, De Magnete, Engl. transl. Dover Publications, New York 1958) described how iron loses its magnetic (like magnesian stone) properties when it is made red-hot. He also introduced the word “electric” (like amber) for bodies that seem to be charged (but differently than the magnetic ones) or may even produce sparks. It was Faraday who divided the magnetic materials into the groups diamagnetic, paramagnetic, and ferromagnetic and he also (1839) introduced the word dielectric for materials through which electric forces are acting (which could get polarized). Confirming Gilbert’s observation, Faraday (M. Faraday, Experimental Researches in Electricity, Vol. 1-3, Taylor & Francis, London 1839-1855) also found that iron loses its ferromagnetic properties at high temperature (770 “C) and becomes paramagnetic. This temperature is today called the Curie point, or Curie temperature, for iron. In his thesis (1895) Pierre Curie [64] investigated the magnetic properties of a number of different materials. He found that the magnetic susceptibility is independent of temperature for diamagnetic materials. For paramagnetic materials (like oxygen) he found a relation of the type in Eq. (5). A different relation
x=-
C T-T,
2.4 Developments in the Understanding of Polar Effects
17
was found for such bodies that showed ferromagnetism, on approaching the ferromagnetic state from above. That is, the susceptibility does not diverge at T=O (in which case one would not observe a divergence), but at a finite temperature T,, which is the temperature below which the body is ferromagnetic. When Curie’s friend and colleague Paul Langevin in 1905 deduced the formulae corresponding to Eqs. (3), (4),and ( 5 ) it was the first microscopic physical theory relating to macroscopic phenomena [65], in this case to the measured magnetization M and magnetic susceptibility x=dM/dH. Thus Langevin found for C, the Curie constant, the value Np2/3k where p is the magnetic moment of the atom. The electric case reflected in our Eqs. (3), (4),and ( 5 ) with p instead of p, is of course modeled in exact analogy to the magnetic case. One year later, in 1906, Weiss [66], at that time professor at the ETH in Zurich, attacked the problem of ferromagnetism by a phenomenological theory similar to that presented in 1873 by van der Waals in his thesis on the condensation of a gas to a liquid (J. D. van der Waals, On the Continuity of Gas and Liquid State, Engl. trans]. in Physical Memoirs, London 1890). Weiss introduced the notion of a “molecular field”, the average field on a certain magnetic moment caused by its magnetized environment, and set this field proportional to the external field H. In doing this, he could account for the spontaneous magnetization of ferromagnets, and he was able to deduce a relation for the susceptibility corresponding to Eq. (lo), which since then has been called the Curie- Weiss law. The divergence of at the Curie temperature T, corresponds to the divergence of the compressibility of a fluid at the critical point. The approach by Weiss has been very successful and a longstanding model for descriptions of condensed matter phenomena which go under the name of “mean field theories”. The word dielectric is analogous to diamagnetic. Physically, however, dielectric does not correspond to diamagnetic in any other sense than that the effects are very general, whereas paraelectric and paramagnetic effects can only be found where we have orientable electric and magnetic dipoles. In fact, there is no electric effect corresponding to diamagnetism, the origins of which (in the classical description) are induced atomic and therefore lossless currents (“superconductivity on an atomic scale”) and which therefore give a small magnetization against the applied field, corresponding to a negative susceptibility. As, in contrast, the general dielectric effect and the more special paraelectric effect do not differ in sign, the reason to make a distinction between them is not generally felt as very strong, except when we want to express the fact that in the dielectric effect (in the narrow sense) there is no temperature dependence, because the dipole does not exist prior to the application of the field, but is only created by the field, just as in the diamagnetic case. Dielectric and diamagnetic effects are therefore also always present in paraelectric and paramagnetic materials, but can on the other hand usually be neglected there. While electric phenomena connected to matter seem to have been observed, like the magnetic ones, even in ancient times, their investigation and comprehension is of a more recent date. Curiosity about these phenomena was raised in about 1700,
x
18
2
Polar Materials and Effects
when Dutchmen in foreign trade brought tourmaline from Ceylon to Europe. They observed how these rodlike crystals acquired a force to attract light objects, like small pieces of paper, when heated. The first scientific description [67] was given in 1756 by the German Hoeck (Aepinus) worlung in Berlin and then mainly in Russian Saint Petersburg. Brewster in Scotland, who made a systematic study on a number of minerals [68], coined the name pyroelectric in 1824. However, it was necessary to wait until the present century to really get an understanding of the phenomenon, and the historical name (pyro etymologically related to fire) is a very unfortunate one. It is true that a change AT in temperature causes the appearance of a polarization P, and pyroelectricity is often defined this way. Still, it is not really the heart of the matter. Brewster's colleague in Glasgow, Lord Kelvin, who worked out (1878- 1893) the basic thermodynamics of the pyroelectric effect and its converse, the electrocaloric effect (heat dissipation on applying an electric field) suggested [69] that pyroelectric materials have a permanent electric polarization. This is the basic concept: a pyroelectric has a spontaneous macroscopic polarization, i.e., it is an electric dipole in the absence of an external field. The pyroelectric effect, however important it may be for applications, is only a secondary effect, just a manifestation of the temperature dependence of this polarization. The charged poles in a pyroelectric cannot normally be observed, because they are neutralized in a rather short time by the ubiquitous free charges which are attracted by the poles. (As there are no free magnetic charges, such a screening does not occur in a magnet, which makes magnetic phenomena simpler in many respects than the corresponding electric phenomena.) However, the magnitude (and in rare cases even the direction) of the polarization is generally a function of temperature, P = P ( T ) .Hence a change AT in temperature causes an additional polarization A Z' which is not compensated by the charges present and can be observed (normally in a matter of minutes) until the redistribution of charges to equilibrium has been completed. This is also one of the standard methods to measure polarization: the pyroelectric current is integrated through an external circuit after firing a heat pulse from an infrared laser. This gives aP/aT, which is determined as a function of T. From this, P(T ) is deduced. Charles Friedel, the father of George Friedel, had studied pyroelectric phenomena in crystals since 1860 and in the 1870s published a number of papers on the topic with Jacques Curie, who was his assistant in the mineralogy laboratory at the Sorbonne (and later went to Montpellier as a lecturer in mineralogy). When the brothers Jacques and Pierre Curie discovered piezoelectricity in 1880, their conjecture [70] was at first that when creating surface charges by compressing the crystal they had invented an alternative method for making a pyroelectric, because they believed that there was a correspondence between thermal and mechanical deformation. This was objected to by Hankel, professor of physics in Leipzig, Germany, who pointed out that the two effects must be fundamentally different in nature and proposed the name piezoelectricity [711, which was immediately adopted by the Curie brothers. Basically we can say that temperature, being a scalar, can never change the symme-
2.4 Developments in the Understanding of Polar Effects
19
try. Therefore the polarity must already exist and be intrinsic in a pyroelectric crystal. A strain, on the other hand, being a tensor, can alter the symmetry and thereby provoke the polarity. The further theory of pyro- and piezoelectricity was later essentially worked out by Voigt (the man who also coined the word “tensor”) and summarized in his monumental Lehrbuch der Kristallphysik (Treatise of Crystal Physics), which appeared in 1910 [W. Voigt, Lehrbuch der Kristallphysik, Teubner, Leipzig 19101.Voigt, who had been a student of Franz Neumann in Konigsberg and then professor of theoretical physics in Gottingen from 1883, was able to show in detail how these polar phenomena were related to crystal symmetry. In particular he showed that among the 32 crystal classes, 20 have low enough symmetry to admit piezoelectricity and, within these 20 classes, 10 have the still lower symmetry to be polar, i.e., pyroelectric. We can therefore divide dielectric materials into hierarchical categories, and this is illustrated in Fig. 6. In this figure, the higher, more narrow category has the special properties of the lower ones, but not vice versa. Thus, for instance, a pyroelectric is always also piezoelectric. We have also inserted in the figure the very important subclass of pyroelectrics called ferroelectrics, to whch we will return in detail below. How the 32 crystal classes are distributed among dielectric, piezoelectric, and pyroelectric classes is shown in Fig. 6a. For 35 years piezoelectric materials were rather a laboratory curiosity, but in 1916, during the first World War, Langevin started his epoch-making investigations in the harbour of Toulon in southern France, to range submarines and to develop navigation and ranging systems based on his technique of producing and detecting ultrasound by means of piezoelectric (quartz) crystals. He thereby became the founder of the whole industry of sonar and ultrasonics with very important military as well as civil applications. At the end of the war, Cady, in the United States, initiated the applications of piezoelectric crystals for oscillators and frequency stabilizers, which subsequently revolutionized the whole area of radio broadcasting, and set a new stan-
Figure 6. The hierarchy of dielectric materials. All are of course dielectrics in a broad sense. To distinguish between them we limit the sense, and then a dielectric without special properties is simply called a dielectric; if it has piezoelectric properties it is called a piezoelectric, if it further has pyroelectric but not ferroelectric properties it is called a pyroelectric, etc. A ferroelectric is always pyroelectric and piezoelectric, a pyroelectric always piezoelectric, but the reverse is not true. Knowing the crystal symmetry we can decide whether a material is piezoelectric or pyroelectric, but not whether it is ferroelectric. A pyroelectric must possess a so-called polar axis (which admits no inversion). If in addition this axis can be reversed by the application of an electric field, i.e., if the polarization can be reversed by the reversal of an applied field, the material is called ferroelectric. Hence a ferroelectric must have two stable states in which it can be permanently polarized.
dielectric piezoelectric pyroelectric
20
2 Polar Materials and Effects
Figure 6a. Dielectric, piezoelectric, and pyroelectric crystal classes in crystallographic (left) and Schoenfliess notation (right). These relations were first expounded in Voigt’s Lehrbuch from 1910. The modem version of Voigt is the treatise by Nye, reference 1241.
dard for the measurement of time prior to the arrival of atomic clocks (up until 1925, when quartz oscillators took over; the frequency of the electronic master clocks was still set by tuning forks). In the meantime Debye, who the year before had been appointed professor of theoretical physics at the University of Zurich, in 1912 introduced the idea of “polar molecules”, i.e., molecules with a permanent electric dipole moment (at that time a hypothesis) and worked out a theory for the macroscopic polarization in analogy with Langevin’s theory of paramagnetic substances. He found, however, that the interactions in condensed matter could lead to a permanent dielectric polarization, corresponding to a susceptibility tending to infinity for a certain temperature, which he proposed to be the analog to the Curie temperature for a ferromagnet. In the same year, Schrodinger in Vienna, in his habilitation thesis [72] applied the model to solids and concluded that all solids ought to become “ferroelectric” at sufficiently low temperatures. Thereby he coined the word ferroelectric even before such a similar state was to be found eight years later. (The state in Schrodinger’s anticipation does not quite cover the concept in the sense it is used today.) After the first World War, the strategic importance of piezoelectric materials was fully recognized and research was intensified on Rochelle salt, whose piezocoefficients had been found to be much higher than those of quartz. It became the thesis subject for Valasek of the University of Minnesota. Valasek recognized the electric hysteresis in this material as perfectly analogous to magnetic hysteresis [ I ] and also used the word Curie point for the transition at 24 “C. Thereby the phenomenon of ferroelectricity was discovered (although it was not quite recognized in the beginning). After some time, it was found that ferroelectric materials were not only extremely interesting as such, but also had the most powerful piezoelectric and pyroelectric properties. Important research projects were started in Leningrad headed by Kurchatov who founded the Russian school and wrote the first treatise on ferroelectricity (or “seignette electricity” as it was previously often called after the French name for Rochelle salt) [73]. In Zurich, Busch and Scherrer (the latter initially stud-
2.4 Developments in the Understanding of Polar Effects
21
ied both in Konigsberg and with Voigt in Gottingen) discovered [74] the important ferroelectric KDP (Curie point 123 K) in 1935. When Landau in 1937 presented his very general framework based on symmetry changes in phase transitions [75], it would gradually come to dominate the whole area of phase transitions and collective phenomena. Mueller at ETH, independently, worked out a similar theory [76] and applied it successfully to KDP. Finally, wartime research efforts during the second World War led to the discovery in 1945 of the very important perovskite class starting with barium titanate (BaTiO,, with the conveniently located Curie point of 120“C) by Wul and Goldman at the Lebedev Institute in Moscow [77,78] and, probably independently, in 1946 by von Hippel and his group at MIT in the U. S. A. [79]. In the same year, Ginzburg applied the Landau theory for the first time to the ferroelectric case (of type KDP) [go]. Today, a unified microscopic theory is still not available for these different ferroelectrics. This is a good argument for using phenomenological theories, which we will do in the following. These theories also have the advantage that they can be applied to ferroelectric liquid crystals. Finally, a comment has to be made about the polar materials called electrets. The word itself was introduced by Heaviside at the end of the last century [81] to designate a permanently polarized dielectric, in analogy with the word magnet. It has, however, hardly been used in that general sense, but has instead acquired a special meaning. In current terminology, an electret is a dielectric that produces a permanent or rather quasi-permanent external field, which results from the ordering of molecular dipoles or stable, uncompensated surface or space charges [82].The first electrets were prepared by Eguchi [83, 841 at practically the same time as the discovery of ferroelectrics by Valasek. Normally electrets are made by orienting dipoles, or separating, or injecting charges by a high electric field (for instance, in a corona discharge) acting on materials like natural waxes or resins or, in modern times, organic polymers, and then cooling down the material in the presence of the field such that the charges, or dipoles, stay trapped (thermoelectrets). A powerful modern method is to inject charge in the material directly by an electron or ion beam (charge implantation). Amorphous selenium and other inorganic materials can also be used as electrets in which charges are separated by illumination in the presence of the field (photoelectrets). Common to all electrets is the fact that the polarization or space charge is a non-equilibrium condition. These materials are therefore in a “glassy”, metastable state and their polarization in principle slowly decays with time, more rapidly at higher temperatures. However, for practical purposes their long-term stability can be very high and their industrial importance is enormous both for electroacoustic devices (electrets are used in almost all small earphones, microphones, and loudspeakers), sensors and transducers (ultrasound and touch devices). The typical electroacoustic electret is a thin foil of polyimide, polyethylene, or teflon (polytetrafluoroethylene), 5-50 pm thick, with one side metallized and charged to about 20 nC cm-’ (corresponding to 10 V pm-’) and in extreme cases up to 500 nC cm-2. Although liquid-crystalline electrets are presently a subject of some research activ-
22
2
Polar Materials and Effects
ity, especially in connection with NLO applications, electrets will not be considered further in this book. The development of ferroelectric materials and concepts has been reviewed by Fousek [ 3 ] and, with particular emphasis on historical details, by Busch [85] and Kanzig [86].
2.5 The van der Waals Attraction and Born’s Mean Field Theory In Sec. 2.2 we said that the nematic order is a result of the anisotropic van der Waals interactions. We would like to somewhat qualify that statement. It is clear that there are also simply steric (packing) effects due to the shape of the molecules, in the same sense that the matches in a matchbox are happier if they are aligned parallel. This is the “excluded volume” contribution to the order. In certain cases like long hard-rod macromolecules (tobacco mosaic virus in solution) [86a] this contribution alone may suffice to stabilize nematic order. In normal nematics, however, the van der Waals interaction is the dominating one. The two typical structures of Fig. 7 show the conjugated bonds leading to a high degree of delocalization of the n electrons along the elongated molecule. This means that there are large anisotropic charge fluctuations along the molecular axis. If we again idealize the molecules to the shape of rods, a charge fluctuation as illustrated in Fig. 7 a, constituting a dipole 6p, induces a corre-
‘’
o‘CHj
Figure 7. The conjugated k electron system in typical mesogenic structures means delocalized electrons, which leads to strong anisotropic charge fluctuations along the molecular axis. (The dotted line illustrates the flexibility of the carbon chains, which promotes the liquid crystalline state as their thermal mobility counteracts crystallization.)
Figure 7a. A charge fluctuation along the molecular axis creates a dipole, the field of which induces a dipole of opposite sign in a neighboring molecule. This leads to a van der Waals attraction. The force falls off very rapidly with distance and is strongly orientationally dependent.
2.5 The van der Waals Attraction and Born’s Mean Field Theory
23
sponding dipole 6p’ of opposite sign in a parallel neighbor molecule, leading to an interaction potential
Sp . Sp’ U ( R )- ____ R3
(5.1)
where the induced dipole 6p’ is proportional to the electric field from the first dipole, which itself is proportional to 6p but falls off like R-3, where R is the distance from the dipole. The proportionality constant is the polarizability a, hence
Sp‘=aE
--6P R3
yielding
U(R)
- R6a
(5.3)
-
equivalent to a force
F--1 R7
(5.4)
This is the weak van der Waals force which not only falls off very rapidly but here is orientationally dependent, because it obviously does not work at all if the delocalized n electron systems are not aligned. This indicates that the nematic phase requires a certain minimum amount of parallel order of the molecules to persist. In other words, if we describe this order by a parameter S, called the order parameter, with the property that S = 1 for perfect order and S = 0 for complete disorder, the nematic phase will not persist until S+O but collapse into the isotropic state for some S#O, when we increase the temperature. In other words, from this microscopic point of view, we should expect the nematic-isotropic transition to be of the 1st order. The convenient order parameter in this case is the second Legendre polynomial P2 (this P has nothing to do with polarization), taken at its space-time average
s = (p2(cosei)) = 1 (3 cos2 e, - 1) -
2
(5.5)
where eiis the angular deviation of the axis of the ith molecule from the director, cf. Fig. 7b. For perfect order 6,=0, thus S = 1, whereas for the completely disordered (isotropic) state S = 0, because then cos2ei averaged over three dimensions is equal to 1/3. In 1916 Max Born advanced the first theory for the liquid crystal state [86b]. As this theory is every now and then referred to it is interesting to see what it contains.
24
2 Polar Materials and Effects
n
Figure 7b. The angular fluctuations Oiof individual molecules relative to the directorn in a nematic characterize the degree of nematic order at a given temperature. 1 A convenient measure of this order is the scalar order parameter S=-(3cos2 0-1)
2
/
where the brackets indicate time-space average. S is invariant under n + -n, is equal to one for perfect order along n and zero for complete disorder (isotropic
The theory is a failure (a rare thing in the scientific life of this giant), but it is quite instructive to review it. It is clearly inspired by Weiss’s theory of ferromagnetism from 1906. Born assumes that the molecules interact via their permanent dipole moments p . Each dipole is considered to be inside an effective field created by all other dipoles in its surroundings. This “molecular field” or “mean field” technique advanced by van der Waals and Weiss is a very important technique to describe the interaction in a large system of identical particles: if we consider a specific dipole, it then only interacts with one entity, the effective field or the mean field E . If the external field is B,, the effective or local field felt by the dipole is
which is equal to P/3&,in the absence of any external field. The probability for the dipole to deviate by 8from the direction of E is given by the Boltzmann factor e-“‘kT where
The probability for the dipole making an angle 8 relative to E is then expressed by
(5.8)
K
with a = - p p and Z = ~ d c o s 6 e u C o S e 3 ~ kT 0 0
2.5
The van der Wads Attraction and Born's Mean Field Theory
25
The total dipole moment per unit volume is equal to the polarization
I
P = N p (cos0) = NP d cos0 e" ZO
cos 0
This means that (cos 0) = L (a), the Langevin function. We will also need the second moment according to the Boltzmann distribution 1
j dxeaxx2 (cos20)= -1
I
2 = 1- - L ( u )
j heax
a
(5.10)
-1
For small a, L ( a ) = (cos 0) is equal to
(5.11)
If we were to pursue even the next term we would find
a -a 3 + ... L ( a )= 3 45
(5.12)
We can now sum up our calculations. According to Eq. (5.9) the polarization is (5.13) This corresponds to the earlier Eq. (3) with E replaced by P/3&,. If we multiply both sides by p / 3 & , kT and put Np2/9&,k = T, we get
a = 3 Tc L ( a ) ~
T
(5.14)
26
2 Polar Materials and Effects
or (TIT,).a = L ( a ) 3
(5.15)
This is a transcendental equation, the solutions of which are most easily obtained as the cuts between the two functions U
Y = ( T / T , ) .-
(5.16)
Y=L(a)
(5.17)
3
and
as illustrated in Fig. 6e. If the temperature is low enough there are solutions corresponding to a = p P/3&,kT f 0, i.e., corresponding to P f 0. For T > T,, however, the only solution corresponds to P = 0. Obviously T, is the phase transition temperature at which the nematic state is supposed to transform into the isotropic state. It is clear from Fig. 7c that as we increase T, P decreases monotonically to the value zero. This model thus describes a continuous transition between nematic and isotropic. To see how P varies near T, we rewrite Eq. (5.15) as a u (TIT,).= L ( a )= 3 3 ~
a3 --
45
(5.18)
or U
a3
(1 - TIT,)= 3 45
-
(5.19)
giving U*
- p 2 - (1 - TIT,)
(5.20)
a
Figure 7c. The solutions to Eq. (5.15) are obtained as the cuts between the two functions of Eq. (5.16) and (5.17).There are solutions with a#O if T < T , .
2.5
The van der Waals Attraction and Born’s Mean Field Theory
27
or (5.21) This variation is depicted in Fig. 7d and is typical for a phase transition of second ordei: It is also interesting to see how the scalar order parameter S from Eq. (5.5) varies near T,. From our calculation of (cos28),Eq. (5.10), we find (5.22) For T -+0, a --+DO, L ( a ) + 1 and thus S approaches the value 1 which is reasonable. But near T, (5.23) This means that the nematic order would fall off to zero in a linear fashion as Tapproaches T,. Born’s theory describes the nematic as an electric analog to a magnet. It describes the local dipole moments to be aligned in parallel, thus the nematic phase would be pyroelectric or ferroelectric. The nematic-isotropic transition would be second order. Both results are manifestly in conflict with experiments. Further, the order parameter S would go to zero in a linear way when T, is approached. This can be tested by measuring the birefringence which is proportional to S. Also here the result is in complete disagreement with the model. In hindsight it is easy to see what is wrong with the assumptions. The local field E in Eq. (5.6) is a superposition of the applied field and the Lorentz fieldP/3~,which describes the response of the matter itself being polarized by the applied field. However, in the absence of an external field there is no reason to believe that there would be a local field due to a tendency for dipoles to order in parallel. In fact, the oppo-
Figure 7d. The dipole moment per unit volume in Born’s hypothetical nematic phase below the clearing point.
28
2 Polar Materials and Effects
site is true, as we have illustrated in Fig. 3. Thus neighboring dipoles rather tend to align in an antiparallel fashion. But then, how come that this assumption works in the case of Weiss’s molecular field theory? The answer is that this is a historical accident. The idea that local magnetic dipoles would spontaneously arrange in parallel would be equally wrong as in the electric case. However, the collective phenomenon of ferromagnetism is not due to magnetic dipole-dipole interaction. The phenomenon of spontaneous magnetization in Weiss’s elementary domains could not, in fact, be understood until 1928 when Heisenberg showed that it is due to the exchange interaction between electrons (which is spin dependent) and thereby traced it back to coulombic interaction. Thus magnetism in essence is an electrostatic phenomenon. Already in the year following Born’s communication to the Prussian Academy of Sciences in Berlin, Franqois Grandjean communicated a similar mean field description, based on the analogy with the Weiss theory, to the French Academy of Sciences in Paris [86c], but this time he had the good physical sense to replacing the polar interaction by a quadrupolar interaction. In essence, what he did was to replace 8 in Eq. (5.7) by 28. He motivated this by saying that this pure hypothesis was chosen to conform with the properties of liquid crystals. Grandjeans model gives a first order nematic-isotropic transition and is the first valid molecular-statistical theory of liquid crystals. It has, however, remained practically unknown until today and therefore unfortunately did not have any impact upon subsequent developments. Only in 1958 came the breakthrough in the description of nematic order by the Maier-Saupe theory [86d]. Whereas Grandjean does not actually care about what sort of potential function he is using (it does not actually matter), Maier and Saupe start with a careful discussion of the potential and its background on a molecular level, which correctly reflects the symmetry of the van der Waals forces. This means that their theory also proceeds considerably beyond the framework of a mean field theory. At the same time as Born, C. W. Oseen tried to work out a theory for liquid crystals. A difficulty with his early attempts was that he tried to develop a thermodynamic theory and an elastic theory at the same time, which was just too much and probably delayed the whole theory. Eventually, he was successful with the elastic part, but not with the thermodynamic. In the elastic theory he identifies the elementary deformations (cf. Sec. 4.1) and gives them each an elastic constant. The theory does not depend on temperature except for the consideration that the elastic constants K do. Their temperature dependence is a matter for a thermodynamic theory and, as the Maier-Saupe theory shows, this is traced to the temperature dependence of the scalar order parameter S , in that K S2. We will not pursue the thermodynamic part in this book but work a great deal with Oseen’s elastic theory, which has an interesting relation to the thermodynamic theory of Landau and Ginzburg. (Every elastic theory is basically thermodynamic.) We will conclude only by repeating what we said about Grandjean’s treatment: by replacing 8 with 2 8 in Eq. (5.7) the order of
-
29
2.6 Landau Preliminaries. The Concept of Order Parameter
the transition turns from second to first. This gives a fascinating hint that deep connections may be involved relating the order of the transition to the symmetry of the parameters characterizing the phase. We will introduce these considerations in the next section.
2.6 Landau Preliminaries. The Concept of Order Parameter The well-known shape of the phase diagram for a single substance which exists as a solid (s), liquid (1) and gas (g) depending on the values of pressure ( P ) and temperature ( r ) is shown in Fig. 8. The substance could for instance be carbon dioxide, CO,. We note the triple point Tt at which all three phases coexist, and equilibrium lines between gas-liquid, gas-solid, and liquid-solid. When we cross such a line a phase transition takes place, for instance the gas condenses to a liquid. This is a discontinuous transition or a transition of first order, recognized by the fact that heat is liberated or absorbed. We see that the higher the temperature the higher pressure we have to apply to the gas to condense it to a liquid, but we also see that above a certain temperature, T,, the equilibrium line suddenly stops, i.e., ceases to exist. For carbon dioxide this temperature is 31 "C. What it means is that for T > T, you cannot condense carbon dioxide to a liquid form whatever pressure you apply. But it also means that you can circumvent the critical point, as T, is called, along the trajectory indicated in the figure, and pass from gaseous form to liquid in a continuous way, without going through a phase transition. The point T, is indeed very special, there is no more latent heat because the gaseous state can no longer be distinguished from the liquid state, as it can for T < T, and there are huge fluctuations of liquid-like and gas-like domains in each other, hence density fluctuations, leading to the phenomenon of critical opalescence, which is a strong anomalous scattering of light. On the other hand the line between solid (s) and liquid (1) never stops. How can we know this? The answer was given by Lev Landau who noticed that the symmetry, Figure 8. The gas-liquid-solid phase diagram. Two phases coexist along the transition lines and all transitions are discontinuous, involving latent heat. When we move from the triple point Tt toward T, the latent heat for the liquid-gas transition diminishes monotonically and the transition becomes continuous at T,. the critical point. For carbon dioxide this point lies at 3 1"C. Above this temperature you cannot condense CO, to liquid form whatever pressure P is applied. On the other hand, you can go from gas to liquid without passing a phase transition, as indicated.
I+-
never stops
I
p
I
(s)
n /i93 T
30
2 Polar Materials and Effects
apriori, can only change discontinuously. Between the solid state and the liquid state there is always a symmetry change and you cannot go from one state to the other without a transition from one symmetry to the other. But for liquid and gas this is not so: both states of matter have the same symmetry. This is the reason why we can have a critical point. Such points are rare, but other examples exist, for instance in liquid crystals. Recognizing the importance of symmetry, Landau in 1937 could dispose of the then existing vague notions of “sharp” and “gradual” transitions. Although different measurable parameters may change more or less smoothly with temperature, a phase transition involving symmetry change is always sharp, and thus takes places at a distinct temperature, in the sense that the symmetry change is sharp. Thus the phases on either side of the transition temperature may be characterized by their symmetries, and the whole transition by the symmetry change. Then the first part of the Landau theory is really a discussion of symmetry, involving the methods of group theory, from which emerges the extremely important concept of the order parameter. The second part consists mainly of working out the thermodynamic consequences of the physical character of the order parameter. As an example, let us compare the transition from the paramagnetic to the ferromagnetic state in a spin ensemble with the transition from a normal (isotropic) liquid to the nematic state in a liquid crystal (see Fig. 8a). In both cases the disordered (high temperature) state has full spherical rotational symmetry whereas the condensed state is anisotropic, with an axis of cylindrical symmetry pointing along some direction in space. The first property required of the order parameter is that it is zero in the disordered phase but nonzero where the structure is ordered. Furthermore, the order parameter must reflect the symmetry of the ordered phase - a vector quantity in the ferromagnetic state. The order parameter in this case is the magnetization M which vanishes at the Curie point. Its direction along the cylindrical axis leads to two physically distinguishable states, + M and - M . In the liquid crystal case, the molecular rods have no “up” and “down”, and if we characterize the nematic state by the unit vector n,the director, we have to add that -n describes exactly the same physical state. This possibly subtle difference has other consequences, including that the order parameter will vanish in a different way at the transition point T, for the two
‘t C J 4 I T
,
III 1 I 1 1 I (I I I I I I
’’
T>T,
”
Figure 8a. A comparison of the ferromagnetic-paramagnetic and nematic-isotropic transitions. The ordering shows similarities, but the fact that in the first case it takes place in a vector medium (put an arrow on the liquid crystal “rods” in the lower part to go to the magnetic system in the upper part) makes the phase transition continuous or second order. The nematic medium has the symmetry of a tensor of rank two, which makes the nematic-isotropic transition discontinuous or first order.
\ \----‘ I ’ \’ -1, \-
2.6 Landau Preliminaries. The Concept of Order Parameter
31
systems: the nematic transition is first order while the magnetic transition is second order. We now turn our attention to the free energy G of the system. In a ferromagnet with homogeneous magnetisation M , the free energy may be expressed as a power series in the order parameter
G = G o + u , M + u 2 M 2 + a 3 M 3 + ...
(6.1)
where the coefficients uj are temperature dependent. Landau argued that, in the absence of an external field, the free energy cannot depend on the direction in which M points: more particularly, it must be the same if M points in the opposite direction. Thus G must be invariant under a change of sign in M . This implies that all odd powers of M must vanish, i.e.,
G = Go + u 2 M 2+ u4M4 + ...
(6.2)
He then developed a stability condition for u2 which could be satisfied most simply if a2=uo(T-T,), which has the desired consequence that the quadratic term changes sign at the transition. The linear approximation is the simplest which is analytical and implies that a. is independent of temperature and u4 only weakly dependent, furthermore that uo, u,>O. Thus
G=Go+a o( T- T, ) M2+ u4M4+ ...
(6.3)
Minimizing G with respect to M ( d G l d M = 0) gives one solution for finite M
M
- (T, - T)”*
(6.4)
i.e., parabolic variation of the order parameter near T, (see Fig. 8b), valid below T,, and another solution, M = 0, valid for T > T,. In Fig. 8 b we have also indicated that while the order parameter M goes continuously to zero when we approach the transition point T, from below, the magnetic susceptibility above T, diverges when we approach T, from above. Both things are characteristic of a second order phase transition. At T, the paramagnetic phase and the ferromagnetic phase do not coexist, as they would do at a first order transition. Rather, they cannot be distinguished from each other - just like the situation at the gas-liquid transition at T, described in Fig. 8. Now let us look at the nematic-isotropic case from Landau’s point of view as was first done by de Gennes. What is the symmetry change and what is the correct order parameter? We have already introduced the scalar parameter S to describe the nematic order. As an order parameter it is limited because it does not quite embrace all of the symmetry characteristics of the nematic and it is not really capable of express-
32
2 Polar Materials and Effects
Tc
Figure 8 b. Variation of the order parameter and its susceptibility on either side of a second order phase transition. In this case the order parameter is M and its susceptibility a M / a H , expressing the ease of magnetization in an external field. The divergence of means that an ever weaker field is sufficient to obtain a certain magnetization, which becomes spontaneous at T,. This is also indicated by the growth of ferromagnetic fluctuations in the paraelectric phase, given by their coherence length <(7‘). The variation of M and can be described by the critical exponents fi and y, for which the Landau theory gives the values 1/2 and 1, respectively. While the value T, is different for every system, the critical exponents are essentially independent of the system.
x
x=
x
ing the isotropy of the isotropic phase. Nevertheless, it describes some of the features: it has quadrupolar character in the nematic phase and it is zero in the isotropic phase. (S is a lund of “reduced” order parameter of which we later will give several examples.) As for the symmetry properties of S we notice that it is “not at all symmetric”. If we put 8 = rc/2 in Eq. (5.5) -a somewhat unphysical situation because it means that the long axes of the molecules would lie in planes perpendicular to the director - we find S=-1/2. On the other hand, S is equal to +1/2 for 3 cos28=2, which gives an average 8 of about 30”. So a change of sign in S does not by any means give two descriptions of the same state. Hence, if we were to expand the free energy G in powers of S, G(-S) would not be equal to G(+S) and we would not just have even powers in the expansion. Let us pursue this question a little further with a more correct choice of order parameter. If we start from the isotropic phase and decrease the temperature, what happens at T, is that the liquid becomes anisotropic, and as we continue to lower temperatures it becomes more and more anisotropic. We could follow the evolvement by concentrating on a parameter giving both the direction of anisotropy and its size. Such a parameter would be the dielectric permittivity E (the refraction index n or the birefringence An could not be used because they have no tensor description). An even better choice would be the magnetic susceptibility because it does not change character with frequency, which E often does. In diagonal form can be written
x,
x
where in the second form we have turned the z axis parallel to the director and used the fact that there is cylindrical symmetry around the director = x, = Could
(x,
xL).
2.6 Landau Preliminaries. The Concept of Order Parameter
33
x
this be a good order parameter? Obviously not, because in the isotropic phase its value is not zero but the average of the three elements, or
that is. one third of the trace. This we must subtract, i.e., we create
1 1 0 XI - 3 xi1 3 1 1 0 3 XI - 3 XI1
-
0
0
0 0
2 32 XI1 - 7 XI
There are several equivalent ways to write this tensor. If we introduce the susceptibility anisotropy, A x = it can be written
x,, x1,
-~ l
3 0
Q’=Ax
-l
o
3 2 0 -
0
2
=SAX
o o 2 0 - -l o 2 0 0 1
We could use this as an order parameter, especially if we normalized it (to equal unity for perfect order, which is done just by adjusting the scalar part), but then we could choose
Q=S
3 0
-l
o
0
0
2 -
3
which is even better and already normalized. Here S is the nematic scalar order parameter. Often the tensorial order parameter is expressed in the director components and is then written
(
Qij= ninj - 31
&J
(6.10)
34
2
Polar Materials and Effects
Diagonalized along x, y , z this means
(6.11)
This of course is also traceless, as confirmed (omitting the scalar factor S) by
TrQ = n,2 + ny2 + n,2 - 1 = 0
(6.12)
The nematic order can be characterized by a rotational ellipsoid as in Fig. 8 c, where we illustrate how the anisotropy spontaneously appears at T = T, and grows larger when T decreases. More precisely the order parameter, as we have seen, is a traceless second rank tensor, the tracelessness implying that it only describes the deviation from spherical symmetry (dark in the figure), because it is this deviation that vanishes at T, when the liquid becomes isotropic. Can we say anything about the expansion for G near the nematic-isotropic transition by looking at the symmetry properties of the order parameter Q? Would G be unaffected by a change in the sign of Q ? It would not, because changing sign means changing the signs of all tensor elements, which means going from a prolate to an oblate form in the representation quadric of the basic x tensor, cf. Eq. (6.5). The effect is shown in Fig. 8d. This means going from one kind of anisotropy to another (much like going from positive to negative birefringence). In other words +Q and -Q do not at all represent the same state, so G(-Q) f G ( Q ) .Therefore if we expand G in powers of Q both odd and even powers must appear.
T>T,
TS,
TaT,
Figure 812. Tensor order parameter of the nematic-isotropic transition. The order parameter expresses two things - the degree of anisotropy and the local direction of the anisotropy axis. At T, a discontinuous transition takes place, from a state in which all directions are equivalent, to a state with afinite value of the anisotropy - the transition turns out to be first order - which then increases as we decrease the temperature. For T > T, we have a case of spontaneously broken rotational symmetry, as now a local axial direction (the director) exists everywhere in the liquid. The axial direction has been drawn differently in the figure to indicate that this is a local property, which fluctuates in the liquid. In fact, the lack of preferred direction in space leads to giant fluctuations responsible for the anomalous light scattering in bulk nematics.
2.6 Landau Preliminaries. The Concept of Order Parameter
35
Figure 8d. Representation quadric of a symmetr i c second rank tensor with positive eigenvalues
x
as represented by the susceptibility of Eq. (6.5). The normalized traceless tensor Q in Eq. (6.9) represents the deviation from spherical symmetry, illustrated in black. l
iQ
A precision should now be made. In Eq. (6.1) we have expanded the free energy G, which is a scalar, in powers of M . At least it looks like this and is very often expressed that way. But it is not quite that simple because the order parameter here is a vectorM while the expansion in the order parameters can only contain scalar terms. The expansion therefore does not contain powers of the order parameter but rather scalar invariants of first, second, third order and so on. In later sections we will also include other terms, for instance field terms, which have the required rotational invariance. Returning now to the nematic-isotropic transition, we have to expand G in the scalar invariants of Q. These are Qaa, QaPQaP,Q a p Q ~ r Q F... and so on. The first of these is the trace, which is zero, so there will be no term in the first order. The second is proportional to S2, the third to S 3 and so on. We can therefore write the expansion either using S, or we can use a scalar Q, and write
G = Go + ao(T-To) Q 2 -a3Q3 + a4Q4+ ...
(6.13)
Here we have written To instead of T, because it turns out that To does not now acquire the meaning of transition temperature. We have also given the Q3 term a negative sign, which is necessary if we consider a3>0, like a, and a4. The difference between this free energy and that of Eq. (6.3) is the Q3 term and it is important, as illustrated in Fig. 9 where we have compared G as a function of Q in the absence and in the presence of this Q3 term. In the first case we see that G is symmetric with respect to sign of Q: +Q and -Q describe the same state. We also see that the equilibrium state (illustrated by a ball which comes to rest where G is minimum) has Q = 0 for T > T,, Q#O for T
To such that the minimum at Q = 0 will be followed by a maximum and a decrease of G until finally the positive Q4 term raises its value again. Thus we now have two
36
2 Polar Materials and Effects NO Q3TERM
\& [&<Ti
Figure 9. The shape of the free energy function G according to a Landau expansion without (top) and with (bottom) a cubic term, representing a second order and a first order transition, respectively. Supercooling is possible down to To,superheating up to T , , in a first order transition but do not exist in a second order transition. The nematic phase is metastable for values of the order parameter below Qc.
WITH Q TERM
minima, one for Q = 0 and another for a finite Q # 0. The two minima will exist until T equals Towhen the G function becomes horizontal at the origin and the ball will be forced to roll down into the deep minimum created by the Q 3 and Q4terms alone. At Tothus the isotropic phase ( Q = 0) becomes completely unstable and Tois the lowest temperature to which the isotropic phase can be supercooled. At T, the two minima represent the same free energy. This is the phase transition temperature at which both phases would coexist indefinitely. At a slightly lower or higher temperature one of the phases is metastable and a transition occurs in which the order parameter changes discontinuously between zero and a value near Q,. The Q 3 term thus gives afirst order transition. Because of the energy barrier between the states the transition may not take place at T, but is slightly shifted in temperature. In practice essentially supercooling is observed. It may be quite appreciable in liquid crystals. The temperature variation of the order parameter Q, corresponding to the various minima in the free energy with Q3 term is shown in Fig. 9a. If the expansion contains a cubic term the transition is first order. If there is no cubic term the transition might still be first order but will always have a “second order character” (i.e., quasi-continuous transition with a very small amount of latent heat). In this case the character of the transition can only be disclosed by going to the sixth order term. If we decrease the temperature in the nematic phase we will in many materials especially if the molecules have more than two aromatic rings or if they have long flexible alkyl chains - observe a new “condensation”, to a lamellar order as illustrated in Fig. 9 b. The liquid crystal now becomes periodic along one dimension. In these
2.6 Landau Preliminaries. The Concept of Order Parameter Figure 9a. Variation of the order parameter for a first order phase transition; to be compared with Fig. 6i. The equilibrium transition temperature T, is here the temperature of infinite coexistence of the low- and high-temperature phases. The discontinuous behavior of the order parameter at T = T, is accompanied by such phenomena as latent heat and supercooling/superheating, absent from any second order transition. The nematic phase may be slightly superheated before it breaks down whereas the isotropic phase may be noticeably supercooled before the nematic order appears.
Figure 9 b. Molecular organization corresponding to nematic (N), orthogonal (A), and tilted (C) liquid-like smectic order, from left tonght.TheN-Atransitionisgenerally first order, as is the direct N-C transition. The A-C transition is generally second order (drawing after Y. Bouligand).
37
I_ 1
/‘
Tc
N
A
layered or smectic materials, of which only the two most common varieties smectic A and smectic C are shown, the nematic order is in a way preserved in each layer, indeed, the order parameter S is slightly higher than in the nematic phase at higher temperature. The three phases may appear successively in many materials (NAC materials) and normally the N-A transition will be discontinuous (first order) and the tilting transition A-C continuous (second order), i.e., we will observe how the tilt starts from zero at T, and then continuously grows to higher values of typically 20-30”. The tilt angle 8 will therefore seem to be a natural choice for the order parameter describing this transition. As we will work a great deal with the smectic C phase in this book it might be good to look into the layer of tilted molecules to get a slightly more realistic picture of the molecule than the one corresponding to a rigid rod. We will then see that the tail units generally are less tilted than the core, as illustrated in Fig. 9c. In our example we have chosen a non-chiral, very symmetric molecule of the 4,4’-phenyl benzoate kind and drawn the conformation with the alkyl chains in their fully extended form. As the core contains the x electron part it essentially determines the optical properties and its tilt is roughly what we observe as the optical tilt. This tilt will then be larger than the crystallographic tilt measured by x rays. The rotation of the core around the 4,4’-axis leaves the system relatively unchanged, as shown to the right. It can therefore be considered to take place fairly freely and be unbiased in the rotation angle. That the core tilts more than the end chains is the general picture but is not always true. The first counter example was discovered by Goodby and Chin [86e].
38
2
Polar Materials and Effects
Figure 9c. Typical molecular shape in a tilted srnectic phase, like SmC, illustrated on a4,4’-C8 phenyl benzoate molecule. The central core is generally more tilted than the end chains and relatively free to rotate around the 4,4’-axis. The flexibility in the alkyl chains can be considerable and a number of different conformations will contribute to the time-averaged picture.
As we pointed out, the fact that the trace of the order parameter is zero for the nematic-isotropic transition, TrQ=O
(6.14)
means that the linear term in the expansion of G ( Q ) is identically zero. On the other hand, from our discussion of G ( Q ) ,it is evident that G ( Q ) canna have any linear term in Q if G should have a minimum for Q = 0. Thus every Landau expansion starts with a second order term if it describes a transition between one state and another characterized by Q = 0, regardless of the character of the order parameter. Terms linear in the order parameter will appear in two cases. First, exemplified by the Landau expansion in an external magnetic field, we will have a free energy term -M .H . This scalar term has of course the required rotational invariance and will lead to a minimum of the free energy corresponding to a finite magnetization M . The second example is the case of so-called Lifshitz invariants with the consequences that the ground state of the medium corresponds to a certain finite value of chiral order parameter instead of the zero value.
2.6 Landau Preliminaries. The Concept of Order Parameter
39
To sum up, we have seen how the symmetry change at a phase transition may determine the character of the transition. In this fashion Landau arrived at a simple set of rules for predicting whether a certain phase transition is first or second order. By expanding the free energy in a convenient order parameter he could also develop the essential physics of new phenomena which were not accessible, or not yet accessible, to any microscopic theories. One such well-known example was superconductivity. The Landau theory is in principle equivalent to mean field theories and could be described as the most powerful of all mean field theories. Once developed, the method has become a paradigm in physics and due to its simplicity it has been applied widely beyond its origin in condensed matter physics, as a kind of avantgarde method to penetrate into unknown domains where theories have not yet acquired any shape. Thus it has become a common phenomenological tool in elementary particle physics and cosmology. We will use it extensively, beginning in the next section. In fact, every description of ferroelectric phenomena throughout this book will be based on this formalism. This is because it offers a language which is more universal, yet easier to handle than any other description presented so far. That the creation of this language has required considerable mental efforts might at first be a surprise. But Landau’s pupil and long-time collaborator, Vitaly Ginzburg, himself a theoretician beyond all categories, has witnessed this. As he recalls [86f]: “L. D. Landau told me once that his attempts to solve the problem of the second order phase transitions had demanded greater effort than any other problem he had worked upon.” Landau himself was never quite successful in this but his work opened up one of the most fascinating domains of modem physics. To give an illustration of the special fascination with the second order transition, one may contemplate the physical content of Fig. 6i: only at a second order transition is it possible to observe how matter behaves down to the point where the collective order vanishes, to give rise to a new form of organization. With modern tools, for instance light scattering, it has been possible to probe matter to the closest vicinity of T, and discover the strange phenomena in both space and time which occur when the constituents of matter cannot decide which of two symmetries it wants to join. In this vicinity the Landau theory breaks down. We can already understand this by noting that the theory ignores fluctuations (the order parameter only occurs as a mean value in the theory) whereas these fluctuations grow without limit when we approach the transition point T, and the order parameter itself is considered to become zero. While the Landau theory has this limitation it is still extremely useful. It can be added that the attempts, both experimental and theoretical, to explore the phenomena in this region in the vicinity of T, have been quite successful and belong to the most beautiful chapters of modern physics. Magnetic and liquid crystalline systems have here played an important role, in exploring the relation between symmetries and properties in media of different space and order parameter space dimensionality. Before these most modern developments, Landau’s collaborators Lifshitz and Ginzburg had already extended the original treatment in several important respects.
40
2
Polar Materials and Effects
Both extensions take account of the effect that the order parameter may not be homogeneous in the medium. We might then extend the expansion, in the form G = Go + a 2 M 2+ a4M4+ ... + ~(vA.4)~
(6.15)
The last term, quadratic in the gradients of the order parameter, is a fundamental term in the description of liquid crystals. If we use the director description for a nematic, at constant S, this term turns out to be the free energy due to the inhomogeneity of the director field,
This is the Oseen expression of 1928 which introduces the Oseen elastic constants K,,, K22and K33.It can today formally be included as a part of the Landau expansion. The major part of this book will be formulated in the language of Landau and Oseen.
2.7 The Simplest Descriptions of a Ferroelectric As mentioned above, Debye concluded that the interaction between local dipoles could lead to a permanent dielectric polarization. The mechanism can be described as follows: As long as the polarization is small, it is directly proportional to the applied field according to Eq. (4). However, as P becomes large, it begins to contribute to the field at the site of the dipoles. If we consider one of them, it will be in a field E superposed by an average field created by the other dipoles, which will be proportional to P . If we call the proportionality constant A, we can replace E in Eq. ( 4 ) by ( E + A P ) to give
P=
Np2 ( E + ilp) 3kT
Solving for P gives
p = - NP2 . 3k
E T-ANp2/3k
and for the susceptibility C O X = amaE =
N p 2 / 3k T -ANp2/3k
2.7 The Simplest Descriptions of a Ferroelectric
41
We see that the susceptibility diverges and becomes infinite when T approaches the value
which means that at this temperature we can have a nonzero polarization P even if the external field E turns to zero. T, is therefore the Curie temperature at which the dielectric goes from the paraelectric to the ferroelectric state. In Eq. (5) we introduced the Curie constant C equal to Np2/3k. With this inserted in Eq. ( l l ) , the susceptibility can be written as
C &OX = -
T-T,
This corresponds to the Curie- Weiss law for magnetic phenomena and has the same name for polar materials. In fact, Eqs. (1 1) to (15) are just the mean field description by Weiss, applied to polar materials. It may be pointed out that, although Eq. (1 1) correctly connects the ferroelectric state to an instability, the numerical value required for the phenomenological coefficient ilturns out (just as in the Weiss theory of ferromagnetism) to be quite unrealistic. Most polar materials should also be expected to be ferroelectric according to this model, in disagreement with experience. On the contrary, the collective interaction of an assembly of dipoles may not mean that the lowest state is one where all dipoles are directed in the same manner. The general problem is still unsolved, but in 1946 Luttinger and Tisza [87] were able to show that an assembly of dipoles forming a simple cubic structure has a lower energy in antiparallel fashion (antiferroelectric order) than in one where all dipoles are in the same direction. As a matter of fact, antiferroelectrics are much more common in nature than ferroelectrics! The paraelectric -ferroelectric instability is thus considerably more complex than indicated by the ideas behind Eq. (11). We may also notice that the instability described by this formalism corresponds to the very general concept of positive feedback, describing how an instability appears in a system. For instance, the diverging susceptibility describing the transition from paraelectric to ferroelectric at the instability point T, corresponds to the diverging gain when an electronic amplifier makes the transition to a self-sustaining oscillator at a certain feedback coefficient /3 (the fraction PV of the output voltage is fed back to the input voltage Vi). These analogous cases are compared in Fig. 10. It may be pointed out in advance that in liquid crystals there is no such instability connected with polar interactions. The Landau description, although in essence a mean field theory, is a phenomenological description of far greater generality and penetration than our discussion so far. Even from its simplest versions, deductions of high interest can be made. We be-
42
2 Polar Materials and Effects gain CP )
Figure 10. Phase transition from paraelectric to spontaneously polarized state in a ferroelectric, compared to the state transition in an amplifier when
ferroelectxic phal
paraelectric
the gain V/Vi=
A I-PA diverges for
/3= 1/A. A is the amplification at zero feedback. Tc
T
1/A
P
gin by introducing an order parameter, and in this case we naturally choose the polarization P, describing the order in the low temperature phase (P#O) and the absence of order in the paraelectric phase (P=O). The order parameter is a characteristic of the transition and should conform to the symmetry of the high temperature phase as well as to that of the low temperature phase. We expand the free energy in powers of P 1 a(T)P2 G ( T ,P ) = Go(T)+ 2 + -1b P 4 + - c1P 6 +... 4 6 Often, the high temperature phase has a center of symmetry (for instance, the cubic paraelectric phase of BaTiO,). If the high temperature phase is piezoelectric (as in the case of KH,PO,, abbreviated KDP), the free energy nevertheless cannot depend on the sign of strain and polarization. It is also true that the two states of opposite polarization in the low temperature ferroelectric phase are energetically equivalent. G is therefore invariant under polarization reversal (P+-P) and only even powers can appear in the Landau expansion. It may be noted that no such invariance is valid for a pyroelectric material, and that a corresponding Landau expansion cannot be made. Sometimes expressions of this type (Eq. 16) are called Taylor series expansions, which is rather misleading because the coefficients bear no relation to Taylor coefficients. A further point is that a Taylor series works very well in the limit of a vanishingly small value for the expanded variable. This is not the case for the Landau expansion: it fails in the limit when the order parameter goes to zero. Although we nevertheless take for granted that the order parameter is small, it is sometimes surprising to see that the expansion works fairly well even for rather large values of the order parameter. For instance, in a nematic, the order parameter is never small. In fact (and we will see examples of this later) the Landau expression is of a far more general nature and, in addition to powers of several order parameters and their conjugates and coupling terms, may contain scalar invariants of their components as well as their derivatives of different order. Generally speaking, the terms in-
43
2.7 The Simplest Descriptions of a Ferroelectric
cluded in the Landau expansion have to be invariant under all symmetry operations of the high temperature phase. (Then they are also automatically invariant under those of the low temperature phase, whose symmetry group is here assumed to be a sub-group of that of the high temperature phase.) As for the coefficients, we have indicated a temperature dependence of a, which we will express with the simplest straightforward choice
It is further assumed that b and c in Eq. (16) are small and both positive in the first instance. The term Go(Vis often included in G on the left side. Equation (17) means that the coefficient a changes sign at the temperature T,. For T > T,, a is positive and the free energy (now with Go included) will be a simple parabolic function at the origin, according to Eq. (16), thus with a minimum for P=O. For T< T,, the free energy will have a maxinium instead (see Fig. 11a). When P grows larger, however, the P4 term takes over and will yield two symmetrically situated minima, corresponding to a nonzero polarization which can take the opposite sign. At T,, the P2 term giving a curve which is very flat at the origin (see vanishes and G becomes -p, Fig. llb). The equilibrium value of P is obtained by minimizing the free energy with respect to this variable, thus
which gives, if we disregard the fifth power,
P [ a ( T - T,) + bP2] = 0
(19)
For T> T,, the only real solution is P=O, describing the paraelectric phase. For T < T,, we get 112
tion of the order parameter according to the Landau expansion Eq. (16) with Eq. (17). For T > T,, G has a minimum for order parameter value zero (P=O, corresponding to the paraelectric phase); for T
(20)
T>Tc
Tc
T G P
I
44
2 Polar Materials and Effects
describing how the spontaneous polarization increases smoothly from zero in a parabolic fashion when we decrease the temperature below the Curie point T,. This is characteristic for a second-order phase transition and qualitatively conforms well with real behavior in most ferroelectric materials. If b were negative instead and c positive, we would have had to include the original P6 term and would have found a (weakly) first-order transition with P malung a (small) jump to zero at the Curie point, but we will not pursue this matter further. If we apply an electric field, we have to include the term -PE in the Landau free energy. The field E is here the thermodynamic intensive conjugate variable to P. In the presence of a field, we have of course a nonzero polarization even in the paraelectric phase for T > T,. However, as P is small, we only keep the P2 term from Eq. (14) and write
1 a ( -~ G =c ) p 2- PE 2
for T > T,
Minimizing the free energy now gives
i.e., a polarization proportional to the applied field according to P=
E a ( T - T,)
and a corresponding susceptibility ap
1
&OX=aE= a(T-T,)
x
i.e., the Curie- Weiss law describing the divergence of when we approach the Curie temperature from above. Let us compare this to the situation in the ordered (ferroelectric) phase (T< T,). As P is now not small, we have to include the @ term in G and therefore write 1 G=-1 ( x ( T - T C ) P 2 +-bP 2 4 for T < T , .
4
-PE
Minimizing with respect to P gives ap
= a ( T - T , ) P + bP3- E = 0
(25)
2.7
45
The Simplest Descriptions of a Ferroelectric
and, differentiating with respect to E (remembering that Q X = aP/aE)
a ( T - T,)q)X + 3bP2EQX= 1
(27)
If we insert P*=(db>(T,-T), according to Eq. (20), into this expression we get 1 &OX = 2a(T,- T )
Our results are summarized in Fig. 12. Analytically, we can express them as
P
- 1 T - T, 1 a
X - I T - T, 1
X-'TX+ with the so-called critical exponents /I= 1/2 and y= 1, for the order parameter and susceptibility characteristic of the Landau description or, more generally, of mean field models. On approaching the Curie point from below, the susceptibility diverges just as on approaching from above, with the difference that x(T T,). This is a general mean field result: the susceptibility in the condensed 2 phase is half of that in the disordered phase. The reason for their difference is evident: the effect of an external field on the polarization must be smaller in the condensed phase, due to the existing internal field (molecular field) in that phase, which has a stabilizing effect on the polarization. We may also write this as
because it also means that the Curie constant above and below the transition differs by a factor of two. Graphically this is most easily seen if we plot the inverse of the
(.:;I
Figure 12. The growth of the order parameter P ( n , in this case representing ferroelectric polarization below the Curie point T,, 1 ap together with the corresponding susceptibility = -- above and below the phase transition. EO a E
jl XP(T)
x
TC
T
46
2 Polar Materials and Effects Figure 12a. The inverse dielectric susceptibility following a Curie law with different Curie constants (C+lC-=2) above and below the Curie point T,.
x-1
t\
susceptibility as a function of the temperature, as in Fig. 12a. In passing it may be mentioned that no such critical behavior (no divergence in would be observed if the low temperature phase has antiferroelectric order, corresponding to the fact that such a phase has zero macroscopic polarization. Finally, let us look a little closer at the behavior in an external field both above and below the Curie point. We will rewrite the free energy from Eq. (25) in a slightly more expanded form
x)
1 G=Go+-aP 2
2
1 +-bP
4
4
1 +-cP 6
6
+...- PE
(33)
Minimizing the free energy with respect to P at constant external field E
($1,
=O
E = aP
+ bP3 + cP5 + ...
(34)
yields
Sometimes this is written (including .so in
E=1P ~
X
(35)
x)
+ 4 P3 + 5 P5
where the values of 4 and 5 are given as measure of the dielectric nonlinearity. Let us again neglect the highest power and write
E = a ( T - T,) P
+ bP3
When T > T, we may also discard the P 3 term and see that E
- P (paraelectric state)
(37)
2.7 The Simplest Descriptions of a Ferroelectric
47
whereas for T= T, we get a purely cubic relationship
E - P3
(39)
A graphic illustration of Eq. (37) is given in Fig. 11. The P - E curves are analogous to van der Waals isotherms. For the isotherm at T=T,, instead of Eq. (39) we write
introducing a third critical exponent 6, with the value 6=3 for the critical isotherm of the order parameter. This value, S= 3 , is also a characteristic of mean field theories. For T T,, and much more dramatically at T= T,, P does not change sign at first below the Curie point. But P is now not a single-valued function of E, and below the value -E,, the decrease in P no longer follows the upper curve but continues, after a jump, along the lower curve. We also see that below T, there are two stable states for E=O. In Fig. 12c the free energy as a function of polarization according to Eq. (33) is sketched (the P6 term can here be neglected) for three different values of the exter-
Figure 12b. Graphic representation of E=aP+bP3. This strongly nonlinear relationship P(E)describes the ferroelectric hysteresis (emphasized) appearing for T< T,. Two stable states of opposite polarization exist at E =O. A field superior to the coercive field E, has to be applied in order to flip one polarization state into the other.
E
P
Figure 12c. The free energy as a function of polarization for different values of the external field. The symmetrical curve for E=O for which the two polarization states ?Po are stable has been emphasized and E, >E, corresponds to the coercive field E, which destabilizes the -Po state.
48
2
Polar Materials and Effects
nal field. The stable states for P correspond to the minima of G. In a ferroelectric sample these are seen in the spontaneous domains. For a sufficiently high field, E2 (corresponding to E, in Fig. 12b), the oppositely directed polarization state becomes metastable and the polarization is reversed.
2.8 Improper Ferroelectrics So far it has been assumed that the polarization is the variable that determines the free energy, but it may happen that the transition is connected to, i.e. occurs in a completely different variable, let us call it q, to which a polar order can be coupled. A polarization P appearing this way may affect the transition weakly, and it can only have a secondary influence on the free energy. We will therefore call q and P, respectively, the primary and secondary order parameters, and the simplest Landau expansion in those two parameters can be written
G = -1a ( T ) q 2 +-bq 1 4 +-xo 1 - 1 P2 -AqP 2 4 2 We note, already from the presence of the coupling term -Aq P , that the Landau expansion is not a “power series”, cf. the previous comments made with reference to the expression (1 6). If q and P have the same symmetry, i.e., if they transform in the same way under the symmetry operations, then a bilinear term of the form -Aq P must be admitted since it behaves like the other terms of even power. The minus sign means that the free energy is lowered due to the coupling between q and P. The coupling constant A is supposed to be temperature-independent. The fact that q appears to the fourth power, but P now only to the second, reflects their relative importance. Instead of using c as the coefficient in front of the P2 term, it is convenient to introduce the symbol xi’.We assume that xi’,in contrast to the coefficient a ( T ) in front of q2, does not depend on temperature, which also reflects the fact that P is not as a transition parameter. In fact, it can often be shown that xO’-T, which in practice means that it is temperature independent compared to a coefficient varying like T-T,, where T, lies in the region of experimental interest. As will be discussed in detail later, takes the form of a special kind of susceptibility. The equilibrium values of q and P are obtained by minimizing G with respect to these variables. Let us begin with P . From
xo
we see that P will be proportional to the primary order parameter
2.8
Improper Ferroelectrics
49
Continuing with
aq
+
= a ( T - T,)q bq3 - ?LP= 0
(44)
and inserting P from Eq. (43), we get
[ a( T - T,) - ?L2xo] q + bq3 = 0
(45)
or
a(T- TL)q+ bq3 = 0 with
a2xo
T,= T, +-
(47)
a
Equation (46) is the same relation as Eq. (19), hence q in the ordered phase will grow in the same way as the primary order parameter P in our previous consideration, i.e., according to Eq. (20), or
Equation (43) shows that this is also true for P as the secondary order parameter. However, we have a new Curie temperature given by Eq. (47), shifted upwards by an amount AT= TL- T, A T = -L2X0
(49)
a
Thus the coupling between the order parameters will always raise the transition temperature, i.e., stabilize the condensed phase. As we can see, the shift grows quadratically with the coupling constant A. Let us now look at the dielectric susceptibility in the high temperature phase ( T > TL).Both the order parameters q and P are zero in the field-free state, but when an electric field is applied it will induce a polarization and, by virtue of the coupling between P and q, also a nonzero value of q. In the high temperature phase we will thus have a kind of reverse effect brought about by the same coupling mechanism: whereas in the low temperature phase a nonzero q induces a finite P, here a nonzero P induces a finite q. The values of P and q above TL are small, so we can limit ourselves to the quadratic terms for G and write
C = - 1a q 2 + - x1o 2 2
-1
P2 -AqP-PE
(50)
50
2
Polar Materials and Effects
We then find
giving q=
~
P for T > T l
Because P is proportional to E, in the high temperature phase we induce a nonzero value of the order characteristic of the low temperature phase by applying a field, and this order is proportional to the field. In liquid crystals, the electroclinic effect is an example of this mechanism. This is not a normal pretransitional effect, because it is mediated by the kind of order that shows no translational instability and by its susceptibility = (aP1dE). Sometimes it will also be convenient to introduce a corresponding susceptibility for q, Xq=aq/dE. If we minimize G with respect to P in Eq. (50)
%x
and insert q from Eq. (52),we find
Therefore
(55) or T - T_
(56)
with T ~ - T , = A T = A 2 ~ o lThis a . may be written
x
If the coupling constant A is sufficiently large, we see that will satisfy the CurieWeiss law. Then the dielectric susceptibility will indeed show a divergent behavior,
2.9 The Piezoelectric Phase
51
although P is not the transition parameter. If ilis small, the susceptibility will instead be practically independent of the temperature except in the vicinity of T,, where we might observe some slight tendency to a divergent behavior. In conclusion, the case where q and P have the same symmetry in many respects resemblesthe case where Pis the primary order parameter. When there is a strong coupling between the two order parameters we might not see too much of a difference. In the case where q and P do not have the same symmetry properties, the term -Aq P is not an invariant and cannot appear in the Landau expression. Therefore it must be replaced by some other invariant like -q2P (depending on the actual symmetries) or similar terms like q2PiP,.The coupling will then take on a different character. In the case where we have two order parameters q, and q2,which couple to P , we may have a term -ql q2P, and so on. In this last case, Blinc and 2ek3 found ([30], pp. 45, 58) that P will not be proportional to q1 and q2, but to their squares, which means that P will grow linearly below the transition point To
instead of parabolically [Eq. (20)], as in normal ferroelectrics. Furthermore, the susceptibility is found to be practically independent of temperature above To,as well as below it, and makes a jump to a lower value when we pass below To.The nondivergence of the susceptibility is due to the fact that the fluctuations in q l and q2 are not coupled to those of P in the first order in the high temperature phase 1301, hence does not show a critical behavior. Ferroelectrics in which the polarization P is not the primary order parameter are called improper ferroelectrics. The concept was introduced by Dvofak [88] in 1974. As we have seen in the last example, improper ferroelectrics may behave very differently from proper ferroelectrics. The differences are related to the nature of the coupling. If we write the coupling term -AqnP, our first example showed that for n = 1, the behavior may, however, be very similar to that of proper ferroelectrics. This class of improper ferroelectrics for which n = 1 is called pseudoproper ferroelectrics. Ferroelectric liquid crystals belong to t h s class.
x
2.9 The Piezoelectric Phase The paraelectric phase has higher symmetry than the ferroelectric phase. However, it might have sufficiently low symmetry to allow piezoelectricity. In the case where the paraelectric phase is piezoelectric, it will become strained when electrically polarized. In addition to the term -PE there will appear a term -so in the free energy, where s designates the strain and o the applied stress. The stress o is a “field” corresponding to the field E and in a thermodynamic sense the intensive conjugate var-
52
2
Polar Materials and Effects
iable to the extensive variable s, just like E is to P (or H to M , or T to S, etc.). If we suppose that the elastic energy is a quadratic function of the strain, we can write (for materials in general)
G=-s 1
2 --SO
(59)
2x0
so that we deduce the equilibrium strain from
or s = xoo
Thus the strain is a linear function of the applied stress (Hooke's law). The Landau expansion for a paraelectric phase with piezoelectric properties will now contain this new term in addition to a coupling term -ASP, of the same form as in our previous case
G = -1a P 2
2
+-s 1
2x0
2
-ASP-so-PE
Now, aG/aP=O and aG/as=O give the equilibrium values of P and s, according to
or
E = a P - As and 1 -s
xo
-AP-0 =0
or
o = - s1- A P xo If we arrange the experimental conditions such that we apply an electric field at the same time as we mechanically hinder the crystal's deformation (this is called the "clamped crystal" condition), then s=O, and Eq. (64) gives
E=aP
(67)
2.9 The Piezoelectric Phase
53
Hence
1 = uapx
whence
Thus the Curie law is fulfilled. In this situation a stress will develop to oppose the effect of the field and its induced polarization. Its value is, from Eq. (66) O=
-AP
(70)
If instead we have the "free crystal" condition, there can be no stress and, with 0=0 in Eq. (65),we get
The insertion of s in Eq. (64) now gives
or, differentiating with respect to E 1 = ( u - i l x0)- ap aE
(73)
Thus
(74) or 1 = a ( T - T;)
with a new Curie point
T,=T,+-
a
> T,
(75)
54
2 Polar Materials and Effects
Thus we see again that the coupling raises the transition temperature. As an example, the paraelectric -ferroelectric transition point in a “free” crystal KDP lies 4 “C higher than for a clamped crystal. The reason that the Curie point is lower in the clamped crystal is that the mechanical clamping eliminates the piezocoupling. Finally, we want to see how the piezocoefficients behave at the transition. These coefficients dijkare tensors of rank three defined by
but as the tensorial properties do not interest us for the moment we continue to use a scalar description and write
Due to the appearance of the two conjugate pairs soand P E in the free energy expression (Eq. 62), a Maxwell relation
(79) exists, whch allows a very simple calculation of d. Rewriting Eqs. (64) and (66) to eliminate P in order to form d = (JslaE), yields
AE-ao =
(22 ) -
s
and
This gives
If we compare this with Eqs. (74) and (7.3, it is evident that it can be reshaped to
d ( T )=
xo a(T-T,)
(83)
2.9 The Piezoelectric Phase
55
Hence the piezocoefficient diverges just like the dielectric susceptibility at the paraelectric- ferroelectric transition. This divergence has other consequences: because of the piezoelectric coupling between polarization and elastic deformation, it will influence the elastic properties of the crystal. If we look at the coefficient relating the strain to an applied stress, it can be measured under two different conditions, either keeping the electric field constant or keeping the polarization constant. In the latter case the crystal behaves normally, it stays “hard” when we approach the transition. In the former case the elastic coefficient diverges just like d, i.e., the crystal gets anomalously soft when we approach the transition. The formalism developed in this section was first used to describe the piezoelectric effect. It was later employed to treat the electroclinic effect in orthogonal chiral smectics. As will be seen in Secs. 5.7-5.8 there are very close analogies, but also some characteristic differences between the two effects.
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3 The Necessary Conditions for Macroscopic Polarization 3.1 The Neumann and Curie Principles When we are dealing with the question of whether a material can be spontaneously polarized or not, or whether some external action can make it polarized, there are two principles of great generality which are extremely useful, the Neumann principle and the Curie principle. Good discussions of these principles are found in a number of books, for instance [24,36, 891. The first of these principles, from 1833 and named after Franz Neumann, who founded what must be said to be the first school of theoretical physics in history (in Konigsberg), says that any physical property of a medium must be invariant under the symmetry operations of the point group of the medium. The second, from 1894, after Pierre Curie, says that a medium subjected to an external action changes its point symmetry so as to preserve only the symmetry elements common with those of the influencing action. Symbolically we may write the Neumann principle as
K c P
(84)
i.e., the crystal group either coincides with P , the property group, or is a subgroup of P . The Curie principle is expressed
i.e., the point symmetry K changes to K,the highest common subgroup of K (the crystal group) and E (the external-influence symmetry group). This means that, in particular, if
then K=K
(87)
i.e., if K is a subgroup of E, then the crystal symmetry is not influenced by the external action. The Neumann and Curie principles have long been the dominating symmetry principles in condensed-matter physics. They were also the first symmetry principles to
58
3 The Necessary Conditions for Macroscopic Polarization
be applied in physics. Both can be formulated in a number of different ways. For instance, the Neumann principle says that the symmetry of a medium is the symmetry of its least symmetrical property. It may also be stated as follows: ‘the symmetry elements of an intrinsic property must include the symmetry elements of the medium’. This formulation stresses that every physical property may and often does have higher symmetry, but never less than the medium. A well-known example of this is that cubic crystals are optically isotropic, which means that the dielectric permittivity has spherical symmetry in a cubic crystal. Another example is that the thermal expansion coefficient of a cubic crystal is independent of direction. In fact, if it were not, the crystal would lose its cubic symmetry if it were heated. Thus, as far as thermal expansion is concerned, a cubic crystal “looks isotropic” just as it does optically. Since, according to Neumann’s principle, the physical properties of a crystal may be of higher symmetry than the crystal, we will generally find that they range from the symmetry of the crystal to the symmetry of an isotropic body. A more general example of higher symmetry in properties is that such physical properties characterized by polar second rank tensors must be centrosymmetric, whether the crystal has a center of symmetry or not, cf. Fig. 27. For, if a second rank tensor T connects the two vectors p and q according to
and we reverse the directions of p and q, the signs of all the components pi and qj will change. The equation will then still be satisfied by the same Tu as before. How can we know that the Neumann principle is always valid? Consider, for the sake of argument, that the crystal group K had a symmetry operation that was not contained in the property group P. Then, under the action of this operation, the crystal would on the one hand coincide with itself, and on the other, change its physical properties. This inherent contradiction proves the validity of the principle. This principle is often used in two ways, although it works strictly in only one direction. Evidently, it can be used to find out if a certain property is permitted in a medium, the symmetry of which is known. However, it may and has also been used (with caution) as an aid in the proper crystallographic classification of bodies from the knowledge of their physical properties.
3.2 Franz Neumann, Konigsberg, and the Rise of Theoretical Physics The time span 1833 - 1894 between the Neumann and Curie principles is a dynamic chapter in the history of crystallography and the beginning of solid state physics, which can be dated back to 1801 when abbC Hauy described the morphology and
3.2 Franz Neumann, Kiinigsberg, and the Rise of Theoretical Physics
59
symmetry of crystals and introduced the unit cell. In 1824 Neumann distinguished between point and space lattice symmetry and in 1830 Hessel for the first time deduced the 32 crystallographic point groups (classes). Neumann introduced the spherical projection, since then applied universally. The rational index nomenclature (hkl) introduced by Weiss and Neumann was ameliorated by Miller into the more convenient reciprocal indices (hkl) in 1840. In 1850 Bravais derived the 14 symmetrical space lattices but also discovered the elegant geometrical device later known as the reciprocal lattice (the name coined by Ewald 1913). Finally, in 1890 Fiodorov and Schoenfliess independently deduced the 230 space groups. The Neumann principle thus dates back to the very beginning of the recognition of crystal symmetry; the Curie principle apppears at the end. During this same time there was a fascinating development in electromagnetic theory in which Neumann played an important role, and in which the most spectacular event is Maxwell’s electromagnetic theory of light, published 1861. Faraday discovered his law of induction in 1831but its mathematical formulation was given by Neumann and finally Hertz (in 1890) in the deceptively simple form e = -d@ldt
To appreciate the deep and complicated physics behind this formula one can remember that historically it was the inherent difficulties in the mathematical formulation of the induction law - and not the Michelson-Morley experiments - which brought Einstein to the discovery of his theory of relativity. Or to cite his own words: “What led me to the special theory of relativity was the conviction that the electromotive force acting on a body in motion in a magnetic field was nothing else but an electric field”. In 1845 Neumann introduced the concept of vector potential A for a source-free field B
B=VxA
(89 a)
a concept which has acquired eminent importance in modem physics. Lagrange had introduced the scalar potential in 1773. In both cases the potentials were introduced as a convenient mathematical expedient; today they lead their very independent lives. An elegant use of the vector potential for the magnetic field is in deducing the mutual induction of two circuits, an expression given in every textbook on electricity and magnetism under the name of Neumann’s formula. Franz E. Neumann (1798-1895) first studied mineralogy under C . S. Weiss in Berlin, receiving his doctorate in 1825. In 1826 Neumann came together with the mathematician Jacobi as Privatdozent to Konigsberg were he was promoted by the astronomer Bessel and became professor of mineralogy and physics in 1829. In the preceding century the Konigsberg university had been dominated by the philosopher
60
3 The Necessary Conditions for Macroscopic Polarization
Kant (1724- 1804) and now continued with its blossoming period. More than by anything else Neumann had been impressed by Fourier’s Analytical Theory of Heat which appeared in 1822, and he was to spend a great deal of work in developing the theory of orthogonal functions discovered by Fourier. Thus in Koningsberg he advanced the theory of spherical harmonics and discovered the Bessel function of the second kind (the Neumann function) which is required for the general solution of Bessel’s differential equation. In addition he made important contributions to elasticity, crystal physics, optics, and electromagnetism. Together with Jacobi he set up the Konigsberg Mathematics-Physics Seminar. It is interesting to note that while Neumann’s principle expresses symmetry and invariance of objects and properties, these concepts almost immediately went over to more abstract forms like equations. In fact, the analytical mechanics founded by Hamilton in Dublin in 1827-1 833 were developed by Jacobi in Konigsberg into the Hamilton-Jacobi formalism in 18371840. In his lectures he showed that within this formalism, invariance under displacements in time, position, and angle give rise to the conservation of energy, linear, and angular momentum, respectively. These invariance principles thus were not introduced by elementary-particle physicists, as a present-day student might believe. Indeed, Voigt, in 1887 was the first to discover (before Larmor, Lorentz, and PoincarC) that the coordinate transformation in 4D space-time, later known as “Lorentz transformation”, is the one that leaves the form of the wave equation invariant. Neumann’s experimental work is also considerable, but of less importance historically. But in particular, Neumann’s time in Kongigsberg was the time of the rise of Theoretical Physics, a discipline which did not exist outside of Neumann’s school. Whereas at the time of Euler and Lagrange no distinction was made between mathematics and physics, one hundred years later there were chairs in mathematics and in physics, but theoretical physicists at that time were considered assistants to the experimentalists, i.e., to the physics chairs and could not be promoted to full professors. The influence of Neumann’s school gradually changed this, first in Germany by his many students, for instance Voigt who went to Gottingen and Kirchhoff who went to Berlin, and later in other countries. The first full professors of theoretical physics in history were in Konigsberg, Gottingen (Voigt), and Berlin (Plank, who followed Kirchhoff in 1889). Neumann’s monumental “Lectures on Mathematical Physics” in seven volumes were edited at the end of his life (by his son, the mathematician Carl Neumann, later professor in Halle and Leipzig) and became the model for the teaching of theoretical physics, adapted by Sommerfeld in Munich (who as a student in Konigsberg heard Hilbert’s lectures on mathematics) and which for a long time was the standard at all German universities. This model has also in particular continued with the Landau-Lifshitz Course on Theoretical Physics, now in ten volumes. That the recognition of theoretical physics as a discipline of its own did not proceed without resistance, however, is illustrated for instance by the fact that in Italy, the first chair of theoretical physics was created only in 1925 - for Enrico Fermi !
3.3 Neumann's Principle Applied to Liquid Crystals
61
3.3 Neumann's Principle Applied to Liquid Crystals The general application of the Neumann principle in condensed matter is normally much more formalized (for instance, involving rotation matrices) than we will have use for. Few textbooks perform such demonstrations on an elementary level, but there are exceptions, for example, the excellent treatise by Nussbaum and Phillips [90]. For a liquid crystal, we illustrate the simplest way of using the symmetry operations of the medium in Fig. 13. (The same discussion, in more detail, is given in [46].)We choose the z-direction along the director as shown in Fig. 13a and assume that there is a nonzero polarization P = (P,, Py, P,) in a nematic or smectic A. Rotation by 180O around the y-axis transforms P, and P, into -P, and -Pz, and hence both of these components must be zero, because this rotation is a symmetry operation of the medium. Next, we rotate by 90 ' around the z-axis, which transforms the remaining Py into P,. If therefore Py were nonzero, we would see that the symmetry operations of the medium are not symmetry operations of the property, in violation of Neumann's principle. Hence Py = 0 and P will vanish.
Nematic
Smectic A
smectic C
a
Rotation 180" around 9
Symmetry operation: (b)
rotation 180 around
9
P1=0,P,=O Rotation 90" around z^
Figure 13. Neumann's principle applied to ( a ) the nematic and smectic A, (b)the smectic C and smectic C* phases.
62
3 The Necessary Conditions for Macroscopic Polarization
In Fig. 13b we have illustrated the C phase, normally occurring at a secondorder phase transition A +C at a certain temperature T,, below which the molecules start to tilt, and with an order parameter that can be written as Y=8etq, where 8 is the tilt angle with respect to the normal and cp is the azimuthal angle indicating the direction of tilt in the layer plane. The variable q is a phase variable (the transition is helium-like) with huge fluctuations; the director in the C phase therefore has a large freedom to move on a cone with half apex angle 8 around the layer normal. In the figure, the tilt is chosen to be in the xz-plane. Hence, a 180” rotation around the y-axis is still a symmetry operation so P can have a component in the y-direction, whereas a 90” rotation around the z-axis is no longer a symmetry operation. However, if the xz-plane is a mirror plane, then Py must be zero, because a nonzero Py goes to -Py on reflection. On the other hand, if the medium lacks reflection symmetry, i.e., is chiral, such a component must be admitted. The important concepts chiral and chirality were introduced by Lord Kelvin, who derived them from the Greek word for hand. After having contemplated such symmetry-dissymmetry questions for at least a decade, he used them for the first time in his Baltimore lectures in the fall of 1884, published twenty years later [91]. He there stated, “I call any geometrical figure or group of points chiral and say it has chirality if its image in a plane n ~ o ideally r realized, can not be brought to coincide with itself’. These concepts are used, and should be used, in exactly the same manner today. Thus, in order to let Py survive, we can put “small propellers” on the molecules, i.e., make them chiral. We could for instance synthesize molecules with one or more asymmetric carbon atoms, but it would, in principle, also be sufficient to dissolve some chiral dopant molecule in the nonchiral smectic C in order to remove the mirror plane. This can, symbolically, be written c+*+c*
(90)
The smectic is now chiral and may be denoted C* (which is independent of whether there is any observable helix or not). The first demonstration of the symbolic equation (90) was made by Kuczynski and Stegemeyer [91a] who added chiral dopants to non-chiral hosts (today a standard technique for device materials) and verified the linear dependence of P on concentration (for low values) as well as the linear relation between P and 0. Whether the constituent molecules are chiral or whether we make the medium chiral by dissolving chiral molecules to a certain concentration c >O, in both cases the only symmetry element left would be the twofold rotation axis along the y-direction, and a polarization along that direction is thus allowed. The symmetry of the medium is now C, (or 2), which is lower than the symmetry of the property. The polarization P,like the electric field, is a polar vector, hence with the symmetry Cmv (or mmm in the crystallographic notation).
3.4 The Surface-Stabilized State
63
The fact that P, if admitted, must be 90 degrees out of phase with the director, could have been said from the very beginning. The fundamental invariance condition in Eq. (6) means that if there is a polarization P in the medium, it cannot have a component along n,because the symmetry operation (6) would reverse the sign of that component, and thus P must be perpendicular to n P l n It was the Harvard physicist Robert Meyer who in 1974 first recognized that the symmetry properties of a chiral tilted smectic would allow a spontaneous polarization directed perpendicular to the tilt plane [61]. In collaboration with French chemists, he synthesized and studied the first such materials [62]. These were the first polar liquid crystals recognized and as such something strikingly new. As mentioned before, substances showing a smectic C" phase had been synthesized accidentally several times before by other groups, but their very special polar character had never been surmised. Meyer called these liquid crystals ferroelectric. In his review from 1977 [43] he also discussed the possible name antiferroelectric, but came to the conclusion that ferroelectric was more appropriate. We will call the polarization Ppermitted by the symmetry argumentjust discussed, the spontaneous polarization. Leaving out its vectorial property, it is often indicated as P,. The spontaneous polarization is so far a local property, not a macroscopic one.
3.4 The Surface-Stabilized State According to the symmetry argument already given, the spontaneous polarization P is sterically connected with the molecule, lying along +zx n. (The plus or minus sign is a characteristic of the material.) However, the prerequisite for the existence of P was that the molecules are chiral, and the chiral interactions between them lead to an incommensurate helical superstructure, as shown in Fig. 14. Actually there are two interactions involved in the helix, one chiral leading to a spontaneous twist and one nonchiral (steric) leading to a spontaneous bend deformation in the director field (see Sec. 4).The phase angle q describes a right- or left-handed helix with a period of typically several micrometers, whereas the smectic layer periodic length is several nanometers. The helical arrangement makes the macroscopic polarization zero for the bulk sample at the same time as satisfying V .P=O everywhere. It is therefore an alternative to the domain formation in a solid ferroelectric. However, there is no coercive force in such a structure, and if we apply an electric field perpendicular to the helix axis we will wind up the helix and turn more and more of the local polarization into the field direction (see Fig. 14a). The P-E response is dielectric - in fact it has the shape of the curve shown at the bottom of Fig. 5 , corresponding to an antiferroelectric with an infinitely thin hysteresis loop.
64
3 The Necessary Condtions for Macroscopic Polarization Figure 14. The helical configuration of the sterically coupled variables n -P in the unperturbed helical C* state. The phase angle 9 changes at a constant rate if we move in a vertical direction.The variables n,P,and 9 are all functions of z, the vertical coordinate. The twist is exaggerated about ten times relative to the densest possible twist occurring in typical smectic C* materials.
Figure 14a. The helical configuration of the directorpolarization couple is unwound by a sufficiently strong electric field E. The increasing field induces a macroscopic polarization (which is thus not spontaneous) and finally polarizes the medium to saturation (all dipolar contributions lined up parallel to the field), as shown to the right.
But if the loop is infinitely thin it is also true that the case is very similar to a ferroelectric with an infinitely thin hysteresis loop. Hence the helical C* state, in a way, is something that lies between ferroelectric and antiferroelectric. Blinov has pointed out [43a] that, if we compare the SmC* with a solid, it would be more appropriate to regard it as pyroelectric. There is a good deal of truth in this observation. By this he means that the local polarization is sterically fixed relative to the “lattice”, it cannot be changed in a given substance. When we change the direction of P by applying an electric field, we do not change the direction of P relative to the lattice but we actually distort the lattice itself. And this we can do because the lattice is so weak, practically fluid. This view is illuminating but the electro-optical properties of the bulk should be considered more important. Also, as has been pointed out by Michelson, Cabib and Benguigui [43b], the helicoidal order is not just something
3.4 The Surface-Stabilized State
65
accidental, because the phase happens to be chiral, but really more fundamental. We will return to their considerations below. The helicoidal structure should appropriately be called helielectric. It is instructive to compare the smectic C* with the best known solid state antiferroelectric which is sodium nitrite, NaNO,. The high temperature phase ( T > 165.5 "C) of this compound, orthorombic with symmetry mmm,is paraelectric with Na+ and NO, groups disordered. At T,, slightly below 164 "C, the Na+ and NO, groups order to a polar arrangement and the structure loses one of the mirror planes to become orthorombic mm2.It is now ferroelectric and has a P , value of 8500 nC/cm2 at room temperature. If we heat this ferroelectric up to just below T,, in a small range of 0.3 K, an incommensurate antiferroelectric phase appears. At T, the hysteresis loop vanishes but the phase is still polar within a small temperature range of about 1.7 K. In this range the polarization precesses in a helical way as we move along one of the crystallographic axes. This sinusoidally modulated phase was the first electrically ordered incommensurate antiferroelectric to be discovered [43c]. The helical period is about eight to ten times the unit cell and depends on the temperature. It does not have any rational relation with the lattice constant - this is the meaning of incommensurate - which is another way to say that the crystal has long rang order but lacks translational symmetry (it has translational symmetry in four dimensions). The helical arrangement causes the macroscopic polarization in the crystal to vanish. This order is called helicoidal antiferroelectric. A shorter useful name is helielectric. The phase diagram of NaNO, in the presence of an electric field is sketched in Fig. 15. The incommensurate antiferroelectric phase is found between the ferroelectric and the paraelectric phase. If we apply an electric field perpendicular to the helical axis it will start to unwind the helix. The critical field for complete unwinding varies with
electric field applied Ihelical axis
Lifshitz point
A sinusoidally modulated phase helix unwinding
ferroelectric
Ti
i
~
TP
paraelectric i
T~
Figure 15. Phase diagram in the presence of an electric field applied perpendicular to the helical axis of the incommensurate antiferroelectric NaNO,. The incommensurate phase is found between the ferroelectric low temperature phase and the paraelectric high temperature phase. T , and T, are 163.8 "C and 165.4 "C, respectively. This phase is a helicoidal antiferroelectric with a helical period of about eight to ten times the unit cell. The electric field unwinds the helicoidal structure and on approaching T , (T,) the critical unwinding field vanishes, but is otherwise of the order of 0.3 V/pm.
~
66
3 The Necessary Conditions for Macroscopic Polarization
temperature and the modulated structure eventually disappears, hardest in withstanding the field in a so-called Lifshitz point. A typical value for the critical field is about 3 kVkm or 3 V across a sample of 10 pm. It has also been reported that the piezoelectric effect vanishes in the incommensurate phase, which is the behavior of an antiferroelectric. However, this can be questioned, for symmetry reasons, cf the discussion to Fig. 40e. In fact, one shear piezo mode ought to be left in the quite unique case of a solid helicoidal antiferroelectric. The helielectric smectic C* has zero macroscopic polarization (as all antiferroelectrics), no hysteresis, no threshold, and no bistability. However, by an artifice it can be turned into a structure with very different properties. This is illustrated in Fig. 16. If the smectic layers are made perpendicular to the confining glassplates, there is no boundary condition compatible with the helical arrangement. Let us assume that we can prepare the surface in such a way that the boundary condition imposed on the molecules it such that they have to be parallel to the surface, but without specified direction. As we make the distance between the surfaces smaller and smaller, the conflict between the helical order and the surface order will finally elastically unwind the helix via the surface forces, and below a certain sample thickness the helix cannot appear: the nonhelical configuration has now a lower energy. The physical problem has one characteristic length, which is the helical period z. We can therefore expect that for a sample thickness d < z , i.e., of the order of one micrometer, the only allowed director positions will be where the surface cuts the smectic cone, because here both the intrinsic conical constraint and the constraint of the surface are simultaneously satisfied. There are two such positions, symmetrical around the cone axis and corresponding to polarization up and down, respectively. Energetically these states are equivalent, which leads to a symmetric bistability. Indeed, when very thin samples of d= 1 pm are made with the appropriate boundary conditions, spontaneous ferroelectric domains all of a sudden appear in the absence of any ap-
Figure 16. Elastic unwinding, by the surfaces, of the helical twist in the bookshelf geometry of smectic C*. The helical bulk state is incompatible with the surface conditions and therefore can never appear in sufficiently thin cells. The surface acts as an external symmetry breaking agent, reducing the degeneracy of the bulk to only two selected states. In the most attractive case these are symmetric, which also leads to a symmetric bistability of the device. The two memorized director states, n, and n2 (n;and n; in the case of a pretilt different from zero) represent polarization states of opposite (or nearly opposite) direction.
3.4 The Surface-Stabilized State
67
Figure 17. Spontaneous ferroelectric domains appearing in the surfBce-stabilized state.
plied field (Fig. 17). This second step was realized five years later than Meyer’s first paper [93]. By applying an external field we can now get one set of domains to grow at the cost of the other and reverse the whole process on reversing the field. There are two stable states and a symmetric bistability; the response has the characteristic form of a ferroelectric hysteresis loop, as represented by the second curve of Fig. 5. We might therefore call this structure a surface-stabilized ferroelectric liquid crystal (SSFLC). The surface stabilization brings the C* phase out of its natural crystallographic state and transfers macroscopic polarization to the bulk. We may note that the polarization in the helical state of Fig. 14 is denoted P and not P,. This is because the local polarization in this state does not correspond to P,, but has a second contribution due to the flexoelectric effect. This contribution, which will be discussed in detail later, is of the same order of magnitude as Ps. There are several different methods to measure Ps, most of them involving the application of an electric field to saturate the polarization between the two extreme states, as represented to the right of Fig. 14a, and the corresponding state with the field reversed. In such a state there is no flexoelectric contribution, hence P coresponds to P, except for a contribution due to the electroclinic effect. This is an induced polarization, Pi,connected to an extra tilt 68 of the director caused by the field. The electroclinic tilt is negligible compared to 8 for T 4 T,, thus Pi is normally quite small, but may be observable at high fields near T = T,, and may then slightly change the shape of the observed polarization curve, as illustrated in Fig. 18. The helical smectic C* state has the point symmetry D , (w22), illustrated in Fig. 19, which does not permit a polar vector. It is therefore neither pyroelectric nor ferroelectric. Nor can it, of course, be piezoelectric, which is also easily realized after a glance at Fig. 14: if we apply a pressure or tension vertically, i.e. across the smectic layers (only in this direction can the liquid crystal sustain a strain), we may influence the pitch of the helix but no macroscopic polarization can appear. On the
68
3 The Necessary Conditions for Macroscopic Polarization
IPy-
Figure 18. Polarization as a function of temperature, as often observed in a measurement. P corresponds to the spontaneous polarization P , except in a small region around T, where an induced polarization due to the electroclinic effect may influence the measurements.
~
:,Jp,
T T,
Figure 19. The point symmetry of the helical smectic C* state is D, (~-322)illustrated by a twisted cylinder. The principal rotation axis is along the smectic layer normal and there are an infinite number of twofold rotation axes perpendicular to this axis, one of them illustrated to the right. The symmetry does not allow pyroelectricity.
/
-
------
Figure 20. The surface-stabilized smectic C* has C, symmetry. It has a single polar axis. A pressure applied across the layers increases the tilt angle and thereby the polarization ( P 0) along the twofold axis which is perpendicular to the paper plane.
-
other hand, if we do the same to the surface-stabilized structure (see Fig. 20), we change the tilt 8 and thereby the local polarization P, hence in conformity with the statement according to the scheme of Fig. 6 that if a structure is ferroelectric it also, by necessity, has to be piezoelectric. That the SSFLC structure with symmetry C, is also pyroelectric, as it must be, is also evident, because 8 and thereby P is a function of the temperature T. As there is now a macroscopic polarization in this state, we have a strong coupling with the external field leading to a high-speed response according to
where y is a characteristic viscosity for the motion around the cone. The difference 28 between the two optic axis directions (see Fig. 16) leads to an electrooptic effect of high contrast. The high available contrast and the very high (microsecond level) speed in both directions at moderate applied voltages, together with the inherent memory, make this a very attractive electrooptic effect for a variety of applications.
3.4 The Surface-Stabilized State
69
While the basic idea behind the work of [93] was to get rid of the helical space modulation of the tilt direction (i.e., of the infinity of polar axes) and achieve bistability by fitting the director to satisfy the cone condition and the surface condition at the same time, it was soon recognized in subsequent papers [94,95] that the bistable switching demonstrated in [93] and the observed threshold behavior, the symmetric bistability, and the shape of the hysteresis curve (see Fig. 21) all came very close to the properties of materials previously classified as ferroelectrics, whereas these properties are absent in the helical smectic C* state. Therefore the concept surface-stabilized ferroelectric liquid crystal (SSFLC) was coined. It was first used in print three years later in [96] from 1983. This was also the first paper in which acceptable monodomain samples were demonstrated, awakening the interest in the display industry and pointing to the future potential of the surface-stabilized smectic C* state. With the developing skill in alignment techniques the first simple device prototypes could also be demonstrated [96a] even before room temperature mixtures were available. Finally, the study of the very characteristic defect structures which showed up in monodomain samples led to the discovery of the chevron local layer structure, which will be discussed in detail in Sec. 8. As already pointed out, one condition for achieving the fast bistable switching and all the additional characteristic properties is to get rid of the “antiferroelectric helix” (to use the expression employed in the 1980 paper [93]). Therefore, achieving the same properties by adding appropriate chiral dopants to the smectic C* could be imagined, in order to untwist the helical structure but, at the same time, keeping a residual polarization. This is illustrated by the structure to the left in Fig. 22. However, such a hypothetical bulk structure is not stable in the chiral case. It would transform to a twisted state where the twist does not take place from layer to layer, but in the layers, in order to cancel the macroscopic polarization. For this reason, surfacestabilization requires not only that d < ~ but , that the sample thickness is sufficiently small to prevent the appearance of this different twist state, by forcing it to appear on such a small length scale that its elastic deformation energy is higher than the
Figure 21. Oscilloscope picture showing photodiode response (change in optical transmittivity of the cell) to triangular pulses of opposite polarity (4 V peak-topeak), demonstrating threshold, saturation, memory, and symmetry bistability; cf. Fig. 12b. The material is DOBAMBC at 60 “C. Horizontal scale 1.5 V/div (from [95]).
70
3 The Necessary Conditions for Macroscopic Polarization
I l l l l l l l l l l l l l l l l ~ I I I I I I I I I I I I I I I I I I
(a)
(b)
Figure 22. Director configuration for (a) a nonchiral smectic C and (b) a chiral smectic C* in its natural (helicoidal) structure (after [97]). The fust could also be imagined as the bulk structure for a chiral smectic C* with infinite pitch, which would then correspond to a macroscopic polarization P pointing towards (or from) the reader. However, such a structure is not stable. The reason is that the spontaneous bend and twist inherent in the chiral smectic are not compensated at the same time. This is discussed in Chap. 10.
energy of the untwisted state. The first analysis of these questions was made by Handschy and Clark [98]. We will postpone the discussion till Sec. 12.3. The natural proposal of calling the helical SmC* state “helielectric” was first made by Brand and Cladis [98a] and should be adopted. At the same time Brand and Cladis suggested that a helix-free structure were “truly ferroelectric”, a statement which would be meaningful only if one could demonstrate stable macroscopic polarization states. No doubt, when one goes to the more ordered solid-like smectics where the crystalline structure prevents a helix, such states might occur, though they have not yet been observed. Whatever the polar order, it would, however, no longer belong to a liquid crystal but to a soft solid. Unfortunately, no investigation seems as yet to have been performed of the polar order in the crystal phase of the typical FLC compounds. A guess is that this order has a high probability of turning out, in crystal smectics and even more in the crystalline phase, to be antiferroelectric. The concept of surface stabilization has frequently been misunderstood or misinterpreted and various descriptions have been given which are physically incorrect [98b-d]. The most common mistake is to mix it up with “hysteresis”, which is a word that can have a very wide usage, especially in various kinds of pinning effects, but should not be used in that sense for equilibrium states. Hence reference [93] has repeatedly been cited as a confirmation of hysteresis or even bistability and macroscopic polarization predicted in [43] as a result of suppressing the helix, without caring for what that article really contains [98b-d]. It might therefore be of interest to cite what actually is said in this review from 1977, and even do it in its context. What Meyer says about this matter is (page 39): “Another aspect of the ferroelectric response to an applied field is hysteresis. In crystalline ferroelectrics, in which there may be only a few easy axes for polarization, domain walls can have a high energy, due mainly to crystalline anisotropy. The pinning of domain walls, or the dificulty of nucleating new ones, is a major cause of hysteresis effects. In a single crystal chiral smectic C, there is no easy axis forpolarization in the smectic layers, and thus there are no spontaneous domains; only
3.4 The Surface-Stabilized State
71
line defects are allowed. Therefore in principle hysteresis is not possible. However; in polydomain samples, or very thin ones contained between sugaces with strong alignment anchoring, there can be pinning efects whichproduce at least partial hysteresis. This is easily observed in experiments involving unwinding of the helix. When E is reduced below E,, the helix reappears non-uniformly by the generation of discrete twist walls which nucleate on defects. At E = 0, the equilibrium pitch may not be achieved after such an experiment.” As we see, Meyer is referring to the helix unwinding by an electric field and in a different geometry, the layers being parallel to the glass plates, and the reappearing of the helix associated with hysteresis. He can observe this by the motion, reappearance, and coalescence of unwinding disclination lines. These effects have nothing to do with either the bistability or the hysteresis in helix-free bookshelf cells in which the unwinding lines are permanently eliminated. That symmetric bistability (the concept of which did not exist in any form prior to 1980) appears in this case, has nothing to do with whether the alignment anchoring is strong or weak, it is even essentially independent of what the surface conditions actually are. Therefore [93] is not a proposal to use a kind of pinning hysteresis [98b, c], but rather a demonstration of bistable switching together with an explanation of its origin, describing for the first time the significance of surface stabilization. As stated above, the expression SSFLC itself was not coined until 1983 in reference [96]. We will return to the SSFLC concept, in a slightly different context, in Sec. 5.9. We want to stress again that the surface-stabilized smectic C* state has symmetry and properties which on decisive points distinguish them from the properties of the helical smectic C* state and it is crucial that these properties cannot be achieved in bulk. The helical C* state is, on the other hand, more deeply intrinsic to the bulk than would correspond to just adding chirality. Shortly after the first paper by Meyer et al. [62], Michelson, Cabib and Benguigui [43b] undertook to carefully analyze the symmetry changes in smectic A to C phase transitions of second order and from Landau theory draw the conclusions relative to possible kinds of polar order in the C phase. In fact, the scope of their study is even wider and amounts to investigate whether there can exist some liquid crystal phases exhibiting dipolar ordering. As a first result they conclude that this is only possible in a smectic of the general type smectic C. As for the hypothetical brick-like type of smectic C called C, [43d] where the angle between the long molecular axis and the layer normal z is zero but the rotation around z ”frozen” or biased, leading to quadrupolar symmetry (not yet observed) they find that a transition from an A phase of symmetry D,, (i.e. a normal non-chiral A phase) to a true ferroelectric (bulk polarized) non-chiral phase of symmetry D,, is possible. On the other hand, a transition from an A phase of symmetry D,, i.e. the chiral SmA*, to a uniformly polarized state of the tilted SmC* is not possible. In this case only a helicoidal arrangement of the dipoles is permitted, with a local symmetry C, and with the local dipole perpendicular to the tilt plane. They call this phase dipole-ordered and object to the name ferroelectric as not very appropri-
72
3 The Necessary Conditions for Macroscopic Polarization
ate since it can produce confusion [43e], [43f]. It is noteworthy that they also predict, in the C, case, a possible transition to an antiferroelectric state, with herrringbone structure, i.e. with molecules in alternate layers tilting in opposite directions and corresponding to a doubling of the lattice period in the z direction. Michelson et al. in [43b] and in their further elaboration of the theory in [43e] introduce and use a two component order parameter for the A to C transition
which we will meet later in the text. This order parameter, or essentially the same, was also introduced independently by Pikin and Indenbom [ 1971 at the same time and is commonly referred to under their name. Let us finally focus on a remark made by Michelson et al. in ref [43b] (page 969) after having ruled out the possibility of a uniformly polarized smectic C*. They conclude that such a sample ”can have bulk polarization only if other interactions are present”. Such an interaction is the surface interaction in the suface-stabilized smectic C* state.
3.5 Chirality and its Consequences Chirality is a symmetry concept. We note from Kelvin’s definition that being chiral is a quality and that this quality is perfectly general, regardless of the nature of the object (except that it must be three-dimensional, if we speak about chirality in three dimensions as Kelvin did). Thus we may speak of a “chiral molecule” and of a “chiral phase” or medium made up of chiral molecules or of chiral or nonchiral molecules ordered in a chiral fashion. Examples of nonchiral molecules ordered in a chiral fashion in the crystal state are a-quartz and sodium chlorate where, respectively, Si02 and NaC10, molecules are arranged in a helical order. We thus find right- and left-crystals of both substances. The reason for this is simply that the energy is lower in the helical state. Other examples are sulfur and selenium, which are found to form helical structures, thus ordering to a chiral structure although the atoms themselves are nonchiral. Sometimes the designation “structure chirality” or “phase chirality” is used to distinguish this phenomenon where nonchiral objects build up a chiral structure but, because both of these terms are ambiguous, a better name is superstructural chirality. In these examples we see the important one-wayness of chirality: a phase, i.e., an ensemble of molecules or atoms, may be chiral whether the constituents are chiral or not, but a phase built from chiral objects is always chiral, i.e., it can never possess reflection symmetry. When we here talk about chiral objects, we do not consider the trivial case which might be taken as an exception, when there are exactly as many right-handed as left-handed objects, which are each others mirror images (racemic mixture). We may also remember that when talking about
3.5 Chirality and its Consequences
73
a chiral phase, we consider its static properties for which all molecules are in their time-averaged configurations (organic molecules would otherwise have very few symmetry elements). This corresponds to considering atoms to be situated on regular points in a lattice, without considering individual displacements due to thermal motion. In the case where only the building blocks are chiral, the optical activity is relatively weak (as in liquid solutions), whereas if there is a helical superstructure, this activity can be large, as it is in quartz and NaClO,, or immense as in liquid crystals, because the effect sensitively depends on d/A, the ratio of the stereospecific length to the wavelength of light, which is appreciable in the latter cases. When the building blocks have higher symmetry than the structure, e.g., in NaClO,, there is, as we just mentioned, an inherent tendency to arrange them to a structure without a mirror plane and a center of symmetry. However, if we take a pure left-NaC10, crystal, dissolve it in water and let it crystallize from this solution, there is an equal chance to build up both mirror image arrangements. Thus from a solution of a leftcrystal NaClO,, right- and left-crystals grow with equal probability. This is a local breaking of a non-chiral symmetry to chiral or, otherwise expressed, this is a local breaking of mirror symmetry, although the mirror symmetry is preserved on a global scale. A considerable number of such cases exist in solid crystals, although they are, relatively speaking, not very frequent. In principle, chiral “domains” or “crystallites” could be expected to be created in the same way in liquid crystals from nonchiral mesogens; see the discussion in Sec. 5.6. At this point we might also comment on the striking lack of mirror symmetry in the universe and in living systems. The 20 amino acids are all lev0 ( L ,left), the sugars are all dextro (D, right), double helices are right-handed, etc. A cultivator of snails in Bourgogne knows that there is about one left-handed snail in one million righthanded snails, and so on. Thus there is no mirror symmetry in these domains; our life is largely homochiral. The functional molecules have the same handedness in all organisms. For a long time the possible origin of biomolecular chirality has been an intriguing question. Has there been a global brealung of mirror symmetry, not entirely unlikely if life is a unique accident, or could it be referred to some external chiral action? This question is at least as old as the thoughts and reflections by Pierre Curie. Aplausible model has been forwarded by F. C. Frank [99] according to which the balance between left and right is just unstable due to chirally autocatalytic competing reaction mechanisms. If a metabolic product of species L is an inhibitor of species D , and vice versa, we will have an instability. A very interesting and surprisingly simple experiment to this has been performed by Kondepudi and collaborators (99a] as late as 1990. They studied crystallization of NaC10, from aqueous solution and found that if the solution is not stirred, statistically equal numbers of L and D crystals were found. When the solution was stirred, however, almost all of the crystals in a particular sample had the same chirality, either lev0 or dextro. The “handedness” of stirring has no influence, it is only a question of bringing the process out
74
3 The Necessary Conditions for Macroscopic Polarization
of equilibrium. Under equilibrium conditions there is no autocatalytic production of crystallization nuclei; far from equilibrium, rapid production of secondary nuclei from a primary nucleus is chirally autocatalytic and leads to an instability, because the formation of crystals of a particular handedness is accompanied by depletion of solute and suppression of nucleation of crystals of opposite handedness. Achiral phase normally shows optical activity. Similarly, if we observe optical activity we would perhaps be tempted so say that there are chiral building blocks or chiral centers somewhere. But neither of these are true. Therefore the statement cannot be formulated more strongly in this direction either. Thus, optical activity does not necessarily mean that the phase is chiral. We give examples of this in the next section. Whereas chirality is a property that does not have a length scale, optical activity is a global, bulk property and it can only be measured on a large ensemble of molecules. It is therefore less correct to speak about “optically active molecules”. Optical activity and chirality are of course strongly related, because the optical activity is one of the consequences of chirality. We might, however, have a chiral substance that does not show any optical activity, because different parts in the molecule give rise to antagonistic contributions. If we now mix this substance with a nonchiral substance, the result may very well be that we find an easily measurable optical activity, and we may even see immediately, in the case of liquid crystals, that the phase gets twisted, thus a clear manifestation of chirality. Descriptions of such a case as “the chiral molecule does not show any chirality in its pure state” are not uncommon, but should be avoided, apart from their lacking logic, because they confuse cause and effect. Optical activity (or more precisely optical rotatory dispersion) is a quantitatively measurable property, like the helical twisting power, and both are therefore very useful to quantify the manifestations of chirality. However, expressions like “more or less chiral” are just as devoid of sense as “more or less cubic”. A structure can have cubic symmetry, or else it does not, but there can be nothing in between. Likewise it is not the chirality that “goes down by a factor of five on raising the temperature” (to cite one of many similar statements), but rather the twisting power or a similar quantitative measure. Nevertheless, this common misuse of the word “chiral” points to a real problem. Now, let us go back to the example of the chiral smectic C, written C* whenever we want to emphasize the chirality. The symbolic relation [Eq. (90)] means that if we have added a concentration c#O of chiral dopant to our smectic layer then a local polarization P is permitted perpendicular to the tilt plane. If c=O, no polarization is permitted. This corresponds to the qualitative character of chirality. Lack of reflection symmetry is thus a prerequisite for polarity. The argument does not say anything about the size of the effect and could not, because it is a symmetry argument. We have to take recourse to other arguments in order to say anything about the size of the effect. For instance, if we only add one dopant molecule, only the rotational motion of this molecule will be biased if we disregard its weak interaction with its
3.5 Chirality and its Consequences
75
neighboring molecules (see further discussion in Sec. 5.2), and a fraction of its lateral dipole moment will show up in P. If we increase the concentration, it is highly probable that P will be proportional to c, at least for small values of the dopant concentration. This is also borne out by experiment. This case is very similar to the case of dissolving noncentrosymmetric molecules in a liquid solution. Such a solution shows an optical activity that is, at least for low concentrations c, proportional to c. The sense of rotation depends on the solvent and is thus not just specific of the noncentrosymmetric molecule. We can expect the same thing to happen in the smectic C* case: the sense of optical rotation, the sense of P, and the sense of the helical twist will all depend on both the solute and the solvent, and cannot (so far) be predicted from first principles but have to be measured, at least until sufficient empirical data have been collected. We will define the polarization as positive if it makes a righthanded system with the layer normal and the director, and hence lies in the same direction as z x n (see Fig. 23). A priori, there are two classes of chiral smectics, those with P>O and those with P
director, and local polarization for the two possible classes (+/-) of chiral smectics.
PO>O
76
3 The Necessary Conditions for Macroscopic Polarization
’& 1
Figure 23a. Examples of ferroelectnc cornpounds. The value of the spontaneous polarization P, is increasing from 1 to 6. In nC/cm2 the P, values are 3, 15, 42, 170, and 220. All are negative in sign.
2
3
4
5
3.6 The Curie Principle and Piezoelectricity If we want to investigate the conditions for an elastic stress to induce a macroscopic polarization, we have to turn to the Curie principle, which in a sense is a generalization of Neumann’s principle. It cannot be proven in the same way as that principle; in fact it is often the violations (or maybe seeming violations) that are the most interesting to study. Among these cases are the phase transitions (spontaneous symmetry breakings) where the temperature is the “external force” in Curie’s language. As mentioned before, Curie’s principle can be stated in different ways and the earliest formulation by Curie himself is that “when a cause produces an effect, the symmetry elements of the cause must be present in the effect”. This means that the produced effect (the induced property) may have higher symmetry, but never less symmetry than the cause. In particular in this formulation, the principle has to be used with care. An example of this is related by Weyl in his book Symmetry [loo]. Weyl here tells about the
3.6
The Curie Principle and Piezoelectricity
77
intellectual shock the young Mach received when he learned about the result of the 0rsted experiment (see Fig. 24 taken from the book). The magnetic needle is deflected in a certain sense, clockwise or anti-clockwise when a current is sent in a certain direction through the conducting wire, and yet everything seems to be completely symmetric (magnet, current) with respect to the plane containing the needle and the conductor. If this plane is a mirror plane, the needle cannot swing out in any direction. The solution to this paradox is that this plane is not a mirror plane, because of the symmetry properties of the magnetic field. The problem is that while we easily recognize the reflection properties of geometric objects, we do not know a priori the corresponding properties for abstract things, e.g., physical quantities. In fact, the magnetic field has reflection properties such that it is rather well illustrated by Magritte’s well-known surrealist painting where a man is looking into a mirror and sees his back. A lesson to be learned from this is that we cannot rely on appearances when we judge the symmetry of various fields and physical properties in general. Pierre Curie was the first person to study these symmetries in a systematic way and, in order to describe them, he introduced the seven limiting point symmetry groups (also called infinite or continuous point groups), which he added to the 32 crystallographic groups. With this combination he could classify the symmetry of all possible media and all possible physical properties, illustrating the continuous symmetries with drawings related to simple geometric objects as, e.g., in Fig. 25. Obviously, these continuous symmetries have a special relevance for liquid crystals, liquid crystal polymers, and liquids in general, being continuous media without a lattice. Let us apply a stress to a general medium and ask under what conditions it could cause an electric displacement. The basic problem we then have to sort out is the proper description of the symmetry of stress. Clearly a stress cannot be described by a polar vector - there must be at least two. A simple illustration of this is given in Fig. 26. We see that in two dimensions a homogeneous tensile stress as well as a pure shear stress has two perpendicular mirror planes, one twofold rotation axis, and one center of symmetry (center of inversion). In three dimensions we have analogously three mirror planes, three twofold axes, and the center of symmetry, which we may enumerate as m, m, m,2,2, 2, Z (see Fig. 27). These are the symmetry elements of
78
DO
3 The Necessary Conditions for Macroscopic Polarization
mmm
0022
WJrn
-hmm
OOJOO
DOJcornm
Dmh
SO(3)
O(3)
alternative notation
CDO
Coo,
DDO
Ccoh
Figure 25. Pierre Curie’s seven continuous point groups illustrated by geometric “objects”. Among them we might distinguish -mm representing a polar vector (like the electric field), and -lm representing an axial vector (like the magnetic field). Three of these symmetries, m, -22, and -/-, can appear in a right- handed as well as in a left-handed form (enantiomorphic). This is not so for -lm (the magnetic field is not c h i d ) . The ”sense” in the mlm object should not be mistaken for the sense of a helix. The difference is that a helix does not have a mirror plane perpendicular to its axis. Several equivalent ways of expressing the symmetry are in common use, for instance, the one in Fig. 19. Pyroelectricity (i.e., a macroscopic polarization) is only permitted by the first two groups (- and wm).They represent a chiral and a non-chiral version of a longitudinal ferroelectric smectic. If the continuous medium in question can sustain the mechanical strain, piezoelectricity would be allowed by the three groups -, -mm, and -22. The third of these (also written D-) represents the symmetry of the cholesteric (N*) phase as well as the smectic A* phase and the helicoidal smectic C* phase, and the effects are, respectively, inverse to the flexoelectrooptic effect, the electroclinic effect and deformed helix mode - all oneway effects. If we polymerize these phases (i.e. crosslink them to soft solids), the effects would become truly piezoelectric (two-way effects). The crystallographic as well as the Schoenfliess symbols have been given. As no Schoenfliess symbols exist for the two spherical groups, the mathematical symboi O(3) is used for the complete orthogonal group in three dimensions (containing all rotations and reflexions) and SO(3) for the special orthogonal group containing only rotations.
.‘ I
k a
.
--.!
!
,.
.
b
C
Figure 26. The symmetry of stress: (a) pure tensile stress, (b) pure shear stress.
Figure 27. Characteristic surfaces of second-rank polar tensors: (a) ellipsoid, (b) hyperboloid of one sheet, (c) hyperboloid of two sheets. (After reference 36.)
the point group mmm (or D2J. This is the orthorombic point group, which can be illustrated by a matchbox or a brick. Now apply the stress in the most general way, i.e., such that its three axes of twofold symmetry and its three planes of reflection symmetry do not have the same directions as the axes and planes in the unstrained medium. Curie’s principle then says that the stressed medium will retain none of its symmetry axes and planes.
3.7 Hermann’s Theorem
79
However, if the unstressed medium has a center of symmetry 2, it will retain that center under stress because the stress also has a center of symmetry. Therefore, in a medium with inversion symmetry, no effect representable by an arrow can be induced by the stress, and hence no electric displacement or polarization, whatever stress is applied. In other words, a medium with inversion symmetry cannot be piezoelectric. We could also have reasoned formally in the following way: If a property P (polarization in this case) should appear in the medium K as a result of the external action E (stress CT in this case), then P must be compatible with the symmetry of the strained medium, according to Neumann’s principle PZK
but, according to Curie’s principle
If we now insert ~m for P and mmm for o,we get from Eqs. (92) and (93) mrn 2 K n mmm The symmetry elements to the right (mrnm)contain a center of inversion, but those to the left (corn) do not. Therefore, if Eq. (94) should be satisfied (i.e., the Neumann and Curie principles together), K must not have a center of inversion, i.e.,
Out of the 32 crystallographic groups, 11 have symmetry elements including a center of inversion. Piezoelectricity should therefore be expected to appear in the other 21, but it appears only in 20. The exception is 432 (=Oj, the octahedral group. However, this exception is sufficient to give the very important insight that the Neumann and Curie principles, and in fact all symmetry principles, only give necessary (never sufficient) conditions for a certain phenomenon to appear. The principles can only be used in the affirmative when they prevent things from happening.
3.7 Hermann’s Theorem It is often said that group 432 is “too symmetric” to allow piezoelectricity in spite of the fact that it lacks a center of inversion. It is instructive to see how this comes about. In 1934 Neumann’s principle was complemented by a powerful theorem proven by
80
3 The Necessary Conditions for Macroscopic Polarization
Hermann (1 898- 196l), an outstanding theoretical physicist with a passionate interest for symmetry, whose name is today mostly connected with the HermannMauguin crystallographic notation, internationally adopted since 1930. In the special issue on liquid crystals by Zeitschrijl f u r Kristallographie in 1931 he also derived the 18 symmetrically different possible states for liquid crystals, which should exist between three-dimensional crystals and isotropic liquids [ 100aI. His theorem from 1934 states [IOOb] that if there is a rotation axis C, (of order n), then every tensor of rank r < n is isotropic in a plane perpendicular to C,, as shown in Fig. 28. For cubic crystals, this means that second rank tensors like the thermal expansion coef, be isoficient aiJ,the electrical conductivity oij,or the dielectric constant E ~ will tropic perpendicular to all four space diagonals that have threefold symmetry. This requires all these properties to have spherical symmetry, as already mentioned above for the optical wave surface and for the thermal expansion. It also means that crystals belonging to the trigonal, tetragonal, or hexagonal systems are all optically uniaxial (most liquid crystal cases can be included in these categories). The indicatrix is thus an ellipsoid of revolution, as illustrated in Fig. 28 a, with the optic axis along the threefold, fourfold, or sixfold crystallographic axis, respectively. If we have a fourfold axis C4,then the piezocoefficients dijkwill be isotropic perpendicular to C4. Finally, if we have a sixfold axis C,, the elastic constants cijklwill be isotropic perpendicular to C6.Thus in a smectic B the layer can also be considered isotropic with respect to the elastic constants. We see that, in many cases, Hermann’s theorem allows a certain precision to be added to the general statement following from Neumann’s principle, that properties may be more symmetric than the medium. There are now three cubic classes that do not have a center of symmetry.We should therefore expect these three - 23 (T), 43m (Td), and 432 (0)- to be piezoelectric,
I cn
I cm
rotation axis.
Material
0.a.
Negative Positive
Figure 28a. Media with a threefold, fourfold, or sixfold crystallographic axis must be optically uniaxial. Thus crystals belonging to the trigonal, tetragonal, or hexagonal system, including smectic B, must be uniaxial. The picture shows the indicatrix with its optic axis in the case of positive as well as negative birefringence.
3.8 The Importance of Additional Symmetries
81
but only the first two are. The last has three fourfold rotation axes perpendicular to each other, making the piezoelectric tensor d,, isotropic in three dimensions. The piezoeffect would not then depend on the sign of the stress, which is only possible for all d,,rO. It is clear that similar additional symmetries would occur frequently if we went to liquids, which are generally more symmetric than crystals.
3.8 The Importance of Additional Symmetries A property that may be admitted by noncentrosymmetry may very well be ruled out by one of the other symmetry operations of the medium. As an example we will finally consider whether some of the properties discussed so far would be allowed in the cholesteric liquid crystals, which lack a center of inversion. A cholesteric is simply a chiral version of a nematic, abbreviated N*, characterized by the same local order but with a helical superstructure, which automatically appears if the molecules are chiral or if a chiral dopant is added (see Fig. 29). Could such a cholesteric phase be spontaneously polarized? If there were a polarization P,it would have to be perpendicular to n,because of the condition of Eq. (6), and thus along the helical axis direction m.However, the helical N* phase has an infinity of twofold rotation axes perpendicular to rn and the symmetry operation represented by any of these would invert P. Hence P=O. A weaker requirement would be to ask for piezoelectricity. (Due to the helical configuration, the liquid has in fact some small elasticity for compression along m.)Nevertheless, as illustrated in the same figure, a compression does
Figure 29. The symmetry of the cholesteric phase (a, b) and the effect of a shear (c).
(C)
82
3 The Necessary Conditions for Macroscopic Polarization
not change the symmetry, and hence a polarization P#O cannot appear due to a compression. Finally, it is often stated that a medium that lacks a center of inversion can be used for second-harmonic generation (SHG). This is because the polarizability would have a different value in one direction compared with that in its antiparallel direction. Could this be the case for the cholesteric phase, i.e., could it have a nonlinear optical susceptibility ~ ‘ ~ ’ # The 0 ? answer must be no, as it cannot be in the n-direction, and the rn-direction is again ruled out by the twofold axis (one of the infinitely many would suffice). For the unwound (non-helical) N*, the twofold axis In (or the n axis) works in the same way. The cholesteric example shows that it is not sufficient for a medium to lack a center of inversion in order to have SHG properties. Questions like these can thus be answered by very simple symmetry arguments, when we check how additional symmetries may compensate for lack of inversion symmetry. In contrast to the cholesteric phase, the unwound smectic C* phase does have a direction along which a second harmonic can be generated. This is of course the C2 axis direction in the SSFLC geometry. Even if initially small, the existence of this effect was soon confirmed [ 101, 1021 and studied in considerable detail by several groups [103]. The SSFLC structure is the only liquid (if it can be considered as such!) with SHG properties. No other examples -they can only be looked for among anisotropic liquids - are known, the externally poled electret waxes left aside (materials out of thermal equilibrium in which the polar axis is not an intrinsic property). Walba et al. were the first to synthesize C* molecules with powerful donor and acceptor groups active perpendicular to the director (thus along the C2 axis), which is a necessity in order to achieve X(2)-valuesof interest for practical applications [ 104, 1051. However, there is in fact really no point in having a surface-stabilized ferroelectric liquid crystals as an SHG material. A pyroelectric material would be much better, in which the polar axis is fixed in space and nonswitchable, but this cannot be achieved in the framework of liquids. The first such materials were recently made by Hikmet and Lub [lo61 and by Hult et al. [106a], using an SSFLC as a starting material and crosslinking it to a nonliquid-crystalline polymer. It is thus a soft solid. As we have seen, most liquid crystals have too high a symmetry to be macroscopically polar if they obey the n +-n invariance (which all “civilized” liquid crystals do, that is, all liquid crystal phases that are currently studied and well understood). The highest symmetry allowed is C2 (monoclinic), which may be achieved in materials which are “liquid-like’’at most in two dimensions. Even then external surfaces are required. Generally speaking, a polar liquid crystal tends to use its liquid translational degrees of freedom so as to macroscopically cancel its external field, i.e., achieve some kind of antiferroelectric order. For more “liquid-like’’liquids, piezo-, pyro-, ferro-, and antiferroelectricity are a fortiori ruled out as bulk properties. These phenomena would, however, be possible in crosslinked polymers (soft solids). A simple example may illustrate this. If we shear the cholesteric structure shownin Fig. 29c, Curie’s principle tells us that a polarization is now permitted along the twofold axis
3.9 Optical Activity and Enantiomorphism
83
shown in the figure. If the liquid crystal will not yield, this would constitute a piezoelectric effect. Hence liquid-crystalline polymers are expected to show a piezoelectric effect both in the N* phase, the A* phase, and the C* phase (as well as in chiral phases with lower symmetries). However, as already pointed out, these materials are not really liquid-like in any direction, thus not liquid crystals but soft solids (like rubber). It may finally be pointed out that the symmetry operation n + -n describes a bulk property of liquid crystals. At surfaces, no such symmetry is valid. Therefore SHG signals can be detected from nematic surface states and have, indeed, been used to probe the order and directionality of such surface states [107]. As a general rule, a surface is always more or less polar, which certainly contributes to the complexity of alignment conditions at the interface between a surface and a liquid crystal with the special polar properties of the materials treated in this chapter. In this case, therefore, we have in principle a surface state with a polar property along n as well as perpendicular to It.
3.9 Optical Activity and Enantiomorphism As we have emphasized, a phase - be it amorphous, crystal och liquid crystal - built from individual chiral building blocks is always chiral. The converse is not true: chiral phases exist built from non-chiral objects. Further, if a phase is chiral, it is necessarily optically active (although, as we have said, the optical activity might in instances be unmeasurably low). Nor in this case is the converse true. A phase built of non-chiral objects can be optically active. We have used quartz and sodium chlorate to illustrate this. But perhaps a more surprising statement is that a phase which is optically active does not even have to be chiral. A chiral object and its mirror image are enantiomorphous, right- and left-handed crystal forms are called enantiomorphs (greek enantios opposite, they are of opposite form). When the symmetry consideration is applied at the molecular level, the word enantiomers is preferred (greek meros, part). Thus molecular species are said to be enantiomeric. Optical activity and enantiomorphism are very often confused, or it is believed, because they most often appear together, that one is the condition for the other. It is thus often believed that if a structure contains a mirror plane, it cannot be optically active. However, this is not true. Optical activity puts a less stringent condition on dissymetry than enantiomorphism. This is shown in Fig. 30. If the medium is chiral we can have optical activity in any direction, which makes it very easy to detect if the medium is not birefringent. Thus, in NaClO,, which is cubic and therefore optically isotropic, the eigenmodes are circularly polarized in all directions and the optical activity is evident. If the medium is linearly birefringent the optical activity is often completely overpowered by this part of the birefringence. If we have
84
3 The Necessary Conditions for Macroscopic Polarization
Figure 30. The quality of handedness, right or left, can be preserved even when light is propagating through a medium possessing mirror symmetry, i.e., a non-chiral medium. No optical activity is, however, possible in directions where the propagation direction is parallel or perpendicular to the mirror plane, because in these cases the mirror inverses the handedness.
a mirror plane we cannot have optical activity in just any direction. If the propagation direction is along a mirror plane or perpendicular to it, optical activity is not possible because in either case the reflection inverts the handedness, which violates the assumption that the reflection is a symmetry operation. But if the propagation is in a direction which is oblique to the mirror plane, the handedness is preserved and we can make a distinction between left- and right-handed circularly polarized light, as shown in the figure. The optical activity has the same size but opposite sign for waves which propagate symmetrically with respect to the mirror plane. The operation of inversion (x -+ -x, y -+ -y, z -+ -z) transforms a right-handed object into a left-handed and vice versa. Therefore, if a structure is centro-symmetric it does not permit enantiomorphism, because inversion is a symmetry operation of the structure. Likewise it does not permit optical activity because the inversion operation transforms right-handed circular polarized light to left-handed propagating in the same structure. So both phenomena are excluded in structures having a center of symmetry (or three mirror planes). But optical activity is permitted in structures having one mirror plane and even in those having two mirror planes. Such structures by definition are not chiral. This is reflected by the fact that among the 32 crystal classes 11 are chiral, i.e., permit enantiomorphic forms, whereas there are 15 classes permitting optical activity (14 and 18, respectively, if we include the continuous groups). The chiral groups are
I, 2, 3,4,6, 222, 32,422, 622, 23,432 [-, -22, d-] (crystallographic notation) or
C1, C2, C,, C,, c6, D2, D,, Dd, D6, T, 0 CC-7 Dm, sO(3)l (Schoenfliess notation). In addition optical activity is permitted in m, mm2,4, and 42m
or
C,,
GV, S4, and D2,
3.9 Optical Activity and Enantiomorphism
85
These four groups are thus optically active but not enantiomorphous. The first, m, is monoclinic, the second, 2mm, orthorombic and the last two, 4 and 42m tetragonal. Two of them (mm2and 42m) contain two mirror planes. To exemplify how we can have non-chiral structures which are optically active let us further imagine that we take a right-hand glove and fit it together with a left-hand glove by such a joint that they have a mirror plane in between. Or take one righthanded and one left-handed screw or any two enantiomorphic objects and fit them together in a corresponding way, like in the example of Fig. 30a. If the joint 1-1 is very short, the electromagnetic wave, having a spatial structure of the order of 500 nm, will integrate both contributions from the stereospecific centers and the optical activity may be vanishingly small ( d l A 6 l), but with larger separation it might be measurable. The structure is, in principle, optically active. It is non-chiral, its symmetry is ?= m and there cannot be any enantiomeric forms of such a molecule. But we do not even have to start from two chiral parts and make a non-chiral object. In Fig. 30 b we have an object of symmetry 42m, thus having two mirror planes, showing optical activity. An example of a molecule with this symmetry is H,C=C=CH,, allene. An example of a crystal structure with this symmetry is silver thiogallate, AgGaS2, which is optically active. For an example of a molecule in the case of Fig. 30a we choose a liquid crystal, the three-ring Shiff-base R,S-MHTAC, with the structure shown in Fig. 30c, first synthesized by Keller, Liebert, and Strzelecki [ 1081. Where-
Figure 30a. Non-chiral but optically active object composed of two stereospecific tetrahedra (1,2, 3 , 4 are assumed to he different ligands) being fitted together by a joint 1-1 such that m is a mirror plane.
Figure 30b. Non-chlral but optically active object being composed of nonchiral parts in such a way that there are two mirror planes perpendicular to each other (going through the diagonals of the upper and lower squares of the circumscribing figure) An example of molecular species with this symmetry 15 allene, H,C=C=CH, Consider a right-handed circularly polanzed wave propagating upwards along the main axis of this model structure, though slightly obliquely to the axis, a\ required by our discussion of symmetry For the sake of the argument let the wavelength be such that at a certain moment the electnc field vector E is directed from 1 to 1’ at the lower end whereas it 15 from 2 to 2’ at the higher end Then the corresponding left-handed wave would have the same field from 1 to 1‘ but the oppohite direction, from 2’ to 2 on the top The polanzation induced in the medium by the two waves is 1 therefore different and the dielectric comtant is different for the two waves
3
86
3 The Necessary Conditions for Macroscopic Polarization
cC
H,
(S)
L-OOCHGHQ-N=HQXH=QH=CHCO I=
I
A
(R/
Figure 30c. The structure of R,S-MHTAC.This is a specific isomer of a compound that can have two chiral and two non-chiral forms. The first two are the S,S-and the R,R-isomers which are enantiomeric, optically and electro-optically active. (They are antiferroelectric.) The third is the racemate S,S/R,R, inactive. The fourth is the isomer R,S, which is non-chiral but optically active. Because S is the mirror image of R, the molecule has a mirror plane going through the central benzene ring. The formula has been written such as to emphasize the mirror symmetry.
as the 5,s- and R,R-isomers of this compound are antiferroelectric [108a-c], the R,S-isomer, in contrast to the racemate, is an optically active compound in spite of being non-chiral. This is an illustration that optical activity is not synonymous with chirality, and that the classes of optical activity have a wider scope than the chiral classes. The compound in Fig. 30c also illustrates that a compound may have “chiral centers” without itself being chiral. This, together with other inconveniences, has caused many scientists today to prefer the expression “stereospecific center” instead of chiral center. In order to take a further example from the liquid crystal world, let us consider the periodic splay-bend structure of Fig. 30d. The pattern is supposed to represent the director field in a non-chiral nematic. Such “arcs” can be observed when we cut a cholesteric obliquely to the helix axis as shown in Fig. 38a, b of Sec. 4.7, where we will discuss this feature in a different context. The difference between the pattern in Fig. 30d and the arcs in Fig. 38 is that in the latter figure the pattern is created by the projection of the director on the plane of the cut, while here we assume that the director itself is lying in the plane of the paper. We will not discuss how such a pattern could be created but just assume that it could. Then the paper plane is a mirror plane and we have a periodic sequence of mirror planes perpendicular to it. Hence, if we consider the point symmetry, we have a structure with two mirror planes. Now let a plane-polarized light wave be incident towards the horizontal plane drawn in the figure but obliquely to the surface normal of the paper for instance slightly from the left. A comparison with Fig. 38 will persuade the reader that the light wave will now see a helical periodic structure which will affect the plane of polarization, i.e., which will have turned it in a certain sense after passage of the medium. If instead the light is incident from the right, the sense of the helix will have changed and the
I
I
I
Figure 30d. Periodic splay-bend structure in an achiral nematic liquid
3.9 Optical Activity and Enantiomorphism
87
rotation of the polarization plane will be in the opposite sense. Thus the achiral medium with two perpendicular mirror planes of Fig. 30d is an optically active medium turning the polarization plane oppositely for plane-polarized light of opposite incidence. From this example we can see that, in general, achiral structures may have directions in which the structural order has the character of a screw. An example of this from the crystal world is sodium nitrite, NaNO,, already mentioned in Sec. 3.4. In its ferroelectric phase ( T < 162.5 "C) it belongs to the class mm2, which means that it has a two-fold axis (this is the polar axis) and two mirror planes at right angles. Optically it is biaxial with an angle between the two optic axes of 66", cf. Fig. 30e. Along these two axes the atoms are arranged in a helical structure of opposite handedness with one of the mirror planes bisecting the angle between the axes. For light travelling along these two axes, the optical activity is of the same magnitude and opposite sign, and Fig. 30e, like Fig. 30, illustrates that helical structures are not incompatible with mirror planes. The spontaneous polarization of the ferroelectric has to lie in both mirror planes, thus along the direction of their cut. Finally, let us stress that neither the chiral nor the optically active classes are identical to the pyroelectric or piezoelectric classes, cf. Fig. 3 1. What they have in common is that they lack a center of symmetry. Thus they are all contained in the wide family of the piezoelectric classes. Among other things Fig. 31 shows that there are six classes, 3m, 4mm, 6mm, 43m, 6 and 6m2, which are not optically active in spite of the fact that they lack a center of symmetry. Thus, absence of a center of symmetry is a necessary but not a sufficient condition for optical activity. Here again, we have an example showing that a property might have more symmetry than the medium. We note that the six classes in question all have rotation axes C, with n 2 3 . Now, the gyration tensor responsible for the optical activity is an antisymmetric tensor of rank two (it represents the imaginary part of the general dielectric tensor connecting the vector D and E ) . Therefore, according to Hermann's theorem (Sec. 3.7)
Figure 30e. The two optic axes of the mm2 structure of NaNO, make an angle of 66" with a mirror plane in between. This is an example of a non-chiral structure showing optical activity. The optical activity is the same for propagation in opposite directions but opposite for the two axes. This illustrates that the symmetry of a screw lacks a center of symmetry but has an infinite number of two-fold rotation axes perpendicular to the screw axis, and finally that a pair of left- and right-handed screws are connected through a mirror plane (from reference [108d]).
I
88
3 The Necessary Conditions for Macroscopic Polarization
c,
Dzh C2h
C4h D4h
co,
Figure 31. The 21 non-centrosymmetric classes in crystallographic (left) and Schoenfliess notation (right). These include the 15 optically active classes, which in turn include the 11 chiral classes. The piezo-and pyroelectric classes are contained in the dashed circles.
the gyration tensor is isotropic, and thus the optical activity is isotropic, around this axis. Together with the effect of the other symmetry elements this requires the gyration tensor to have a center of symmetry, whence its elements must all be zero.
3.10 Non-Chiral Polar and NLO-active Liquid Crystals Chirality is of course not, in principle, a condition for a liquid crystal to be polar. The easiest way to see this is to consider the ten polar groups in solids (Fig. 31). Of these only five are chiral, the other five have mirror planes and, except m, they also have a symmetry axis of different order. These groups are illustrated in Fig. 3 1 a together with their corresponding limiting group mmm with axis of infinite order. As we see a polarization is always admitted along the symmetry axis. These groups thus have a single polar axis. The symmetry operations are rotations and reflections and it is immediately seen that none of these can reverse the symmetry axis. The con&tion for polarity is obviously that the polar direction lies in the mirror plane. Hence in the group rn it can be in any direction within that plane, and in the groups with two or more mirror planes it must lie along the rotation axis which is the only direction contained in all mirror planes. Now, the orthorombic 2mm is the symmetry of a biaxial smectic A and of (crystal) smectic E, 6mm is the symmetry of (crystal) smectic B, and m m m is the symmetry of both the nematic phase and the smectic A
89
3.10 Non-Chiral Polar and NLO-active Liquid Crystals
2mm
3m
4mm
6mm
mmm
c2v
c 3v
c4v
c6v
Cmv
Figure 31a. Polar groups containing a rotation axis in combination with mirror planes, having two-, three-, four-, six- and infinite-fold rotation axis. The symmetry 2mm correspond to a biaxial smectic A and to smectic E, 6nzm corresponds to smectic B, while w n m corresponds to nematic and uniaxial smectic A. In principle all symmetries would admit polar order but empirically no such order appears in liquid crystals.
phase, It is clear that the all-important condition which prevents these phases to be polar is the n -+-n invariance, which seems to be a very basic microscopic symmetry condition reflecting the character of being liquid. In fact, this condition is so basic that it would seem convenient to regard liquid crystals as that most important subclass of all mesomorphic states for which this invariance condition is valid. If it is not valid in a certain phase this phase could be called a mesophase but not a liquid crystal. And as we have seen, the origin of the invariance condition is connected to the free energy: for a given allowed global symmetry, the question is which state minimizes the free energy. If we admit n 3 -n invariance there is a gain in entropy which is one part of lowering the energy which is easily realized if we have fluidity. The fact that neither nematic, nor smectic A, B or E are polar is due to this invariance and it demonstrates the liquid-like character of the “crystal” phases B and E. Thus the polarity of liquid crystals is not just a question of symmetry alone. When Michelson, Cabib and Benguigui (43b] conclude that the nematic phase cannot be polar their argument is not based on symmetry but on energy considerations. As already discussed, non-chiral molecules may form chiral liquid crystal structures in the same way that quartz and sodium chlorate form chiral superstructures out of non-chiral elements. These structures may then be ferroelectric or antiferroelectric, including the special case helielectric. In this case the molecules are not chiral but the liquid crystal is. But we may perhaps also have polar liquid crystals which are non-chiral, like in the crystal example of NaN02 from the previous section (and like almost all other solid ferroelectrics and antiferroelectrics). If we have a mirror plane then the polarization of course has to lie in that plane as we just said. It cannot have a component at right angle. Obviously the general condition for a macroscopic polarization is the same as the general condition for optical activity: the structure must be non-centrosymmetric and contain at most two perpendicular mirror planes. However, because this condition is necessary but not sufficient it does not
90
3
The Necessary Conditions for Macroscopic Polarization
mean that these two phenomena will occur in the same symmetry classes. Consider, for example, the optically active structure in Fig. 30b. A possible P vector would have to be directed along the main vertical axis because it then lies in both mirror planes. However, any of the twofold axes perpendicular to this axis and parallel to the sides of the circumscribed figure would reverse this polarization. Thus P has to be zero. As Fig. 31 confirms the class 42m is not a polar (pyroelectric) group. However, if we deform the structure according to Fig. 3 l b, the horizontal two-fold axes vanish and the vertical four- fold inversion axis is reduced to a two-fold axis along which a non-zero P is permitted. This structure has symmetry mm2 and is a non-chiral polar structure just like NaNO,. It is optically active and it also has non-linear optical properties along the polar axis. As is clear from the previous discussion the optical activity and the NLO activity do not have the same directionality. The other non-chiral group permitting both optical activity, NLO activity and polarity is m. Moreover, the non-chiral groups 3m, 4mm, 6mm and wmm will permit polar materials with NLO activity. So within the groups m, 3m, mm2,4mm,6mm and mmm one might expect a macroscopic P in non-chiral liquid crystals - but only if we drop the n -+-n invariance! The case m at first seems to be different so let us look at it. We can imagine such a smectic structure as in Fig. 31 c, to the right, where bent shape molecules are considered to be frozen or, more realistically, their rotation about an inertial axis is supposed to be biased with only the paper plane as the mirror plane. The generating symmetric case of this is shown to the left where the director symmetry has degenerated to a symmetry axis of only twofold symmetry which is certainly the minimum symmetry we could require of a “director”, n. Anyway the C2
Figure 31 b. Non-chiral structure that is polar, optically active and NLO-active, obtained from reducing the symmetry in Fig. 30b from 4 2m (uniaxial) to mm2 (biaxial).
i
%
Figure 31 c. Non-chiral bent-shaped molecules may hypothetically still be considered to have director symmetry, if n has at least preserved a two-fold rotational symmetry in combination with a C2 axis perpendicular to the paper plane, which is a mirror plane. But in order to have a net in-plane polarization, this director symmetry has to be dropped (right). This reduces the symmetry from mm2 to rn (C2”to Cs),i.e. the C , axis is also dropped.
91
3.10 Non-Chiral Polar and NLO-active Liquid Crystals
axis is preserved and it would still be meaningful to talk about the IZ + - n invariance. Even in this case a lateral dipole would cancel out. But a net in-plane polarization would require a biased situation like the one to the left of the figure. This is equal to saying that a P i t 0 in this non-chiral case requires that we drop the C, symmetry (C2+ C,) and the IZ +-n invariance. But as soon as we drop this invariance, a net polarization is admitted in all liquid crystal phases in an almost trivial way. The unique thing with the symmetry of the chiral tilted smectic is that it admits a polar phase in the presence of the basic n 4 - I invariance. Z As soon as the invariance is dropped polarity may appear in all structures. Whether the order will turn out to be polar or not can only be answered by experiment but from a symmetry point of view it is permitted. A number of efforts have been made to synthesize non-chiral polar smectics, beginning with Tournilhac and collaborators [ 108el. Experiments have so far not been quite conclusive as to what kind of polar order exists in these materials, but recently clear-cut antiferroelectric order has been demonstrated in non-chiral tilted smectics by Soto Bustamante et al. [ l o s f ] where the local polarization is in the tilt plane, which is also a mirror plane, cf. Fig. 3 1 d. The subject of non-chiral polar liquid crystals is very much in flow and has recently been reviewed by Blinov [ 108g]. We will return to this topic in Section 5.6.
P f-
Figure 31 d. Tilted smectic polymer liquid crystal structure with antiferroelectric properties. The tilt direction in each bilayer can be turned around by an electric field to the ferroelectric state, which relaxes back to the antiferroelectric state when the field is turned off (from reference [ losf]).
P
+
This Page Intentionally Left Blank
4 The Flexoelectric Polarization 4.1 Deformations from the Ground State of a Nematic Let us consider a (nonchiral) nematic and define the ground state as one where the director n is pointing in the same direction everywhere. Any kind of local deviation from this direction is a deformation that involves a certain amount of elastic energy I GdV, where G is the elastic energy density and the integration is over the volume V
of the liquid crystal. As n ( r )varies in space, G depends on the details of the vector field n (r).Now, a vector field, with all its local variations, is known if we know its divergence, V . n , and curl, V x n , everywhere (in addition to how it behaves at the boundaries). It was an advance in the theoretical description of liquid crystals (the continuum theory) when in 1928 Oseen, who had introduced the unit vector n , which de Gennes later gave the name “director” [38], showed [ 1091 that the elastic energy density G in the bulk (i.e., discarding surface effects) can be written in the diagonal form
This relation expresses the fact that the elastic energy is a quadratic form in three curvature deformations or strains, now called splay, twist, and bend, which we can treat as independent. They are sketched in Fig. 32. The splay is described by a scalar (a pure divergence), V - n ,the twist is described by apseudoscalar (it changes sign on reflection in a plane parallel to the twist axis or when we go from a right-handed to a left-handed reference frame), which is the component of V x n along the director, IVxnlII, whereas the bend is described by a vector with the component of V x n perpendicular to the director, 1 Vxnl,. Oseen was also the first to realize the importance of the n +-n invariance [ 1lo], which he used to derive Eq. (96). This expression was rederived thirty years later by Frank in a very influential paper [ 1 111, which led to a revival in the international interest in liquid crystals. The denominations splay, twist, and bend stem from Frank.
c-
\
, //=.‘ /--\\
-\
Figure 32. The three elementary deformations splay, twist, and bend. None of them possesses a center of symmetry.
\\II/
05 /
: ‘
94
4
The Flexoelectric Polarization
,
The Oseen constants K , , K2*, and K33 (in the common careless jargon, though unjustified, called Frank constants) are components of a fourth rank tensor,just like the ordinary (first order) elastic constants in solids. The question we will ask and answer in this section is whether these deformations will polarize the nematic. The reason for such a conjecture is, of course, the analogy with piezoelectric phenomena: as we have seen in Sec. 3.7 an elastic deformation may polarize a solid. Do we find the same phenomenon in the liquid crystal? We recall from the discussion in Sec. 3.7 that a necessary condition for the appearance of a polarization was that the medium lacks a center of symmetry. The reason for this was that since at equilibrium the stress as well as the strain will be centrosymmetric, the piece of matter cannot develop charges of opposite sign at opposite ends of a line through its center if it has a center of symmetry, in accordance with Curie’s principle. On the other hand, inspection of Fig. 32 immediately reveals that none of the three strains splay, twist, and bend has a center of symmetry. Hence Curie’s principle allows a local polarization to appear as a result of such local deformations in the director field, even if the medium itself has a center of symmetry. We see from Fig. 32 that the splay deformation violates then +-n invariance, whereas twist and bend do not. Therefore a polarization may appear along n in the case of splay, but has to be perpendicular to n in the case of twist and bend. In fact, as a result of a general local deformation in a nematic, a local polarization density P will appear in the bulk, given by
P = e,n ( V n ) + ebn x( V x n )
(97)
This polarization is called flexoelectric and the phenomenon itself the flexoelectric effect.
4.2 The Flexoelectric Coefficients Equation (97) consists of two parts, one of which is nonzero if we have a nonzero splay, the other of which is nonzero for a nonzero bend. The coefficients e, and eb for splay and bend are called flexoelectric coefficients and can take a plus or minus sign (a molecular property). In the case where e, and ebare positive, the polarization vector is geometrically related to the deformation in the way illustrated in Fig. 33. For a splay deformation, P is along the director and in the direction of splay. For a bend, P is perpendicular to the director and has the direction of the arrow if we draw a bow in the same shape as the bend deformation. An important feature of Eq. (97) is that it does not contain any twist term, although the Curie principle would allow a polarization caused by a twist. However, as P would have to be perpendicular to n, it should then lie along the twist axis. Now, it is easy
95
4.3 The Molecular Picture Figure 33. Geometrical relation between the local polarization density and the director deformation in the case of positive values for the flexoelectric coefficients. For negative values the direction of the induced dipole should be reversed. The size and sign of e, and eh (alternatively called e , and e 3 )are a molecular property. P
P
Figure 33a. The symmetry of twist. If we locally twist the adjacent directors in a nematic on either side of a reference point, there is always a twofold symmetry axis along the director of the reference point. Therefore a twist deformation cannot lead to the separation of charges. Thus a nematic has only two nonzero flexoelectric coefficients.
to see (see Fig. 33a) that there is always a twofold rotational symmetry axis along the director in the middle of a twist. Thus any P along the twist axis would be reversed by a symmetry operation and must therefore be zero. In other words, the symmetry of twist does not permit it to be related to a polar vector (see our earlier discussion of additional symmetries in Secs. 3.8 and 3.9). Hence a twist deformation cannot lead to a separation of charges. This property singles out the twist from the other deformations, not only in a topological sense, and explains why twisted states are so common. The well-known fact that a nematic which is noninvariant under inversion (a chiral nematic) is unstable against twist and normally adopts a helical (cholesteric) structure, would not have been the same if the twist had been connected with a nonzero polarization density in the medium. This also means that a “twisted nematic” as used in displays is not polarized by the twist.
4.3 The Molecular Picture Recognition of the flexoelectric effect is due to Meyer [18], who in 1969 derived an expression equivalent to Eq. (97). He also had a helpful molecular picture of the effect, as illustrated in Fig. 34a. To the left in this picture, the unstrained nematic structure is shown with a horizontal director in the case of wedge-shaped and crescentshaped molecules. To the right the same molecules are shown adjusted in their dis-
96
4 The Flexoelectric Polarization
Figure 34. (a) The flexoelectric effect. A polarization is coupled to a distortion (from [18]). (b) The inverse flexoelectric effect. An applied field induces a distortion (from [ 1121). With our sign conventions, (a) corresponds to e,>O, eh
(b) E =0, P = 0
Bend
Splay
Bend
Splay
Bend
tribution corresponding to a splay and a bend distortion, respectively. In either case the distortion is coupled to the appearance of a nonzero polarization density. As we can see, this is a steric (packing) effect due to the asymmetry of the molecular shape. The inverse effect is shown in Fig. 34 b. When an electric field is applied it induces a distortion. In this figure we have illustrated a hypothetical compound splay-bend deformation and also used a slightly more general shape for the model molecules. It is clear that the observable effects will depend on the molecular shape as well as the size and distribution of the dipoles in the molecules. The inverse flexoelectric effect simply offers another mechanism for polarizing the medium, since the distortion will imply a polarization parallel to E . It is therefore a special kind of dielectric mode in-
Figure 35. Orientational polarization in an applied field is generally coupled to a director distortion in the liquid-crystalline state. The drawn direction of the dipoles corresponds to the case q < O .
4.5 The Importance of Rational Sign Conventions
97
volving orientational polarization. In this respect, the inverse effect bears a certain resemblance to the field-induced lining up of dipoles in a paraelectric material, as the dipoles are already present in the molecules in both cases (see Fig. 35). Flexoelectric polarization leads to a separate contribution to the general term G = - E . P in the free energy. With Eq. (97) this contribution takes the form
Gf = -e,E . n ( V . n )- ebE . [n X ( V Xn ) ]
(98)
4.4 Analogies and Contraststo the Piezoelectric Effect The flexoelectric effect is the liquid crystal analogy to the piezoelectric effect in solids. To see this we only have to make the connection between the translational variable in solids and the angular variable in liquid crystals. For both effects there is a corresponding inverse effect. However, the differences are notable. The piezoeffect is related to an asymmetry of the medium (no inversion center). The flexoelectric effect is due to the asymmetry of the molecules, regardless of the symmetry of the medium. We have seen how the elementary deformations in the director field destroy the center of symmetry in the liquid crystal. Therefore the liquid may itself possess a center of symmetry. In other words, the situation is just the opposite to the one in solids! The fact that the flexoelectric effect is not related to the symmetry of the medium, in particular means that it is not related to chirality. It means that the effect is common to all liquid crystals. Therefore expressions like, for instance, a “flexoelectric nematic”, which are encountered now and then, are nonsensical.
4.5 The Importance of Rational Sign Conventions The flexoelectric effect is still a “submarine” phenomenon, which has so far mostly been of academic interest. One of the reasons is its complexity when we go beyond nematics, but another reason is that, even in nematics, the size and sign of the flexocoefficients are largely unknown. A condition contributing to this is that Eq. (97) can be written in a number of ways, not only in terms of the vector operators, but also corresponding to different sign conventions (which are mostly not stated). This leads to great confusion and difficulties in extracting correct values from published data (see the discussion in [ 1121).We may write the flexoelectric polarization (Eq. 97) in the form
P = e,S
+- e,B
(99)
98
4 The Flexoelectric Polarization
As far as the director gradients are concerned we have chosen to use the form B = n x (VX n )
( 100)
in the bend polarization density, corresponding to the most common form
in the Oseen expression for the bend elastic energy. Often this last expression is, however, written in the equivalent form Gb = 1 IT3, [ ( n. V)n]’
or Gb = 1 K33 ( n . V n ) 2 ~
2
(102a)
for instance, found with Oseen [lo91 and Nehring and Saupe [ 1131, which then corresponds to a change of sign in B
B = -e,(n . V ) n
(103)
This is due to the equality
nx(Vxn)=-(n V)n
( 104)
It should be noted that Eq. (104) is not a general equality for vectors, but is only valid for a unit vector. In that case we may use the expansion rule for a triple vector product
nx
vx n = V ( n
*
n)- (n . V ) n
(105)
which is equal to -(n . V ) n ,since n .n = 1. If we use Eq. (103), the expression for the flexoelectric polarization (Eq. 97) changes to
On the other hand, the signs in the expression could also change due to a different geometric convention adopted for the direction of P relative to the deformation. Thus
4.6 Singularities are Charged in Liquid Crystals
99
Meyer in his first paper uses the opposite convention (Fig. 34) to ours (Fig. 33) for the positive direction of the P arrow, which leads to the form
P = e,n (V.n ) + eb( V x n ) x n
(107)
P = e,n (V . n ) - ebn x ( B x n )
( 107 a)
or
although in later work he changes this to conform with Fig. 33 as well as Eq. (97). This geometric convention for the direction of the P arrow was first proposed by Schmidt et al. [114]. The different sign conventions are discussed at length in [ 1121. It is important that a unique sign convention be universally adopted for the flexoelectric effect. Our proposal is to write P in the form given by Eq. (97), which corresponds to the commonly used form for bend in the elastic energy, combined with the geometric convention as expressed by Fig. 33, which is most natural and easy to remember. This will be applied in the following.
4.6 Singularities are Charged in Liquid Crystals The drawings in Fig. 33 mean that one of the consequences of the flexoelectric effect is that disclinations and, in general, singularities, i.e., also dislocations, are charged in liquid crystals. We can illustrate this by some cases of disclinations in a nematic, shown in Fig. 36. We look at simple cases where the singularity line is perpendicular to the paper with its two-dimensional director field around it. In (a) we have a case of pure splay, in (b) a case of pure bend and in (c) a compound splay-bend deformation. As the splay and bend vectors are directed outwards, we have in either case the core of the singularity negatively charged if the flexocoefficient e is positive. If the distance from the core is r, the P field falls off like llr, i.e., it is divergence-free (V.P=O) which means that the deformation is not connected to any polarization charge density in the region outside of the core. In (a), (b), and (c) the strength of the disclination is the same, s = + l . For s = -1 the director field has the shape as in (d) and we see that now the core is positively charged if e,> 0, eb>0, because now the splay and bend vectors are both directed inwards. The case of half-integral disclinations, s = 1/2, is shown in (e) and (f). For s=-1/2 the core is positively charged, for s=+1/2 it is negatively charged, if e,>O and eb>O. If the flexocoefficients were all negative, the core charge would just change sign everywhere. Anyway, with our chosen sign convention the splay and bend deformations act together when e, and eb have the same sign while they counteract for the opposite sign. This is important in judging the local PO-
100
4 The Flexoelectric Polarization
larization field in compound deformations such as (c), (d), (e), and (f). Another compound field, to which we will soon direct our attention, is shown in Figs. 38, 38a and 38 b. It is well known that disclinations like those in Fig. 36 behave like charges in a two-dimensional space, attracting or repulsing each other, for opposite or same sign of the strength s, with a force
F
- s2/1
(107b)
where I is the distance between the singularities. For instance, the +1/2, -1/2 dipole in Fig. 37 may annihilate to s = 0 (“vacuum”) with a force inversely proportional to the length of the “string” in between. Of course this force is a manifestation of the elastic torques in the nematic, which vanish only in a homogeneous director field.
s=+ 1
s=+ 1
s=+l
e,>O eb>O
Figure 36. Disclinations in a two-dimensional nematic. The core carries an electric charge due to flexoelectric effects. The sign is indicated for e,>O, e,>O. (a), (b), (c) show s = + I disclinations with splay, bend, and splay-bend. (d) This shows an s = -1 disclination which has splay-bend character like the s = 1/2 disclinations in (e) and (f).
4.7 The Flexoelectrooptic Effect
101
Figure 37. The director field around a dipole of +1/2, -112 strength singularities just before (top) and just after charge annihilation giving a singularity-free nematic further relaxing towards a state of homogeneous director field (bottom).
Our knowledge about flexoelectric effects now permit the conclusion that there is also a true coulombic charge interaction between these singularities. This interaction is always attractive in the case of Fig. 37 and may be appreciable if e, and f?b have the same sign. In nematic droplets in which the boundary condition is not homeotropic, i.e., where there is at least a component of n along the surface, there must, by topological necessity, be a total disclination strength of +2 integrated over the surface, just like there is for instance in the field of meridians over the earth. It follows that we will also have charge singularities. The corresponding flexoelectric effects are of course also present in smectics where the singularities are both disclinations and dislocations. Charge effects will also accompany local splay and bend deformations across walls between different director domains.
4.7 The Flexoelectrooptic Effect The periodic deformation in Fig. 34b cannot be observed in a nonchiral nematic because it does not allow for a space-filling splay-bend structure. Instead, such a pattern would require a periodic defect structure. However, we can continuously generate such a space-filling structure without defects in a cholesteric by rotating the director everywhere in a plane containing the helix axis. In one of his early classic papers from 1969, Bouligand [ 1151 showed that if a cut is made in a cholesteric structure at an oblique angle to the helix axis, an arc pattern of the kind in Fig. 34b will be observed as the projection of the director field onto the cut plane, as illustrated in Fig. 38. We will call this oblique plane the Bouligand plane or the Bouligand cut. Evidently, if we turn the director around an axis perpendicular to the helix axis until it is aligned along the Bouligand plane, we will have exactly the pattern of Fig. 38, and therefore a polarization along the plane. This con-
102
I
I
4 The Flexoelectric Polarization
--a
a
0 w
0
-
I
no
tb)
I
I
I
Figure 38. (a) Oblique cut through a cholesteric structure showing the arc pattern produced by the director projection onto the cut plane (Bouligand plane). In (b) the same right-handed twist is seen looking perpendicular to the twist axis, and in (c) looking perpendicular to the cut plane. Here the splaybend pattern becomes evident, even if the directors are not lying in this plane (after Bouligand [ 1161). If an electric field is applied along no, which corresponds to the n direction at the top and bottom in (b), the directors will swing out around no into a Bouligand plane corresponding to the value of the field.
11 \ \ €l
Q
>
H H (4
sideration will facilitate the understanding of the flexoelectrooptic effect. This new linear dielectric mode was reported [ 1171 by Pate1 and Meyer in 1987. It consists of a tilt 9 of the optic axis in a short-pitch cholesteric when an electric field is applied perpendicular to the helix axis, as illustrated in Fig. 38a. The optic axis is perpendicular to the director and coincides with the helix axis in the field-free state. Under an applied field, in its polarized and distorted state, the cholesteric turns slightly biaxial. The physical reason for the field-induced tilt is that the director fluctuations, which in the field-free state are symmetrical relative to the plane perpendicular to the helix axis, become biased in the presence of a field, because for tilt in one sense, keeping E fixed, the appearing splay-bend-mediated polarization lowers the energy by -E 6P, whereas a tilt in the opposite sense raises the energy by +E .6P. The mechanism is clear from a comparison of Fig. 38 (b), (c). If we assume that the twist between top and bottom is 180”, the director at these planes is no, as indicated in Fig. 38(b). Viewed perpendicular to the Bouligand plane, the cholesteric structure corresponds to “one arc” in the splay-bend pattern, but the directors do not yet lie in this plane. However, if they all turn around no as the rotation axis, also indicated in Fig. 38 (a), they will eventually be in the Bouligand plane and the medium will thereby have acquired a polarization (see the lower part of Fig. 38a) along no. Hence, if we apply a field along no,increasing its value continuously from zero, the optic axis will swing out continuously, as shown in Fig. 38b, and the Bouligand plane (perpendicular to the axis) will be more and more inclined. At the same time,
4.7 Figure 38a. An electric field E applied perpendicular to the helix axis of a cholesteric will turn the director an angle $ ( E ) and thereby the optic axis by the same amount. The director tilt is coupled to the periodic splay-bend director pattern shown below, which is generated in all cuts perpendicular to the new optic axis. In this inverse flexoelectric effect, splay and bend will cooperate if e, and eh have the same sign. The relation between E and 4 is shown for a positive helical wave vector k (right-handed helix) and a positive average flexoelectric coeffi-
The Flexoelectrooptic Effect
103
k >O 'O
E=O t
1
cient e = -(e,+e,). When the sign of E is reversed, 2 the optic axis tilts in the opposite direction (@+-$).
the polarization P increases according to Eq. (97) with the growing amplitude of the splay- bend distortion. The flexoelectrooptic effect is a field-sensitive electrooptic effect (it follows the sign of the field), which is fast (typically 10- 100 ps response time) with two outstanding characteristics. First, the induced tilt $ has an extremely large region of linearity, i.e., up to 30 O for materials with dielectric anisotropy AE=O. Second, the induced tilt is almost temperature-independent. This is illustrated in Fig. 39 for the Merck cholesteric mixture TI 827, which has a temperature-independent pitch but not designed or optimized for the flexoelectrooptic effect in other respects. It can be shown [ 117, 1181 that the induced tilt is linear in E according to
eE Kk
$1-
where
1 (e, + %) e =2 and
and k is the cholesteric wave vector. The response time is given by I,
104
4 The Flexoelectric Polarization
Optic axis
Figure 38b. Field-induced tilt (left) and corresponding splay -bend distortion when looking at the Bouligand plane along the field direction (righf).The same director pattern will be found in any cut made perpendicular to the tilted optic axis, whereas the director is homogeneous in any cut perpendicular to the nontilted axis.
M
where yis the characteristic viscosity. We note, in advance of discussing the same feature in the electroclinic effect, cf. Sec. 5.8, that the response time does not depend on the value of the applied electric field. The temperature independence is seen immediately from Eq. (log), because both e and K should be proportional to S2, the square of the scalar nematic order parameter. Therefore a mixture with temperatureindependent pitch (not too difficult to blend) will have a @ ( E )that is independent of temperature. Since the original work of Patel and Meyer [ 1171, the effect has been further investigated [119- 1221, in the last period [112, 118, 123- 1261 with a special emphasis on ruling out the normal dielectric coupling and increasing the linear range, which is now larger than for any other electrooptic effect in liquid crystals. With the available cholesteric materials, the applied field is quite high but can be estimated to be lowered by at least a factor of ten if dedicated synthetic efforts are made for molecules with convenient shapes and dipoles.
105
4.8 Why Can a Cholesteric Phase not be Biaxial? TI 827
*
i 0
(4
$(E) T=25"C
0 0 0 0 00000000000000 0
0 0000000,
$(E) T=35"C @(E) T=44T
-
-
B
B E=5OV/pm
D 0
10
10
20
-
30
40
Applied electric field (V/pm)
50
n
25
60
-
30
35 Temoerature
40
45
(b)
Figure 39. The tlexoelectrooptic effect measured on the Merck mixture TI 827. (a) Induced tilt as a function of E . (b) Induced tilt as a function of temperature.
The flexoelectrooptic effect belongs to a category of effects that are linear in the electric field, and which all have rather similar characteristics. These include the electroclinic effect, the deformed helical mode in the C* phase, and the linear effect in antiferroelectrics. They all belong to the category of "in-plane switching" with a continuous grayscale and stil have a great potential to be exploited in large scale applications.
4.8 WhyCana Cholesteric Phase not be Biaxial? The nonchiral nematic is optically a positive uniaxial medium. A cholesteric is a nematic with twist. The local structure of a cholesteric is believed to be the same as that of the nematic except that it lacks reflection symmetry. This means that the director and therefore the local extraordinary optic axis is rotating around the helix axis making the cholesteric a negative uniaxial medium with the optic axis coinciding with the twist axis. The question has been asked as to why the nematic with twist could not be biaxial, and attempts have been made to measure a slight biaxiality of the cholesteric phase. In other words, why could the twist not be realized in such a way that the long molecular axis is inclined to the twist axis? Why does it have to be perpendicular? Section 4.7 sheds some light on this question. As we have seen, as soon as we make the phase biaxial by tilting the director the same angle out of the plane everywhere, we polarize the medium. Every fluctuation 6n out of the plane perpendicular to the helix axis is thus coupled to a fluctuation 6P raising the free energy, -(Sf')'. The energy of the state is thus minimized for #=O, which is the ground state. In
106
4 The Flexoelectric Polarization
other words, the N* state cannot be biaxial for energetic reasons, just as the N state cannot be polar for the same reason (see the discussion in Sec. 2.2). In other words, the cholesteric phase is uniaxial because this is the only possible nonpolar state.
4.9 Flexoelectric Effects in the Smectic A Phase In smectics such deformations which do not violate the condition of constant layer thickness readily occur. In a smectic A (or A*) this means that V x n = 0 and the only deformation is thus pure splay, in the simplest approximation. We will restrict ourselves to this case, as illustrated in Fig. 40. Special forms are spherical or cylindrical domains, for instance, the cylindrical structure to the right in the figure, for which a biologically important example is the myelin sheath which is a kind of “coaxial cable” around a nerve fiber. Let us consider such a cylindrial domain. The director field in the xy plane is conveniently described in polar coordinates p and 0 as
from which we get the divergence
This gives us the flexoelectric polarization density
(bj
P = e , n ( V . n )= e,yn
~
Figure 40. The easy director deformation in a smectic A is pure splay which preserves the layer spacing everywhere.
4.10 Flexoelectric Effects in the Smectic C Phase
107
The polarization field strength thus falls off as llp. In the center of the domain we have a disclination of strength one. Because the disclination has a finite core we do not have to worry about P growing infinite. It is interesting to take the divergence of this polarization field. We get
V . P = -i- (ap P P aP
)=O P
Thus the polarization field is divergence-free in this geometry, which makes it particularly simple, because a nonzero V . P would produce polarization charges and long range coulomb interactions. Hence, a myelin sheath only has charges on its surface from the flexoelectric effect. It is further a well-known experience when working with both smectic A and smectic C samples, under the condition that the director prefers to be parallel to the boundaries, and that cylindrical domains are quite abundant. In contrast, spherical domains do generate a polarization charge density in space as easily checked, V .P is nonzero, falling off as -l/r2.
4.10 Flexoelectric Effects in the Smectic C Phase Flexoelectric phenomena in the smectic C phase are considerably more complicated than in the smectic A phase, even in the case of preserved layer thickness. Thus under the assumption of incompressible layers, there are not less than nine independent flexoelectric contributions. This was first shown in [ 1271. If we add such deformations which do not preserve the layer spacing, there are a total of 14 flexocoefficients [128]. If we add chirality, as in smectic C*, we thus have, in principle, 15 different sources of polarization to take into account. In a smectic C, contrary to a smectic A, certain bends and twists are permitted in the director field without violating the condition of constant smectic layer spacing. Such deformations we will call “soft” in contrast to “hard” distortions which provoke a local change in layer thickness and therefore require much higher energy. As in the case of the N* phase, soft distortions may be spontaneous in the C* phase. Thus the chiral smectic helix contains both a twist and a bend (see Fig. 41) and must therefore be equal to a flexoelectric polarization, in contrast to the cholesteric helix which only contains a twist. This may even raise the question of whether the spontaneous polarization in a helical smectic C* is of flexoelectric origin. It is not. The difference is fundamental and could be illustrated by the following example: if starting with a homogeneously aligned smectic C you twist the layers mechanically (this could be done as if preparing a twisted nematic) so as to produce a helical director structure identical to that in a smectic C*, you can never produce a spontaneous polarization. What you do achieve is a flexoelectric polarization of a certain strength and sign, which is
108
4 The Flexoelectric Polarization
Figure 41. Model of the director configuration in a helical smectic C*. To the left is shown a single layer. When such layers are successively added to each other, with the tilt direction shifted by the same amount every time, we obtain a space-filling twist-bend structure with a bend direction rotating continuously from layer to layer. This bend is coupled, by the flexoelectric effect, to an equally rotating dipole density.
always perpendicular to the local tilt plane, just as the spontaneous C* polarization would be. However, the flexoelectric polarization is strictly fixed to the deformation itself and cannot, unlike the spontaneous one, be switched around by an external electric field. On the other hand, it is clear from the example that the two effects interfere: the flexoelectric contribution, which is independent of whether the medium is chiral or nonchiral, will partly cancel or reinforce the spontaneous polarization in a smectic C* in all situations where the director configuration is nonhomogeneous in space. This is then true in particular for all chiral smectic samples where the helix is not unwound, and this might influence the electrooptic switching behavior. The simplest way to get further insight into the flexoelectric polarization just described is to use the “nematic description” [46] of the C* phase. We then look, for a moment, at the smectic as if it were essentially a nematic. This means that we forget about the layers, after having noticed that the layers permit us to define a second vector (in addition to n),which we cannot do in a nematic. The second vector represents the layers and is the layer normal, z (or k ) , which differs from the n direction by the tilt angle. The implicit understanding that z is constant in space represents the condition of undeformed smectic layers. We use the Oseen elastic energy expression [Eq. (96)] for a nematic medium as a starting point. Now, as pointed out by Frank, if the medium is chiral, and an ever so slight chiral addition to a nematic by symmetry transforms the twist term [ 1111 according to
[n . ( v x n)I2 + [n . ( v x n ) + qI2
(116)
The simplest way to see this is to calculate the value of n = V x n for a homogeneous twist deformation of wave vector q. Such a twist is depicted in Fig. 41a where the end point of n describes a right-handed helix as we move in the z direction. With 4 ( z )= q z , the director has the components ~t = (cos qz, sin qz, 0)
(116a)
4.10 Flexoelectric Effects in the Smectic C Phase
109
Figure 41a. A twist of the director n in a frame of reference xyz. If the phase angle projection 4 in the xy plane is a linear function of z, $ ( z ) = qz, the variation of n is a pure twist of wave vector q, with z as the helical axis.
The scalar q represents a right-handed helix if q > 0, a left-handed if q < 0. The twist field has the curl
In other words Vxn=-qn
(116c)
n.Vxn=-qn . n = - q
(116d)
Hence
This means that in the elastic energy expression the twist term to the right in (1 16) is minimized (is zero) for a pure twist of wave vector q, i.e., when the twist deformation has the value -4, according to Eq. (1 16d). The ground state now corresponds to a twisted structure with a nonzero value of n . ( Vx n ) given by a wave vector q, the sign of which indicates the handedness. Note that the reflection symmetry is lost but the invariance condition [Eq. (6)] is still obeyed. Chirality thus here introduces a new scalar quantity, a length characteristic of the medium. If the medium is also conjectured to be polar, it might be asked if it is possible, in a similar way, to introduce a true vector (n and z are not true vectors, since there is one symmetry operation that changes the sign of both). A look at the first term of Eq. (96) clearly shows that this would not allow such a thing. In fact, it is not possible to add or subtract any true scalar or vector in the splay term without violating the invariance of ( V . n ) 2under the operation n + - n . Thus no ground state can exist with a spontaneous splay. Is there a way to introduce a vector in the bend term? There is. The bend transforms, obeying Eq. (6), as follows
[ n x (Vx n)I2 + [ n x ( V Xn ) - B]'
(1 i7)
110
4 The Flexoelectric Polarization Figure 41b. For undeformed smectic C layers any bend curvature must be perpendicular to the tilt plane. As the drawing shows, no component of bend can be in the tilt plane without changing 0 and thereby the layer thickness.
-
XI::::
where B is a vector. Only a vector B parallel to n x ( V x n ) can be introduced in the bend term. For undeformed smectic layers the bend must be perpendicular to the tilt plane as explained in Fig. 41b. Moreover, it has the direction opposite to z x n and can therefore be written
B = -pz x n
(118)'
where the positive scalar p describes the tendency for spontaneous bend and is zero in the non-chiral case. If p is nonzero, the medium is characterized by the local vector B and the reflection symmetry is lost. The form of the bend expression in the presence of a local polarization then corresponds to a constant spontaneous bend in the local frame of the director. The converse of this is the flexoelectric effect. Note that Eq. (117) conforms to Eq. (100). From the above reasoning we see two things: First, that this description permits the smectic C* to be polar and requires the polarization vector to be perpendicular to the tilt plane, a result that we achieved before. Second, that the chiral and polar medium will be characterized by both a spontaneous twist and a spontaneous bend. The smectic C* is, in fact, such a medium where we have a space-filling director structure with uniform twist and bend. This nematic description has been very helpful in the past and permitted rapid solutions to a number of important problems [96, 981. We will return to it in Sec. 5.5 and in Chap. 10. If we insert the new twist term according to Eq. (1 16) into Eq. (96), the free energy attains its minimum value for n . V x n = - q if splay and bend are absent (the cholesteric ground state). It means that we have a spontaneous twist in the ground state. This is connected to the fact that we now have a linear term in the free energy. For, if we expand the square in the twist term we could write the free energy as
GN"= GN+ I K22q2+ K22 qn - ( V xn ) 2 ~
(119)
The last term is not reflection invariant and secures the lowest energy for right-handedness, which therefore characterizes the ground state for positive q.
4.10 Flexoelectric Effects in the Smectic C Phase
111
In the same way, if we expand the bend term (Eq. 117) we get the linear term -K33B [n x ( V Xn ) ] in the free energy. With both these linear terms present, the liquid crystal thus has both spontaneous twist and spontaneous bend. Let us now resume the polar characteristics of the helical C* state. Equation (1 17) introduces a vectorial quantity ( B ) in the SmC* medium. This vector can only be parallel ton x V Xn which is the bend deformation vector. The ground state for SmC* is the one with spontaneous bend n x V x n = B . For a given bend curvature, the direction of V x n and n x V x n is illustrated in Fig. 41 c. If eb> 0 in Eq. (99), the flexoelectric polarization has the same direction as the bend vector. This also conforms with our sign convention in Fig. 33. Hence, in Fig. 41 c Pf = ebB = ebn x V x n. In the helicoidal SmC* the bend curvature is always perpendicular to the tilt plane (cf. Fig. 41b), thus the bend changes direction, and also Pf,as we move in space from layer to layer. This is illustrated in Fig. 41 d, which enhances the features from the model of Fig. 41. In a tilted smectic phase there are many more deformations in the director field which are connected with the appearance of a flexoelectric polarization field. In order to find them, however, we would have to know much more about smectic C elasticity. As can be imagined, this is quite complex and, for the time being, we will only state the results of tracking the deformations coupled to polarization in smectic C. We will later develop the theory of smectic C elasticity (Chap. 10). Actually we will do this by extending our quasi-nematic description into a local frame of
vx n
nxvxn
o - + Figure 41c. If the director bends to the left as in the drawing, its curl i s towards the reader and n x V x n lies in the plane, in the direction shown.
Figure 41d. The local polarization P is everywhere along the direction of local bend curvature which is always perpendicular to the tilt plane. Both components of P,the spontaneous polarization P, and the flexoelectric polarization P,must be along this direction. If f o < O and e,>O both contributions co-operate in the outward direction as shown in the drawing.
112
4 The Flexoelectric Polarization
reference which follows the variations of the local director everywhere in the material. As for the flexoelectric terms in smectic C we find, as already stated, in the case that we assume the layers to be incompressible as many as 9 different terms, derived in reference [ 1271. In the general case of compressible layers, five new contributions appear. All 14 terms are given in reference [128]. In the following we only want to illustrate the results in the simplest way possible. This is done in Fig. 42, where we describe the deformations with regard to the reference system, k , c,p, where k is the local layer normal (along the direction of the wave vector for a helical smectic C*), c is the local tilt direction, i.e., n tilts out in this direction (hence k , c is the tilt plane), and p is the direction of k xc (corresponding to the direction along which the spontaneous polarization has to be in the chiral case). It should be stressed that k , c, and p are all considered unit vectors in this scheme, thus c does not give any indication of the magnitude of the tilt, only its direction. There are thus 14 independent flexoelectric coefficients, 10 of which describe five distinct deformations generating dipolar densities in the tilt plane, (k,c ) . By the converse effect, these deformations are generated by electric field components lying in the tilt plane. A field component perpendicular to the tilt plane will generate the other four deformations (3, 5, 6, 9), which, consequently, themselves generate dipole densities along thep direction. In the figure, the deformations have been divided into three categories: those with gradients in k (1, 2, 3), those with gradients in c
au i.e., gradients in layer spacing (7, 8,9), (4,5,6), and those with gradients in y- -, aZ
which are hard deformations. However, it is illuminating to make other distinctions. Thus deformations (l), (2), (4), (7), and (8) give quadratic terms in the free energy and are not influenced by whether the medium is chiral or not. As we have repeatedly stated, the flexoelectric effect is not related to chirality. However, a certain deformation which gives rise to a flexoelectric polarization, whether the material is chiral or not, may turn out to be spontaneous in the chiral case, like the C* helix. Thus the four deformations ( 3 ) , (3,(6) and (9) turn out to give linear terms (like Eq. 119) in the free energy in the chiral case. Therefore we may expect them, at least in principle, to occur spontaneously. They all give contributions to the flexoelectric polarization in the direction along p . In the incompressible limit, deformation (9) may be omitted. The remaining three deformations (3), (3,and (6) are coupled to each other and have to be considered as intrinsic in the smectic C* state. Of these, the only space-filling structure is (6) (the same as illustrated in Fig. 41) and will therefore give rise to the dominating flexoelectric effect. Of the other two, deformation (3) means an inherent spontaneous tendency to twist the flat layers in the chiral case. This deformation actually only preserves a constant layer thickness very locally, but not in a macroscopic sample, and will therefore be suppressed. In cyclindrical domains it will tend to align the c director 45 O off the cylinder axis [ 1271. Finally, deformation ( 5 ) amounts to a spontaneous bend in the c
4.10
Flexoelectric Effects in the Smectic C Phase
(3)
f k
113
(4)
e
(c.vk.c)c (c.Vk.c)k
(7)
tkfP
Figure 42. The six soft and the three hard deformations in a smectic C; these are coupled to the appearance of local polarization. Below each deformation is stated the covariant form of the independent vector field corresponding to the deformation. Five of the distortions (1,2,4,7,8) create a dipole density along the c (tilt) direction as well as along the k direction. The four other distortions (3,5,6,9) create a dipole density along the p direction, which corresponds to the direction of spontaneous polarization in the case of a chiral medium (C*). By the inverse flexoelectric effect, the distortions will be provoked by electric fields along certain directions. In more detail they can be described as follows: (1) is a layer bend (splay in k ) with the bending axis alongp, (2) is the corresponding layer bend with bending axis along c . Both deformations will be provoked by any electric field in the tilt plane (c, k ) . ( 3 ) is a saddlesplay deformation of the layers, with the two bending axes making 45 ' angles to c andp. This deformation will be caused by a field component alongp. (4) is a splay in the c director (corresponding to a bend in the P field if the smectic is chiral). It is generated by any field component in the tilt plane. ( 5 ) is a bend in the c director (in the chiral case coupled to a splay in P)generated by ap-component in the field. (6) is a twist in the c director. It generates a dipole in thep direction (and is itself thus generated by a field in this direction). As this distortion is the spontaneous distortion in the helicoidal smectic C*, it means that we have a flexoelectric polarization along the same direction as the spontaneous polarization in such a material. Whereas these six deformations do not involve compression, the last three are connected with changes in the interlayer distance, described by the variable y=dul&, where u is the layer displacement along the layer normal (z).Thus (7) is a layer compression or dilatation which varies along k and thereby induces a bend in the n director, connected with a dipole density, in the tilt plane, see Fig. 41 b. (8) is a layer splay inducing a splay -bend deformation in then field, and thereby a dipole density, in the tilt plane. (7) and (8) are thus generated by field components in that plane. Finally, (9) is a layer splay perpendicular to the tilt plane, which induces a twist-bend in then field alongp. The bend component then means a dipole alongp. Distortions ( I ) , (2), and (4) are coupled among themselves as are (3). ( 5 ) , and (6).The latter three, and in principle also (9), occur spontaneously in chiral materials.
114
4 The Flexoelectric Polarization
director (corresponding to a splay in P ) . This means that P has the tendency to point inwards or outwards at the edges of the smectic layers. It means, on the other hand, that the effects are transformed to a surface integral so that the term does not contribute to the volume energy. Thus the general contributions to the flexoelectriceffect are exceedingly complex and, at the present state of knowledge, hard to evaluate in their relative importance, except that the dominating contribution by far is the twisted (helicoidal) deformation. The consequences of the inverse flexoelectric effect are even harder to predict, especially in the dynamic case, without any quantitativeknowledge of the flexocoefficients.When strong electric fields are applied to a smectic C (or C*) any of the nine deformations of Fig. 42 will in principle be generated, if not prohibited by strong boundary conditions. This consideration applies equally well to antiferroelectric liquid crystals as to ferroelectric liquid crystals. The fact that sophisticated displays work well in both cases seems to indicate that the threshold fields for these detrimental deformations are sufficiently high in practice. A final remark to chirality and spontaneous bend. The fact that p in Eq. (1 18) has a chiral character does of course not mean that the bend vector n x V x n would be chiral; only that this bend B is caused by a chiral interaction. Thus one has to distinguish between the spontaneous bend B in Eq. (118) (which has a special value of n x V x n ) and the expression n x V x n for the general bend vector.
5 The SmA*- SmC* Transition and the Helical C* State 5.1 The Smectic C Order Parameter By changing the temperature in a system we may provoke a phase transition. The thermodynamic phase that is stable below the transition generally has a different symmetry (normally lower) than the phase that is stable above. The exceptions to this are rare, for instance, the liquid-gas transition where both phases have the same symmetry. In a number of cases of transitions between different modifications in solids there is no rational relation between the two symmetries. In such cases it may be difficult to construct an order parameter. In the majority of cases, however, the transition implies the loss of certain symmetry elements, which are thus not present any longer in the low temperature (condensed or ordered) phase. The symmetry group of the ordered phase is then a subgroup of the symmetry group of the disordered phase. In such cases we can always construct an order parameter, and in this sense we may say, in a somewhat simplifying manner, that for every phase transition there is an order parameter. Thus, in liquid crystals, the transitions isotropic ++nematic ++ smectic A H smectic C tjsmectic F, etc. are all described by their different specific order parameters. (There may be secondary order parameters, in addition.) The order parameter thus characterizes the transition, and the Landau free energy expansion in this order parameter, and in eventual secondary order parameters coupled to the first, has to be invariant under the symmetry operations of the disordered phase, at the same time as the order parameter itself should describe the order in the condensed phase as closely as possible. In addition to having a magnitude (zero for T> T,, nonzero for T < Tc),it should have the same symmetry as that phase. Further requirements of a good order parameter are that it should correctly predict the order of the transition, and that it should be as simple as possible. As an example, the tensorial property of the nematic order parameter
correctly predicts, as we have already seen in Sec. 2.6, that the isotropic-nematic transition is first order. Very often, however, as already discussed, only its scalar part S is used as a reduced order parameter, in order to facilitate discussions. In the case of the smectic A-smectic C transition, the tilt 8 is such a reduced order parameter. The symmetry is such that positive or negative tilt describes identical states in the C phase (see Fig. 43), hence the free energy can only depend on even
116
5 The SmA*-SmC* Transition and the Helical C* State
Figure 43. Simple diagram of the SmASmC transition. Positive and negative tilt describe identical states.
powers in 8 and we may write, in analogy with the discussion in Sec. 2.7
G= 1 2
T ~ ) O +* +1 b e 4 4
1 6 +-ce
4
For the time being, this simple expansion will be sufficient to assist in our discussion. A first-order SmA-SmC transition occurs if k O . The equilibrium value of the tilt is the one that minimizes the free energy. For b>O and T-T,=O, 8 will be a G = O then gives small and the Q6 term can be dropped. Putting -
ae
a(T-To)8+b8'=0
( 122)
with two solutions
8=0
(123)
which corresponds to the SmA phase (a check shows that here G attains a maximum value for T < To,but a minimum value for T > To),and
corresponding to the SmC phase (the extremal value of G is a minimum for T< To). The transition is second order and the simple parabolic function of AT= To-T at least qualitatively describes the temperature behavior of the tilt for materials having a SmA-SmC transition. However, the director has one more degree of freedom and we have to specify the tilt direction. In Fig. 43 we had chosen the director to tilt in the plane of the paper. By symmetry, infinitely many such planes can be chosen in the same way, and evidently we have a case of continuous infinite degeneracy in the sense that if all the molecules tilt in the same direction, given by the azimuthal angle q, the chosen value will not affect the free energy. The complete order parameter thus has to have two components, reflecting both the magnitude of the tilt 8 and its direction q in space, and can conveniently be written in complex form
5.1
The Smectic C Order Parameter
117
With a complex scalar order parameter, the SmA-SmC transition is expected [ 1291 to belong to the 3D X Y universality class, which does not have the critical exponent p equal to 1/2. Nevertheless, experiments show [ 1301 that it is surprisingly well described by mean field theory, although with an unusually large sixth order term. As all tilt directions are equivalent, the free energy can only depend on the absolute value squared of the order parameter. Actually, the first intuitive argument would say that there could be a linear term in the absolute value. However, physical descriptions avoid linear terms in absolute values, since these mean that the derivatives (with physical significance) would have to be discountinous (see Fig. 44). Therefore, we assume that the free energy can only depend on
The Landau expansion then has to be in powers of I YI2= Y* Y and we see that the free energy is independent of cp, as it has to be. This is just equivalent to our first expansion. Generally speaking, G is invariant under any transformation
This is an example of gauge invariance and the azimuthal angle is a gauge variable. The complex order parameter Yis shown schematically in Fig. 45 illustrating the conical degeneracy characteristic of the SmC phase. The gauge variable cp is fundamentally different in character from 8. The latter is a ''hard" variable with relatively small fluctuations around its thermodynamically determined value (its changes are connected to compression or dilation of the smectic layer, thus requiring a considerable elastic energy), whereas the phase cp has no thermodynamically predeter-
Figure 44. Functions of the kind G- 1771, i.e., linear dependence in an absolute value, are with few exceptions not allowed in physics, because they imply discontinuities in the first derivative, which are mostly incompatible with the physical requirements. rl
Figure 45. Illustrations of the two-component order parameter Y = 8eiqdescribing the SmASmC transition. The thermodynamic variable 8 and the gauge variable cp are completely different in their fluctuation behavior. The fluctuations in cp may attain very large values and, being controlled by an elastic constant scaling as 8'. may actually become larger than 2n at a small but finite value of 8.
fluctuations in 0
118
5
The SmA*-SmC* Transition and the Helical C * State
mined value at all. Only gradients V q in this variable have relevance in the energy. The result is that we find large thermal fluctuations in q around the cone, of long wavelengths relative to the molecular scale, involving large volumes and giving rise to strong light scattering, just as in a nematic. The thermally excited cone motion, sometimes called the spin mode (this is very similar to the spin wave motion in ferromagnets), or the Goldstone mode, is characteristic of the nonchiral SmC phase as well as the chiral SmC* phase, but is of special interest in the latter because in the chiral case it couples to an external electric field and can therefore be excited in a controlled way. This Goldstone mode is of course the one that is used for the switching mechanism in surface-stabilized ferroelectric liquid crystal devices. The tilt mode, often, especially in the S m A phase, called the soft mode (although “hard” to excite in comparison with the cone mode, it may soften at a transition), is very different in character, and it is convenient to separate the two motions as essentially independent of each other. Again, this mode is present in the nonchiral SmAphase but cannot be detected there by dielectric methods, because a coupling to an electric field requires the phase to be chiral. In the S m A * phase this mode appears as the electroclinic effect.
5.2 The SmA*- SmC* Transition If the medium is chiral the tilt can actually be induced in the orthogonal phase by the application of an electric field in a direction perpendicular to the optic axis, as shown in Fig. 46. The tilt is around an axis that is in the direction of the electric field. This is the electroclinic effect, presented in 1977 by Garoff and Meyer [21]. A tilt may also be induced by the polarity of a surface (surface electroclinic effect). Soft mode fluctuations occur in the SmAphase just as in the S m A * phase, but only in the SmA* phase can we excite the tilt by an electric field. This is because a tilt fluctuation in the medium with chiral symmetry is coupled to a local polarization fluctuation, resulting from the transverse dipoles being ever so slightly lined up when the tilt disturbs the cylindrical symmetry of the molecular rotation. Superficially, it might be tempting to think that chiral and nonchiral smectic A (SmA* and SmA) would be very similar or even have the same symmetry, because they are both orthogonal smec-
Figure 46. If an electric field E is applied perpendicular to the optic axis in a smectic A* (or any chiral orthogonal smectic), the optic axis will swing out in a plane perpendicular to the plane E , n defined by the optic axis and the field (electroclinic or soft-mode effect).
5.2 The SmA* - SmC* Transition
119
tics with the same organization of the molecules in the layers and with the same cylindrical symmetry around the layer normal. Nothing could be more wrong. As we have seen (in one striking example - several more could be given), their physics is very different, due to the fact that they have different symmetry. Reflection is not a symmetry operation in the SmA* phase, and the electroclinic effect does not exist in the SmA phase just because the SmA phase has this symmetry operation. We may also apply the Curie principle to this phenomenon. The SmA phase has D,, (or -lm) symmetry, with one C, axis along the director (optic axis), infinitely many C, axes perpendicular to this axis, and in addition one horizontal and infinitely many vertical mirror planes. The mirror planes are absent in the S m A * phase (0, or -22). If we apply an electric field E (of symmetry C-”) perpendicular to the C, axis, the only common symmetry element left is one C2 axis along E , which permits a tilt around C,. In Fig. 46 we also see that, in particular, the plane ( E , n) is a mirror plane in the nonchiral SmA phase, and consequently neither of the two tilting directors shown in the figure are allowed. In the SmA* phase (E, n) is not a mirror plane, hence one of these tilt directions will be preferred. (Which one cannot be predicted as this is a material property.) An interesting aspect of this phenomenon is that it means that an electric field acting on the chiral medium actually exerts a torque on the medium. This is further discussed in Sec. 5.8. The torque is of course inherent in the chirality. The axial symmetry character is provided by the medium. The electroclinic effect is a new form of dielectric response in a liquid crystal. If we increase the electric field E, the induced polarization P will increase according to the first curve of Fig. 5. With a small field, P is proportional to E, and then P saturates. As the coupling between tilt and polarization is also linear at small values of tilt, we get a linear relationship between 8 and E. With 8 = e*E, the proportionality factor e* is called the electroclinic coefficient. It has an intrinsic chiral quality. (We have used an asterisk here to emphasize this and also to clearly distinguish it from the flexoelectric e-coefficients.) In nonchiral systems, e * is identically zero. With present materials, 8 is quite small (I 15 O), but this may change with new dedicated materials. The electroclinic effect is also appropriately called the soft-mode effect, because the tilt deformation is a soft mode in the SmA phase, the restoring torque of which softens when we approach the SmA* -SmC* transition, at which the deformation starts to “freeze in” to a spontaneous tilt. When we have such a spontaneous tilt as we have in the SmC* phase, we will also have a spontaneous polarization because of the rotational bias. It is important to note that the molecular rotation is biased in the nonchiral SmC phase as well as in the SmC* phase. In principle, this bias will be different in the two cases but the difference might not be essential or very relevant for the polarization, the origin of which is due to the combination of only two basic things: the rotational bias brought about by the tilt, and the tilt plane not being a mirror plane. Only the latter requires chirality. It can
120
5 The SmA*-SmC* Transition and the Helical C* State
nevertheless be worthwile to contemplate what the rotational bias could be like, because this evidently is strongly related to the shape of the molecule. A quite extreme case has been considered by h k s , Filipii. and Carlsson [204] in regarding the molecule to be a brick of sides a,b,c, with a < b < c, cf. Fig. 46 a. In spite of the crudeness of this model it gives some interesting insights. Such brick molecules would, by sterical interactions alone, tend to be oriented in the situation to the top left of the figure because this gives the smallest excluded volume. If we now rotate them by 180 degrees aound their inertial axes, we arrive at the same situation. If we denote the angular rotation variable by y, taking y = 0 along the C , symmetry axis, we see that the angular distribution is not only symmetric with respect to the tilt plane (w= k d 2 )
*I
EXCLUDED VOLUME V,=a'b
tan 8
E X C L U D E D VOLUME V2 = a b ' t n n
1=0
e
Figure 46a. The brick model liquid crystal in the tilted smectic phase and its consequences for the angular distribution of transverse dipoles. The top left situation is preferred to the top right situation because of lower excluded volume. The latter case gives a larger smectic layer thickness for the same 0, which is not considered in the figure. A molecular rotation around the director is thus not actually allowed in this model if a and b are different, as it is coupled to a change in layer thickness. However, a more realistic model giving essentially the proposed potential is shown in Fig. 9c. In the non-chiral case, the angular distribution is quadrupolar, in the chiral case polar. This is, however, a consequence of the assumed shape of the molecule. From reference [132a].
5.2 The SmA*-SmC"Transition
121
but also with respect to the C , direction ( y =O). The potential U describing this distribution therefore has quadrupolar symmetry. This is illustrated in the figure as the second part form the top. Because the brick is non-chiral we have to imagine some perturbation of its shape, such that the tilt plane is not any longer a mirror plane, in order to proceed to the chiral case. This by necessity adds a new part to the potential which is not symmetric around kn/2, i.e. it has polar symmetry, shown in the third part of the figure. As the first part of the potential must be accepted as a contribution also in the chiral case, the full potential in that case is the sum, which is illustrated in the bottom part of the figure. In the chiral case transverse dipoles in the molecule will cancel statistically except along the C2 axis ( y = 0, fn)where a net local polarization will appear. The brick model is interesting but leads to the incorrect conclusion that the angular distribution generally is quadrupolar in the non-chiral (C) case, and that the polar angular potential is something specific for the chiral (C*) case. In fact, this is not a characteristic of the phase - C or C* - but of the molecular shape. This is easily seen if we take another extreme model, the ice hockey stick, in this case symmetrized such that it corresponds to the invariance under n + -n, cf Fig. 46b. To the left in this figure we see two very schematic chiral molecules in a configuration corresponding to a tilted smectic. The C, symmetry corresponds to the n + -n invariance, i.e. that ( 1 ) and (2) are equally represented statistically. This symmetry also rules out any net polarization lying in the tilt plane because, regardless how the mo-
w=
Figure 46b. Model molecules which are not brick-like in a tilted smectic structure. To the left a chiral molecule in two equivalent orientations 1 and 2, corresponding to the required phase invariance under n +-n. Orientations 1 and 2 are related by the I80 degree rotation around the C 2 axis, which has to be a symmetry operation. This symmetry operation is intrinsically built into the middle structure. Although i t is unclear, a priori, how these molecules will behave under rotation, in part or wholly, around any assumed long axis, they will almost certainly not behave as the bricks in Fig. 46a, although some other models would [ 131a1, in terms of the rotational potential. Hypothetical double-hockey stick model, to the right, rhough non-chird, will show a pronounced polar bias in its c director distribution. This bias is thus essentially related to the molecular structure and not first-hand to chirality.
122
5
The SmA*-SmC* Transition and the Helical C* State
lecular dipoles are distributed, all in-plane components are reversed by the C2 operation. This is of course also in accordance with Hermann’s theorem: a vectorial property Pi in a plane perpendicular to a C2 axis has to be isotropic in that plane. Hence P cannot have any component in the tilt plane. Another way to express the same thing is to make the hypothetic molecule itself symmetric in this respect, as shown in the middle. If we focus on this dummy molecule it is not evident how exactly the ditribution of ywould look. However, for the non-chirul hockeystick model to the right, which contains the same C2 axis, it is absolutely clear that the situation (l,l), corresponding to y=O,cannot have the same probability as (l’,l’), corresponding to w= k ~Thus . the angular distribution is polar, corresponding to one of the two potentials V(y) on the bottom of Fig. 46a, and not to the quadrupolar potential on top. Hence, this has nothing to do with, at least qualitatively, whether the phase is chiral or not. This is evident from the corresponding discussions in [131] and [ 131 a] and has recently been emphasized by Photinos and Samulski and coworkers [ 131b]. On the other hand, the polar angular distribution does not mean that the phase is electrically polar. This requires loss of mirror symmetry. To make this more clear, let us sketch the possible qualitative cases of rotational distribution. We may start with the “unit cell” of cylindrical symmetry of the SmA (or SmA*) phase and the corresponding cell of monoclinic symmetry of the SmC (or SmC*) phase, as in Fig. 47. We put a molecule in each one and ask whether the molecular motion (which, like the shape of the molecule, is much more complicated than here indicated) is directionally biased in its rotation around the optic axis (only roughly represented by the core part of the molecule). One simple way of representing this bias is to draw the surface segments corresponding to some interval in the rotation angle. In contrast to a corresponding figure in [ 132dl we have here chosen to let the angular distribution represent the c director, representing the local tilt direction. Let us imagine that the molecule has a dipole perpendicular to its average rotation axis. This dipole also has to be perpendicular to c. We then find that the probability of finding the dipole in any particular angular segment is same in all directions in the S m A (SmA*) phase, as it has to be, because the phase is uniaxial. The layer normal, which is also the optic axis, is a C, axis. In the nonchird C phase, the rotation cannot have this circular symmetry, but it must be symmetrical with respect to the tilt plane, as this is a mirror plane. If the molecule corresponds to the brick model, this means that the rotational bias has a quadrupolar symmetry, which we can describe by a quadrupolar order parameter. This is no doubt a special case, but on the other hand, it would be wrong to believe that it is “ridiculously” incorrect. For instance, a look at Fig. 9c will make clear that even for quite reasonable molecular shapes, the rotational bias might not deviate to far from being quadrupolar. However, in the general case this cannot be true as we have seen. In this general non-chiral case the only symmetry plane is the tilt plane and now the azimuthal distribution of the c director is asymmetric except for the symmetry around this plane.This means that the angular distribution of the c director is
An
5.2 The SmA*-SmC* Transition
123
phase
chiral molecule in A or A* matrix
phase is uniaxial; unbiased rotation
brick molecule in C matrix
rotational bias has quadrapolar character, symmetric with respect to both tilt plane and C2 axis
I
tilt plane -
rotational bias has uolar general shaue ;on-chiral molecule character but no dipolar order is permitted since the in C matrix tilt plane is a mirror plane
chiral molecule in C or C* matrix I
C2 axis
with the mirror plane removed, there is a fixed sterical relation between the c director and the transverse dipole; hence the polar distribution of c implies a corresponding polar distribution of the dipole
Figure 47. The origin of the spontaneous polarization. The polar diagrams show the probability of the c director to be in a certain angular segment. The local dipole has to be perpendicular to this c director. In the A or A* phase the c distribution has cylindrical symmetry. In the non-chiral C phase it may have quadrupolar symmetry under certain conditions (second diagram) but generally has less symmetry (third diagram). In spite of this, however, the asymmetric (polar) angular distribution of c only then gives a non-zero polarization when the mirror plane quality of the tilt plane is removed. The common statement that the origin of polarization is a “hindered rotation” is quite misleading because, hindered or not, a rotation will only result in nonzero polarization for a specific polar directional bias. if moreover, mirror symmetry is absent.
124
5
The SmA*-SmC* Transition and the Helical C* State
Figure 48. Origin of the spontaneous polarization in a smectic C* phase. The illustration shows the directional bias for a lateral dipole in the rigid rod model of a molecule, with the directionality diagram for the short axis when the molecule is rotating about its long axis (from Blinov and Beresnev
J
polar and not quadrupolar in the general case. But in spite of the fact that c has a polar distribution, there can be no local polarization, since there can be no steric relation between c and a transverse dipole as long as the tilt plane is a mirror plane. Hence the mirror plane has to be removed, and this is done by the chirality. By necessity now the whole angular distribution becomes asymmetric, as indicated in the bottom azimuthal diagram of Fig. 47. In reality the shift along the C, axis is probably insignificant in most cases, the bias from the tilt is actually what matters, as illustrated in Fig. 48 where the angular distribution refers to the dipole. The result is a nonzero polarization density in a direction perpendicular to the tilt plane. Which direction cannot be predicted a priori but it is a molecular property. It is fairly obvious that the bias will increase as we increase the tilt angle, and hence the polarization will grow when we lower the temperature below the SmA* -SmC * transition. This transition occurs when the tilt 8 becomes spontaneous and is only weakly influenced by whether the phase is chiral or not. However, chirality couples 8 to P (which is a dramatic effect in a different sense) and brings P in as a secondary order parameter. There might also be a coupling between P and the quadrupolar order parameter, which will, however, be ignored at this moment. In an attempt to quantify the ideas about the angular distribution, Zeks and coworkers [ 132al introduced the simple potential function for the rotational angle w around the director axis
U ( w )= -al 8 cos ty - a,@ cos 2ty
(127a)
These two terms correspond to the polar and quadrupolar distributions of Fig. 46a and their sum to the sum potential shown at the bottom of that figure. At first one might think that this potential is completely unrealistic because it describes the properties of the brick model. However, as we already pointed out with reference to Fig. 9c, molecules with this kind of rotational bias in their environment are not inconceivable nor improbable. Hence the expression might be used as a general ansatz for the rotational potential. As the shape of the bias depends on the ratio a,/a2,the potential can be given a fairly wide meaning and it seems reasonable to split up the potential in a polar and a quadrupolar part. Only the first term gives a dipolar order.
5.2 The SmA*-SmC*Transition
125
However, the statement that the first term is chiral and the second non-chiral is obviously wrong since this is an artifact of the brick model. Thus both terms have to be admitted as soon as we have a tilted phase. None is thus of chiral origin. Both terms contribute to a minimum in U for y = 0. When the molecule rotates out of y = 0 in either + or - direction U ( v )passes a maximum for y=+7d2 and then goes to a minimum for y = ?TC which is the opposite state to ly = 0, but U ( + z ) is less deep than U ( 0 ) .Even if the difference U(+T) - U ( 0 )is not very large it means a bias and leads to a nonzero local polarization in the chiral case. If the lateral dipole is given by pLI then this local polarization is given by
where 2n
(cos ly) =
(127c)
2n
I e-ul(kT)d y 0
and N is the number of molecules per unity of volume. This microscopic model has been extended by Meister and Stegemeyer who showed [132b] that under certain structural conditions also the quadrupolar ordering (cos 2 yf) may contribute to Ps, which is also born out by phenomenological Landau models. We will later deal with such models and find that in the free energy we have to introduce not only a bilinear term - P 8 but a biquadratic term -P2 fI2.These terms correspond to the two terms in ( 127a). Depending on the sign, the P2 82term will change the shape of P ( T )such that it may deviate considerably from the ideal parabolic dependence: if the coefficient is positive the term will add to P at large tilt angle 8, and P ( T ) will then increase more like a linear function, whereas for negative coefficient, the quadrupolar contribution counteracts P which then saturates to an almost constant value. Such effects have been observed, in particular the former, and can thus be judged to be a consequence of a certain shape of the molecule or, at least, a certain rotational bias originating from the interaction with the molecule and its surrounding matrix. The symbolic equation (90) giving the effect of adding a chiral dopant to a smectic C host contains a number of interesting and important scientific questions. We have already pointed out that this is now the standard method of preparing FLC materials for use in displays. The first evident questions to ask is then how the spontaneous polarization depends on the concentration c , how it depends on the dopant, how it depends on the host matrix. It might also give some information, at least qualitatively, on the distribution function for the angle Adding the dopant also induces a twist in the tilted C structure. This has to be controlled at the same time as the
v.
126
5
The SmA*-SmC* Transition and the Helical C * State
desired P,. Generally, a given P, value is aimed at in combination with a long pitch in the C* phase, but also important (or even more) is a long pitch in the higher lying cholesteric phase which is normally used for aligning the sample into the desired bookshelf geometry. With some exceptions [ 1 3 2 ~ few 1 investigations of industrial relevance have been published. However, systematic investigations have been performed by the Stegemeyer school and we will briefly refer to some of their results. The twisting power of a particular dopant in combination with a given host is given by the inverse pitch of the induced helical structure and is found to be a linear function of the concentration in both the cholesteric and smectic phase. In the smectic phase, however, there is a threshold that has to be exceeded in order to get a twist, which is not the case in the cholesteric phase, cf. Fig. 48 a. One may note that in the smectic case there can be no twist in the layers, hence the twist has to develop by interaction between already created layers, whereas in the cholesteric case the local twist develops in any direction which is not topologically blocked, which only happens at the boundaries. Anyhow, with regard to the polarization, there is no such threshold, hence it is in principle possible to induce a nonzero P, in the C* phase in combination with a twist vector k (- inverse pitch) which is zero. The next question regards the growth of P, as a function of concentration and host for a certain dopant. The Paderborn results show that it depends on the type of do-
0
1 X;:
0.06
012 x<.
-
Figure 48a. The twist wave vector depends linearly on concentration in both cholesteric and tilted smectic phase of the same compound, albeit with a concentration threshold in the latter case (from Kersting, reference [132d]).
127
5.2 The SmA*-SmC* Transition
pant, as illustrated in Fig. 48b. Type I, which represents almost all dopants used so far, has the chiraUpolar part outside of the core. It turns out that in this case P, is remarkably linear in the concentration and, in addition, independent of the host material, cf. Fig. 48c. On the contrary, for type I1 where the chiral/polar functions are situated in the core, 9, may be quite nonlinear outside of the region of low concentrations and strongly depends on the host material. The dopant can for instance give rise to Ps>O in one host matrix, while to Ps
Figure 48b. Dopant molecules of two kinds: type I has the chiral/polar part in one of the end chains, while this part is situated in the core for type I1 (after reference [132e]).
30
25
y Figure 48c. Spontaneous polarization as a function of dopant concentration for a dopant molecule of the first kind, shown on the top of the diagram. P, is represented by Polsin@ i.e., normalized per unit of tilt angle and the concentration is given by the mole fraction x . Dots, triangles, and crosses represent different host materials. Note the linearity of P ( x ) and the independence of host material (after reference [ 132fl).
20
5 Y 1s \
ao 10
5 0
0
0.2
0.6
0.4 XG
0.8
1
128
5 The SmA*-SmC* Transition and the Helical C* State -
Figure 48d. Spontaneous polarization as a function of dopant concentration for a dopant of the second type, shown on the top of the diagram. The polarization is now strongly dependent on the host materials represented by the different curves (after reference [132g]).
C,Y.
0
80 70
60
50
N I
E
U
Y \
43
30 20
0
LL
10 0 -10
0
0.05
0.1
0.15
0.2
0.25
xG
functionf(y) than the tail which seems to sense only the magnitude of the tilt which is given by the host. A microscopic theory has been worked out which seems to reasonably well account for these experimental data [ 132el. The first estimation of the spontaneous polarization was made by Meyer himself in the very first investigations on DOBAMBC and HOBAMBC by measuring the field for complete unwinding of the helix [62], [43]. Shortly thereafter it was measured by Pieranslu, Guyon and Keller [ 1331 and by Petit, Pieranski and Guyon [ 133a] in an elegant series of experiments which are also very illuminating from a physical point of view. The smectic C* is confined between two glass plates with the layers parallel to the plates, i.e. with the helix axis in homeotropic condition, cf. Fig. 48e. A shear is applied along a certain direction parallel to the layers as shown in the figure. The experiment is very similar and corresponds to the piezoelectric action on a solid. But it also corresponds to the unwinding of the helix in an electric field. The symmetry is broken by the shear, a lateral net P appears, uniaxiality is changing to biaxiality. The shear will distort the helix towards a limiting unwound state (the one shown in the figure although not realistic) at very high shear rates where all molecules are supposed to tilt in the same direction. The liquid crystal cannot sustain a constant shear, but when an alternating shear is applied alternating polarization charges will appear at lateral electrodes in the cell, growing linearly with shear. As pointed out by Prost [133b], it is illustrative to analyze this situation according to the Cu-
5.3 The Smectic C* Order Parameters
Figure 48e. If subjected to a shear along the layers the helical smectic C* structure will be distorted and a finite polarization will appear at a right angle to the shear. In the figure a substance with P,,>O is chosen, giving a P vector directed into the paper, and the right figure, for simplicity, corresponds to the completely unwound state. In reality that state is unobtainable in this kind of experiment: because the medium is a liquid the shear cannot he static, and only a slight structural distortion can be achieved in a dynamic experiment.
129
+
rie principle. The situation is analogous to that of Fig. 29. The undisturbed medium (the helical SmC*) has the symmetry 0022 (or Om)with no mirror symmetry but with an infinity of two- fold axes at right angle to the helix axis. The shear has symmetry 2/m (or C2h)with a two-fold axis perpendicular to the paper and a mirror plane normal to this axis. The only common symmetry element is the C2 axis out of the paper. Hence this is the symmetry of the “effect” caused by shearing the medium. This symmetry is polar and allows a polarization along the axis which is also observed. This method of determining the value of P, is in practice too complicated and is not free from flexoelectric influence. It has therefore been replaced by field reversal methods, to be discussed later. Alikewise unpractical (for routine work) but elegant method of great physical interest for measuring P s , has been demonstrated by Hoffmann and Stegemeyer [ 1 3 3 ~ 1as , they manage to handle free-standing smectic C* films in a way that microelectrodes can be inserted directly into the film such that they go through the whole film thickness. We will return to experiments performed on freely suspended films in a later section.
5.3 The Smectic C* Order Parameters If we keep the tilt angle 8 in Fig. 48 fixed (just by keeping the temperature constant), we can move the molecule around the tilt cone, and we see that the rotational bias stays sterically fixed to the molecule and the tilt plane. The direction of the resulting P lies in the direction z x n . If we change the azimuthal angle by 180”, this corresponds to a tilt -8 (along the -x direction), and results in a change of P direction fromy to -y. This means that we could tentatively write the relation between the sec-
130
5 The S d * - S m C * Transition and the Helical C* State
ondary and primary order parameters as
or more generally that P is an odd function of 8
P = a O + b @ + ... As sin8 is an odd function, an expression that has this form is
where we have introduced Po (with its sign, see Fig. 23 in Sec. 3 3 , the magnitude of spontaneous polarization per radian of tilt. With 0 = z x n we may write the tilt as a vector 0 = 8 O to compare its symmetry with that of P. 0 is then an axial vector, while P is a polar vector. As inversion is not a symmetry operation in chiral phases, this difference does not cause any problem. P and 0 thus transform in the same way under the symmetry operations of the medium, and we may make a Landau expansion according to the improper ferroelectric scheme of Sec. 2.8. If we take the scalar part of Eq. (130), it takes the form
This expression is often used but does not have the correct symmetry because it is not invariant under the C2 symmetry operation. For, if we change 8 to 8 + x , sin8 changes to -sin8 although P cannot change, because this is the same state (see Fig. 49). Therefore Eq. (13 1) should be replaced by
1 p0 sin28 P =2 which is a C , invariant expression. In practice the expressions do not differ very much; P just increases somewhat slower at large tilts according to Eq. (132). For small values of tilt the expressions do of course give the same results, i.e., P=P,8.
Figure 49. Under the action of the twofold symmetry axis in the smectic C* state the tilt changes from 0 to 0 + K.
5.4 The Helical Smectic C* State
131
Figure 50. (a) Schematic temperature dependence of the tilt angle 19and polarization P at a second-order SmA*-SmC*transition. (b) Tilt angle versus temperature for DOBAMBC below the SmA*-SmC*transition at 95 "C (from Dumrongrattana and Huang [134]).
Experimentally, this simple linear relation between P and 8 often holds quite well, as shown in Fig. 50. However, there are exceptions, for instance, cases where P more resembles a linear function of temperature rather than parabolic. Although the tilt angle has certain limitations, we will continue to use 8 as the order parameter for the time being. Finally, however, we will construct a different order parameter (Sec. 5.11) to describe the tilting transition.
5.4 The Helical Smectic C* State We did not change anything in the nonchiral order parameter (Eq. 125) when we proceeded to the chiral case, except that in addition we introduced a secondary order parameter P which does not exist in the nonchiral case. Now the normal state of the smectic C* is the helical C* state, i.e., one in which the tilt has a twist from layer to layer. At constant 6, cp is therefore a function of z (see Fig. 5 l), with cp (z)=qz, q being the wave vector of the helix 2.n q = y
(133)
where the helix is right-handed for q> 0 and Z is the helical periodicity. For this case there is an obvious generalization of the nonchiral order parameter (Eq. 125) to
132
5
The SmA*-SmC* Transition and the Helical C* State
j cx[r y
I
\ \ \ , I
__-
Y
X
Figure 51. (a) In the helical smectic C* state the constant tilt 0 changes its azimuthal direction cp such that cp is a linear function in the coordinate z along the layer normal. (b) The c director is a two component vector (cx,c v )of magnitude sin 0, which is the projection of n on the smectic layer plane. (c) P makes a right angle with the tilt direction. The P direction here corresponds to a material with P,>O.
X
(b)
(a)
with
where we have introduced a new length q-’ as an effect of chirality. Yqis now chiral by construction but does not make any change in the Landau expansion since Yq*Yqis still equal to 82, and we will still have the SmA* -SmC* transition as second order. We can also write Yqas Yq= B(cosqz + i sinqz)
(135)
and as a vector
or
Yq= oc
(136a)
where we have introduced the “c director”, the projection of the director n on the smectic layer plane (see Fig. 5 1b). Note that the c director in this description (which is the common one) is not a unit vector, as it is in the movable reference frame k , c, p used, in contrast to the space-fixed reference frame x, y, z, in Sec. 4.10 to describe the flexoelectric deformations.
5.4 The Helical Smectic C* State
133
As the polarization vector is phase-shifted by 90 O relative to the tilt vector lvq,we see from Eq. (136) that it can be written (see Fig. 51c) as
P = (-Psinqz, Pcosqz, 0)
(1 37)
It is interesting to form the divergence of this vector. We find
This is an important result as it means that the helicoidal C* state is a divergencefree vector structure P(r)in space. Hence we have no appearance of space charges anywhere. This means that not only does the helix cancel the macroscopic polarization and thereby any external coulomb field, although we have a local polarization P everywhere, but the fact that V-P = 0 also secures that there are no long range coulomb interactions in the material itself. A non-zero divergence of P,on the other hand, is equivalent to a polarization charge density ppr
pp =-v.P
(138a)
which would cost an additional electrostatic energy proportional to ( V .P)2,a term which we then would have to consider in the free energy. The helical state is therefore “low cost” and “natural” from several points of view. A homogeneous state of the director, as can ideally be realized by surface stabilization, of course also has V.P=O, although it has an external coulomb field if not matched by electrode charges and therefore is not stable by itself. However, as we will see later, the condition V.P=0cannot be maintained everywhere if we have a local layer structure of chevron shape (see Sec. 8.3). Independently of chevrons, we will also have to consider this electrostatic term in some interesting device structures (see Sec. 12.5). For the sake of completeness, but also for distinction, let us finally make a comment on an entirely different kind of chiral order parameter. Let us imagine that the substance we are dealing with is a mixture of two enantiomers (R) and (S). We can then define a scalar quantity K=- R - S
R+S
(139)
where R and S stand for the relative concentration of (R) and (S) enantiomers. This quantity can in a sense be regarded as an order parameter. K = 1 for “all R’, -1 for “all S”, and zero for a racemic mixture. Thus in general, 0 I IKI I 1. This quantity is nothing other than what organic chemists introduced long ago, but in a completely different context. It is sometimes called “optical purity”, but is today rather called
134
5
The SmA*-SmC*Transition and the Helical C* State
enantiomeric excess. It is an extremely useful quantity in its right place, but it is not relevant as an order parameter because it does not depend on the temperature T, only on the concentration, which is trivial in our context. It does not help us in introducing chirality and we will not have use for it in the physical description.
5.5 The Flexoelectric Contribution in the Helical State We note from Fig. 5 1 that the projection of the director on the smectic layer plane is n sin 8= sin 8. If we write the components of n in the x,y,z system, we therefore have n, = sin8 cosq ny = sin8 sinq n, = cosB
If we insert q (z) = qz for the helical state this gives,
n, = sin8 cosqz ny = sin8 sinqz n, = cos8 This is a twist-bend structure, as we have stated above, and is therefore connected to a flexoelectric polarization of size (see Eq. 97)
Pf=e,n x ( V x n )
(142)
We will now calculate this contribution. Forming the curl as
we get
V x n = x sinB(qcosqz)-ysin8(qsinqz)+O = -q sin8(cosqz, sinqz,O)
and
Y Z n x ( V x n )= -qsin8 sinBcosqz sin8sinqz cos8 0 cos qz sin qz X
(145)
135
5.6 Nonchiral Helielectrics and Antiferroelectrics
This is a vector in the layer plane which is antiparallel to the P, vector for a smectic C* material with Po>O. Its magnitude is
In x ( V x n ) 1 = q sin8 cose = 1 q sin28
(146)
-
2
Therefore we can write the flexoelectric polarization due to the helical deformation as
Pf = - e b . - q1 s i n 2 8 2
(147)
Note that this polarization grows with the tilt according to the same function as the spontaneous polarization P, in Eq. (132). We may then write the total polarization density P = P , + P, as 1 P = -(Po 2
-
eb q ) sin28
(148)
or, for small angles
P = (Po - e,q) 8
( 148 a)
If Po and eb have different signs, the two contributions will cooperate, otherwise they tend to cancel. (This is not changed if we go from a certain chiral molecule to its enantiomer, as q and Po will change sign simultaneously.) In the absence of a smectic helix, q = 0, and the flexoelectric contribution will vanish. The sign of Po has been determined for a large number of compounds, but sign (and size) determinations of flexoelectric coefficients are almost entirely lacking. Such measurements are highly important and should be encouraged as much as possible. However, recent measurements by Kuczyriski and Hoffmann [ 134al show that the flexoelectric polarization is of the same magnitude as the spontaneous one. Equation (148) may also be a starting point for a new question. In the absence of chirality, thus with P,=O, could the flexoelectric effect lead to a helical smectic state in a nonchiral medium? This will be the topic of our next section.
5.6 Nonchiral Helielectrics and Antiferroelectrics As we have repeatedly stressed, flexoelectricity is a phenomenon that is a priori independent of chirality. But we have also seen that some flexoelectric deformations do have a tendency to occur spontaneously in a chiral medium. All except the helical C* state are, however, suppressed, because they are not space-filling. A flexo-
136
5 The SmA*-SmC* Transition and the Helical C* State
electric deformation may of course also occur spontaneously in the nonchiral case, namely, under exactly the same conditions where the deformation is space-filling and does not give rise to defect structures. In other words, in creating the twist-bend structure which is characteristic of a helielectric. Imagine, for instance, that we have mesogens which have a pronounced bow shape and, in addition, some lateral dipole. Stericallythey would prefer a helicoidal structure, as depictedin Fig. 52, which would minimize the elastic energy, because the spontaneous bend B would cancel the bend term (Eq. 117) in the elastic energy. The resulting local polarization from Eq. (148) with P,,=O may still not be too costly because the external field is cancelled and V.P= 0. This would have an interesting result: because the starting material is nonchiral, we would observe a spontaneous breaking of the nonchiral symmetry leading to equal regions with left-handed and right-handed chirality. It would correspond to the well-known examples of SiO, and NaC10, discussed in Sec. 3.5. It seems that finally this kind of phenomenon may have been observed in liquid crystals [135]. However, in contrast to the cases of SiO, and NaClO,, the helical structures are probably possible on a molecular level as well as on a supermolecular level. Thus we may expect domains of nonchiral molecules in different conformations, right and left-handed, which behave as if they belong to different enantiomeric forms. The possibility that a space-filling flexoelectric deformation will be spontaneous for certain molecular shapes and thereby create a chiral structure out of non-
, I
I
Figure 52. Space-filling twist-bend structure of strongly bow-shaped achiral molecules equally split into right-handed and left-handed helical domains. (Here a right-handed domain is indicated.) A bow-shaped molecule of this kind is illustrated in Fig. 53. The kind of domain depicted in this figure would correspond to the normal helielectric organization in a smectic C*. As a result of the two-dimensional fluidity of these smectics, the cancellation can, however, be expected to occur on the smallest possible space scale.
Figure 53. Bow-shaped nonchiral molecule which may create chiral C* domains (from [135]).
5.6 Nonchiral Helielectrics and Antiferroelectrics
137
chiral molecules will be much enhanced if the deformation can take place in the layer rather than in the interlayer twist-bend structure of Fig. 52, and may then lead to antiferroelectric (rather than helielectric) order similar to that in antiferroelectric liquid crystals made of chiral compounds. The polarization may very well be switchable, because it is not connected to a supramolecular director deformation. In other words, the deformation represented to the lower right of Fig. 33 now applies to the single molecule and this can be flipped around by the electric field. Further investigations of these new materials are important and will shed light on a number of problems related to polarity and chirality. Many of these molecules can be made with a very strong dipole attached to the bow. Whether they will show some macroscopic polarization is hard to say because as long as there is fluidity in the system there are many ways to escape such a polarization. This is illustrated in Fig. 53 a. So far some kind of antiferroelectric order seems to be realized in a way similar to what is found in the polymer case mentioned in Section 3.10. However it is now clear that the phenomenon of spontaneous optical resolution has been observed. Thus we have achiral molecules which form chiral smectic layers. This has been most strikingly demonstrated by the Boulder group investigating freely suspended films of the compound in Fig. 53. As Link et a1 were able to show [135b] the bulk states are either antiferroelectric-racemic with polarization direction and chiral handedness alternating from layer to layer, or they are antiferroelectric-chiral with uniform handedness in the layers. Both states can be switched into the ferroelectric state on application of a field of the order of 5 V/pm and they then relax back into the AF state on
Figure 53a. Four ways of escaping macroscopic polarization. From upper left, clockwise: polar domains, smectic layers with antiferroelectric order, twisted layers (twist-grain-boundary phase) and twist from layer to layer. After Watanabe [ 135al.
138
5
The SmA*-SmC* Transition and the Helical C* State
field removal. These experiments recall the classic discovery in 1848, thus 150 years ago, by Pasteur, who could separate crystallites of tartaric acid into right-handed and left-handed pieces, although the separation here, in the liquid crystal, is on a microscopic scale - from layer to layer. We also already know this phenomenon to appear in the semicrystalline polymer state. For instance, the polypropylene chains are helixes which form helical structures on a macroscopic scale. The mesoscopic arrangement contains two layers of orthogonal right-handed helixes followed by two layers of left-handed helixes. Each “chiral domain” thus consists of two homochiral layers in which the chains are packed orthogonal to each other. This reminds indeed of the liquid crystal case where only monomers are involved. It is interesting that even if these monomers themselves are achiral, the polar phenomena are here still related to chirality, although we should rather turn it around and say that chirality is related to, or a consequence of, polarity! Similar observations have now been made on a number of related compounds [ 135c] - [ 135el, in one case [ 135c] on a compound closely related to the wellknown antiferroelectric prototype molecule TFMHPOBC. The textures are normally very interesting, with helicoidal periodicity on several length scales. It is also spectacular to watch the helicoidal phase growing from the isotropic phase in form of long straight helicoidal strands, the symmetry of which might be due to the tendency for immediate local cancellation of the polarization.
5.7 Mesomorphic States without Director Symmetry When we vary the shape of the mesogen in a more or less arbitrary way it is not certain, as we have discussed in Sec. 3.10, that the phase will have director symmetry, i.e. be invariant under n +-a, nor might it even be meaningful to define a director. If we turn back to Fig. 3 , we are reminded that this symmetry is related to the shape of the molecule and the distribution of its surrounding neighbors. If instead of the elongated rod we transform our model molecule into a circular disc and define the director n as being the rotational symmetry axis as before, we get a different result. If the disk would have a dipole along n, then the different disks would like to order in a polar way, one after the other in a long row because their neighbors are now found predominantly along the dipolar direction, cf the dipole field distribution in Fig. 3. The same effect would appear if we deform the disk slightly into a bowl. This is a kind of “steric polarity” and the bowls would tend to pack together like saucers, i.e. in a polar way. In the world of real molecules these two things, electric and steric polarity, would probably happen together. Anyway it is clear that for non-flat disks there is no director symmetry. It is not clear whether for such molecules the director really is a useful concept. That for such (“bowlic”) molecules the mesophase in general has to be polar - in principle - is trivial. But if it is going to be polar not only on the microscopic scale
139
5.7 Mesomorphic States without Director Symmetry
depends on the long range order, and it is not evident that such phases even will have long range order. It is also not clear if they will ever crystallize to an ordered state or rather cool to a polymer-like glassy state. Less trivial is the fact that even a state with long-range nematic order might turn out to be polar. With reference again to Fig. 3: the condition for the director symmetry was not only that the molecules are rodlike but also that they can flip around such that, in dynamic equilibrium, they always can be in the lowest energy situation relative their neighbors. This, however, presupposes that the rods are not too long. Very long and rigid rods will have a tendency for polar order in the nematic phase, if the rods are at the same time dipoles. This is illustrated in Fig. 53 b. Now n +-n is not any longer a symmetry operation. Under this assumption, Khachaturyan in 1975 investigated the stability condition for such a polar nematic phase [ 13Sfl. He found, not surprisingly, that a homogeneously polarized nematic state is unstable with respect to a helical perturbation and therefore transforms into a helical structure of lower energy in which the local polarization turns together with the director, cf. Fig. 53 b. The structure is of course degenerate with respect to handedness. Thus in the case of a non-chiral material, domains of right- and left- handed helix should be expected. The structure in each domain is a polar cholesteric and also another example of a helielectric liquid crystal. It is certainly unlike a normal cholesteric. The helical pitch (a in the Figure) does not only depend on the material properties, as twist
a
I
Figure 53b. A polar cholesteric, which is a new form of a helielectric liquid crystal is possible in case of very long and stiff rods in solution (a-helix structures, biopolymers etc). This state was first considered by Khachaturyan. The homogenously polarized nematic is unstable which leads to a local spontaneous breaking of mirror symmetry.
140
5
The SmA*-SmC*Transition and the Helical C * State
-
elastic constant and polarization (a K22/P2),but also grows with the volume of the sample. It is interesting to note that again, like in the last example of the previous section, this is not a case where chirality entails polarity but, on the contrary, it is the polarity which entails the breaking of mirror symmetry. That is, breaking this symmetry is the natural way for the nematic to do away with the macroscopic polarization and can take place owing to the fluidity. There are now indications that such a phase has actually been observed for the first time. It is found in high-viscous solutions of certain very stiff biopolymers [ 135gl.
5.8 Simple Landau Expansions In order to sharpen the observance for inconsistencies in a theory, it is sometimes instructive to present one which does not work. For instance, let us see if we can describe the transition to a polar nematic phase by a Landau expansion. It might also be thought of as describing the transition isotropic-polar smectic A or nematic-polar smectic A. (In either interpretation it will suffer from serious deficiencies.) We write the free energy as G = - a1 S 2 +-bP2-cSP2 1 2 2
(149)
where S is the (reduced) nematic order parameter and P the polarization. The same abbreviation a =a( T - To)and sign conventions are used as before. The form of the coupling term is motivated by the fact that S and P have different symmetries, quadrupolar and polar, respectively, see the discussion at the end of Sec. 2.8. (It is easy to persuade oneself that a term -SP would lead to absurdities.) G has to be minimized with respect to both order parameters: putting dG/aP and dG/dS equal to zero gives b P - 2~S P = 0 a s - c P* = 0 from which we deduce S=- b 2c 2 a ab P =-S=--,(T-To) c 2c
(153)
Whether it is already not clear physically how a non-zero S-value would couple to a non-zero polarization, we see that Eq. (153) describes a P(T) increasingbelow the transition point only if b
5.8 Simple Landau Expansions
141
In order to give S>O, we have then to require that c
P2-PE
G=2x0 Eo
Why we have chosen to write the coefficient in front of the P2term in this form will be evident in a moment. dG/aP=O gives
L P - E = O
xo Eo
(155)
or
xo
where E~ is the dielectric susceptibility, by definition. For the chiral smectic phase we now write the free energy with the tilt 8 as the primary order parameter 1 1 P2-c*PPe-PE G=-a82+-b84+2 4 2x0 E0
(157)
where the 8-P coupling coefficient is written c* to emphasize that it is chiral in its nature, while all other coefficients are non-chiral. We have disregarded any coupling with the nematic order parameter S . This is totally justified. Although it is true that Sincreases to some extent when 8 increases, this is certainly of no significance around the A*-C* transition. At this stage we also neglect the complications of an eventual helical structure. This is admissible because the helix is not a necessary condition for a chiral tilted smectic. We are first interested in the spontaneous polarization, i.e. the case E=O. The variable P means just “polarization”, i.e. the total polarization, even if a number of sources may contribute - we cannot have different P variables for different “sorts” of polarization. We may however ask: what is the origin of the P2 term and what does it signify? Answer: take the coupling away, i.e. put the coupling coefficient c* =O. Then minimizing G with respect to P just gives
142
5
The SmA*-SmC* Transition and the Helical C* State
i.e.
P=O
(159)
In other words, the third term in the expansion counteracts polar order. It secures that P vanishes, unless there is a coupling between 8 and P. Its origin is entropic and has nothing to do with dielectric effects. The third term is just not “electric” and therefore is often referred to as a “generalized” susceptibility. What this P2 term means is that other factors being equal, polar order increases the order, hence decreases the entropy and increases the free energy: a non-zero polarization is always connected with an entropic cost in energy - whence the plus sign in Eq. (157) - and therefore will not occur spontaneously if there is no other contribution that diminishes the energy even more for P+O. We might therefore expect that the result depends on the strength of the chiral coupling constant c*. With c*#O Eq. (157) gives in the field-free case (E=O)
xo
or
Perhaps to our surprise, this means that however weak the coupling, any non-zero coupling constant c* actually leads to a non-zero value of spontaneous polarization P. This polarization, at least for small values of 0, is proportional to 0
i.e. the polarization increases with increasing tilt, and the proportionality factor xoq,c* only vanishes for c* strictly equal to zero. This means that however weak the coupling, it dominates the counteracting entropy factor. This is also consistent with our discussion in Sec. 3.9: the coupling is via chirality which brings the decisive change in symmetry. We could symbolize the strength c* of the coupling constant by the concentration of chiral dopant which we add to a non-chiral smectic C host. Any non-zero concentration, however small, is sufficient to change the symmetry. If we now allow a non-zero external field along the layers (P will then turn into that direction), Eq. ( 157) in the same manner gives
Hence
5.8 Simple Landau Expansions
I43
Now we see that the total polarization has two origins. On one hand a polarization is induced by an external field, the proportionality factor being the susceptibility xOq,and on the other, there is a tilt induced polarization xoq,c*B independent of any external field E. If we would have applied the external field along just any direction, the E in Eq. (164) would correspond to the component perpendicular to the tilt plane. Therefore we could also write Eq. (164) as
expressing the two contributions to P and emphasizing that the susceptibility is the one perpendicular to the tilt plane. Without changing anything else, we can make c" = O by replacing the chiral substance by its racemate. Eq. (1 65) then reduces to
xo~O,
Hence our X ~ E or~ , is the susceptibility of the racemate perpendicular to the tilt plane. Let us now continue with the case E=O, that is with the spontaneous polarization below T,. Introducing the structure coefficient s" we can write the relation Eq. (1 6 1 ) as
The structure coefficient
is the dielectric susceptibility of the racemate multiplied by the chiral coupling coefficient and thus another related chiral parameter, which we can expect to be essentially temperature independent. We can expect it to grow very slowly towards lower temperatures, like -l/kT, originating in the entropic character of the P2 term (xi' kT) in Eq. (157). The linear relation Eq. (167) and the temperature independence of s* for reasonable temperature ranges have been experimentally confirmed in dielectric measurements, especially by Bahr and Heppke [ 1881. The quotient P/B= s* is free of divergences and a characteristic for each molecular species. Its meaning is the dipole moment per unit volume for unit tilt angle. In rare cases of strong conformational changes in the molecule it may behave anomalously and even change sign. It can be looked upon as a kind of susceptibility like and we will understand all these and similar susceptibilities more strictly as response functions in the limit of small tilts or small applied fields, for instance = (dP/aE)E=,,. The relation Eq. (167) may look trivial but hides something fundamental that we should just not let go unnoticed. The polar order expressed by P is a result of a rota-
-
x
xoq
144
5
The SmA*-SmC*Transition and the Helical C* State
tional bias expressed by 0. It is counteracted by thermal disorder, represented by the 1 (xo~)-1P2 in the Landau expansion which is porportional to kT. Thus in term 2 Eq. (167) s*- 1/T. But this is nothing else than the old Langevin balance, just the same in principle as between a magnetic field H wanting to align magnetic dipoles and thermal motion wanting to destroy the alignment, leading to a magnetization
M - -H
T
or between an electric field E wanting to align electric dipoles against thermal motion, leading to a polarization
We already discussed this in Chap. 2 (cf. Eqs. 4 and 5). Now, with xo- l/kT we can write Eq. (167) in the form P - -0 T The tilt here acts like a (biasing) field corresponding to H and E in Eq. (169) and (170). This is thus nothing else than the Curie law, only unusual in the sense that the field variable is a tilt. As we already noticed, this law indicates that the local dipoles are non-interacting. The collective behavior in the medium is in 0, not in P. So far, except for discussing the physical character of the terms in the Landau expansion, we have mainly derived a very simple relation (Eqs. 167 and 168) between P and 0 in the case that both appear spontaneously, that is, in the C* phase. But what is their relation if P is a polarization induced by an electric field? When we consider the A* phase, i.e. T> T,, P and 0 can be nonzero only if E#O, thus the -PE term is now important. And now we have a situation which in a sense is opposite to the one below T,: in the C* phase we have a non-zero tilt which gives rise to a non-zero polarization; here we have a non-zero polarization causing a non-zero tilt. How do we get the equilibrium value of 8 for a certain constant value of E? We here recall the starting expression Eq. (157).
and want to proceed by forming dG/& and putting it equal to zero. But the sensitive reader might react to this: we cannot vary 0 while keeping both E and P constant. In (dG/dO),, we can keep E constant but we cannot vary 0 while keeping P constant.
5.8 Simple Landau Expansions
145
A variation of 8 involves a variation of P, or does it not? The answer to this question reveals the very special character of a Landau expansion. This is an expansion around equilibrium. The variables 8 and P appearing in the expression are pre-thermodynamic variables in the sense that only the minimization of G will give the relation between those variables in equilibrium. It is quite possible to imagine that we tilt the director without changing P , but then we disturb the statistics which costs energy. Conversely we may imagine that the rotational bias changes in such a way that P increases (we decrease the rotational entropy) without changing 8, but then we are not in equilibrium. In order to go back to equilibrium without changing 8we would have to apply an electric field counteracting the 8 change (which costs energy), and so on. Hence, the variables in the Landau expansion have to be considered independent, and only when we have determined the equilibrium values by dG/aP= 0, aG/d8 = 0, we may look upon them as dependent, interrelated variables. We will now take that step and see which physical results can be derived from the expansion. Putting the two derivatives equal to zero gives, respectively,
and a 8 + be, - c * P = 0
(1 7 3 )
Writing the first one (as previously Eq. 164)
and inserting P in the second, we find a 8 + bs? - x ~ ~ c * ~ ~ - ~ ~ ~ c * E = ~
(175)
For E=O the first solution 8 =0 corresponds to the A* phase. In the second solution a-Xoqc*2+b@=0 we rewrite the first two terms according to
with
T,=T()+ xo EO c *2 a
( 176)
146
5 The SmA*-SmC* Transition and the Helical C* State
T C-A
\\\\\\\\\\\\\\\\\\\\\\\ \ I \\\\\\\\\\\\\\\\\\\\~~ I
\\\\\\\\\\\\\\\\\\\\\\\\I
C*-A*
\‘////I IIIIII
Figure 54. The chiral coupling between 0 and P shifts the tilting transition to a higher temperature than for the non-chiral C +A transition. The shift IS proportional to the coupling constant squared. Doping a non-chiral SmC host we should therefore expect a shift AT proportional to the square of the concentration of the dopant.
Thereby Eq. (134) is reshaped into
a ( T - T,)
+ be2 = 0
(179)
giving the wellknown parabolic increase of 8 below T,. The reader may now compare with Eq. (121) and see that in the Landau expansion Eq. (157) the coefficient a(T) is equal to a(T-To) where To is the temperature at which the tilt goes to zero in the nonchiral system (the racemate). Therefore we see from Eq. (178) that the phase transition temperature has been raised to a slightly higher value than the transition temperature Tofor the racemate, cf. Fig. 54. This increase in transition temperature AT corresponds to the additional energy needed to break the coupling between polarization and tilt in the chiral case. Conversely, a spontaneous tilt appears earlier on cooling a chiral material than in the corresponding nonchiral material. We note that the offset in the transition point
AT= xo EO c **
a
is proportional to the chiral coupling coefficient squared and inversely proportional to the nonchiral thermodynamic coefficient a, the latter fact being reasonable because a is proportional to the restoring torque density (-a6)trying to establish the 8 = 0 value of the orthogonal A* state, thereby diminishing the rotational bias. We have, so far, considered the non-helical C* case. With the helix present we would have found the transition C* -+A* pushed even slightly further upwards in temperature because now, in addition, we have to bring up energy to unwind the helix at the transition. We would then have found another contribution to ATbeing proportional to q2, where q is the value of the helical smectic C* wave vector at the transition. In any case, however, AT still turns out to be quite small (like all chiral perturbations), in practice often less than a degree.
5.9 The Electroclinic Effect
147
With the same reshaping of a-Xo&oc*2to a(T -T o ), which we just used in Eq. (179), our previous Eq. (175) will look like
With this we now specifically turn to the case T > T,. As the tilt induced in the A* phase by an external electric field is quite small we first discard the O3 term and get
This gives the induced tilt angle
It is thus linear in the electric field in this approximation, but strongly temperature dependent, diverging as we approach T, from above. In the A* phase the relation between 8 and E is almost as simple as that between P and 0 in the C* phase, while the relation between P and 8 is not as simple. Equation (183) describes the electroclinic effect. We may write it
where e* is the electroclinic coefficient
showing the close relationship between the chiral material parameters e*, c* and s*. Specifically we see that the structure coefficient s* is the non-diverging part of e*.
5.9 The Electroclinic Effect If we first assume (which is not quite true, but empirically fairly well justified) that the same simple relation P = s * 8 is valid also for T > T,, the combination
is equivalent to P=s*e*E
148
5 The S d * - S m C * Transition and the Helical C* State
With e* taken from Eq. (185) we see that
i.e., that part of the polarization which corresponds to the induced tilt, diverges in the same way as the primary order parameter. (There is of course also a non-diverging part of P as in all dielectrics.) While 6 s* in Eq. (184), P - s * ~ which , has to be the case as P , unlike 8,cannot depend on the sign of s*. We note, furthermore, that as e* and s* only differ in the temperature factor they always have the same sign. This means that the electroclinically induced tilt or polarization will never counteract a spontaneous tilt or polarization, except in the sense that in a switching process -because the electroclinic effect is so fast - the electroclinic contribution may temporarily get out of phase from the ferroelectric contribution. In order to get the correct relation between P and 6 for T > T, we have to start from the generally valid Eq. (174) which we write as
-
P = s * 8+
(5) E
If we assume the linear relationship 8= e* E to hold - which is not true near T, - then
P=s*6+s*6
c*e*
and inserting e* from Eq. (1 85)
P = XO EO c * + a (T -. T,)
(191)
-
e
C
(xo-
The first term slowly decreases with increasing temperature 1/T ) whereas the second slowly increases. This explains the empirically found result that P/Bis essentially the same (=s*) in the C* phase as well as in the A* phase. The ratio is thus practically the same whether P and 6 are spontaneous or induced. Within the approximation of small tilt, Eq. (1 82), we now insert the value for 6 from Eq. (183) in the expression Eq. (189) for P. This gives
or
5.9 The Electroclinic Effect
149
The second part is the electroclinic contribution to the dielectric susceptibility corresponding to Eq. (188). The first is the “background’ part corresponding to s* =0, i.e. to a racemic or non-chiral substance. The electroclinic effect is linear only if we do not approach too closely to T,. At the transition, T = T,, the first term in Eq. ( 18 1) vanishes and the relation between 8 and E becomes b e 3 = S* E
( 194)
or
This non-linear behavior with saturation is well known experimentally. It means that the electroclinic coefficient becomes field-dependent and falls off rather strongly at high fields. It also means the same for the electroclinically induced polarization which acquires relatively high values at low applied fields but then shows saturation. It explains the shape of the measured P ( T ) curves near T, as illustrated in Fig. 18 (dashed line). For T# T, but near T, we have to keep all the terms in Eq. (18 1). We may write this expression in the form
with
A=
a
(7-- T,)
and
Generally it can be said that Eq. (196) very well describes the experimental results. A particularly careful study has been made by Kimura, Sako and Hayakawa, confirming both the linear relationship Eq. (197) and the fact that B, and therefore b as well as s*, are independent of temperature [ 1891. The electroclinic effect is intrinsically very fast. The torque reacting on the director, and giving rise to a change in tilt, is obtained by taking the functional derivative of the free energy with respect to the tilt,
150
5
The SrnA*-SmC* Transition and the Helical C* State
Equation (157) combined with Eq. (164) then gives
-re= a0 + be2 - c*P = a ( T - T,)8 + be3 - S* E = ae-XoEoc*28+ be3 - x ~ % c * E
As evident from the middle one of these equations, we confirm, by Eq. (194) that this torque vanishes when we approach T = T,, where 8 shows a diverging tendency and we can expect a critical slowing down in the response. The viscous torque T" counteracting any change in 8 is, by definition,
r
2)
=-ye-
ae at
where ye is the electroclinic or soft mode viscosity. In dynamic equilibrium the total torque r=T e + P is zero. This gives
a ( T - T , ) O + be3 + s * E +yo- ae = 0 at
(202)
This equation describes the dynamics of the electroclinic effect in a material where we have a lower-lying phase ( T < T,) characterized by a spontaneous tilt of the optic axis. If we divide all terms by s*, it can be written A 6 + Be3 +($)6 = E
(203)
This is the dynamic equation corresponding to Eq. (1 96). For small induced tilts, however, we skip the O3 term in Eq. (202). If we further put E=O, we will see how the optic axis relaxes back to its state along the layer normal from a beginning state of nonzero tilt. Equation (202) in this case simply reads
which is directly integrated to
with the characteristic time constant
151
5.9 The Electroclinic Effect
If instead a constant field E, is applied at t=O, the growth of 8 will be given by
which is integrated to
with the saturation value of 8 equal to
which of course also corresponds to Eq. (183). If we finally apply a square wave, where we reverse the polarity from -E, to +E, at t=O, the optic axis will change direction by an amount 29, according to O ( t ) = 8,(l - 2e-‘”)
(2 10)
corresponding to the initial and final values O(0)= -O0, 8(m) = +O,. We note that the response time given by Eq. (206) is independent of the applied electric field. In comparison, therefore, the response time due to dielectric, ferroelectric and electroclinic torque is characterized by different powers of E, as
The reason for this apparent peculiarity in the electroclinic effect is of course that if the field E is increased by a certain factor, the saturation value of the tilt is increased by the same factor. The angular velocity in the switching is doubled if we double the field, but the angle for the full swing is also doubled. Thus, electrooptically speaking, the speed of a certain transmission change in a modulator certainly does increase with increased field. But one mystery remains to be solved. How come, that the characteristic time z in Eq. (205) describing the relaxation back to equilibrium, is the same as the “response time” z in Eq. (208) describing the change of 8 under the influence of the electric field? Should not the relaxation back be much slower? This is what we are used to in liquid crystals: if a field is applied to a nematic, the director responds very quickly, but it relaxes back slowly when we take the field off. The answer is: the electric field exerts no torque at all on the director in the electroclinic action. Its effect is to shift the equilibrium direction in space of the optic axis. If the
152
5 The SmA*-SmC* Transition and the Helical C* State
field is high the offset is large and the angular velocity in the director motion towards the new equilibrium state will be high. Equation (207) describes this motion and tells that the rate of change of 6 is proportional to the angular difference between the initial and final director state. In fact Eqs. (204) and (207) are the same equation and can be written
ae - e-e, at
The difference is only that the final state So in Eq. (204) is zero, whereas in Eq. (207) 0, is the value given by the electric field according to Eq. (209). Note that in Eq. (214) the electric field does not appear at all. The electric field E does, however, exert a torque on the whole medium during the electroclinic action. This torque, by conservation of angular momentum, is opposite in sense to the torque exerted by the “thermodynamic force” on the director. As already mentioned in Chapter 1, the electroclinic effect was first announced as a “piezoelectric”effect [20], but shortly thereafter renamed. The similarities are interesting but the differences sufficient to characterize the electroclinic effect as a new dielectric effect. The variable 13in which the “distortion” takes place is here an angular variable, cf. Fig. 55. If E is reversed, P and 0 are reversed. But there is no converse effect. (One cannot choose k6’in the distortion variable to induce a P of a given sign.) The electroclinic tilt is connected with a mechanical distortion, a shrinking of the smectic layer thickness d. In the simplest model, assuming tilting rigid rods instead of molecules, thus exaggerating the effect, the thickness change is
hence 6d - E2, not -E and we have an electrostrictive, not a piezoelectric distortion. This is also consistent with the fact that no dilatation is possible in the A* phase, thus the effect is inherently unsymmetric. (Note that this is not so in the C* phase; with
**
7t
7 C’
j/
____
1 ~~
-,--. 6
!/
1
7 i
Figure 55. Analogy between the piezoelectric and the electroclinic effect. A translational distortion in the former corresponds to an angular distortion in the latter. If the sign of the applied field E is reversed, the sign of P and 0 is also reversed. But there is no converse effect in the electroclinic case. The mechanical deformation is electrostrictive, i.e. proportional to E2. In the smectic C* case there is also a component proportional to E.
153
5.9 The Electroclinic Effect
Figure 56. Modulation of the transmitted light intensity I for all electro-optic effects describing a tilt of the optic axis in the plane parallel to the cell plates (in-plane switching), for the case that the initial axis direction is along one of the two crossed polarizer directions. The curve is drawn for a cell thickness corresponding to /u2 plate condition, for which a tilt sweep of 45 degrees gives full modulation (from zero to 100 percent transmitted light). If this condition is not met, the degree of modulation will be reduced.
\I/ 450
-22.5'
0"
22.5"
450
a finite 6'dilatation as well as compression is allowed.) The electrostriction may be a problem in the practical use of the electroclinic effect, because the field-induced layer shrinking, at least sometimes, leads to the appearance of layer kink defects, which may be observed as striations in the texture [190]. However, very little is known today of how the electrostrictive properties vary between different materials. The electroclinic effect is the fastest of the useful electro-optic effects so far in liquid crystals, with a speed allowing MHz switching rates in repetitive operation. Its electro-optical performance is discussed and compared with other modes in the review of reference [ 1911.A number of device applications are discussed in reference [140]. The effect appears in all chiral orthogonal smectics but has been studied in a very restricted class of materials. It suffers so far, except from the fairly strong temperature sensitivity, in particular from a limitation in the value of induced tilt angle (12 to 15 degrees). This means that only a very small part of the typical V-shaped transmission curve can be utilized for a modulator using this kind of material, cf. Fig. 56. In this respect short pitch C* or N* materials are much better (exploiting the deformed helical and flexoelectro-optic mode, respectively) with a sweep in either direction of more than 22 degrees in the first and more than 30 degrees in the second case. However, new A* materials may very soon change the scene. The ideal material for the electroclinic effect would be a so-called de Vries compound [ 191a] in which the molecules have a large but unbiased tilt in the A phase (adding up just as in the short pitch C* or antiferroelectric case, to an optic axis along the layer normal) but a biased tilt in the C* phase. We will retum to such materials in Sec. 9.5 because they are also of eminent interest for high resolution displays. They are still hypothetical although seemed to have been found not very long ago [ 1921.Even if this turned out to be incorrect, nothing prevents, from first principles, that such materials would be possible and could be found one day. Such de Vries materials would be ideal even if they have to be somewhat slower (the torque counteracting tilt fluctuations would not be so strong for such compounds) because first of all the electroclinically induced tilt would be very large, namely up to the full value of unbiased individual tilt, and this for an applied field that can be expected to be very low. Furthermore the tilt biasing does not involve any electrostrictive effect or, at least this
154
5 The S d * - S m C * Transition and the Helical C* State
would be minimal. Finally the temperature sensitivity can be expected to be very low as it is in the absence of collective behavior, essentially T-' instead of (T- T J ' . The first theoretical analysis of the electroclinic effect was given in the original papers by Garoff and Meyer [21]. Shortly thereafter Michelson and Benguigui extended the analysis [43b], taking into account that the underlying C* phase is incommensurate (helicoidal). Very recently Beldon and Elston [ 192al have analyzed available experimental data and found that all presented phenomenological descriptions so far are inadequate because they neglect the elastic part of the energy involved in the compression of the smectic layers. Several other slightly different theoretical elaborations in combinations with experiments [ 192b], [ 1 9 2 ~ have 1 now contributed to a sufficient understanding, such that one would hope that this powerful effect, essentially unexploited so far, will find important applications in the next coming years.
5.10 The Deformed Helix Mode in Short Pitch Materials An interesting electro-optic effect utilizing the helical SmC* state of a short pitch material was presented in 1980 by Ostrovskij, Rabinovich and Chigrinov [193]. In this case the twisting power was so high that the pitch came below the wavelength of light. In such a densely twisted medium the light wave averages out the twist to feel an optic axis directed along the helical axis, i.e. in the direction of the layer normal z (Fig. 57). When an electric field is applied perpendicular to this axis it starts to partly untwist the helix and in this way perturb the spatial direction of the average. The result is a linear effect very similar to that of the electroclinic effect and the flexoelectro-optic effect: the angular deviation (0) of the effective optic axis is proportional to the applied field. Like the other two mentioned effects this deformedhelix mode (DHM) has no memory but a continuous grey scale. Though not as rapid as the electroclinic effect it has at least two distinctive advantages. First, (0) can attain much higher values than the soft-mode tilt. Second, the apparent birefringence (An) is quite low, allowing a larger cell thickness to be used and still matching the A/2 condition. The short-pitch materials have shown another highly interesting property. When they are used in the SSFLC mode, i.e. when a field sufficient for complete unwinding is applied and then reversed, in order to actuate the switctung between the two extreme cone positions, a new mode of surface-stabilization is found, in the limit Z S d , where 2 is the pitch or helical periodicity and d is the cell gap thickness. It may at first seem surprising that surface stabilization can be achieved not only for d l Z but also for d S Z , in fact the smaller the value of 2 the better. In this limit it could be a pinning phenomenon which would make it distinctly different from the
5.10 The Deformed Helix Mode in Short Pitch Materials
155
E-0
E>O
Figure 57. Partial untwisting of the helical axis in short-pitch materials (pitch in the range of 0.1 pm to 0.5 pm)has the effect of turning the averaged optic axis away from the layer normal as shown by the direction of the space averaged indicatrix. The twist also has the effect of averaging out part of the inherent large birefringence n,-no to a much smaller value (ne)-(no)(averaged over a length>pitch) (from Beresnev et al. [194]).
E
first case of surface-stabilization. In other words, whereas for Z > d the non-helical state is the equilibrium state, for Z e d it could be a metastable state. The helix would thus tend to rewind but, for topological reasons (the back-transition is defect-mediated) it may stay for a considerable time. However, as later shown by Shao, Zhuang, and Clark [ 194al also in this short pitch bistable mode, the surfaces rather than the stripe defect texture hinder the formation of the helix in the cell. In any case shortpitch materials are very interesting alternatives to pitch-compensated materials for display applications. Their demonstrated inconvenience so far has been the difficulty to align well, generally giving somewhat scattering textures. One further advantage of this effect is that the response time is only weakly temperature dependent. A general discussion of this and other features is found in chapter VI of reference [41]. The DHM has been developed further by the Roche and Philips groups and applied in active matrix displays with grey scale, cf references [195] and [196]. A comprehensive review of short pitch and other novel device structures with high application potential has been given by Funfschilling and Schadt [ 196al.
156
5 The SrnA*-SrnC* Transition and the Helical C* State
5.11 The Landau Expansion for the Helical C* State The Landau expansion Eq. (157) that we have used so far G = -1a 0 2 + -1b e 2 4
4
+-
2x0 EO
P 2 - c * P O - PE
(157)
refers to the non-helical C* state. Even if the non-helical state is somewhat special - the general case is one with a helix - and if it is true that for many purposes we may even regard the presence of the helix as a relatively small perturbation, it is now time to take the helix into account. The first thing we have to do then is to add a socalled Lifshitz invariant in the expansion. This invariant is a scalar
permitted by the local C, symmetry which describes the fact that the structure is modulated in space along the z direction. If in Fig. 49 we let the x direction be along the C2 axis, we see that the C, symmetry operation involves y+-y, z+-z, n,+-n, and n,+-iz,, of which the first and last do not interfere and the middle two leave Eq. (216) invariant. We remember from Fig. 29 that the cholesteric structure has infinitely many C, axes perpendicular to the helix axis. The expression Eq. (216) is therefore also an invariant for the N* phase. If q is the cholesteric wave vector, we write the director as
n = (cos 4, sin 4) = (cos qz, sin qz)
(217)
Inserting this in Eq. (216) we find that the Lifshitz invariant (which has a composition rule slightly reminiscent of angular momentum, cf. L, = x p y -y p,) in the cholesteric case has the value equal to q, the wave vector. In fact we can gain some familiarity with this invariant by starting from an expression we know quite well, the Oseen expression for the elastic free energy Eq. (96). Because of its symmetry, this expression cannot describe the cholesteric state of a nematic which lacks reflection symmetry and where the twisted state represents the lowest energy. Now, if there is a constant twist with wave vector q, the value of n. V x n in the K2, term equals -4. The expression Eq. (96) therefore has to be “renormalized” to
which, if there is no splay and bend present, becomes zero for n . V x n =-q. If we
5.11 The Landau Expansion for the Helical C* State
157
expand the square in the K22 term we can write the energy in the form GN*=GN+ K 2 2 q n .V x n
+ 21 K22q2 -
where G, corresponds to the energy expression Eq. (96) for a non-chiral nematic. Two new terms have appeared in the energy as a result of dropping the reflection symmetry. The first is a linear term which is the Lifshitz term, the second is a new elastic term quadratic in the wave vector. In fact, a simple check shows that
We could therefore alternatively write
or use
instead of Eq. (216). We may finally note that if we go from a right-handed to a lefthanded reference frame then the Lifshitz invariant (like angular momentum) changes sign because V x n does. We can also see this in Eq. (216) because in addition to the C2 operation the inversion involves that x -+ -x and nx +-nx. In the helical smectic C* case, again with q ( z ) = q z , the director components were stated in Eq. (141) n, = sine cos$ ny = sine sin$ nz = cose
Inserting this in Eq. (216) we find that the Lifshitz invariant equals
which is also often written
a9 . L=-sin
az
8
158
5
The SrnA*-SmC* Transition and the Helical C* State
for the more general case that the twist is not homogeneous. We may also note, that in the smectic C* case we can equally well write Eq. (216) in the form
i.e. using the two-dimensional c director, as the yz, component does not appear in L. Before we add the Lifshitz invariant to our Landau expansion Eq. (157) another consequence of the helix may be pointed out. It is the fact that, as we have already seen in Sec. 5.5, with a helical deformation we also have a flexoelectric contribution to P. This is opposite in sign to P, for e,>O, cf. Eq. (148), and proportional to the wave vector q. These two contributions must now be separated in the Landau expansion because we have to regard q as a new independent variable. In that case we not only have a coupling term -c* P8 but also one corresponding to the flexoelectric contribution which we could write eqP8. With this convention, Eq. (157) would be extended, for the helical C* case, to G = - a e 2 + - b 8 4 +P2-c*Pe+eqPo-a*qe2 2 4 2x0 €0 +-K2,q 1 2 +-K33q282 1 2 2 where the -PE term has been skipped since we are here only interested in the fieldfree case. For the Lifshitz invariant we have used Eq. (223) in the limit of small 8, but chosen the minus sign, corresponding to A* > 0, after comparing with Equations (219) and (222). We have also added terms corresponding to the elasticity of the helix, as in Eq. (219). However, as the smectic helix is a combined twist-bend deformation there has to be a term depending on the tilt angle as well as on q. By symmetry this dependence has to be -02 (or, actually proportional to the square of sin 28). But it is illustrative to examine these last terms in a less ad hoc way. Thus, if we go back to our “nematic” description, introduced in Sec. 4.10, we can write 1 1 Gc* = - K22 ( n . V x n + q ) 2+ - K33(nx V x n - B ) 2 (227) 2 2 1 1 = Gc + K22q2 + K33B2 + K22qn.V x n - K33B . n x V x n 2 2 ~
~
where G, corresponds to the expression (96) without splay term. As the spontane1 sin 2 8 =: q8 for small tilts, cf. Eq. (147), the third term on ous bend B is equal to -q 2 the right side is equal to the last term in Eq. (226). From Eq. (145) we further know that n x V x n =-q sin8cos8 z x n for the helical C* case, hence the last term in Eq. (227) can be written K3,Bq 8 for small tilts. With P=e,B for the flexoelectric
159
5.11 The Landau Expansion for the Helical C* State
contribution we write this term eqPB, with the abbreviation K33/eb=e.This term was already included in Eq. (226). Before continuing with the Landau expansion we note that with II.V x n =-q and n x V x n = B inserted in Eq. (227), the free energy in the helical C* state can be written in the form 1 Gc. = G(8, P ) + e q P 8 - -(K22 + K3,B2)q2 2
(228)
With the new terms in Eq. (226), minimizing with respect to P, 8 and q gives
ae
=a 0
+ b03 -c*
P + e q P - 2 1 " q O + K330q2= 0
(230)
The first of these yields
P = xo.q,(c* - e q ) 0
(232)
Comparing this with Eq. (148) which, for small tilts we can write
P = (Po - qe,) 8
(233)
we find that Po corresponds to ~~.q-,c*, equal to s* in Eq. (167), and that eb corresponds to Xo.q,e, where e is the coefficient in the Landau expansion. We will now insert P from Eq. (232) into Eq. (23 1) which gives
We have here introduced the renormalized coefficients
which contain the flexoelectric contributions. The wave vector is then obtained as
160
5 The SmA*-SmC*Transition and the Helical C* State
. . 0Cr)
T TC
Figure 58. The measured helical pitch Z at the transition C* +A*. The pitch should diverge according to Eq. (238).
For a non-chiral material, Af* = O (both A* and c* are chiral coefficients in Eq. (236), and hence q = 0. The pitch of the smectic helix Z equals 2 d 9 , thus Z=2n
K22 + Kf O2
a: e2
indicating that the pitch should diverge at the transition C* +A*, cf. Fig. 58. Experimentally, the situation is still not very clear regarding Z(T). While some measurements do indicate that a pure divergence exists if the layers are parallel to the cell plates, the majority of measurements give a result that the pitch goes through a maximum value slightly below the transition and attains a finite value at T= T,. If we write the pitch dependence
z=zo+gs’
(239)
there should be a saturation value Z0=27rKf/k.*fat low temperatures. Eq. (239) is more simply obtained from Eq. (228) on assuming that PI8 = const (= S*). 2, as well as < = ~ z K ~ ~ shows / A ; the balance between the non-chiral elastic coefficients ( K ) wanting to increase the pitch and the chiral coefficients (A*)wanting to twist the medium harder. When we finally insert the value of P from Eq. (232) into Eq. (230) we get
After skipping the solution 8= 0, we are left with a-
x0 E~ c * -~2 (A* - xoE~ ec*) q + ( K -xO c0e 2 )q2+ b e 2 * q+ Kf 42 + b e 2 = o
= a - ~ o ~ o C-2af *2
5.12
The Pikin-Indenbom Order Parameter
161
Together with Eq. (237) this leads to a complicated expression for O ( T ) , which we will not pursue further here. However, in the approximation that we take the low temperature limit for the wave vector
A:
40 =Kf
(corresponding to Z,) we see that Eq. (241) simplifies to a-
x0%c * -~Kfq02 + be2
(243)
which is reshaped in the usual way to
a ( T - T,) + be2
(244)
with a phase transition shift T , - T , equal to
A T = xo Eo c *2 +Kf 402
a
(245)
While AT is still expected to be generally small, it is certainly true that the contribution from unwinding the helix may raise the transition temperature more than the contribution from the chiral coupling, at least in the case of hardtwisted media like short-pitch SmC* where the pitch is smaller than the wavelength of light.
5.12 The Pikin-Indenbom Order Parameter A group theoretical symmetry analysis by Indenbom and his collaborators in Moscow [197, 1981 led, around 1977, to the introduction of a very attractive order parameter for the A*-C* transition which usually is referred to as the Pikin-Indenbom order parameter. Independently, Michelson, Cabib and Benguigui from similar symmetry considerations at the same time used what is essentially the same order parameter in their phenomenological theory of the A to C transition [43b], [43e] already referred to in Sec. 3.4. This order parameter is attractive because it presents, in the simplest form possible, the correct symmetry and a very lucid connection to the secondary order parameter P . It was adopted by the Ljubljana group around Blinc for the description of both static and dynamic properties of the C* phase [ 199-2041. The basic formalism has been described in particular detail by Pikin [40] and by Pikin and Osipov [41].
162
5 The SmA*-SmC*Transition and the Helical C* State
In the smectic C* phase, the layer normal z and the director n represent the only two natural directions which we can define. That is, z and n are the only natural vectors in the medium. Using these we now want to construct an order parameter which is a vector, is chiral, has C, symmetry, leads to a second order A* -C* transition and is proportional to 8 for small values of 8. We can do this by combining their scalar and vector products, multiplying one with the other,
We then obtain a vector in which the first factor is invariant, while the second changes sign on inversion. N thus has the two first properties which we required above. Performing the multiplication we find
which only has two components, hence is written
The n,ny and nzn, combinations do not change on inversion but the x and y directions do, thus we see that N changes sign on inversion. The angle between n and z is 8. Therefore, taking the absolute value in Eq. (246) 1 N = cos8sin8 = -sin28 2
(249)
We then recognize that this order parameter automatically reflects the C2 symmetry, cf. the discussions around Eqs. (131), (132) and Fig. 49, where we had to introduce this symmetry in an ad hoc way. Clearly N = 8 is valid for small tilts according to Eq. (249). Finally, we can check that N and -N describe the same state from Eq. (248), by virtue of the fact that n and -n describe the same state. Thus, if we make a Landau expansion in N , only even powers can appear and the transition will be one of second order. Before going on to this expansion we should, however, make the connection to the secondary order parameter P.For a positive material (P>O) the direction of P is the same as that of z x n. This means that, according to the construction of N in Eq. (246), N has the opposite direction. Thus, for a positive material, P -N. If we introduce the so-called tilt vector 6 by
-
5.12 The Pikin-Indenbom Order Parameter
163
Figure 59. N , n and z (or N , 5 and z ) form a right-handed system. N therefore points in the polarization direction for a negative material. In case of a positive material ( P > 0) the polarization direction corresponds to - N .
we can write the Pikin-Indenbom order parameter N in terms of the as
5 components
Consequently, the relation between the polarization components and the tilt vector components can be expressed, for a positive material, as
This is the relation between the secondary order parameter P=(P,, P,) and the primary order parameter N=(n,n,, -nzn,), when the latter is expressed in the tilt vecThe tilt vector 4 has the same direction as the c director, i.e. lies in tor g=(51, the direction of the tilt. It is identical to the order parameter Q introduced by Michelson, Cabib and Benguigui [43b], cf. Sec. 3.4. The order parameter N is very similar to but orthogonal to the tilt direction, hence collinear with P . It can also be regarded as an axial tilt vector. We may now proceed to ask what invariants we may form in N and P , and their 1 combinations, to include in the Landau expansion. N 2 and N 4 give Ta({:+{;)
c2).
1 and -b(5:+5;)2 4 same way the
corresponding to our earlier O2 and O4 terms, P 2 gives in the
(P:+P;)
~
term. But because P and N transform in the same
2x0 EO
way under the symmetry operations, cf. Eq. (252), their scalar product P . N will also be an invariant,
164
5 The SmA*-SmC* Transition and the Helical C* State
This corresponds to our previous PO and gives a term -c* (P,t2 - Py 5,) in the Landau expansion. In contrast to PO the form P, 52- P, directly shows the chiral character. Thus, this invariant changes sign when we change from aright- to a left-handed reference frame, which means that the optical antipode of a certain C* compound will have a polarization of opposite sign. Further, we have the Lifshitz invariant
corresponding to Eq. (225) and, because both 5 and P behave like the c director, also one in P.
ap, Px--P
aZ
ap, ~
az
(255)
Both have the same chiral character as the form in Eq. (253). Only one of these can be chosen, because they are linearly dependent, according to Eq. (252). This results in a Landau expansion
In this expansion the last two terms describe the flexoelectric (previously eqPO) energy and the elastic energy due to the helix. The wave vector q in these terms is and hidden in the deriatives The expression Eq. (256) is the “classical” Landau expansion in the formulation of Pikin and Indenbom. We now have to insert convenient expressions for tl, P, and Py.This can be done in several ways if we only correctly express the spatial relationship of 6 and P.For a positive material (Po>O), P is always 90 degrees ahead. If the director tilts out in the x direction, this will cause a polarization along y , symbolically expressed by
at1/& a<&.
t2,
and similarly, with tilt along y
5.12 The Pikin-Indenbom Order Parameter
165
Generally, with q = qz, and small tilts, =n7n,=cosOsinOcosrp= Ocosqz
t2= n,ny = cos0 sin0 sinq = 0 sinqz
(259)
corresponds to P, = -P sinqz
Py = Pcosqz Inserting these in Eq. (256) immediately gives G=-a0 2
+-be 4
(26 1) P 2 c * P O + e q P O i l * q 0 2 + K 1 3 j q 2 0 2 +2 xo &o 2
1 K,,q2 is lacking. While 2 minimizing with respect to P gives the same equilibrium value as before, dG/dq = 0 gives
This is the same as Eq. (226) except that here the term
eP6- A*02 + K3,q02 = 0
-
(262)
Since P - 0, all terms will be proportional to O2 and the &dependence simply cancels out. This means that the wave vector, according to this model, will simply be q = @Kf, equal to the low temperature value of Eq. (242). Thus the pitch Z is temperature independent, equal to Z,. This is of course in serious disagreement with experiment and the reason for this deficiency is the failure to recognize the twist component of the helix. We can learn from this that a good order parameter will not by itself take care of all the physics. But our previous Landau expansion is likewise far from perfect, even if it gives a more reasonable behavior for the pitch. We should therefore stop for a moment to contemplate how capable the theory is, so far, to reproduce the experimental facts. As we already discussed, the pitch has a diverging behavior as we approach T, (this is a priori natural, as no helix is allowed in the A* phase) but seems to change its behavior (within about 1 K from the transition) to decrease to a finite value at the transition. Another important fact is that the ratio P/O = s* experimentally is constant, as predicted by theory, up to about 1 K from the transition after which it decreases to a lower value [ 1331. A third fact, which we had no space to elaborate on, is that the dielectric response, according to the Landau expansion used so far, should exhibit a cusp at the transition [43e], which does not correspond to experimental facts, see page 374 of b k s and Blinc [242]. What is then the deficiency in the theory? Did we make an unjustified simplification when taking sin 0 = 0 in Eq. (259)? Probably not since, as the problem seems to
166
The SmA*-SmC* Transition and the Helical C* State
5
lie in the very vicinity of T,, where 8 is small, it would not do any good to go to the more accurate expressions valid for large values of 8. Then how do we go further? Should we extend our expansion to include O6 beyond 04,or P 4 beyond P2? Instead of such somewhat formal procedures, keks in 1984 argued physically in the following way [201]. Our Landau expansion so far has failed to take into account a nonchiral contribution which may be important. At the tilting transition a rotational bias appears, which has both a polar and a quadrupolar component, whether the material is chiral or not, which is quadrupolar in character, cf. Fig. 47. But if there is a strong quadrupolar order it will certainly contribute to increasing the polar order, once we remove the reflection symmetry. This leads to a coupling term in P and 8 which is biquadratic, -P202. If we now only consider those terms in the free energy which contain P and q, we can write
+ 1 K33q2 O2 - v* 404 ~
2
1 The biquadratic coupling term is - ~ l 2 ( P & P , e , ) ~ using the Pikin-Indenbom order parameter. The v* go4 term is the higher invariant -v*(c:+e:) which gives a coupling between tilt and twist. The P4 term has been added to stabilize the system. aG/aP=O gives
For small 8 we can skip O2 and P3 and get the same result as before, P=xoq,c*8 or P/e= s*. For large P we get 0 P 3 --PO
17
2
=o
(265)
or
p=e
K -
Thus the larger constant value of P/8 is f l q , which drops to s* near T, where the rotational bias is not so efficient, as we expect for small tilts. Finally, dG/aq gives
5.12 The Pikin-Indenbom Order Parameter
167
from which follows
If 0 is not too small, P / 0 in the second term is constant and q increases with increasing 8, i.e. Z decreases on lowering the temperature. Near T,, q increases again with increasing T since Pl0 decreases. An interesting feature is that the flexoelectric coefficient e has to be positive in order to yield a maximum in the Z ( T )curve. It is therefore conceivable that certain materials show a divergence, while others do not. In general the coupled equations have to be solved numerically but give, at least qualitatively, the correct temperature dependence for the different parameters. We would like to close this section with some remarks, in particular concerning the different order parameters. First of all we note a surprising thing about the very smart Pikin - Indenbom order parameter. It has been very important in the basic symmetry discussion, but it has never really been used as such in practice. Perhaps because it is too sophisticated in its simplicity. Everybody instead uses the tilt vector Lj as order parameter. But g=({,, is non-chiral, hence does not have the correct symmetry, whereas N = ( { , , -tl>is chiral by construction. Thus the tilt vector Lj should not be confused with the chiral vector (ninyr- n z n x ) . Having said this, however, it is clear that 5is easiest to handle, if we only combine it with (-P2, Pl).Note further that 5is not the projection of the director on the smectic plane, Lj#c=(c,, cv) = ( n x ,ny).Although 5 and c are parallel they are not identical: Lj=n,c. The 8 dependence of 5 and c is different but for small tilts the difference between the vectors is not big. After having defined 6, the first thing we did in using it in Eq. (256) was in fact to go to the small tilt limit. Thus in practice we have not used anything else than the c director as order parameter. Which means that we have in essence, all the time, in different shapes, used the two component complex order parameter from the beginning of Chap. 5 , cf. Eq. (125)
c2)
As primary and secondary order parameters we have thus used two two-component vectors lying in the smectic plane, perpendicular to the layer normal. The number of degrees of freedom, i.e. the number of components, is evidently more important than subtle differences in symmetry, if the order parameters are used with sound physical arguments.
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6 Electrooptics in the Surface-Stabilized State 6.1 The Linear Electrooptic Effect The basic geometry of a surface-stabilized ferroelectric liquid crystal is illustrated in Fig. 60. The director n (local average of the molecular axes) may be identified with the optic axis (we disregard the small biaxiality) and is tilted at the angle 8 away from the layer normal (z). In the figure the director is momentarily tilted in the xz plane with a tilt angle 13,which is fixed for a fixed temperature. Its only freedom of motion is therefore described by the azimuthal variable cp indicated in the xy plane. If the phase is chiral, no reflections are symmetry operations and therefore a local polarization P is permitted in the (positive or negative) y direction. If electroded glass plates are mounted as in the figure, such that the layer normal z runs between them in the middle, we see that this most peculiar geometry with P perpendicular to the
Figure 60. Local symmetry of the smectic C* phase. The direction of P is shown for a positive material (P>O); by definition, one in which z, n, and P make a right-handed system. If electrodes are applied as shown, P will follow the direction of the electric field generated between them.
Figure 61. Originally assumed FLC cell structure. The field E is applied between the electroded glass plates e. The smectic layers are perpendicular to these glass plates and have a layer thickness d(T) of the order of 5 nm, which is temperature-dependent. The thickness of the smectic slab is 1-2 pm and equal to L, the distance between the plates. The shaded area illustrates the achievable switching angle of the optic axis. The symbols below illustrate that the polarization vector can be actively switched in both directions (1 -2). Strangely enough, this simple geometry, which corresponds to the original concepts of [93-953, has only been realized in recent times, thanks to a devoted synthetic effort by chemists over the last 15 years. Before these materials are coming into use (not yet commercial) the layer structure is much more complicated. These problems are discussed in Sec. 8.
L
170
6 Electrooptics in the Surface-Stabilized State
optic axis is ideal for a display: by applying the electric field across the thin slice, we can move the optic axis back and forth between two positions in the plane of the slice. This so-called in-plane switching cannot be achieved in a solid ferroelectric (nor in a nematic) without the use of a much more complicated electrode configuration. A typical cell structure is shown in Fig. 61. It illustrates the fact that we have two equivalent stable states; this is the reason for the symmetric bistability. The coupling is linear in the electric field and described by a torque
driving n (which is sterically coupled to P ) both ways on reversal of the field direction and, as we will see below, leading to a microsecond response speed according to
dominating over any quadratic (dielectric) terms at low fields. NON-TRANSMITPINGSTATE
TRANSMITTING STATE
ANALYZER
GLASS
GLASS
4
nc
GLASS
POLARIZER
L\l
Figure 62. Comparison between twisted nematic and FLC switching geometry.
6. I
The Linear Electrooptic Effect
171
Figure 63. The presentation of the first Canon monochronie A4 size panel in 1988 not only demonstrated the high resolution capability based on speed and bistability, it was also a definite breakthrough in LCD optics, demonstrating the essentially hemispherical viewing angle connected with in-plane switching.
In Fig. 62 this linear electrooptic effect is compared with the quadratic effect controlling the state of a twisted nematic device. With a torque -E2 the latter effect is insensitive to the direction of E and is driven electrically only in the direction from twisted to untwisted state, whereas it has to relax back elastically in the reverse direction. The difference is also striking optically. The important feature of the in-plane switching is that there is very little azimuthal variation in brightness and color, because the optic axis is always reasonably parallel to the cell plane. If the optic axis pointed out of this plane (see Fig. 63), which it often does in the field-on state of a twisted nematic, a considerable variation with viewing angle will occur, including reversal of contrast when the display is viewed along angular directions nearly coinciding with the optic axis. The basic FLC optical effect is also illustrated in Fig. 64. In-plane switching has another advantage than the superior viewing angle, which may be equally important but has so far passed fairly unnoticed. This lies in its color neutrality. In the top part of Fig. 62, it is seen that when intermediate values of the electric field are applied to a twisted nematic in order to continuously vary the transmission, the tilt of the optic axis and thus the birefringence are changed at the same time. Thus the hue is influenced at the same time as the grey level in a twisted nematic, which makes the color rendition less perfect in a display where the colors are
Figure 64. The basic FLC optic effect is illustrated by the 1988 Canon monochrome prototype in which the front polarizer sheet has been removed; this also makes the drivers along the bottom visible. The optical contrast between the two polarization states is given by the small hand-held polarizer to the lower left
172
6 Electrooptics in the Surface-Stabilized State
generated either by red-green- blue filter triads or by sequential color illumination. In-plane switching, on the other hand, in principle allows perfect separation of color and grey shades.
6.2 The Quadratic Torque When we apply an electric field across an FLC cell there will always be a dielectric torque acting on the director, in addition to the ferroelectric torque. This is of course the same torque as is present in all liquid crystals and in particular in nematics, but its effect will be slightly different here than it is in a nematic. This is because in the smectic C phase the tilt angle 8 is a “hard” variable, as earlier pointed out. It is rigid in the sense that it is hardly affected at all by an electric field. This is certainly true for a nonchiral smectic C and also for the chiral C*, except in the immediate vicinity of TC*A*,where we may not neglect the electroclinic effect. Hence we will consider the tilt angle 8 uninfluenced by the electric field, and this then only controls the phase variable a, of Fig. 60. In order to appreciate the difference, it is illustrative to start with the effect in a nematic. We will thus study the torque on the director due to an electric field E that is not collinear with n , as illustrated in Fig. 65. If we introduce the dielectric anisotropy
we may write the dielectric displacement D=%EE caused by the field in the form of the two contributions
~ E ,the ~ first and adding it to the second term gives Subtracting E ~ E from
This gives a contribution to the free energy (which is to be minimized for the case
Figure 65. The dielectric torque will diminish the angle a between the director n and the electric field for A&>O.
6.2 The Quadratic Torque
173
of a fixed applied voltage) - j D . d E = - E~
A e j ( n . E ) n . d E - E 0 E I j E .dE
= - 1 E~ AE(n. E )2 - 1 E ~ E ~ E ' ~
2
2
(275)
We discard then the second term as it does not depend on the orientation of n and write the relevant free energy contribution 1 E~ A & ( n .E ) 2= - 1 E" A&E2cos2a G =- 2 2
(276)
where a is the angle between n and E . With positive dielectric anisotropy, A&>0, we see that this energy is minimized when n is parallel to E . If AE 0. The dieelectric torque P i s obtained by taking the functional derivative of G with respect to a
which yields
re= - -1~ ~ A ~ E ~ s i n 2 a 2 The viscous torque r"is always counteracting the motion, and can be written
where fi is the so-called twist viscosity counteracting changes in a. In dynamic equilibrium the total torque is zero. From this we finally obtain the dynamic equation for the local director reorientation in an externally applied field E
-1& " A & E 2 s i n 2 a + y, -d=a0 2 dt Equation (280)describes the dielectric response to an electric field. It is easily integrated, especially if we look at the case of very small deviations of the director from the field direction (a),for which we have sin2a=2a and
174
6 Electrooptics in the Surface-Stabilized State
In this case, a will approach zero simply as
where a,is the initial angular deviation in Fig. 65 and z is given by
r=-
Y1
E2 The characteristic response time is thus inversely proportional to the square of the applied field. What we have derived so far is valid for a nematic, where there are no particular restrictions to the molecular motion. In smectic media there will be such restrictions, and therefore Eqs. (278) and (280) will be modified in a specific way. In any liquid crystal the induced polarization Pi caused by an applied electric field is obtained fromD=q,&E=EOE+Pias P i = D - ~ Eor, from Eq. (274)
Pi = EOA&(n. E ) n + E ~ E-~E E~ =E(
+ q,A&(n. E ) n
E ~ -1)%E
(284)
The first term represents a component parallel to the electric field and does not therefore give any contribution to the torque P x E . From the second we get
r " = P ix E = A&&& . E ) ( n x E )
(285)
In the nematic case we would have, with n .E = E cos a and In x E I = -E sina
where a i s the angle between n and E , which is equivalent to Eq. (278). In the smectic case the rigidity of the tilt angle 8 prevents any other change than in the phase variable p. Consequently, we have to take the z component of the torque working on pand causing the conical motion of the director. From Fig. 51 and Eq. (140) we find, with the external field E applied in the y direction, i.e., E = (0,E, 0)
n . E = E sin0 sinp
(287)
n x E = (-nZE, 0, n, E) = E(-cos0, 0, sin0 coscp)
(288)
and
from which Eq. (285) takes the shape
r"= A&GE2(-sinOcos0sinp, 0, sin28 sinp cosp)
6.3 Switching Dynamics
175
The z component of this torque is thus 1 Tf = A&&"E2 sin20 sinq coscp = -A&% E2 sin20 sin2q 2
(290)
Compared to the nematic case we get a formally similar expression, except for the appearance of the sin20 factor, but the involved angle cp has a completely different meaning than a. The azimuthal angle cp is not the angle between the director and the field, but in the SmC* case the angle between the spontaneous polarization and the field, as seen when we imagine the director to move out in direction cp in Fig. 60. When P is parallel to E (cp = 0 ) there is no ferroelectric torque, but also no dielectric torque on the director.
6.3 Switching Dynamics We are now able to write down the equation of motion for the director moving on the smectic cone and being subjected to ferroelectric and dielectric torques, as well as the elastically transmitted torques K V 2 q ,which tend to make n spatially uniform. Disregarding the inertial term I @(of negligible importance) and the elastic torques from the surfaces (a far more serious omission), we can write the dynamic director equation
acp = - P E sinq + -1 y at
2
E2 sin20 sin2cp + K V 2 q
For E=O this reduces to
which has the form of the equation for heat conduction or diffusion and means that spatial gradients in tp will damp out smoothly with a rate determining quantity Kly, corresponding to the heat conductivity or diffusion coefficient, respectively. With regard to the signs, we note that (see Fig. 60) cp is the angle between the tilt plane (or c director) and x, and therefore also between P and E . The torque P x E = - P E sin9 is consequently in the negative z direction for q > O (see Fig. 66).
Figure 66. If the xz plane is parallel to the electrodes and the electric field is applied in the positive y direction, the ferroelectric torque will turn the tilt plane (given by q ~ )in the negative cp direction.
X
176
6 Electrooptics in the Surface-Stabilized State
Equation (29 1) describes the overdamped approach towards equilibrium given by the field E, together with the material parameters P , 0, y,, K , and A&.Let us divide all terms by PE ~
Y9 a ‘=-sincp+ PE at
1 AEE,,E2 2 sin2esin2q+-V K P PE
2 cp
(293)
Introducing a characteristic time Z= -, yv a characteristic length 5 = PE dimensionless quantity
acp = -sincp zat
x = EOAEE’we can write Eq. (293) as ~
1 sin - 2 + __ 8 sin2q + 5”“ V2 cp
211
(294)
Evidently we can write Eq. (294) in an invariant form using the variables t’=t/z a -- ii -,a etc., Eq. 294 takes the dimensionand -= and rf=r/t. For, with at at 5 less shape
a
a ax
ax
The new variables are scaled with
and Z=c Yp
PE
(297)
which set the basic length and time scales for the space-time behavior of cp (r, t). In addition, the behavior is governed by the dimensionless parameter
The characteristic time t determines the dynamic response towards the equilibrium state when we apply an electric field. If we put in the reasonable values y,= 50 CP (=0.05 Nsm-2) (see Fig. 67), P= 10 nCcmP2,and E= 10 Vpm-’, we get a z value of about 5 ps. The characteristic length 5 expresses the balance between elastic and electric torques. An external field will align P along the field direction except in a
g yj{ 177
6.3 Switching Dynamics
Figure 67. (a, b) An external electric field
E perpendicular to the smectic layer norma1 is rotating the local polarization P as indicated in (a), forcing P to end up parallel to E . The corresponding reorientation of the optic axis takes place through the variable q around a conical surface, i.e., the tilt cone, with no change in tilt 8. In contrast to a pure 8 motion (corresponding to viscosity yo, or the equivalent twist viscosity y, in the nematic case), the rp reorientation has a significant component around the long molecular axis, which leads to low effective viscosity. (c) Arrhenius plot of the viscosity y, in the nematic phase, as well as soft mode and Goldstone mode viscosities yo and y, in the srnectic C* phase of a two-component mixture (from Pozhidayev et al. [148]).
-z
= YN
2
T@
(4
(b)
2.8T
80°C
3.2 1 0 3 1 ~ ( ~ )
3.0 50°C
thin layer of the order of 6,where a deviating boundary condition will be able to resist the torque of the field. As this layer thickness falls off according to 6 E-1'2, it will be much smaller than the sample thickness d for sufficiently high fields. In fact, it also has to be much smaller than the wavelength of light in order for the sample to be optically homogeneous. Only then will Eq. (297) also apply for the optical response time. It is easy to check that this is the case. With the P and E values already used and with K=5 x lo-'' N, we find from Eq. (296) that 4 is less than a nanometer. Finally, the parameter expresses the balance between the ferroelectric and the dielectric torque. As it appears in the dielectric term of Eq. (295) in the combination sin28/2x, which is = 0 . 1 5 / 2 ~for 8-22', would have to be less than about 0.3 in order for us to have to keep this term. With the same P and E values as before, this requires a I A&1 value as high as 24. We will therefore skip this term for the moment, but return to this question later.
-
x
x
178
6 Electrooptics in the Surface-Stabilized State
6.4 The Scaling Law for the Cone Mode Viscosity Figure 67 gives an indication that, for the motion on the cone, the effective viscosity ought to depend on the tilt angle. Evidently, for small values of tilt it approaches the limiting case, where all the motion is around the long inertial axis, which must have the lowest possible viscosity. We also note from the experimental data of that figure that the cone mode viscosity, yp, is even lower than the nematic viscosity at a higher temperature, which at first may seem puzzling. We will look at the viscosity in a more general way in Sec. 6.9, but here only try to illuminate this question. We will treat the viscosity in the simplest way possible, regarding it for a moment as a scalar. To a first approximation, any electric torque will excite a motion around the @preserving cone, i.e., around the z axis of Fig. 60. The essential component of the viscous torque will therefore be I',, which is related to drpldt (E@). We can thus write for the cone motion -rz
= Yp@
(299)
For a general rotational motion of the director n , the relation between the viscous torque and the viscosity has the form
Talung the z component we get
expressed by the two-dimensional c-director, which is the projection of n onto the plane of the smectic layer. Their relation means that c=n sin8 = sine (see Fig. 51). However
Hence
-r, = ysin28@
(303)
Comparing Eqs. (299) and (303) we obtain the important scaling law with respect to the tilt angle for the viscosity y, describing the motion around the cone (Goldstone mode) yQ = y sin26
(304)
6.5
Simple Solutions of the Director Equation of Motion
179
The index cp refers to the cp variable, and the index-free ycorresponds to the nematic twist viscosity. Equation (304) means that the cone mode viscosity is lower than the standard nematic viscosity (yq< It also seems to indicate that the viscosity y, tends to zero at the transition T + TCA(0 +O). This, of course, cannot be strictly true, but we will wait till Sec. 6.9 to see what actually happens, see Eq. (347).
x).
6.5 Simple Solutions of the Director Equation of Motion If we discregard both the dielectric and elastic terms in Eq. (291), the dynamics is described by the simple equation
y -+PEsincp=O acp 9
at
(305)
For small deviations from the equilibrium state, sincp = cp and Eq. (305) is integrated directly to
where
z=__ 79 and qo is the angle between E and P at time t=O. The response
PE time zis inversely proportional to the field, instead of being inversely proportional to the square of the field, as in the dielectric case. Generally speaking, this means that the FLC electrooptic effect will be faster than normal dielectric effects at low voltages, and also that these dielectric effects, at least in principle, will ultimately become faster when we go to very high applied fields. The unique advantage of FLC switching is, however, that it is not only very fast at moderate fields, but that it switches equally fast in both directions; a feature that cannot be achieved if the coupling to the field is quadratic. If q is not small in Eq. (305), we change the variable to half the angle v cos -) q and easily find the analytical solution (sincp = 2 sin 2 2
with the same z= y9 This can alternatively be expressed as PE ~
'
180
6 Electrooptics in the Surface-Stabilized State
6.6 Electrooptic Measurements Equations (307) and (308) are a solution to Eq. (305) if the electric field is constant with time. The expression is therefore also useful when the field is a square wave. Such a field is commonly used for polarization reversal measurements when switching between + P , and -P, states. In order to switch from -P, to + P , we have to supply the charge 2 P, to every unit area, i.e., 2 PsA over the sample, if its area is A. dQ The reversal of the vector P, thus gives rise to an electric current pulse i(t)= dt through the curcuit in which the sample cell is connected. This current is given by the time derivative of the charge due to P , on the electrodes of area A . When P , makes an angle cp with E , the polarization charge on the electrodes is Q = PsA cos cp. The electric current contribution due to the reversal of P, is thus given by
Substituting the time derivative of cp from Eq. (305), the polarization reversal current is directly obtained as
When cp passes 90 O during the polarization reversal, i ( t ) in Eq. (3 10) goes through a maximum. The shape of the function i(t) is that of a peak. T h s is shown in Fig. 68 where Eqs. (308) and (310) have been plotted for the case cpo= 179 O. Figure 68. The polarization reversal v(?)according to Eq. (304) for a half-period of an applied square wave and the resulting current pulse i ( f ) .The current peak appears roughly at r=5T and the reversal is complete after twice that time. The latter part of the rp curve corresponds to the simple exponential decay of Eq. (32). The optical transmission is directly related to v ( t ) .If the state at f = O is one of extinction, the transmission will increase to its maximum value, but in a steeper way than /&p/atl (from Hermann [ 1361).
6.6 Electrooptic Measurements
181
By measuring the area under the polarization reversal peak, we obtain the total charge transferred and can thus determine the value of the spontaneous polarization P,. The time for the appearance of the current peak then allows a determination of the viscosity y , The curves in Fig. 68 have been traced for the case where P makes an angle of 1 O relative to the surface layer normal, corresponding to the same pretilt of the director out of the surface. This pretilt also has to be included as a parameter determining the electrooptic properties. On the other hand, the integration of Eq. (305) giving the current peak of Fig. 68 can only give a crude estimation of the electrooptic parameters, because the equation does not contain any term describing the surface elastic torques. The value of the method lies therefore in the fact that it is simple and rapid. The same can be said about the alternative formalism of [ 1371, which has proven useful for routine estimations of electrooptic parameters, at least for comparative purposes. In [ 1371 a hypothetical ad hoc surface torque of the form Kcos cp has been added to the ferroelectric torque in Eq. (305). With only these two torques present, the equation can still be easily integrated in a closed form. However, the chosen form of the elastic torque means that it cannot describe the properties of a bistable cell. Numerous other approaches have been proposed in order to describe the switching dynamics, for instance by neglecting the surface elastic torques, but including the dielectric torque using the correct representation of the dielectric tensor [ 1381. Common to most of them is the fact that they describe the shape of the polarization reversal current and the electrooptic transmission curve reasonably well. They are therefore easy to fit to experimental data, but may give values to the electrooptic parameters that may deviate within an order of magnitude. (An important exception to this is the polarization value P,, which is essentially insensitive to the model used because it is achieved by integration of the current peak.) Hence no model put forward so far accounts generally for the switching dynamics, and there is a high degree of arbitrariness in the assumptions behind all of them. Experiments in which the parameters in the equation of motion are measured may be carried out with an electrooptic experimental setup, usually based on a polarizing microscope. In the setup, the sample is placed between crossed polarizers and an electric field E(t) is applied across the cell. The optical and electric responses of the sample to the field are then measured. The light intensities Z(t) through the crossed polarizers and the sample are measured by a photodiode after the analyzer. The electric current i ( t ) through the sample is obtained by measuring the voltage u ( t ) across a resistance R connected in series with the liquid crystal cell (see Fig. 69). All three signals E ( t ) ,Z ( t ) , and i ( t ) are fed into an oscilloscope so that the switching dynamics of the liquid crystal cell can be monitored. The oscilloscope is connected to a computer so that the waveforms can be stored and analyzed in detail. From the optical transmission, the tilt angle and the optical response time can be measured. From the electric response the spontaneous (and induced) polarization can be measured, as well as the response time, as dis-
182
6 Electrooptics in the Surface-Stabilized State OPTICAL
D m C O lR
-
I
DIGITAL OSCILLOSCOPE I
BcwmER I
ANALYZER
Optimum optical setting: y-22.5"
Figure 69. (a) The experimental setup for electrooptic measurements. The sample is placed between crossed polarizers. (b) The RC-circuit used in the switching experiments. The current response from the sample is measured across the resistor R (from Hermann [136]).
cussed earlier. If we also include the S m A * phase (the electroclinic effect), the coefficients of the Landau expansion of the free energy density may be derived. Estimations of the viscosities for the 9-and &motions may also be obtained from the electrooptic measurements. In Fig. 69 we see how the quasi-bookshelf (QBS) smectic constitutes a retarder with a switchable optic axis. In the smectic A* phase this axis can be switched continously, but with small amplitude. The same setup is also used to study the DHM effect in short pitch chiral smectics and the flexoelectrooptic effect in the cholesteric phase, in both of which we achieve much higher tilts. Inserted between crossed polarizers, the retarder transmits the intensity
where Y is the angle between the polarizer direction and the projection of the optic axis in the sample plane, A n is the birefringence, A is the wavelength of light in a vacuum, and d is the thickness of the sample. The function Z(Y) is shown in Fig. 70. A very important case is the lambda-half condition given by d A n= A/2.A cell with this value of d A n will permit maximum transmission and flip the plane of polarization by an angle 2 Y. Since the position of the optic axis, i.e., the angle Y, may be altered by means of the electric field, we have electrooptic modulation. As seen from Eq. (3 11) and from Fig. 70, the minimum and maximum transmission is achieved
183
6.6 Electrooptic Measurements
1.0
h / 2 -plate condition
Linear regime
h/S -plate condition \
0.4
-I
v U
0.2
0.0 0
22,5
45
90
67,5
(deg) Figure 70. Transmitted light intensity Z(Y) for a linear optical retarder with switchable optic axis placed between crossed polarizers. Y is the angle between the polarizer direction and the switchable optic axis, and maximum transmitted light intensity is achieved for Y = 4 5 ' combined with a sample thickness fulfilling the L / 2 condition. A smectic C* in QBS geometry and tilt angle 0 = 22.5' could ideally be switched from Y=O to Y = 4 5 in a discontinuous way. DHM and flexoelectrooptic materials could be switched in a continuous way. I(") is proportional to Y in a region around Y = 22.5 which is the linear regime (from Rudquist 11261). O,
for Y = 0 and Y =45 ",respectively. If we adjust the polarizer to correspond to the first state for one sign of E across an SSFLC sample, reversal of sign would switch the polarization to give the second state, if 2 8 = 45". For natural grey-scale materials (electroclinic, DHM, flexoelectrooptic,antiferroelectric),the electrooptic response to a field-controlled optic axis cell between crossed polarizers is modelled in Fig. 7 1. Here Y = Yo+@(E),where Yois the position of the optic axis for E=O and @(E) is the field-induced tilt of the optic axis. The tilt angle in the SmC* phase can be measured by the application of a square wave of very low frequency (0.2- 1 Hz),so that the switching can be observed directly in the microscope. By turning the angular turntable of the microscope, the sample is set in such an angular position that one of the extreme tilt angle positions, say -O0,coincides with the polarizer (or analyzer), giving Z=O. When the liquid crystal is then switched to +$, Z becomes #O. The sample is now rotated so as to make Z=O again, corresponding to +Qo coinciding with the polarizer (or analyzer). It follows that the sample has been rotated through 2 Go, whence the tilt angle is immediately obtained. The induced tilt angle in the SmA* phase is usually small, and the above method becomes difficult to implement. Instead, the tilt can be obtained [ 1401by measuring
184
6 Electrooptics in the Surface-Stabilized State Electrooptic signal / Transmittedlight intensity
4
VI I
Figure 71. Electrooptic modulation due to the field-controlled position of the optic axis between crossed polarizers. I) The optic axis is swinging around Y,=O giving an optical signal with double frequency compared to the input signal. 11) The linear regime. When the optic axis is swinging around Y0=22.5O, the vaFor a liquid crystal L/2 cell with quasi-bookshelf nation of the transmitted intensity is proportional to 0. (QBS) structure, it follows from Eq. (311) that the intensity 1 varies linearly with @ if 0 is small (M,=1/2 + 20(E)), which, for instance, is the case if the electroclinic effect in the SmA* phase is exploited. As electroclinictilt angles are still quite small, certain other materials, though slower, have to be used for high modulation depths.111) Characteristicresponse for @ ( E ) exceeding k22.5 (from Rudquist [ 1261). O
the peak-to-peak value AZpp in the optical response, from the relation
1 arc sin __ NPP Oind = 4
I0
which is valid for small tilt angles. On applying a square wave electric field in the SmC* phase the electric response has the shape of an exponential decay, on which the current peak i(t) (see Fig. 68) is superposed. The exponential decay comes from the capacitive discharge of the liquid crystal1 cell itself, since it constitutes a parallel-plate capacitor with a dielectric medium in between. The electrooptic response i(t)on application of a triangular wave electric field to a bookshelf-oriented liquid crystal sample in the SmC* and SmA* phases is shown as it appears on the oscilloscope screen in Fig. 72, for the case of crossed polarizers and the angular setting Yo=22.5'. Applying such a triangular wave
6.7 Optical Anisotropy and Biaxiality
Figure 72. The electrooptic response when applying a triangular wave over the cell in the SmC* and the SmA* phases. The ohmic contribution due to ionic conduction has been disregarded.
TLU-r A(a)
SmC*phase
I(t)
185
n r
i(t)
(b) SmA* phase
in the SmC* phase gives a response with the shape of a square wave on which the polarization reversal current peak is superposed. The square wave background is due to the liquid crystal cell being a capacitor; an RC-circuit as shown in Fig. 69b delivers the time derivative of the input. The time derivative of a triangular wave is a square wave, since a triangular wave consists of a constant slope which periodically changes sign. The optical response Z ( t ) also almost exhibits a square wave shape when applying a triangular wave, due to the threshold of the two stable states of up and down polarization corresponding to the two extreme angular positions -8 and +8 of the optic axis (the director) in the plane of the cell. The change in the optical transmission coincides in time with the polarization reversal current peak, which occurs just after the electric field changes sign. In the SmC* phase, the optical transmission change in the course of the switching is influenced by the fact that the molecules move out of the plane of the glass plates, since they move on a cone. They thus make some angle c(t) with the plane of the glass plates at time t. The refractive index seen by the extraordinary ray n,(n depends on this angle. The birefringence thus changes during switching according to A n [ ( ( t ) ]=n,[c(t)]-no.Aftercompletion of the switching, the molecules are again in the plane of the glass plates and <=O, so that the birefringence is the same for the two extreme positions -8 and +0 of the director. When raising the temperature to that of the SmA* phase, the polarization reversal current peak gradually diminishes. In the SmA* phase near the transition to SmC*, a current peak can still be observed, originating from the induced polarization due to the electroclinic effect. On application of a triangular wave, the change of the optical response can most clearly be observed: it changes from a squarewave-like curve to a triangular curve, which is in phase with the applied electric field. This very clearly illustrates the change from the nonlinear cone switching of the SmC* phase to the linear switching of the SmA* phase due to the electroclinic effect.
6.7 Optical Anisotropy and Biaxiality The smectic A phase is optically positive uniaxial. The director and the optic axis are in the direction of highest refractive index, nl,-nl=An>O. At the tilting transition
186
6 Electrooptics in the Surface-Stabilized State
SmA+ SmC, the phase becomes positive biaxial, with nll+n3 and nL splitting in two, i.e., nI+nl, n2. The E ellipsoid is prolate and
n3 > n2 > n ,
(313)
A n > 0 , an>O
(3 14)
or
where we use, with some ..esitation, the abbreviations
An = n3 - n l
(315)
an = n2 - n 1
(316)
The hesitation is due to the fact that the refractive index is not a tensor property and is only used for the (very important) representation of the index ellipsoid called the optical indicatrix, which is related to the dielectric tensor by the connections
E l = n 2l , % = n 22 , ~
~2
=
n
~
(317)
between the principal axis values of the refractive index and the principal components of the dielectric tensor. For orthorhombic, monoclinic, and triclinic symmetry the indicatrix is a triaxial ellipsoid. The orthorhombic system has three orthogonal twofold rotation axes. This means that the indicatrix, representing the optical properties, must have the same symmetry axes (Neumann principle). Therefore the three principal axes of the indicatrix coincide with the three crystallographic axes and are fixed in space, whatever the wavelength. This is not so for the monoclinic symmetry represented by the smectic C . Because the symmetry element of the structure must always be present in the property (again the Neumann principle), the crystallographic C 2 axis perpendicular to the tilt plane is now the twofold axis of the indicatrix and the E tensor, but no other axes are fixed. This means that there is ambiguity in the direction of E~ along the director, (which means ambiguity in the director as a concept) as well as in the direction perpendicular to both the “director” ( E ~ and ) the tilt axis (~2). Only the latter is fixed in space, and therefore only E~ can be regarded as fundamental in our . it is well known choice of the principal axes of the dielectric tensor ( E ~E, ~ E, ~ ) Now, (or n2 and n,), that the triaxial character, i.e., the difference between ~2 and E ~ - is very small at optical frequencies. The quantity &is universally called the biaxiality, although this name would have been more appropriate for the degree of splitting of the two optic axes as a result of the tilt. Thus optically spealung we may still roughly consider the smectic C phase as uniaxial, with an optic axis (director) tilting out a certain angle 8 from the layer
a&=
6.8 The Effects of Dielectric Biaxiality
187
normal. However, as has recently been pointed out by Giesselmann et al. [ 1411, due to the ambiguity of the and E~ directions, the optical tilt angle must be expected to be a function of the wavelength of the probing light. As they were able to measure, the blue follows the tilt of the core, whereas the red has a mixture of core and tail contributions. The optical tilt is thus higher for blue than for red light, and the extinction position depends on the color. While this phenomenon of dispersion in the optic axis is not large enough to create problems in display applications, it is highly interesting in itself and may be complemented by the following general observation regarding dispersion. At low frequencies, for instance, lo5 Hz, the dielectric anisotropy A&= c3- E~ is often negative, corresponding to a negatively biaxial material. As it is positively biaxial at optical frequencies Hz), the optical indicatrix must change shape from prolate to oblate in the frequency region in between, which means, in particular, that at some frequency the E tensor must become isotropic.
6.8 The Effects of Dielectric Biaxiality At lower frequencies we can no longer treat the smectic C phase as uniaxial. Therefore our treatment of dielectric effects in Sec. 6.2 was an oversimplification, valid only at low values of tilt. In general, we now have to distinguish between the anisotropy, A&= E~ - E [ , as well as the biaxiality, E ~ E - ~ both , of which can attain important nonzero values typically down to -2 for A& and up to +3 or even higher for JE. The latter parameter has acquired special importance in recent years, due to a special addressing method for FLC displays, which combines the effects of ferroelectric and dielectric torques. As for the background, it was successively discovered in a series of investigations by the Boulder group between 1984 and 1987 that the smectic layers are generally tilted and, moreover, form a so-called chevron structure, according to Fig. 73, rather than a bookshelf structure. The reason for the chevron formation is the effort made by the smectic to fill up the space given by the cell, in spite of the layer thickness shrinking as the temperature is lowered through the SmA-SmC transition and further down into the SmC phase. The chevron geometry (to which we will return at length in Chapter 8) reduces the effective switching angle, which no longer corresponds to the optimum of 28=45 O, thereby reducing the brightness-contrast ratio (we have in the figure assumed that the memorized states have zero pretilt, i.e., that they are lying in the surface). This can, to some extent, be remedied by a different surface coating requiring a high pretilt. Another way of ameliorating the optical properties is to take advantage not only of the ferroelectric torque (-E) exerted on the molecules, but also of the dielectric torque (-E2). This torque always tries to turn the highest value of the permittivity along the direction of the electric field. The tilted smectic has the three principal E values in the directions shown in Fig. 73, E , along the chosen tilt direction, q along
a&=
188
6 Electrooptics in the Surface-Stabilized State
. .. . .
/
2
I
F
\
Figure 73. The simple bookshelf structure w i L essentially zero pretilt would lead to ide:.- Dptical conditions for materials with 2 Bequal to 45“. In reality the smectic layers adopt a much less favorable chevron structure. This decreases the effective switching angle and leads to memorized P states that are not in the direction of the field (a, b). A convenient multiplexing waveform scheme together with a properly chosen value of the material’s biaxiality & may enhance the field-on contrast relative to the memorized (surface-stabilized) value (c), by utilizing the dielectric torque from the pulses continuously applied on the columns (after [ 142, 1431).
a&
the local polarization, and E~ along the director. If is large and positive, it is seen that the dielectric torque exerted by the field will actually lift up the director away from the surface-stabilized state along the cone surface to an optically more favor-
6.8 The Effects of Dielectric Biaxiality
189
able state, thus increasing the effective switching angle. The necessary electric field for this action can be provided by the data pulses acting continuously as AC signals on the columns. This addressing method thus uses the ferroelectric torque in the switching pulse to force the director from one side to the other between the surfacestabilized states (this could not be done by the dielectric torque, because it is insensitive to the sign of E ) , after which high frequency AC pulses will keep the director dynamically in the corresponding extreme cone state. This AC enhanced contrast mode can ameliorate the achievable contrast in cases where the memorized director positions give an insufficient switching angle, which is the case in the so-called C2 chevron structures (see Sec. 9.2). It requires specially engineered materials with a high value of Its main drawback is the requirement of a relatively high voltage (to increase the dielectric torque relative to the ferroelectric torque), which also increases the power consumption of the device. The AC contrast enhancement is often called “AC stabilization” or “HF (high frequency) stabilization”. A possible inconvenience of this usage is that it may lead to a misunderstanding and confusion with surface-stabilization, which it does not replace. The dielectric contrast enhancement works on a surface-stabilized structure. There is no conflict in the concepts; the mechanisms rather work together. In Eq. (291) we derived an expression for the director equation of motion with dielectric and ferroelectric torques included. If the origin of the dielectric torque is in the dielectric biaxiality, the equation (with the elastic term slupped) will be the closely analogous one
a&.
(3 18) In the present case, where the layers are tilted by the angle 6 relative to E, as in Fig. 73a, the effective applied field along the layer will be E cos6, thus slightly modifying the equation to y
a(P = - PE cos6 sinq - &oa&E2cos2 6 sin9 cosq
-
tp at
(3 19)
In addition to the characteristic time z and length 6, we had earlier introduced the dimensionless parameter describing the balance between ferroelectric and dielectric torques. Its character appears even more clearly than in Eq. (298) if we write it as PE
x
x = &ga&E2
If we again replace E by E cos 6, our new P
X=
EgaEECOS6
x will be
6 Electrooptics in the Surface-Stabilized State
SWITCHING
30
-
20
--- _ _ _ - _-
SWITCHING
' ,
.i ... .. .
.
.: .. .:
I
8
I
I
10
.
v.min . :i
-
I I
20
10
30
40
.: .. I
.i ..
.. .. * . :. .i
* :
SWITCHING
.
-
. . . .
I
50
.
60 70 80 90 100
Pulse voltage, V, I V
I\ log 7
log v
From
Figure 74. (a) Typical pulse switching characteristic for a material with parameters giving ferroelectric and dielectric torques of comparable size; (b) Possible routes for engineering materials to minimize both rminand Vmin(from [1441).
x and z= PEyqcos 6 we can form a characteristic time
and from
x and Ed (cos 6 factors canceling in Ed), a characteristic voltage
6.9 The Viscosity of the Rotational Modes in the Smectic C Phase
191
The values of the parameters z, and V, are decisive in situations where the ferroelectric and dielectric torques are of the same order of magnitude, and they play an important role in the electrooptics of FLC displays addressed in a family of modes having the characteristics that there is a minimum in the switching time at a certain voltage (see Fig. 74a), because the ferroelectric and dielectric torques are nearly balancing. This means that the switching stops both on reducing the voltage (due to insufficient ferroelectric torque) and on increasing it (due to the rapidly increasing dielectric torque, which blocks the motion, i.e., it increases the delay time for the switching to take off). The existence of a minimum in the switching time when dielectric torques become important was discovered by Xue et al. [138] in 1987, but seems to have appeared in novel addressing schemes some years later in the British national FLC collaboration within the JOERS/Alvey program. It turns out [I451 that the minimum switching time depends on yq, and P according to Eq. (322), and that this minimum occurs for a voltage that depends on and P according to Eq. (323). At present, considerable effort is given to reduce both z, and V,. Figure 74 b indicates how this could be done: either (1) by lowering the viscosor (2) by increasing both and P by a ity y, whilst increasing the biaxiality relatively large amount.
a&,
a&,
a&
a&
6.9 The Viscosity of the Rotational Modes in the Smectic C Phase In Sec. 6.4 we derived the scaling law for the cone mode viscosity with respect to the tilt angle 8.In this section we want to penetrate a little deeper into the understanding of the viscosities relevant to the electrooptic switching dynamics. Let us therefore first review the previous result from a new perspective. With 8 = const, an electric torque will induce a cone motion around the z axis (see Fig. 75 a). We can describe this motion in different ways. If we choose to use the angular velocity Ci, of the c director, that is, with respect to the z axis, then we have to relate it to the torque component rl
In Fig. 75b we have instead illustrated the general relation, according to the definition of viscosity, between the angular velocity h of the director n and the counteracting viscous torque
r”
192
6 Electrooptics in the Surface-Stabilized State
Figure 75. (a) The variable cp describes the motion of the c director around the layer normal. (b) The relation between the rate of change of the director n, the angular velocity h, and the (always counteracting) viscous torque
r.
(b)
As a is the coordinate describing the motion of the head of the n arrow (on a unit sphere), Oi = n x k and the relation can be written
-r"= ynxri
(326)
We have used this before in Eq. (300). Several things should now be pointed out. The difference between y, and yis, in principle, one of pure geometry. Nevertheless, it is not an artifact and the scaling law of y, with respect to 0 has a real significance. In a nematic we cannot physically distinguish between the 8 motion and the (0 motion on the unit sphere. Therefore y in Eq. (325) is the nematic viscosity. In a smectic C, on the other hand, we have to distinguish between these motions, because the director is moving under the constraint of constant 8. As is evident from Fig. 75 b, however, this motion is not only counteracted by a torque &; instead I'must have nonzero x and y components as well. We might also add that in general we have to distinguish between the 8 and cp motions as soon as we go to a smectic phase, either S m A or SmC. For instance, we have a soft mode viscosity yo in these phases, which has no counterpart in a nematic. Let us now continue from Eq. (326). With sin 8 cos cp n = sinOsincp
G,
1
(327)
193
6.9 The Viscosity of the Rotational Modes in the Smectic C Phase
we have -sin 0 sin cp@ ri = [ i n o c o s p j
1
hence -sin 0 cos 0 cos cp -P= y @ sin0 cosOsincp (sin20
]
(329)
Therefore the z component is
A comparison of Eq. (330) with Eq. (324) then gives y, = y sin2 0
(331)
which is the same as Eq. (304) (y is a nematic viscosity). We may note at this point that Eq. (33 1) is invariant under 0 + 0 + x,as it has to be (corresponding to n +-n). We have still made no progress beyond Eq. (304) in the sense that we are stuck with the unphysical result y, +0 for 0 +0. The nonzero torque components and in Eq. (329) mean that the cone motion cannot take place without torques exerting a tilting action on the director. With 0 constant, these have to be taken up by the layers and in turn counteracted by external torques to the sample. We also note that since 0 = const in Eq. (327), we could just as well have worked with the two-dimensional c director from the beginning
r'
c = (C,,C,) =
0 sin 0 cos cp . sin 0 sin q
r,
(332)
to obtain the result of Eq. (329). One of the drastic oversimplifications made so far has been in treating the viscosity as a scalar, whereas in reality it is a tensor of rank 4. In the hydrodynamics, the stresses oijare components of a second rank tensor, related to the velocity strains Vk.1
(333) in which q is thus a tensor of rank 4 just like the elastic constants C q k [ for solid ma-
194
6
Electrooptics in the Surface-Stabilized State
terials. for which Hooke’s law is written
As is well known, however, the number 34=8 1 of the possibly independent components is here reduced by symmetry, according to
giving only 36 independent components. They therefore permit a reduced representation using only two indices
which has the advantage that we can write them down in a two-dimensional array on paper, but has the disadvantage that cpvno longer transforms like a tensor. The same is valid for the Oseen elastic constants and the flexoelectric coefficients in liquid crystals, which are also tensors of rank 4, but are always written in a reduced representation, Kv and e i j , respectively, where Kij and ev cannot be treated as tensors. The viscosity tensor in liquid crystals is a particular example of a fourth rank tensor. The viscous torque T i s supposed to be linear in the time derivative of the director and in the velocity gradients, with the viscosities as proportionality constants. This gives five independent viscosities in uniaxial ( 12 in biaxial) nematics, the same number in smectics A, but 20 independent components in the smectic C phase [ 1461. With nine independent elastic constants and as many flexoelectric coefficients, this phase is certainly extraordinary in its complexity. Nevertheless, the viscous torque can be divided in the rotational torque and the shearing torque due to macroscopic flow. When we study the electrooptic switching of SSFLC structures, we deal with pure rotations for which macroscopic flow is thus assumed to be absent. Even if this is only an approximation, because the cone motion leads to velocity gradients and backflow, it will give us an important and most valuable description because of the fact that the rotational viscosity has a uniquely simple tensor representation. If we go back to the Eq. (325) defining the viscosity, both the torque and the angular velocity are represented by axial vectors or pseudovectors. This is because they actually have no directional property at all (as polar vectors have), but are instead connected to a surface in space within which a rotation can only be related to a (perpendicular) direction by a mere convention like the direction for the advance of a screw, “right-hand rule”, etc. In fact they are tensors of second rank which are antisymmetric, which means that
6.9 The Viscosity of the Rotational Modes in the Smectic C Phase
195
Figure 76. The principal axes 1,2, and 3 of the viscosity tensor. In the right part of the figure is illustrated how the distribution function around the director by necessity becomes biaxial, as soon as we have a nonzero tilt 8.
with the consequence that they have (in the case of three-dimensional space) only three independent components and therefore can be written as vectors. As y in Eq. (325) connects two second rank tensors, it is itself a fourth rank tensor like other viscosities, but due to the vector representation of I‘and a,the rotational viscosity can be given the very simple representation of a second rank tensor. Thus in cases where we exclude translational motions, we can write yas
Unfortunately, no similar simple representation exists for the tensor components related to macroscopic flow. We will now finally treat y not as a scalar, but as a tensor in this most simple way. In Eq. (338) y is already written in the “molecular” frame of reference in which it is diagonalized, as illustrated in Fig. 76, with the principal axes 1, 2, and 3 . Because the C , axis perpendicular to the tilt plane has to appear in the property (Neumann principle - we use it here even for a dynamic parameter), this has to be the direction for K. As for the other directions, there are no compelling arguments, but a natural choice for a second principal axis is along the director. We take this as the 3 direction. The remaining axis 1 is then in the tilt plane, perpendicular to n. The 3 axis is special because, along n, the rotation is supposed to be characterized by a very low viscosity. In other words, we assume that the eigenvalue y7 is very small, i.e., Y44Y19
75.
Starting from the lab frame x,y, z, y is diagonalized by a similarity transformation
where T is the rotation matrix cos6 0 -sin6 1 0 sin6 0 case
j
(340)
196
6 Electrooptics in the Surface-Stabilized State
corresponding to the clockwise tilt 8 (in the coordinate transformation the rotation angle 8 is therefore counted as negative) around the y (2) axis, and T-' is the inverse or reciprocal matrix to T; in this case (because rotation matrices are orthogonal) T is simply equal to its transpose matrix ?.(i.e., gj=TU).Hence we obtain the viscosity components in the lab frame as
Y=T?T-'
(341)
which gives
y1 0
cos8 0 -sin8
y=[o 1 0 sin8 0 cos8
0 y2 0 0 y3
][o 0
ylcos28+y3sin2e =[ 0
(yl - y3)sin3cos8
cos8 -sin8
0 sin8 1 0 0 cos8
o
(yl -y3)sinBcose
Y2
0 y1sin2 8 + y3 cos2 8
o
(342)
Assuming now the motion to be with fixed 8,the angular velocity a is
a = (0, 0, $)
(343)
and Eq. (325) takes the form
[I] [
y1sin3 cos3@
-r= y
=
o
(yl sin2 3 + y3 cos28)U,
(344)
the z component of which is
-r;=(y, sin28 + y3cos28)(1,
(345)
Comparison with Eq. (324) now gives
y,(e) = yl sin2B+ y3 cos28
(346)
This relation illustrates first of all that the cp motion is a rotation that occurs simultaneously around the 1 axis and the 3 axis, and it reflects how the rotational velocities add vectorially, such that when 8 gets smaller and smaller the effective rotation takes place increasingly around the long axis of the molecule. For B = O we have
6.9 The Viscosity of the Rotational Modes in the Smectic C Phase
197
whereas for 8 = 7d2 we would find (formally) that
(5)
Yq
(348)
= Y1
corresponding to the rotational motion in a two-dimensional nematic. There is another reason than Eq. (347) not to neglect y3: as illustrated to the right of Fig. 76, a full swing of the director around the z axis in fact involves a simultaneous full rotation around the 3 axis. Hence these motions are actually intrinsically coupled, and conceptually belong together. If, on the other hand, we had neglected the small y3 in Eq. (342), the viscosity tensor would have taken the form
!
y1cos2 e
o
y1sinecoso
y1sinocose
o
y1sin2e
(349)
which would have given the same incomplete equation (304) for y, as before. The first experimental determination of the cone mode viscosity was made by Kuczynski [147], based on a simple and elegant analysis of the director response to an AC field. Pyroelectric methods have been used by the Russian school [136,148], for instance, in the measurements illustrated in Fig. 67. Later measurements were normally performed [ 1491 by standard electrooptic methods, as outlined earlier in this book. The methodology is well described by Escher et al. [ 1501,who also, like Carlsson and iekS [ 1511, give a valuable contribution to the discussion about the nature of the viscosity. The measured viscosities are not the principal components y,, y2, y3 in the molecular frame, but y, and f i , which we can refer to the lab frame, as well as y3,for which we discuss the measurement method in Sec. 7. As y2 refers to a motion described by the tilt angle 6, we will from now on denote it yo and often call it the soft mode viscosity (sometimes therefore denoted ys).It corresponds more or less, like yl, to a nematic viscosity (see Fig. 67 for experimental support of this fact). The cone mode viscosity y,, which is the smallest viscosity for any electrooptically active mode, will correspondingly often be called the Goldstone mode viscosity (sometimes denoted yG).As yl and ye must be of the same order of magnitude, Eq. (304) makes us expect y,
198
6 Electrooptics in the Surface-Stabilized State
and yj, as drawn for the uniaxial SmA (SmA*) phase.
As yl and y2 cannot be distinguished in the nematic, the twist viscosity corresponds both to y, and y2(yeye)in the smectic C . The meanings of and ‘/I are illustrated in Fig. 77. If we symbolize the director by a solid object, as we have done to the right of Fig. 76, we see that the coordinates 8, 9,and y, the latter describing the rotation are three independent angular cooraround the long axis, thus corresponding to dinates which are required for specifying the orientation of a rigid body. Except for the sign conventions (which are unimportant here), they thus correspond to the three Eulerian angles. More specifically, 9 is the precession angle, 8 the nutation angle, and y is the angle of eigenrotation. The three viscosities y,, ye, and determine the rotational dynamics of the liquid crystal. Only y, can easily be determined by a standard electrooptic measurement. All of them can, however, be determined by dielectric spectroscopy. We will turn to their measurement in Sec. 7.
x,
7 Dielectric Spectroscopy: To Find the and 6 Tensor Components 7.1 Viscosities of Rotational Modes Using dielectric relaxation spectroscopy, it is possible to determine the values of the rotational viscosity tensor corresponding to the three Euler angles in the chiral smectic C and A phases. These viscosity coefficients (yo, y,, y,) are active in the tilt fluctuations (the soft mode), the phase fluctuations (the Goldstone mode), and the molecular reorientation around the long axis of the molecules, respectively. It is found, as expected, that these coefficients obey the inequality yo > y, 9 y,. While the temperature dependence of yo and y, in the SmC* phase can be described by a normal Arrhenius law as long as the tilt variation is small, the temperature dependence of '/e in the SmA* phase can only be modeled by an empirical relation of the form ye = A eEa'kT+ y j ( T - T&". At the smectic A* to SmC* transition, the soft mode viscosity in the smectic A* phase is found to be larger than the Goldstone mode viscosity in the SmC* phase by one order of magnitude. The corresponding activation energies can also be determined from the temperature dependence of the three viscosity coefficients in the smectic A* and SmC* phases. The viscosity coefficient connected with the molecular reorientation around the long axis of the molecules is not involved in electrooptic effects, but is of significant interest for our understanding of the delicate relation between the molecular structure and molecular dynamics.
7.2 The Viscosity of the Collective Modes Different modes in the dipolar fluctuations characterize the dielectric behavior of chiral smectics. As the fluctuations in the primary order parameter, for instance represented by the tilt vector 1 ' 2 1 = n, n, = - sin 2 8 sin rp 2
= nz n, = - sin28 cosrp
e2
(350)
are coupled to P,some excitations will appear as collective modes. Dipolar fluctuations which do not couple to 5 give rise to non-collective modes. From the temperature and frequency dependence of these contributions, we can calculate the viscosities related to different motions.
200
7
Dielectric Spectroscopy: To Find the ?and E^ Tensor Components
In the standard description of the dielectric properties of the chiral tilted smectics worked out by Carlsson et al. [152], four independent modes are predicted. In the smectic C* the collective excitations are the soft mode and the Goldstone mode. In the SmA* phase the only collective relaxation is the soft mode. Two high frequency modes are connected to noncollective fluctuations of the polarization predicted by the theory. These two modes become a single noncollective mode in the smectic A* phase. There is no consensus [ 1531 as yet as to whether these polarization modes really exist. Investigations of the temperature dependence of the relaxation frequency for the rotation around the long axis show that it is a single Cole-Cole relaxation on both sides of the phase transition between smectic A* and smectic C* [154]. The distribution parameter a of the Cole - Cole function is temperature-dependent and increases linearly (a= a,T +b,) with temperature. The proportionality constant uT increases abruptly at the smectic A* to SmC* transition. This fact points to the complexity of the relaxations in the smectic C* phase. The dielectric contribution of each of these modes has been worked out using the extended Landau free energy expansion of Sec. 5.1 I. The static electric response of each mode is obtained by minimizing the free energy in the presence of an electric field. The relaxation frequency of the fluctuationsin the order parameter is obtained by means of the Landau-Khalatnikov equations, which control the order parameter dynamics. The lunetic coefficients I-, and of these equations have the dimensions of inverse viscosity. The soft mode and Goldstone mode are assumed to have the same kinetic coefficient, I-,, which, however, does not mean that they have the same measured value of viscosity. The viscosity of the soft mode in the smectic A* phase can be written, according to [152]
r,
where c* is the coupling coefficient between tilt and polarization, E, is the dielectric contribution from the soft mode, andiqis the relaxation frequency of the soft mode in the smectic A* phase. In order to calculate the viscosity, a measure of the coefficient c* is needed. It can be extracted from the tilt angle dependence on the applied field, from Sec. 2.5.7
The corresponding expression for the soft mode viscosity in the SmC* phase is not equally simple, and reads (353)
20 1
7.2 The Viscosity of the Collective Modes
where b,/b, is the following combination of parameters from the expansion (263)
c*-eq+2OP8
(354)
xo Eo This ratio can be obtained by fitting the experimentally measured values of E, versus temperature. In the frame of this Landau theory, the soft mode dielectric constant is E,
T:(
=
(355)
4E()a(T-T,)+Eoxo
where the term &dc0is a cut-off parameter. Assuming the ratio b,/b, is independent of temperature, 76 can be calculated in the smectic C* phase. This approximation is valid deep down in the smectic C* phase, where P , 8, and q have their equilibrium values. The Goldstone mode viscosity can be calculated [ 1521according to the expression
Yp=--(-)
1
1
P
2
(356)
47ce0 E G ~ G 8 where P is the polarization, 8 is the tilt angle, and E, andf, are the dielectric constant and relaxation frequency of the Goldstone mode. The polarization and the tilt angle need to be measured in order to calculate the viscosity. Figure 78 shows a plot of the soft mode and Goldstone mode rotational viscosities measured on either side of the phase transition between the smectic A* and 10
Y (Nsec/m*)
1 Figure 78. The soft mode and Goldstone mode rotational viscosities as a function of temperature. The material is mixture KU-100 synthesized at Seoul University, Korea (courtessy of Prof. Kim Yong Bai) (from Buivydas [ 1551).
20
30
40 50 60 Temperature (“C)
70
80
202
7
Dielectric Spectroscopy: To Find the f and d Tensor Components
SmC*. It can be seen that, except in the vicinity of the phase transition, the viscosity ye seems to connect fairly well between the two phases. The activation energies of these two processes are, however, different. This result may be compared to results obtained by Pozhidayev et al. [148], referred to in Fig. 67. They performed measurements of ye beginning in the chiral nematic phase of a liquid crystal mixture with corresponding measurements in the SmC* phase, and have shown the viscosity values on an Arrhenius plot for the N* and SmC* phases. Despite missing data of ye in the smectic A* phase they extrapolate the N* values of ye down to the smectic C* phase and get a reasonably smooth fit. Their measurements also show that ye is larger than yp, and this is universally the case.
7.3 The Viscosity of the Noncollective Modes The viscosity of the noncollective modes can be estimated, as shown by Gouda [ 1541 and Buivydas [ 1551, using the original Debye theory of dielectric response. Assuming that the Stokes law is valid, the viscosity for the long axis rotation around the short axis scales as LK3, where 1 is the length of the long axis. For this transverse rotation we have
kT
Yt =~ I T L3 ~ &
(357)
wheref, is the relaxation frequency of the molecular rotation around the short axis and L is the length of the molecule, as shown in Fig. 79. The viscosity related to the molecular rotation around the long axis has the corresponding expression
kT = 87c2fid 3
(358)
wherefi is the relaxation frequency and d is the cross diameter of the molecule, cf. Fig. 79.
CdJ
Figure 79. The molecular dimensions d and 1. The depicted lengthlwidth ratio corresponds to the substance LCl (see later).
7.3 The Viscosity of the Noncollective Modes
203
The substance used in these measurements is a three-ring compound synthesized by Nippon Mining Inc., which we may call LCI. It has the structural formula 0
and the phase sequence I 105.0 SmA* 68.1 SmC* 8.0 Cr. The relaxation frequencies fl (of the order of 3 MHz) and ft (of the order of 110 MHz) are now measured as a function of temperature, the former in quasihomeotropic alignment (layers parallel to the cell plates), the latter in QBS geometry. With a cross section d taken from a simple molecular model to be about 4.5 P\, the corresponding function y l ( T )is obtained from (358). Now, the length and width of a molecule used in Eqs. (357) and (358)are not selfevident values. Therefore the Landau description may be used to calculate the viscosity related to the molecular rotation around the long axis. In the smectic C* phase, the viscosity is then expressed by
In order to determine the same viscosity in the smectic A* phase, the normalization factor (P/B)2should be preserved, otherwise the viscosity will show a jump at T, [ 1.541. Thus the corresponding equation will read
where (P/B)Tcis the limiting value of the P/B ratio measured at the transition. The viscosity yl calculated by the Landau theory and the y,evaluated by using the Debye formulas are shown for comparison in Fig. 80. We see that they agree fairly well in the limit of small tilt, thus particularly in the A* phase. The non-Arrhenius increase at higher tilts is here an artifact inherent in the Landau method. In order to calculate y,(T) we also need a value for the molecular length 1. This can be obtained by the requirement that we can have only one single viscosity in the isotropic phase. Combining (357) and (358) gives
204
Dielectric Spectroscopy: To Find the
7
Smectic A*
I
"
' 1
' ' '
2,91
1
' J "
2,95
i and 6 Tensor Components
Smectic C*
'
1 ' 3,02
"
299
1
'
1 1
3,06
1000/T (K-')
1
1 N O
'
motions evaluated by two different methods. The substance is LC1 (from Buivydas [155]).
0 0 0
2,62
2,70
2,79 2,87 2,95 1000/T (K-3
3,04
Figure 81. The three components of the viscosity tensor measured by dielectric relaxation spectroscopy. The substance is LC1 (from Buivydas [155]).
On putting %= we find
From this I is obtained as 15 A. When we check the molecular length by searching the lowest energy conformation of LC1 in a simple computer program, we see that this value actually corresponds to the length of the core. We could of course also have slightly adjusted the input value of d such that the Debye and Landau viscosity values coincide exactly in the A* phase. This is found to occur for d=4.7 A, corresponding to I = 16 A. The results from all viscosity measurements on the substrate LC 1 are summarized in Fig. 81. (longitudinal) and The noncollective rotational viscosities are denoted by (transverse). The collective motions have the corresponding designations (for soft
205
7.4 The Viscosity yQ from Electrooptic Measurements
mode) and yG (for Goldstone mode). Equivalently, '/e and y, could also have been used. It is remarkable that the Goldstone mode viscosity is almost as low as the viscosity around the long axis of the molecule. As pointed out earlier, this is explained by the fact that a large component of the angular velocity is around this axis. The soft mode viscosity lies much higher and, as the results show, is essentially the same as the viscosity for the corresponding individual molecular rotation.
7.4 The Viscosity yq from Electrooptic Measurements In the smectic C* phase, the rotational viscosity y, can be estimated by observing the polarization reversal or the electrooptic properties of the cell, as described in Sec. 7.6. The estimation may, for instance, be based on the approximation mentioned there, using the elastic torque [ 1371
pS Esincp- Kcoscp+y,@=O
(363)
Solving this equation and fitting the observed polarization reversal process to the solution, the viscosity can be estimated and compared to the one measured by means of dielectric relaxation spectroscopy. The first method is fast, but less accurate. The polarization reversal measurement is a large signal method requiring full switching of the liquid crystal, and therefore cannot be expected to coincide with the much more precise results from the low signal dielectric relaxation spectroscopy measurements. Basically, a y, value from the polarization reversal technique involves spurious contributions of elastic effects due to the surfaces of the sample. However, the values from the polarization reversal method do not differ by more than typically a factor of two, and the difference tends to be smaller at higher temperatures (see Fig. 82).
2,o
13
_' ' ' '
:
I '
' '''
I
'
u r
1,o :
0,o
8
9
9
I
I
I
I
~
O
r
/
-
:
-0,5 L from dielectric - measurements I " ' I -l,o ' 50,O 53,O 56,O " "
1-
from polarization reversal
0
(Nsec/m2)0,5 1 '
Figure 82. Comparative study of the y, rotational viscosity measured by the polarization reversal technique based on Eq. (363) and dielectric relaxation spectroscopy. The material is LC 1 (from Buivydas [155]).
I
Smectic C"
0
In Y$
0
"
'
'
"
'
I
"
59,O
Temperature ("C)
'
"
If '
62,O
,p,
65,O
206
7
Dielectric Spectroscopy: To Find the f and E^ Tensor Components
7.5 The Dielectric Permittivity Tensor Phases having biaxial symmetry (tilted smectic phases) exhibit dielectric biaxiality in particular. At frequencies of 1 MHz and below, the biaxiality becomes important and critically influences the electrooptic switching behavior of the SmC* phase. It is therefore important to be able to measure the biaxiality at these frequencies. Being a symmetrical second rank tensor, the dielectric permittivity can always be diagonalized in a proper frame and described by three components along the principal directions. The three principal values can then be expressed by a single subscript and can be determined by three independent measurements performed at three different orientations of the director relative to the measuring electric field. In practice, this may not be that straightforward, for instance, in the case of a thick sample (measured on a scale of the pitch) of a smectic C* when the helix is not quenched by the surfaces. In such a case the local frame of the E tensor is rotating helically through the sample.
7.6 The Case of Helical Smectic C* Structures In the molecular frame (Fig. 83) we write the tensor E as
r"
i = 0 E2 O 0
0
:]
(364)
E3
As for y in Sec. 6.9, we may consider 2 being brought to diagonal form (starting from the laboratory frame) by a similarity transformation
where T is a rotation matrix and T-' is the inverse or reciprocal matrix to T, in this case identical to its transpose.
Figure 83. The definitions of the molecular axes and components of the dielectric tensor.
7.6 The Case of Helical Smectic C* Structures
207
In Fig. 83 we may imagine a laboratory frame with the z axis along the layer normal (thus along gllin the figure), such that we have turned the director n a certain angle 8 around the y direction (along E~ in the figure). The x direction is along c. In this laboratory frame xyz, the permittivity tensor is then given by
with the rotation matrix To cose 0 -sin8 To=( 0 1 0 sin6 0 cos6
1
(367)
Performing the multiplication - the same as in Eq. (342) - we find cos2 6 + E~ sin2 8 E(e) = 0 ( E ] - E~ sin 8 cos e
o
- E ~ sine ) cose
0
E2
o
sin2 e + E~ cos2 6
which can be written in the form
E2
-A& sin8 cos6 0
0
Ell
0 E(6)=
-A& sin8 c o d
by introducing the dielectric anisotropy A&=E~ -
1
(369)
and the abbreviations
and E~~= E ,
sin2e + E~ cos26
(371)
If we now further turn the director by an angle cp around the z direction, the permittivity tensor will change to
i'n" :]
As this rotation is performed counterclock-wise (positive direction) in the xy plane, the rotation matrix is now instead
T v = -sincp
si;cp cosq 0
(373)
208
7 Dielectric Spectroscopy: To Find the f and E^ Tensor Components
This gives
or
I
el cos2 cp + e2 sin2 cp
E(8,cp) = -(&I - ~ 2 sincp ) coscp -A&sincpcosOcoscp
c1sin2 cp + E~ cos2 cp
- A&sin 8 cose cos cp AEsin8 cos8 sincp
AEsinOcosOsincp
Ell
-( E ~ c 2 )sin cp cos cp
(375) In the helical smectic C*, the tilt 8 is a constant while the phase angle cp is a function of the position along the layer normal, cp = cp (z).Averaging over one pitch, with (cos2cp) = (sin2 cp) = 1 2
(376)
(sincp) = (coscp) = (sincpcoscp) = 0
(377)
and
E
simplifies to
with the new abbreviations ( E ~=)
( E cos2 ~
e + E2 + E3 sin2 e)
and
+ E~ cos2e
(E,,) = c1sin2e
Equation (378) reveals that the helical state of the smectic C* is effectively uniaxial, if the pitch is sufficiently short to allow the averaging procedure. The shorter and more perfect the helix, the more completely the local biaxiality is averaged out (both dielectric and optical). This is consistent with the empirical evidence, showing almost perfect uniaxiality for helical samples of short pitch materials. Before we can use our derived relations to determine E ~ ~2 , and E ~ we , have to perform one more refinement. The simple measurement geometry where smectic layers are parallel to the glass plates is easy to obtain in practice. On the other hand, it is very
7.7
Three Sample Geometries
209
hard to turn them into being perfectly perpendicular to the plates; normally, the result is that they meet the surface with a slight inclination angle 6. To account for this case we have to see how the tensor ( ~ ( 8 transforms ))~ when we turn the layer normal by a certain angle &around the x axis. We perform the rotation clockwise (in this case the direction does not matter), corresponding to
0 sin6
cos6
and giving
In the case where we assume a helix-free sample and fully addressed or memorized director states lying on extreme ends of the smectic cone, so that we might put cp equal to 0 or n: in Eq. (375), would instead have given the tensor components cos2 8 + ~ sin2 3 8 A&sin 8 cos 8sin 6
A~sin8cos8sin6 cos2 6 + (ql)sin2 6 -A~sin8cos8cos6 (E’ -(ql))sin6cos6 E’
1
-A&sin8cos8sin6 sin 6 cos 6 ~2 sin2 ~ + ( E ~ ~ )6c o s ~ ( E ~ (Q))
(383)
7.7 Three Sample Geometries With the expressions derived in the last section, in the first version worked out by Hoffmann et al. [ 1561 and further developed by Gouda et al. [157], we can calculate E ~ E, ~ and , E~ from three independent measurements, using three different sample geometries. Consider first that we have the smectic layers oriented perpendicular to the glass plates so that the applied field is perpendicular to the layer normal. In the Smectic A* phase we would call this planar alignment, and a dielectric measurement would immediately give the value in the SmA* phase. If we cool to the SmC* phase, the helix will appear in a direction parallel to the glass plates. By applying a bias which is idenfield we unwind the helix and the signal field will now measure E~.,,
210
7
Dielectric Spectroscopy: To Find the
7 and 2 Tensor Components
tical to Q
the permittivity value along the direction of the polarization P. If we take away the bias field the helix will eventually relax. The value of the permittivity measured in the presence of a helix is denoted by &helix. It is equal to &yy in the lab frame, i.e., to ( E ~ in ) Eq. (378) or, by Eq. (379)
1 2
&helix = ~-(E~ cos
2
8 + E~ + c3sin 0) 2
(385)
The third geometry normally requires a second sample aligned with the layers along the glass plates. We may call this geometry homeotropic in the SmA* phase where we measure the value &h = along the layer normal. After cooling to the SmC* phase we may call the geometry quasi-homeotropic and denote the measured value &horn. A quasi-homeotropic SmC* thus has the smectic cone axis perpendicular to the surface. The helix axis ( z ) is now parallel to the measuring field, which means that is E,, in the lab frame, which is (ql) in Eq. (378) or, by Eq. (380) =
sin2@+ E~ cos2e
(386)
From Eqs. (384) to (386) the three principal values of the dielectric tensor can thus , E ~ the , components of the diagonalized be calculated. After solving for E , , E ~ and tensor can be expressed in the measured values as E] =
E3 =
1
[
1- 2sin2 8 (2
1 - 2sin2
-E,,,~
e [(2Ehelix- E,,
cos2 e - &horn sin2
sin
2
01
e - &horn cos2 e]
(387)
(389)
The three measurements require preparation of two samples. By using a field-induced layer reorientation, first investigated by Jakli and Saupe [ 1581, Markscheffel [ 158al was able to perform all three measurements on a single sample, starting in the quasi-homeotropic state, and then, by a sufficiently strong field, turning the layers over to the quasi-bookshelf structure. Generally, one would start with the quasi-bookshelf geometry and then warm up the sample to the smectic A* phase and strongly shear. The advantage of this technique is a very homogeneous, homeotropic orientation. The quality of homeotropic orientation may even improve (without degrading the quality of planar orientation) if the glass plates are coated by a weak solution of tenside.
7.8 Tilted Smectic Layers
211
7.8 Tilted Smectic Layers When the layers make the inclination 6 with the normal to the glass plates, we see from Eq. (383) that G~~(= E ~ Jwill be expressed by ( G = ~ ~~ 2 +~(Ell)sin26 0 ~ ~ 6 whereas
E
~
&helix
~
=(
390)
in~ Eq.~ (382) ~ turns = E out~to be ~ E ~cos2 ) 6
+
sin26
(391)
given by Eq. (386), because of the absence of while stays unaffected, i.e., the layer tilt in this geometry. The value of 6 is often known, growing roughly proportional to 8, and can in many cases be approximated as 6-0.8 8. The principal values of crj can then be extracted from Eqs. f386), (390), and (391). It turns out that the values are not very sensitive to small errors in 6. A special case of tilted layers is the chevron structure, for which the tilt makes a kink either symmetrically in the middle or unsymmetrically closer to one of the surfaces. The chevron structure means that two values +6and -6 appear in the sample. As can be seen from Eqs. (389) and (390), the angle 6 only appears as a quadratic dependence, hence the relations are also valid in the chevron case. Solving for the principal values in the chevron or tilted layers case, we obtain
Typical measured variations of the permittivities E ~ E, ~ and , E ~ as , we pass the phase boundaries from isotropic to tilted smectic phase, are shown in Fig. 84. The values of E , and E* have a similar temperature variation. Therefore their difference, the biaxiality 8&,has a weak but noticeable temperature dependence. The difference between the values and &g is the dielectric anisotropy A&. This depends on temperature much more strongly than the dielectric biaxiality. As a result of the fact that the relaxation frequency of E~ in the nematic phase lies below 100 kHz, the dielectric anisotropy is negative at that frequency. A negative value of A&is a common feature, which may greatly affect the electrooptic switching properties at room temper-
212
7 Dielectric Spectroscopy: To Find the f and E^ Tensor Components
‘i
40
60 80 100 Temperature(“C)
120
Figure 84. The principal values of the dielectric permittivity tensor at 100-kHzmeasured by the short pitch method. The material is the mixture W E 1 2 by BDHlMerck (from Buivydas [155]).
ature. In contrast, the relaxation frequency of c2 lies much above 100 kHz, whence the positive biaxiality 8&(which may rather be considered as a low frequency value). This means that for information about A& we have to go to relatively low frequencies and inversely to high frequencies for corresponding information about 8 ~ .
7.9 Nonchiral Smectics C When the cell has another director profile than the uniform helix, averaging of the angle cp over the pitch length cannot be performed in a straightforward way. It is necessary to know about the director profile through the cell before making simplifications. This profile is hard to get at but may be estimated by optical methods, as described by Sambles et al. [159]. They found the profile to be uniform or nearly so within each surface-stabilized domain. However, other reports show that the surfacestabilized state is more accurately modeled by some triangular director profile [ 1601. When a uniform director profile is assumed, then the smectic director field n can be described by the three angles cp, 6, and 6. These angles are uniform throughout the sample and the dielectric tensor component ~ y corresponds u to the dielectric permittivity ~p measured in the planar orientation. Under certain conditions this may be written Ep=E2-a&-
sin2 6 sin2 e
(395)
where (396) is the dielectric biaxiality.
213
7.10 Limitations in the Measurement Methods
-
Figure 85. The dielectric biaxiality of the nonchiral smectic C mixture M4851 (Hoechst), measured by the long pitch method (from Buivydas [155]).
-
280
30
“
a
’ 40
n
I
.
‘
* . I
50 60 Temperature (“C)
70
, *
* *
80
When measuring in the quasi-homeotropic orientation, we will observe the component &h =,&, and, assuming the layer tilt 6 to be zero, &h takes the form &h = &2
+ A&C O S 2 6
(397)
Again we have to look for an independent measurement of a third value, in order to calculate all three principal components of the dielectric permittivity tensor. At the second order transition S m A +SmC, the average value of E is conserved, thus
With knowledge of the tilt angle 8 and the layer inclination 6, we then can calculate ,c2,and .c3from Eqs. (395), (397) and (398).
7.10 Limitations in the Measurement Methods Each of the models serving as basis for the dielectric biaxiality measurements has been derived using certain assumptions. The method described by Gouda et al. [ 1571 assumes an undisturbed ferroelectric helix, and this assumption is best fulfilled in short pitch substances. Actually, the pitchkell thickness ratio is the important parameter. For measurement convenience, the cells cannot be made very thick, hence the method works well for materials with a helical pitch of up to about 5 pm. This includes practically all cases of single smectic C* substances. The method cannot be applied to nonchiral materials. The method described by Jones et al. [161] (long pitch method) was developed for materials with essentially infinite pitch. it may be used for long pitch materials in cells where the helical structure is unwound by the surface interactions and a uniform di-
214
7 Dielectric Spectroscopy: To Find the
7 and
Tensor Components
rector profile is established. The weakness of the method is that it requires the director profile to be known. On the other hand, it works for the nonchiral smectic C. It also works for materials with @=45"(typically materials with N+SmC transitions, lacking the S m A phase), which is a singular case where the short pitch method fails. The reason for this failure is that for a 45" tilt angle @thereis no difference in average dielectric anisotropy between the planar orientation and the quasi-homeotropic orientation, and thus the two equations in the system (Eqs. 385 and 386) are linearly dependent as far as that they give the same geometric relation between the direction of the measuring field and the direction of the principal axes E, and E~ ( E is ~ not involved in Equations (385) and (386) in this case give
and
respectively, whence it is not possible to split and E ~but , only measure their sum. Using tilted layers (S+O) or knowing the average value E is not remedy for this, bec3appears in all equations. cause only the Combination &, i As a result of this the errors in the calculated values of E , and E~ increase and diverge when @approachesthe value 45".This is also evidentfrom the denominator (1 -2 sin2@) in Eqs. (392) and (394).In Fig. 86 we show how the errors in and ,c3grow with the tilt angle under the assumption that the value used in the calculation formula was was assumed to be exact. measured with a 2% error and that the value of The presence of this singularity limits the usability of the short pitch method to substances with tilt angle @ up to about 35-37'. In practice, however, this is not too serious, as materials with a tilt approaching 40" are rare.
30
-
25
-
9
s
n
v
~
i
Im
.
I
*
I
8
1
,
1
0
,
I
-
El
'
5 ; "
0-
I..
-
Error 15 : 10
1
-
20 :
(%)
8
--p*-q*y-,
/
-
J 0
, ,
I
,
, ,
I
,, ,,
-
Figure 86. The dependence of the errors in the tensor components E , and E? calculated by the short pitch method when the smectic tilt angle 8 approaches 45 '. The input value is assumed to
8 FLC Device Structures and Local-Layer Geometry 8.1 The Application Potential of FLC Almost 15 years of industrial development of ferroelectric liquid crystal (FLC) devices have now passed. This may be compared with more than 25 years for active matrix displays. The potential of FLC was certainly recognized early enough following the demonstration of ferroelectric properties in 1980. In principle the FLC has the potential to do what no other liquid crystal technology could do: in addition to high-speed electrooptic shutters, spatial light modulators working in real time, and other similar hardware for optical processing and computing (all applications requiring very high intrinsic speed in repetitive operation), the FLC (and the closely related and similar AFLC) technology offers the possibility to make large-area, high-resolution screens without the need for transistors or other active elements, i.e., in passive matrix structures using only the liquid crystal as the switching element. Several kinds of such structures (Canon, Matsushita, Displaytech) have now been demonstrated which give very high quality rendition of color images, and one polymer FLC technology has been developed (Idemitsu) capable of monochrome as well as multi-color panels. When the SSFLC structure was presented in 1980, some obvious difficulties were immediately pointed out: the liquid crystal had to be confined between two glass plates with only about 1.5 pm spacing or even less and, furthermore, in a rather awkward geometry (“bookshelf” structure), which meant that both the director alignment and the direction of smectic layers had to be carefully controlled. Luckily, a number of more serious problems were not recognized until much later. In addition, only a few molecular species had been synthesized at that time; in order to develop a viable FLC technology, a considerable effort in chemical synthesis would have had to be started and pursued for many years. (We may recall that the submicrosecond switching reported in the first paper had been observed on a single substance, HOBACPC, at an elevated temperature of 68 “C.)This chemical problem alone would have discouraged many laboratories from taking up the FLC track. Nevertheless, in 1983 excellent contrast and homogeneity in the optical properties were demonstrated on laboratory samples of DOBAMBC, one of the other single substances available (thus not on a room temperature mixture) and sufficiently well aligned for that purpose by a shearing technique [96]. Shortly afterwards, two Japanese companies, Canon Inc. and Seiko Instruments and Electronics, were engaged in R&D. By 1986 about twenty Japanese and five European companies were engaged in FLC research, supported by about ten chemical companies worldwide.
216
8 FLC Device Structures and Local-Layer Geometry
Figure 87. Canon 15 in (37.5 cm) color screen which went into production 1995. High quality rendition of artwork is extremely sensitive to flicker. The FLC freedom of flicker therefore gives an unusual means of presenting art,here illustrated by the turn of the century artist Wyspiahki (self-portrait) and Bonnard (Dining Room with Garden).
8.1 The Application Potential of FLC
Chrono Color Display
217
AMLCD
Figure 88. The first version of Displaytech’s miniature display (ChronoColor), which went into production in early 1997, in comparison with a conventional active matrix display. This microdisplay utilizes sequential color to produce full color on each pixel, resulting in a brighter, crisper image than that of an AMLCD, which uses a triad of red, green, and blue pixels to form a color. In the insets, the pixels of the ChronoColor display and AMLCD are magnified to show the difference in fill factors. The color is what makes the image that you see, and it occupies 75% of the display on the left, but only 35% of the corresponding active matrix display. (Courtesy of Displaytech, I c . )
Common to almost all these ventures was the fact that the FLC teams were very small. When really troublesome obstacles began to show up, one company after the other left the scene, favoring STN or TFT technology. It was clear that the obstacles were not only in manufacturing technology, but there were tremendous scientific difficulties of a very basic nature as well. After having presented the first mature FLC prototypes in 1988 (black and white) and 1992 (color), Canon in Tokyo is now manufacturing the first 15 in. (37.5 cm) FLC panel with 1280x 1024 pixels, where each pixel, of size 230 ymx230 ym, can display 16 different colors because it is composed of four dots or subpixels. This screen (Fig. 87) has a remarkable performance and does not resemble anything else in its absolute freedom of flicker although not driven at a very high frame rate (14 Hz). At the other end of the scale, Displaytech Inc., U.S.A.,is marketing FLC microdisplays, slightly larger than 5 mm by 5 mm in size with VGA resolution (640~480),where every dot is capable of 512 colors (see Fig. 88). Another remarkable display is Idemitsu Kosan’s fully plastic monochrome screen in reflection, of A4 size (15 in) with 640x400 dots (see Fig. 89). This is a passively multiplexed (1 :400) sheet of FLC polymer, thus entirely without glass plates and all in all 1 mm
218
8 FLC Device Structures and Local-Layer Geometry
Figure 89. All-polymer FLC prototype presented by Idemitsu Kosan Co., Ltd., in 1995. This reflective monochrome display is 0.5 mm thick, has a 1:400 multiplex ratio, and a size of 20 cmx32 cm (400x640 pixels each 0.5 mmx0.5 mm), corresponding to a diagonal of 15 in (37.5 cm).
thick. Regarding these achievements, it may be time to look at the real SSFLC structures that are used i n these devices.
8.2 Surface-Stabilized States It is important to point out that for a given material there is a variety of different realizations of surface-stabilized states (states with nonzero macroscopic polarization), on the one hand, and also a variety of other states. Among the former, those with a nonzero P along the applied field direction will have a strong nonlinear dielectric response in contrast to other states, which behave linearly until saturation effects dominate. Only the former may be bistable, i.e., show a stable memory (P#O in the absence of an applied field E ) . We will give two examples to illustrate this. The first is a case with supposedly strong anchoring of the director along a specified azimuthal direction rp, corresponding to parallel rubbing at top and bottom plate of the cell [127]. This case is illustrated in Fig. 90, where successive n-P states are shown across the cell. Clearly we have two completely symmetrical arrangements, which have a certain effective polarization down and up, respectively, and which are, each of them, a stable state. They can be switched back and forth between each other by the application of convenient pulses, and have a symmetrical relationship similar to the two cases of Euler buckling of a bar in two dimensions under fixed boundary
8.2 Surface-Stabilized States
Figure 90. Surface-stabilized configuration with less than optimum efficiency, switchable between two symmetric states with low optical contrast. The surface pretilt angle has been chosen equal to the smectic tilt angle B in this example. For a strong boundary condition with zero pretilt, a different extreme limiting condition with B approaching zero at the boundary is also conceivable, without any essential difference in the performance of the cell.
219
O=o
t_
conditions. The cell is thus surface-stabilized and ferroelectric with a symmetrically bistable reponse. However, it is immediately clear that this configuration may not be the most desirable one, because we have a splayed P state with only a small part of P that can actually be switched by a short pulse, requiring a high voltage for a large optical effect. This is because the optical contrast between the two memorized up and down states is very low. In order to enhance this contrast, we would have to apply a considerable AC holding voltage across the cell after application of the switching pulse, which seems to make this use of ferroelectric structure rather pointless. Clearly the structure is intrinsically inferior to the structure in Fig. 61 where the director profile is supposed to be homogeneous across the cell. However, it not only serves the purpose of illustrating that surface-stabilized structures can be of many kinds and that they might be characterized by their electrical (speed) as well as their optical efficiency, but also this structure actually turns out to be of high practical interest, as we will see in Sec. 8.6. The second example is what we call a twisted smectic [ 1621, and is illustrated in Fig. 91. Ideally a smectic C* with 8=45" should be used as a twisted smectic. In Fig. 91 the anchoring is again assumed to be strong (although this is not a necessity for the working principle), and the alignment directions are at right angles (with zero pretilt at the surface) at top and bottom, introducing a 90" twist across the cell. This also results in zero macroscopic polarization along the direction in which the electric field is applied. With no field applied, the light will be guided in a way similar to that in a twisted nematic, and the cell will transmit between crossed polarizers. On application of an electric field, the twisted structure will unwind, until the whole structure except for one boundary layer has a homogeneous arrangement of the director, giving zero transmittance. The response is linear at low electric field and we have a fast switching device with a continuous grey scale. It has to be pointed out that this, of course, is not a ferroelectric mode. Thus a smectic C* liquid crys-
220
8 FLC Device Structures and Local-Layer Geometry
FIELD OFF (TRANSMISSIVE)
FULL FIELD ON (MIMMUMTRANSMI"
Figure 91. The twisted smectic preferably uses a 45 ' tilt material. It is a fastswitching device giving an electrically controlled, continuous grayscale. This is a smectic C* working in a dielectric mode. It should not be mistaken for a ferroelectric mode (after Pertuis and Pate1 11621).
tal does not have a ferroelectric response per se, but it can be used in many different ways. While it is true that the structure in Fig. 91 represents a surface-stabilized ferroelectric liquid crystal, this structure is used in a dielectric mode. We will, however, postpone the discussion of these rather subtle distinctions to a later stage (see Sec. 13.11) where we will have to analyze this structure in much more detail. Anyway, in the same, somewhat special, sense that we might call the helical SmC* state a helicoidal antiferroelectric we might see the twisted structure as an example of the fact that, as already pointed out, bulk structures of smectic C* tend to have an antiferroelectric-like ordering in one way or another. The point about surface stabilization is that it transfers macroscopic polarization to the bulk, giving it ferroelectric properties in certain geometries. These properties are best recognized by the appearance of spontaneous domains of up and down polarization, and by the fact that the response to an electric field is now strongly nonlinear. Let us finally look back to Fig. 61 and consider the material parameters 8, P, and d, i.e., tilt angle, polarization, and smectic layer thickness. They all depend on temperature: 8 ( T ) ,P ( T ) ,and d ( T ) .The temperature variations of 8 and P, if not desirable, at least turned out to be harmless. On the other hand, the much smaller temperature dependence of d ( T )turned out to be significantly harmful. If the liquid crystal molecules behaved like rigid rods, the layer thickness would diminish with decreasing temperature according to d,cos 8, as the director begins to tilt at the SmA +SmC transition, i.e., we would have a layer shrinkage
in the smectic C phase. In reality, the shrinkage is less (i.e., the molecules do not behave like rigid rods), but still a phenomenon that is almost universal, i.e., present in almost all materials. If, in Fig. 61, we consider the translational periodicity d to be
8.3 FLC with Chevron Structures
22 1
Figure 92. The shrinkage in smectic layer thickness due to the molecular tilt 0(r) in the SmC*phase results in a folding instablity of the layer structure (“chevrons”). Even if the fold can be made to go everywhere in the same direction (in the figure to the right) to avoid invasive zigzag defect structures, the switching angle is now less than 2 0, which lowers brightness and contrast.
imprinted in the surface at the nematic-smectic A transition, and if we further assume that there is no slip of the layers along the surface, the only way for the material to adjust to shrinking layers without generating dislocations is the creation of a folded structure in one direction or the other on entering the SmC phase, as illustrated in Fig. 92. As the layer thickness d, decreases with decreasing temperature, the chevron angle 6 increases according to 42 cos6 = dA
where 6is always smaller than 8. Often Gand 8 have a similar dependence, such that 6= A8, with A= 0.85 -0.90 in typical cases. The resulting “chevron” structure constitutes one of the most severe obstacles towards a viable FLC technology, but its recognition in 1987 by Clark and his collaborators by careful X-ray scattering experiments [163, 1641was one of the most important steps both in revealing the many possibilities of smectic local layer structures and in controlling the technical difficulties resulting from these structures.
8.3 FLC with Chevron Structures The extremely characteristic zigzag defects had been observed for many years before the subtleties of the local layer structure were unveiled. Very often they appear as long “streets” or “lightning flashes” of thin zigzag lines, with the street going essentially along the smectic layers, whereas the zigzag lines themselves are running almost perpendicular to the layers. In Fig. 93 the angle between these lines and the layer normal is roughly 5”. Often these thin lines are “short-circuited” by a broad wall, with a clear tendency to run parallel to the layers, as in Fig. 93a where it is almost straight and vertical, or slightly curved, as in Fig. 93b. It is evident that if the
222
8 FLC Device Structures and Local-Layer Geometry
(a)
(b)
Figure 93. Appearance of zigzag walls in a smectic C structure with smectic layers initially (in the SmA phase) standing perpendicular to the confining glass plates, running in a vertical direction on the micrographs (thin lines to the left, broad walls to the right). As the top figure (a) shows, the zigzag “street”, like the thick line, goes preferentially along the smectic layers. (Courtesy of Monique Brunet, University of Montpelher, France).
layers are not straight in the transverse section of the sample but folded, as in Fig. 92, then where we pass from an area with one fold direction to an area with the opposite direction there must be a defect region in between, and hence it is likely that the zigzags mediate such changes. The question is how. This may be illustrated starting from Fig. 94. In (a) we show the typical aspect of a zigzag tip with y being the small angle between the line and the layer normal (which is often the same as the rubbing direction). In (b) is shown the folded structure making chevrons of both directions, suggesting that at least two kinds of defect structure must be present. In the picture the fold is, as is most often the case, found in the middle of the sample - we will call this fold plane the chevron interface - although its depth may vary, as in (c). We will mainly restrict ourselves to the symmetric case (b) in the following discussion of the general aspects. As we will see, the chevron interface actually acts as a third surface, in fact the one with the most demanding and restrictive conditions on the local director and polarizations fields. It is clear that the transition from one chevron to the other cannot take place abruptly in a layer as sketched in (d), as this would lead to a surface (2D) defect with no
8.3 FLC with Chevron Structures
Figure 94. Thin zigzag lines run almost per pendicular to the smectic layers making a m a l l angle y with the layer normal as shown In (a) In (b) we look at the chevron folds in the plane of the sample, in (e) perpendicular to the plane of the sample Asymmetric chevrons (c) cannot preserve anchoring conditions at both surfaces and are therefore less frequent
#(((/>$$j((
223 +
L 1 (b)
(a)
n 1 + (C)
(4
(el
Figure 95. Section of a thin wall mediating the change in chevron direction. The layers in the chevron structure make the angle G(the chevron angle) with the normal to the glass plates. Along a chevron fold where two surfaces meet, two cone conditions have to be satisfied simultaneously for n, which can be switched between two states. At the two points of the lozenge where three surfaces meet, three conditions have to be satisfied and n is then pinned, thus cannot be switched.
continuity of layer structure nor director field. Instead it is mediated by an unfolded part of width w (this region will be observed as a wall of thickness w ) in (e). looked at from the top as when observing the sample through a microscope. The wall thickness w is of the order of the sample thickness L. A more adequate representation of the three-dimensional structure is given in Fig. 95. The vertical part of the layer has the shape of a lozenge (in the general case, with unsymmetric chevrons it becomes a parallelogram). The sterically coupled fields n (r)and P ( r ) can now be mapped on
224
8 FLC Device Structures and Local-Layer Geometry
Figure 96. (a) At the chevron interface the local polarization P is discon-
(b)
Et
4
Et
4
tinuous, making a jump in direction. (b) When switching from the down to the up state, P rotates everywhere anticlockwise above and clockwise below the chevron interface (time axis fl N - - \\* to the right). The director is locked in ______________________________ the chevron plane and can move k rc betweennlandn,.
-L @ \* ______________________________ # H *K k @
a
this structure. Along the folds, n (r)must simultaneously satisfy two cone conditions (i.e., to be on the cone belonging to a particular layer normal) indicated in the figure. This is always possible for 610 and gives a limiting value for the chevron angle 6. After the discovery of the chevron structures, a number of complexities in observed optical and electrooptical phenomena could be interpreted in these new terms. For a more thorough discussion in this matter, we refer to references [ 166- 1681.Our emphasis will be on the important consequences for the physics due to the presence of chevrons, but even this requires dealing with at least some structural details. First of all, the chevron fold forces the director n to be in the interface, i.e., to be parallel to the plane of the sample in this region, regardless of how n (r)may vary through the rest of the sample and independent of the surface conditions. Although the directorn is continuous at the chevron interface, the local polarization field P cannot be, as shown in Fig. 96. It makes a jump in direction at the interface, nevertheless, in such a way that
V.P=O
(403)
which means that the discontinuity is not connected with a build-up of local charge density. If we now apply an electric field between the electrodes, i.e., vertically as in (a), we see that there is an immediate torque because E and P are not collinear. Hence no nucleation is needed for switching from state 1 to state 2. Neither are fluctuations needed for this. In other words, the switching process is not fluctuation-controlled. On the other hand, the chevron region requires an elastic deformation of the local cone in order to switch: the local tilt 6 must diminish to a smaller value (mid-
8.3 FLC with Chevron Structures
225
way between 1 and 2), which gives a contribution to the threshold for the process. Generally speaking, the bulk switching on either side of the chevron interface preceeds the switching in the interface. The latter contributes to the latching and thus to the bistability. As seen from Fig. 96b, the switching process is unambiguous as regards the motion of n and P (sterically bound to n): on the upper side of the chevron, P rotates counterclockwise, on the lower side it rotates clockwise when we switch from 1 to 2; everything turns around in the reverse switching direction. This explains why there are no twist and antitwist domains like the ones observed in twisted nematics prior to the time when chiral dopants were added in order to promote a certain twist sense. So far we have described the switching concentrating on the chevron interface, completely disregarding what could happen at the two bounding (electrode) surfaces. In fact, if the anchoring condition on the surfaces is very strong, switching between up and down states of polarization will only take place at the chevron interface. At high voltage this will more or less simultaneously take place in the whole sample. At low voltage it will be possible to observe the appearance of down domains as “holes” created in an up background, or vice versa, in the shape of so-called boat domains (see Fig. 105) in the chevron interface (easily localized to this plane by optical microscopy). The walls between up and down domains have the configuration of strength one (or 27c) disclinations in the P field. It should be pointed out that the uniqueness of director rotation during the switching process is not a feature related to the chevron per se, but only to the fact that the chevron creates a certain P-tilt at the chevron interface. If the boundary conditions of the glass surfaces involved a similar P-tilt, this will have the same effect. A glance at Fig. 97 reveals another important consequence of the chevron structure. As P is not along E (applied field) there will always be a torque P x E tending to straighten up the chevron to an almost upright direction. Especially in antiferroelectric liquid crystals, which are used with very high P values, this torque is sufficiently strong for almost any applied field, for instance normal addressing pulses, to raise and keep the structure in a so-called quasi-bookshelf structure (QBS) under driving conditions. In ferroelectric liquid crystals, presently with considerably lower P values, the same effect was previously employed to ameliorate contrast and
Figure 97. The fact that even after switching the polarization P is not entirely in the direction of the applied field will tend to raise the chevron structure into a more upright position, so decreasing the effective 6 but breaking up the layers in a perpendicular direction. This gives a characteristic striped texture from the newly created, locked-in defect network.
226
8 FLC Device Structures and Local-Layer Geometry
threshold properties, by conditioning the chevron FLC to QBS FLC by the application of AC fields [169]. The effect on the switching threshold can be extracted from Fig. 96. When the chevron structure is straightened up, 6decreases and the two cones overlap more and more, leading to an increasing distance between 1 and 2, as well as further compression of the tilt angle 8 in order to go between 1 and 2. The threshold thus increases, in agreement with the findings of the Philips (Eindhoven) group. On the other hand, this straightening up to QBS violates the conservation of smectic layer thickness d,, which will lead to a breaking up of the layers in a direction perpendicular to the initial chevrons, thus causing a buckling out of the direction running perpendicular to the paper plane of Fig. 97.
8.4
Analog Grey Levels
As we just pointed out, in the chevron structure the polarization is no longer collinear with the external field. This can be used (for materials with a high value of P ) to straighten up the chevron into a so-called quasi-bookshelf structure, combining some of the advantages from both types of structure. For instance, it can combine a high contrast with a continuous gray scale. How to produce analog gray levels in an SSFLC display is perhaps not so evident, because the electrooptic effect which we have essentially dealt with so far offers two optical states, hence it is digital. Nevertheless, the shape of the hysteresis curve reveals that there must be small domains with a slightly varying threshold, in some analogy with the common ferromagnetic case. Normally, however, the flank of the curve is not sufficiently smeared out to be controlled and to accommodate more than a few levels. Curve I of Fig. 98 shows the transmission-voltage characteristics for a typical SSFLC cell with the layers in the chevron configuration [ 1651.The threshold voltage is fairly low, as well as the achievable transmission in the bright state, leading to a low brightness -contrast ratio. The position and sharpness of the threshold curve reflect the relatively large and constant chevron angle S, in the sample. If a low frequency AC voltage of low amplitude (6- 10 V) is applied, the smectic layers will be straightened up towards the vertical due to the P-E coupling, so that the Iocal polarization vector increases its component along the direction of the field. This field action, which requires a sufficiently high value of P , breaks the layer ordering in the plane of the sample and introduces new defect structures, which are seen invading the sample. The result is that the chevron angle 6 is reduced, on average, and the threshold smeared out, as shown by curve 11. Lower 6 means a larger switching angle (and higher threshold), and thus higher transmission. Still higher transmission can be achieved by an additional treatment at a somewhat higher voltage (+25 V), giving threshold curve 111, corresponding to a new distribution around a lower 6value and a microdomain texture on an even finer scale.
227
8.4 Analog Grey Levels
T
? Figure 98. Amplitude-controlled gray scale in SSFLC. The chevron structure is transformed to a quasi-bookshelf (QBS) structure by external field treatment. In addition to giving gray shades, the QBS structure increases the brightness and the viewing angle. This method of producing gray levels was developed by the Philips (Eindhoven) group, who called it the “texture method”.
m
7
/
The actual switching threshold is a complicated quantity, not fully understood (no successful calculation has been presented so far), and usually expressed as a voltage-time area threshold for the switching pulse. For a given pulse length it is, however, reasonable that the amplitude threshold increases according to Fig. 98 when the average value of 6 decreases. There are at least two reasons for this, as illustrated by Fig. 96. First, it is seen that the distance between the two positions n , and n2 in the chevron kink level (which acts as a third, internal surface), as well as the corresponding positions at the outer surfaces and in between, increase when 6 decreases. It would therefore take a longer time to reach and pass the middle transitory state, after which the molecules would latch in their new position. In addition, it is seen that the local deformation of the cone i.e., a decrease of the tilt angle 0, which is necessary to actuate the transition from n , to n2, increases when 6 decreases. (A paradox feature of this deformation model is that it works as long as 6#0, whereas 6=0 gives no deformation at all - but also no chevron - at the chevron kink level.) The smectic layer organization corresponding to curves I1 and 111of Fig. 98 is generally characterized as a quasi-bookshelf (QBS) structure, denoting that the layers are essentially upright with only a small chevron angle. The QBS structure has a very large gray scale capacity. This might, however, possibly not be utilized to advantage in a passively driven display (as it can in the AFLC version). Its drawback in this respect is that the shape of the threshold curve is temperature-dependent, which leads to the requirement of a very well-controlled and constant temperature over the whole area of a large display. Furthermore, the QBS structure is a metastable state. Finally, the microdomain control of gray shades requires an additional sophistication in the electronic addressing: in order to achieve the same transmission level for a given applied amplitude, the inherent memory in the microdomains has to be deleted, which is done by a special blanking pulse. Using this pulse, the display is reset to the same starting condition before the writing pulse arrives. As a result of these features, it is not clear whether the microdomain method will be successfully applied
228
8 FLC Device Structures and Local-Layer Geometry
in future FLC displays. A combination FLC -TFT seems to be required for this but, on the other hand, can be quite powerful. We will continue this discussion in Sec. 9.1.
8.5
Thin Walls and Thick Walls
The micrographs of Fig. 93 indicate that the thin wall normally runs with a certain angle y to the smectic layer normal. What determines this angle what determines the thickness of the wall, and what is the actual structure of it? To answer these questions we have to develop the reasoning started for Figs. 94 and 95. This is done in Fig. 99 where we look at the layer kink (the lozenge-shaped part in Fig. 95) from above. By only applying the condition of layer continuity, the following geometric relations are easily obtained, which describe the essential features of thin walls. Starting with Fig. 99a and remembering that the layer periodicity along the glass surface is dA,we obtain a relation between y and the layer kink angle p with the chevron angle 6as a parameter. With dc/dA=cos 6, this is written
For small angles we can solve either for P
p=y+lir2+S2 or for y
This relation shows that y increases monotonously with p. Thus the larger the smectic layer kink, the more the wall will run obliquely to the layer normal. The fact that there is a physical upper limit for fl (see below) explains why we never observe zigzag walls with a large inclination y For ease in computation, when we make estimations Eq. (406) can more conveniently be written
Next we get the obvious relations between the width w of the wall and the width b of the lozenge or the length c of the cross section of the wall cut along the layer normal w=bcos(p- y)=csiny
(408)
229
8.5 Thin Walls and Thick Walls
Figure 99. (a) Chevron structure like the one in Fig. 36, as seen from above. A thin wall of width w joins the region with a chevron bend in the negative z direction (left part of the figure) with another having the chevron bend in the positive z direction (right part of the figure). (b) Section through the wall cut along the z direction, (c) projection along the z direction, and (d) a cut parallel to the xz plane.
I
L
Y
Y
as well as the width b of the lozenge from c tan y = b cos6
(409)
But this width b is also related to the kink angle and the sample thickness L (see Fig. 99c, d and Fig. 94e).
b sin f l = L tan 6
(410)
We may now use Eqs. (408), (410), and (404) to obtain the relation between the wall thickness and the sample thickness
w = L cosy sin 6 sin p
230
8 FLC Device Structures and Local-Layer Geometry
Finally, we can express the angle o between the adjacent surfaces M and N in Fig. 99 by taking the scalar product of the corresponding layer normals u and v [marked in (a) and (d)]. This gives coso = cosp cos6
(412)
Let us now extract the physical consequences contained in all these relations and start with the last one. As numerical examples we will use the data from Rieker and Clark [ 1681 on the mixture W7-W82, for which the tilt angle 8saturates at about 21" at low temperature, and the corresponding saturation value for the chevron angle 6 has been measured as 18". To begin with we see that Eq. (412) sets an upper limit to the layer lunk p, because of the requirement of uniqueness of the director n at each chevron interface. Figure 96 a illustrates the fact that where the two inclined layers meet they can share the director only to a maximum layer inclination of up to twice the tilt angle 8. At this point the two cones touch, and beyond it no uniqueness in the director field can be maintained. Thus o i n Eq. (412) has a maximum value equal to 2 8,which gives a maximum value, in practice only somewhat lower than 2 8,for the allowable layer kink p. This, in turn, gives a limiting value for y. With the data for W7-W82 we get o=42", which gives (Eq. 412), p=39" and (Eq. 407) y= 15". The width of the wall in this case [Eq. (411)] is found to be roughly 0.5 L, i.e., half the sample thickness. This is thus the limiting case for an inclined zigzag in this material. Let us now assume that we observe another wall running at a 5" inclination to the layer normal. In this case Eq. (405) gives p=24" and Eq. (411) gives w=0.8L. Finally, let us check a wall running exactly perpendicular to the layers (y=O). In this case Eq. (404) shows p to be equal to 6, hence B= 18", and w = L according to Eq. (411). This is the other limiting case with a wall of maximum width. Surnmarizing (see Fig. 100) we may say that a thin wall has its maximum width (equal to the sample thickness) when running perpendicular to the layers. The layer kink then has its minimum value, equal to 6, i.e., = 8. Inclined walls are thinner, represent a hgher energy density, and cease to exist when the layer kink approaches = 2 8,which sets the maximum value of obliqueness to about 15-20".
w = Wmax = L
w = wmin = 0 . 5 ~
Figure 100. Thin walls with limiting cases, y=O and y= yman(in this case about 15"). The width may vary by a factor of two, being of the order of the sample thickness L. The layer kink angle is at least equal to the chevron angle and at most slightly more than twice that value.
8.5 Thin Walls and Thick Walls
23 1
Figure 101. Thin wall curving until it finally runs parallel to the srnectic layers, then being a broad wall. The character changes from <<<>>> to >>> <<< as soon as the wall reverses the inclination angle with the angle counted as positive if it is in the same direction as the layer kink p, otherwise it is negative. That the sign of yrelative to the layer normal is irrelevant for the character can be seen if we mirror the picture along a vertical line: the mirror image must have the same character in every part of the wall.
Figure 102. A thick wall in its relaxed state has a width W much larger than the sample thickness L . It can only have the structure >>><<< as shown in (b), not the one in (a). While the chevron angle 6 changes with the smectic C layer thickness, the surface is assumed to maintain the pitch of the layers at the smectic A value, independent of the temperature.
T
i
Let us now imagine that the thin wall of Fig. 99, if we follow it downwards in the positive z direction, will curve so that ychanges sign, eventually to form a closed loop. Any cut along z will have the structure <<<>>> as long as y>O, whereas this configuration changes to >>><<< as soon as y>><<<. With the nomenclature used so far (which we propose to abandon), a thin wall may change between <<<>>> and >>><<<, while a thick wall can only have the structure >>><<<. Figure 102 explains why. In (a) we have assumed, for the sake of argument, the opposite configuration. The translation period at the boundary is d, as before, and below it the layer can tilt in any direction to preserve this horizontal period while shrinking to layer thickness dc. However, when we reach the lozenge, layer continuity would demand the layers to expand not only to d >dc , but even to d>d,. This means that this whole area would stand under great tension. Note that this is entirely different from the seemingly similar configuration in Fig. 99 b. In that case the layers run obliquely to the cut, and hence the periodicity in this cut does not at all correspond to the layer thickness. In contrast, the layers in
232
8
FLC Device Structures and Local-Layer Geometry
Fig. 102 run along the wall, and thus perpendicular to the cut. If we assume the >>><<< structure, on the other hand, as in (b) we see that the thick wall can relax to an equilibrium configuration preserving the layer thickness everywhere. It is easy to see that this gives the equilibrium width W for the thick wall according to
L W
-=
6 tan 2
(4 13)
or, for small values of 6
W Z 2L S
(414)
In a material with 6= 18" this would give an equilibrium thickness of 25 pm for a broad wall in a 4 pm thick sample. Thus a broad wall is much thicker than L , whereas a typical narrow wall is thinner than L. This certainly conforms with observation. Naturally, walls will also be observed that are not in equilibrium. If a thick wall is thicker than the equilibrium value, the smectic layers in the lozenge will be under tension, corresponding to a tilt 8 < Oeq. If it is thinner, the layers will be under compression, corresponding to 8 > Oeq, and thus with a tilt that is larger than the equilibrium tilt angle. Because the layers are upright in the middle of a broad wall, the optical contrast between the two states is particularly high, and may be further enhanced if the width is smaller than the equilibrium value W corresponding to Eq. (413). As is evident from Fig. 102b, a broad wall is thicker the smaller the chevron angle 6. This is also most clearly expressed by Eq. (414). In contrast, the width of a thin wall is not very much affected by 6. However, so far in the literature, a "thin" wall has not been a well-defined but a rather ambiguous concept. We have already seen that the character of a wall changes from <<<>>> to >>><<< when y changes from >O to O) as the wall curves in the other direction, until the layers are perfectly flat when the wall runs parallel to the layers (y=-90"). It has
8.5 Thin Walls and Thick Walls
Figure 103. Inside a closed loop the chevron has a unique direction: (a) downwards or, (b) seen from below, towards the reader. As the lozenges connect two opposite tips in the chevron folds (b), we likewise have a uniquely determined kink direction in the wall. It is relative to this direction that we decide whether yshould be counted as positive (same sense) or negative (opposite sense). Thus note that y>O for both thin walls, even if they have geometrically opposite inclinations.
233
(b)
i
i
i
\
%
been customary to denote a wall in this limiting state a “thick wall”, with the consequence that other walls, called “thin”, can sometimes have the configuration <<<>>> and sometimes >>> <<< in the cross section along the layer normal on either side of the wall. However, it would make much more sense to use the fact that the character of the wall changes when ychanges sign. Thus we propose the designation “thin” for y>O and “thick” for y>> configuration only, and hence that any wall of configuration >>><<< should be called a thick wall. As a consequence, w =L becomes the natural demarcation line between thick and thin walls: any wall with configuration <<<>>> has w < L, whereas any wall with configuration >>><<< has w >L. The proposed criterion should not, and does not, depend on the chevron angle 6, and thus not on temperature. For example, in the same material W7-W82 at a higher temperature, such that 6=5”, w still has the maximum width equal to L when y=O andattainsitsminimumwidth=L/2 when y=4” (ym,,). For y=-5” and-30”, wz2.5 L and w = 12L, respectively. This discussion may be briefly summarized by Figs. 103 and 104. In the first we have indicated a wall in the form of a closed loop, with the corresponding layer kinks and the chevron configuration. In particular, we have marked out that the inclination of the wall relative to the layer normal does not determine the sign of y The thin walls ultimately merging into a fine tip thus have the same character. Figure 99 gives an idea of what happens at the tip: when the walls merge, the kink becomes twice as large (2p), which requires j3 to be less then 8. For 8=21”, this gives a maximum value for y of only 3” at the very tip. In fact, this narrowing of the tip angle can of-
234
8 FLC Device Structures and Local-Layer Geometry
Figure 104. Zigzag wall making a closed loop. The smectic layers are assumed to be vertical in the picture. The chevron fold is always away from the zigzag inside and towards the thick wall. When the wall curves, the chevron character changes from <<<>>> for a thin wall (w>><<< for a thick wall (w>L). The wall attains its maximum thickness (relaxed state) when it runs parallel to the iayers.
ten be observed (see Fig. 93). In Fig. 104 finally, we have indicated how the character of the wall changes around a closed loop with zigzags when the wall curves in different directions. As for the chevron fold, inside the loop it can only be directed away from the zigzag and towards the thick wall.
8.6 C l and C2 Chevrons The presence of chevrons has numerous consequences for optics and electrooptics, above all for the transmission, contrast, bistability, and switching behavior. Some of these have already been hinted at earlier. The effects will be particularly drastic if we combine chevrons with polar boundary conditions that are very common, as most aligning materials favor one or the other direction for P (most often into and less frequently out of the liquid crystal). In contrast, the chevron interface represents a nonpolar boundary, by symmetry favoring neither the up nor the down state (see Fig. 96). These states are distinct and switchable as long as the chevron angle 6 is less than the tilt angle 8 of the director. If the limiting case (6=e> is reached, the two states merge into one and the chevron no longer contributes to the bistability. In Fig. 103(b) we have indicated the P field distribution (one of many possible states) around two adjacent thin walls. The field is predominantly up in one of them and predominantly down in the other. Therefore alternate walls often serve as nucleation centers for domain switching taking place in the chevron plane, i.e., for the motion of walls between up and down domains. Such domains often appear as boatshaped “holes” of up state on a background of down state, or vice versa (see Fig. 105).
8.6 C1 and C2 Chevrons
235
I
c1
c2
Figure 105. A zigzag street (lightning) running between two regions of opposite chevron fold for the case of nonzero pretilt a. This gives a different structure to the left (Cl) and to the right (C2). The two regions then have different properties, like transmission, color contrast, switchability, etc., which get more pronounced the higher a is. Only when a=O are the C1 and C2 structures the same thing. For a#O the director positions are different for C1 and C2, except in the chevron plane where n has to be horizontal. For the purpose of illustrating different possibilities, we have made a very small in the C1 case, whereas we have drawn the C2 case for medium a towards the upper, with high a towards the lower substrate. Thus the boundary conditions to the right and left to not correspond to the same surface treatment.
On the other hand, the zigzags themselves do not normally move in an electric field, but are quite stationary. This is because zigzags normally separate regions with the same average P direction (up or down, likewise illustrated in Fig. 103b). Also, the optical state (transmission, color) is very often practically the same on both sides of a zigzag wall, as in Fig. 93b. Indeed, if the director lies parallel to the surface (pretilt a=O) at the outer boundaries, the chevron looks exactly the same whether the layers fold to the right or to the left. However, if the boundary condition demands a certain pretilt a #O, as in Fig. 105,the two chevron structures are no longer identical. The director distribution across the cell now depends on whether the director at the boundary tilts in the same direction relative to the surface as does the cone axis, or whether the tilt is in the opposite direction. In the first case we say that the chevron has a C1 structure, in the second a C2 structure (see also Fig. 106).We may say that the C1 structure is “natural” in the sense that if the rubbing direction (r)is the same at both surfaces, so that the pretilt a is symmetrically inwards, the smectic layer has a natural tendency (already in the SmA phase) to fold accordingly. However, if less evident at first sight, the C2 structure is certainly possible, as demonstrated in Figs. 105 and 106.
236
8 FLC Device Structures and Local-Layer Geometry
c1 c2
DOWN
UP
c2
U P U P
c1
UP
DOWN
Figure 106. Closed zigzag loop with thin and thick walls and a smaller loop inside. Inside any loop the chevron always points towards the broad wall. With a polymer coating on the inner side of the glass plates the rubbing direction is indicated by the unit vectorr, giving the pretilt a t o the inside. In the lower part of the figure, the C1 structure to the left and the C2 structure to the right of the thin wall are cornpared with their director and polarization fields in the case of switchable surfaces for C1 and nonswitchable surfaces for C2. The figure is drawn for a material with P
The very important discovery of the two chevron structures and the subsequent analysis of them and their different substructures was made by the Canon team [ 1701. The presence of C1 and C2 as distinct structures has obvious consequences. Any wall now becomes a complete configuration consisting of one C1 and one C2 part. Its contrast changes with its running direction. Abroadrelaxed wall may be thick enough that its C1 and C2 parts, and even the small, almost pure bookshelf section in between, can be distinguished by different optical contrast. The regions representing opposite fold direction, separated by a wall, acquire different colors even if the chevron plane is in the middle, and so on. However, the optical differences are small for small pretilts of a. If we take a nonzero pretilt into consideration, leading to C1 and C2 structures, thin and thick walls look as illustrated in Fig. 106. In the lower part of that figure we
237
8.6 C1 and C2 Chevrons
X
;
Figure 107. Polarization switching taking place in the chevron plane at fixed polar Y4 boundary conditions In this case the condition is one of high pretilt a for n with the P vector pointing into the liquid crystal trom the boundary
-/ -
<
z UP
\ \
> J c
\
> J
c
DOWN
have also illustrated the quite important difference in the polarization and director fields across a C1 and across a C2 structure. It is hard to draw these figures to scale and yet demonstrate the characteristic features. In Fig. 106 the situation may roughly correspond to I3 = 30°, 6=20", and a = lo", such that a + 6= 8. We may note that the C2 structure is none other than the chevroned version of the surface-stabilized configuration already discussed for Fig. 90. The same C2 configuration n -P would be found for any case with a+6= 13,for instance, with a=3",6= 15", and I3= 18", as long as 6< 8. We mentioned that V .P=O at the chevron interface, which means that even if P changes direction abruptly, P, (as well as any other component) is continuous across that surface (Fig. 96). As the boundary conditions at the substrates do not normally correspond to the director condition at the chevron, aP,lax is nonzero between the chevron and the substrates, but is small enough to be ignored. That this is not true when we have polar boundary conditions, is shown in Fig. 107. Let us, for instance, assume that the polarization P prefers to be directed from the boundary into the liquid crystal. Whether we have a chevron or not, we will have a splay state P ( x ) corresponding to a splay-twist state in the director. With a chevron the splay is taking place in the upper or lower half of the cell when this is in its up or down state, respectively. These half-splayed states, which are most often just called twist states, occur in both C1 and C2 structures and are then abbreviated C1T and C2T, in contrast to the uniform states C1U and C2U. These twist states are normally bluish and cannot be brought to extinction, hence they give very low contrast in the two switchable states. Moreover, due to the different sign of V .P,there will be an unsymmetric charge distribution between adjacent areas switched to the up and the down state, tending to bum in any already written static picture. For high P, materials, even the electrostatic energy has to be taken into account which, together with the elastic energy, may tend to shift the chevron plane to an unsymmetric state as illustrated to the right of Fig. 107, in order to relieve the high local energy density. Such a state is then completely unswitchable (monostable). It is evident that, whether we choose C1 or C2, the twisted states C1T and C2T have to be avoided.
238
8 FLC Device Structures and Local-Layer Geometry
d
c1
c2
.I/
Figure 108. Stability conditions for chevrons of C1 and C2 type (after [170]).
In their first analysis of C1 and C2 structures, Kanbe et al. [170] put forward the essential criteria for their stability. Both are allowed at low pretilt a.Whereas C1 can exist at high pretilts, the cone condition (n must be on the cone) cannot be fulfilled for C2 if a is larger than (O-S), as illustrated in Fig. 108. This means that for a#0 it cannot be fulfilled at the phase transition point SmA --+ SmC when I3 and 6 are both zero. Hence the chevron that is first created when the sample is cooled down to the SmC phase is always C1. However, C1 is not stable when the tilt angle 8 increases. Therefore C2 appears together with C1 at lower temperatures and the sample is marred by zigzag defects. One of the reasons for this is that C1 has a higher splay- twist elastic deformation energy, which increases with increasing 13.Another reason is that the director is more parallel to the rubbing direction in the C2 case; it has to split more to fulfill the cone condiditon for C 1. This effect will favor C2 under strong anchoring conditions. Finally, as pointed out in the cited Canon paper [170], while a gradual transition C 1 -+ C2 to an almost chevron-free state might be observed on cooling, the once such-created C2 state tends to be quite stable if the temperature is subsequently raised; it is almost necessary to go back to the SmA* phase in order to recreate C 1. Thus C2 is stable over a much broader range of temperature than C1. All this speaks for the C2 state, in spite of the considerably lower effective switching angle between its memorized states. Several FLC projects have also been based on C2 (JOERS/Alvey, ThodCRL, Sharp). The fact that bistable switching only takes place through being latched by the chevron surface is compensated for by a number of advantages. First of all, the surface alignment is relatively simple, as the outer surfaces do not switch but only work together with the chevron. It is also much simpler to avoid twist states for C2 than for C l . This is easily seen from Fig. 106. Here the structure has been sketched for a= (0-6) such that the director is exactly along the rubbing direction and strongly anchored. As already pointed out, the nonchevron version of this is shown in Fig. 90. In practice it is found [171] that a medium pretilt a of about 5" is convenient for achieving this cone surface condition, whereas a values near zero give both C2U and C2T. For several materials, (8-6) is of the order of about 5"; however, observations have been made [ 1721 of chevron-free C2U structures for a=5",although (8-6) seems to be only about IS", thus violating the stability condition for C2.
239
8.6 C1 and C2 Chevrons
C
Figure 109. Effective switching angle In a C2 structure as a function of the amplitude Vd of the applied data pulses for the so-called Malvern-3 scheme. The switching pulse amplitude is 30 v. v d = 0 corresponds to the memorized states as latched by the chevron interface. The P, value of the material is 4.4 nC/cm2 (after 171).
B 10-
,$
-
3
1
1 0;-0
,
7-
5 10 15 AC voltage Vd [V]
Whether this is due to a change of 6 under driving conditions (6decreasing towards QBS case), is unclear. A further advantage with the C2 structure is that zigzags induced by (light) mechanical shock, which causes a local transition to C1, easily heal out by themselves in the C2 structure [170], whereas the opposite is not true. Fortunately, C2 also has a substantially higher threshold to so-called boat-wake defects, which appear on the application of a very high voltage. The main disadvantages with the C2 structure are obviously the low optical contrast in the true memorized state and the quite high voltages applied so far in the electronic addressing of the cell, when the so-called z (V)-min addressing modes are used. This is also combined with a high power consumption. Thus the switching pulse is used to accomplish latching in the chevron plane, whereas the bipolar AC data pulses are used to force the director as much as possible into the in-plane condition between the chevron surface and the outer surfaces (see the lower part of the right of Fig. 106). In this way the effective switching angle can be increased from a typical value of 14”, corresponding to the memorized state, to the more significant value of 24” (Fig. 109), when data pulses of 10 V amplitude are applied [ 1711. This higher switching angle thus corresponds to what we might call “forced memorized states”. A valuable discussion on chevrons and switching behavior in FLCs can be found in the recent review article by Ulrich and Elston [139]. The complicated problem of describing the switching mechanism including the motion of walls and disclinations has been attacked in several papers. One of the most recent and ambitious of these attempts is by Seitz, Stelzer and Trebin [ 139al. In addition to splay, bend and twist they also consider the saddle-splay surface elastic constant K24 introduced by Oseen. However, they neglect the spontaneous bend. The calculations were made for the QBS structure. A valuable discussion of displays in general, including FLCDs, with their structural and addressing problems, has been given by Ian Sage [239].
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FLC Devices 9.1 The FLC Technology Developed by Canon As indicated above, the first Canon prototype from 1988 had a=O giving a hemispherical viewing angle, which was a complete novelty at that time. However, the contrast (- 7:l) was only good enough for a monochrome display. Moreover, the C1 structure could not be made sufficiently stable at low temperatures. To solve this problem the team decided not to abandon C1 in favor of C2, but instead to pursue the much more difficult road to go to a very high pretilt. The idea was to squeeze out C2 completely from the material because its cone condition a <(8-6) can no longer be satisfied. Thus, by going to the rather extreme pretilt of 18" (a technological achievement in itself), Canon was able to produce zigzag-free structures, as the layers can only be folded to C1 chevrons. A considerable development of FLC materials also had to take place, among others, mixtures with (8-6) values between 3 and 5". In this way C2 could be successfully suppressed under static conditions, for instance, in a material with 8 = 15" and 6= lo", which is not surprising, because a i s not only larger than (8-6) but is even larger than Oalone. However, it turned out [ 1731that C2 would appear to some extent under dynamic, i.e., driving conditions. This obviously cannot be explained by assuming that 6 gets smaller under driving conditions. Thus, it seems that the stability criterion can hold at most under static conditions. This indicates that the underlying physics of the C 1/C2 structure problem is extremely complex. In order to avoid C2 at any temperature, several parameters had to be optimized, such that both 8and 6 showed only a small variation as functions of temperature. The obvious advantage of working with C l is that the switching angle is high, at least potentially, between the memorized states. This is particularly true if the cell switches not only at the chevron surface, but also at both outer boundaries. However, surface switching does not normally occur at polymer surfaces with a low pretilt (although it does at SiO-coated surfaces). This is another advantage of going to high pretilt. The situation may be illustrated as in Fig. 110.At high pretilt the surface state not only switches, but the switching angle may be just as large as permitted by the cone angle of the material used. This cone angle is limited by the possible appearance of C2 at low temperatures and also by the fact that the twisted state C l T would appear at higher 8. Thus 8 is limited, quite below the ideal performance value, in the C1 state as well as in the C2 state. Whether the twisted state appears in the C1 case or not is, however, not so much determined by the tilt 8,but preeminently by the polar strength of the surface condition. It is therefore important to realize that this influence may be at least partly neutralized by a suitable cross-rubbing, i.e., obliquely to what is going to be the di-
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9 FLC Devices
Figure 110. Switching angle under different conditions. (a) To the left QBS geometry i s supposed. If the surface cannot be switched (for instance on polyimide), this will lead to a strongly twisted state. In the middle (b) a chevron geometry is assumed. If the surface is nonswitchable, it leads to a smaller twist than in (a), if it is switchable, the switching angle is less than in a switchable version of (a). With high pretilt (c) a polymer surface normally switches and the switching angle may be optimized, similar to the situation in (a).
rection of the smectic layer normal. The idea is quite simple, but can give rise to a number of different interpretations, e.g., suppose that the lower surface demands the P vector to point into the liquid crystal. However, if n at this surface is tilted in a direction clockwise (seen from above) relative to the layer normal, then this n direction corresponds to the same direction of P (for P e O ) , namely, into the liquid crystal. Therefore rubbing in this direction will enhance the polar character of the surface [174]. Hence, if the surface is rubbed in the opposite angular direction, the polar character will be, at least partly, neutralized. Canon, in fact, uses this kind of cross-rubbing in order to increase the symmetric bistability and avoid the twisted state [173, 1751. The rubbing directions deviate 210" from the layer normal, roughly corresponding to the switching angle. The basic geometry of rubbing, pretilt, etc., as applied to the actual Canon screen is illustrated in Fig. 1 11. While the contrast is far better in this version than in the 1988 version, the viewing angle - though still very good - has been compromised due to the fact that the local optic axis is pointing out of the flat surface by 18". The one hundred million chevrons across the screen all have the same direction, making a completely defect-free surface. Even so, Fig. 112 reminds us that on a local scale the order has to be disturbed everywhere where spacers are localized. As should be evident from this account, the development of FLC technology within the Canon group involved a lot of surprises. The group had to discover, and subsequently solve, a number of very complicated problems related to the polar nature of smectic C* materials. Among these problems were the appearance of ghost and shadow pictures (slow erasure of an already written image), and in addition to these was also the problem of mechanical fragility of the bookshelf or chevron layer configuration. The most complicated phenomenon, i.e., the strange electrodynamic flow (backflow net mass transport) that started to occur at high pretilt when electric fields were applied, could not even be superficially dealt with within the frame of this text. Already the materials problems were quite important. The hard to achieve high pretilt condition required several years of dedicated R&D in polymer materials. Finally, the group had to develop their own FLC materials and design manufacturing processes capable of handling large cells [48 in. (120 cm) diagonal] at a cell gap of
9.1 The FLC Technology Developed by Canon
243
0.a.
'
glass plate
on nk
Figure 111. Geometric features in the Canon screen of Fig. 87 showing rubbing directions, smectic layer direction, chevron direction, and director pretilt a, equal to the optic axis tilt out of the screen surface.
Figure 112. Every spacer represents a distortion, which by necessity traps a thick wall. In this picture the chevron kink is to the left everywhere except in the local neighborhood of the spacer (after [ 1761).
\ \ \ \ \ -/lllv\\\\\\\\\\\\\\
.~
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9 FLC Devices
1.1 ym. Considering all this, the development andmanufacturing of the present FLCD is a real tour de force performed by Kanbe and his team. In 1997Canon presented a different FLCD prototype using recently designed materials free of chevrons. We will come back to this new development later (see Sec. 9.6).
9.2 The Microdisplays of Displaytech This is an interesting contrasting case, not only because it is the question of an extremely small display with completely different application areas, but also because the small size permits a different approach in almost every respect. In this case, the FLC layer is placed on a reflective backplane, which is a CMOS VLSI chip providing &2.5 V across each picture element [ 1771. This is thus a case of active driving. Therefore the question of the quality of the memorized states is unimportant, and we can safely use the C2 structure as the one with the simplest aligning technology avoiding any twisted state. (Displaytech uses rubbed nylon, which gives a low pretilt.) Because each pixel is fully driven and not multiplexed, we do not have the disadvantage of high voltage. We may recall the peculiarity of FLC switching in that the threshold is one of voltage-time area. Thus when the switching pulse time is replaced by the frame address time, which is now the relevant time for the applied voltage, the threshold in voltage becomes correspondingly low and well within the CMOS range. This is of course characteristic for all direct or “active” FLC driving. Furthermore, because each pixel is in its fully driven state, we have in-plane switching with its characteristic large viewing angle. A transistor can in principle be used for implementing microdomain grey levels by charge control (a method pioneered by the Philips Eindhoven group), but in the Displaytech case it is only used to write one of two states. The gray levels are instead produced by time modulation, which is quite natural for direct drive. Video color pictures are produced at a frame rate of 76 Hz. Each frame is, however, subdivided into three pictures, one for each color, so that the actual frame rate is 228 Hz, under the sequential illumination from red, green, and blue LEDs. (The LEDs are GaN for blue and green, and AlInGaP for red.) During each of the 4.38 ms long color sequences (corresponding to 228 Hz), a gray level is now defined for every pixel by rapidly repeating the scanning of the FLC matrix a number of times, with a pixel being on or off for the fraction of time corresponding to the desired level. For instance, if the VLSI matrix is scanned 25= 32 times during illumination of one color, then five bits of gray per color are achieved. This requires a scanning time (basic frame rate) of little less than 150 ps for a matrix consisting of 1000 lines. Displaytech has presented several versions of this miniature screen, for instance, one 7.7 by 7.7 mm with 256x 256 picture elements and an aperture ratio of 90%, and some with higher resolution of up to 1280x 1024. In the latter case the dimensions
9.3 Idemitsu's Polymer FLC
245
are 9.7 by 7.8 mm with individual pixel size of 7.5x7.5 pm. The display is viewed through a magnifying glass and may be headmounted. Obvious applications are virtual reality, high resolution viewers, but also projection displays which are good candidates for high definition television (HDTV) using FLC. The first version of this "Chronocolor" display had two bits per color giving 64 colors per pixel. This is the version illustrated in Fig. 88. The second had three bits giving 512 colors, followed by a version capable of six bits or (26)3= 262 144 colors per pixel, generated purely in the time domain. However, to go to full color video, i.e., 8 bits per color with 1024 lines, the time domain alone is not sufficient. This is discussed further in Sec. 10.2.
9.3 Idemitsu's Polymer FLC In 1985, around the same time as Canon started their FLC development project, another Japanese company, Idemitsu Kosan, started to synthesize materials with the goal of making a polymer version of FLC. There is a conflict in a liquid crystal polymer in the sense that a long polymer chain wants to be as disordered as possible, while the liquid crystal monomers want to align parallel to a local director. Nevertheless, it is possible to make both nematic and smectic structures and, in particular, chiral materials having an SmA* - SmC* transition. In one of Idemitsu's successful realizations, the main chain is a siloxane to which side group monomers (which would be normal FLC materials by themselves) are attached via spacing groups. Whereas a nematic polymer reacts too slowly to electric fields to be of any electrooptic interest, the internal cone motion can be activated in the polymer SmC* phase, with its characteristic low viscosity y, giving a (relatively speaking) very fast switching. At high temperatures the switching time can go below a millisecond, i.e., the FLC polymer (FLCP) can be faster than a typical monomer nematic; however, the viscosity increases much more rapidly at low temperatures than in a nematic. The FLCPs have many similarities to the monomer materials. By measuring the polarization reversal current [ 1771 Idemitsu found that the P, value is about the same (typically a little larger) as in the corresponding monomer. As the layer spacing decreases with decreasing tilt angle in the SmC* phase, this also results in chevron structures. In thick samples the smectic helix is present. In thin QBS cells FLCPs show good bistability, and so on. The switching mechanism is in principle a little more complicated in the aspect that the main chain to some extent takes part in the switching. A simplifying aspect is that the polymerization may stabilize the SmC" phase, which may then have a very broad temperature interval (something that in the monomer case has to be made by a multicomponent mixture). The polymer state also seems to stabilize a major chevron direction such that zigzag defects are not so important. Nor has a QBS cell in the polymer version the mechanical fragility that is so characteristic of monomer FLCs.
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9 FLC Devices
-plasticfilm (1W1Un)
(4
Figure 113. (a) Cross-sectional view of the allplastic 0.5 mm thick Idemitsu FLCP display showing the SmC* polymer (black) between the substrates. (b) This shows how in its preparation the display is first laminated and then shearaligned (after [ 1781).
Figure 114. The FLCP display can be bent and shaped to a different curvature. The bending radius can go down to 5 cm without impairing the optical quality. The bottom picture also demonstrates the transmissive mode. (Courtesy of Idemitsu Kosan Co., Ltd.).
On the other hand, the differences are highly interesting. An FLCP display is flexible, light-weight, and only 0.5 mm thick. It is made by spreading the polymer onto an ITO-coated plastic substrate, which is then laminated in a continuous process to a second substrate (see Fig. 113) using rollers, one of which in its bending action shear-alignsthe smectic. The display thus needs neither spacers nor alignment layers,
9.3 Idemitsu’s Polymer FLC
247
and can be made very large, in principle “by the meter”. Its flexibility is demonstrated in Fig. 114. Evident future applications for such panels that can be bent and are extremely lightweight are, for instance, in electrically controllable motorcycle or welding goggles. However, as already demonstrated in Fig. 89, a switching time of 1 ms at room temperature is also sufficient for fairly sophisticated static displays. Although the switching is between a hundred and a thousand times slower than in a monomer FLC, 1 ms is sufficient to give this display an update rate of 2 Hz. Two examples of simple large size color panels are given in Fig. 115. A recent development taking advantage of the 1 ms switching speed (at f 15 V) is a large area shutter for producing 3D TV images [178a]. It is a 32 in (80 cm) diagonal polymer sheet of 16 :9 aspect ratio to be put in front of a corresponding TV. It has one polarizer (next to the TV), and by switching its optic axis 120 times a second, it produces 60 Hz pictures at orthogonal polarization states to the left and right eye equipped with normal passive orthogonal Polaroid glasses. The contrast ratio of this shutter is 100 : 1 and the light transmittance 30 percent.
Figure 115. Two examples of simple color FLCP panels. The sizes are 20cmx32cmand 12cmx60cm, respectively. The color is produced by vertical color stripes giving eight colors per pixel. These 0.5 mm thin displays are mechanicallydurable and work at very low power thanks to the inherent memory. The picture can be changed twice a second. (Courtesy of Idemitsu Kosan Co., Ltd.).
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9.4 The Stuttgart Technology A new FLC technology which has some similarities in appearance with the one described in the last section, yet is basically very different, stems from the university of Stuttgart, Germany. The Institute of Network and Systems Theory at that university, headed by Ernst Luder, is a fairly unique place in the academic world with processing laboratories almost matching those of top industrial companies in the liquid crystal and semiconductor areas. In the last decade Liider and his collaborators have accomplished several remarkable developments, some of which will be described below. It is the question of thin, flexible displays working in reflective mode, controlled by voltages below 3 V and using the bistability in order to have zero consumption of power between the instances when the information is being changed. Being different to the Idemitsu technology, these displays do not use an FLC polymer but a regular monomer FLC mixture. But instead of being confined between glass plates as in conventional LCDs, the liquid crystal is enclosed between 100 pm thin plastic substrates. A conventional LCD the size of a normal letter, which typically would weigh 150 g, in this plastic version would also have the weight of a letter, or about 20 g. These reflective plastic FLCDs neither require a battery for backlight as they work with ambient light, nor do they need power to display their information. However, the additional advantages are no less interesting. Including polarizer and backreflector they are only 0.35 mm thick and can thus easily be integrated in smart cards, ultrathin card calculators, mobile telephones, and other portable equipment, as well as in ultrathin devices displaying book pages or newspapers. Because they are much thinner than displays with glass substrates they are also free from the shadow image usually seen in reflective displays with glass. This point is certainly important for the readability of pages written in high resolution. Even more remarkable - because for a long time FLC displays were notorious for their mechanical fragility - these displays are virtually unbreakable and can be bent and twisted. They can sustain any shocks and very high point pressures. You can walk on a smart card made with this FLCD without damaging it. And you can bend it with a radius of curvature as small as 1.5 cm without damaging its functions, as shown in Fig. 115a. This also means that it can be used in curved panels to suppress certain reflections or designed into bent shape as desired in goggles and eye shields, if used in transmissive mode. As would be expected, the material requirements put on plastics suitable for use in these FLC displays are quite stringent. They are discussed at length in [178b] which gives an overview of liquid crystal displays on plastic substrates, including both passive and active matrix displays. The passively driven bistable displays are described in detail in [ 1 7 8 ~ 1which discusses the two established technologies, using either chiral smectics or cholesterics, and makes a comparison between the two (see below). As for the plastic substrates, there has been a rapid development over the last decade, which is probably far from finished. Among the materials tested have
9.4 The Stuttgart Technology
249
been polyimide (PI), polycarbonate (PC), polyarylate (PAR), polyestersulfone (PES) and polyethylene terephthalate (PET). PET has low absorption in the UV and can be used with UV-cured sealants, while PC cannot. PC is cheap but not very stable chemically and has a glass temperature (155 "C) which is too low to permit but few variants of TFT processing. PES is more expensive but, with a Tg=225 "C, allows process temperatures of up to 180 "C. Plastic foils of these materials are today commercially available already covered with necessary barrier layers and coated with ITO. They have a surface roughness of typically 6 nm and waviness of less than 50 nm. Their transparency must be higher than 90% and, when they are used with polarizers as in the case of FLCD, they have to be practically isotropic, which means that the film has to be manufactured and handled in a stressfree way, such that their optical retardation And is less than 15 nm. PES is available with And = 10 nm. The lower limit d = 100 pm for the film thickness is determined by the fact that, at least presently, thinner films are too difficult to handle. The Stuttgart FLCD prototypes so far are small, typically less than one inch in diameter, in contrast to the Idemitsu large area panels. In most of them the C l chevron mode is used with even higher pretilt than in Canon's screen. However, in small displays it is easy to achieve a high pretilt a by Si02 evaporation under very oblique evaporation direction. In order to give an effective switching angle of 45" the pretilt has to be about 32" which necessitates an evaporation direction almost perpendicular (70-80") to the surface normal. FLCDs of conventional type often require addressing voltages in the range 515 V. However, this very much depends on whether the display is directly or actively driven, if multiplexed or not, and further on addressing schemes and cell parameters. In general, it strongly depends on the pulse length and here different regimes can be distinguished. As is well known, a peculiar thing with the FLC switching threshold is that this threshold is not in the field or voltage but rather in the pulse area, i.e., there is a critical value in the product of voltage and time or, more generally, a critical value in the integral v(t>dt
(415)
This means that if we apply very short pulses the voltage level required to accomplish latching in the bistable state will be high. We will not go into any detail, because it also depends on addressing scheme, pretilt, cone angle, coating material, and many more things. Nevertheless, some of the Stuttgart results [ 178c] may illustrate the point. In a cell of the lund shown in Fig. 115a we can, in multiplex mode, switch between the bistable states with pulses -t3 V of 10 ms length at room temperature. Furthermore, this can be done in a fairly broad temperature range. If we narrow the pulses to 6 ms we find that 6 V would accomplish the switching quite well, but the relative margin is somewhat lower. It works for about 6 -t 2 V, i.e., with about 30% margin and the temperature dependence is slightly more pronounced. For pulses of
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9 FLC Devices
Figure 115a. Photograph of a plastic FLCD working in A14 mode. The cell gap is 0.8 pm and the thickness of the whole display is 350 pm. It is here bent 270" around a bottleneck with a bend curvature radius of 1.5 cm. The contrast is 24 : 1. (Courtesy E. Liider.)
Figure 115b. Example of smart card incorporating a reflective ferroelectric liquid crystal display needing no power supply. The display is the visible interface between the IC electronics on the card and the card-holder and permits communication with bank or telephone terminals. It may, for example show identification codes, error codes, cash value, available telephone units, and other kinds of information. (Courtesy E. Luder.)
1 ms length this multiplex driving works well for about 15 & 3 V, i.e., with a relative margin of about 20%. But now there is a quite pronounced temperature dependence and this driving no longer works at 30 "C.Here the voltage has to be lowered to about 11 V and at 35 "C it has to be centered around 10 V with a margin of k 2 V. Such a driving would therefore require electronics with built-in temperature compensation, which is of course undesirable. On the other hand, if we drive an FLC display directly from a small power source, the broader pulses which can now be afforded bring down the voltages to the range 2-4 V and, especially important, removes essentially the whole temperature dependence. Both things are absolutely necessary in applications like, for instance, smart cards as illustrated in Fig. 115b, which should be compatible with existing card and terminal standards and have to be electrically addressable from the built-in CMOS chips using voltages below 5 V. The Stuttgart demonstrators have been developed both as transmissive and reflective displays. While the former permit higher contrast and have evident applications for light-weight curved goggles and similar things, the reflected type is, together with the corresponding Idemitsu reflective displays, perhaps even more unique as an extreme low-power technology. These plastic FLCDs have been developed in two va-
-
25 1
9.4 The Stuttgart Technology Polarizer
incoming linearly polarized light
[ 0 U
-z=d2 ----- - -- -nd=h/.l
(
+{
eliptically polarized light circularly polarized light elliptically polarized light
mirror
U4-cell
Figure 115c. Optics of reflective FLCD in iV2 mode (left) and iV4 mode. In the first we have crossed polarizers and a cell thickness corresponding to a half-wave plate. In the dark state the optic axis (n)is along the front polarizer. The light is not influenced by the liquid crystal and is extinguished by the analyzer. In the bright state the optic axis is switched to be midway between the crossed polarizer directions. The incoming light then changes its state from linear to elliptic, circular (in the middle), elliptic but now in the analyzer direction and is admitted as linear vibration as it exits through the analyzer (the normal l%C mode). The drawings are only symbolic as the vibration is not in the plane of the paper but perpendicular to it. The cell works in transmission as well as in reflection with a mirror placed behind the analyzer. If the cell gap corresponds to a quarter-wave plate (right) between one polarizer and a mirror, the bright state is now the one with the optic axis along the polarizer. If n is switched out 4.5" (bottom right) the polarization state will change to transverse after the double passage through the cell and the light is extinguished by the polarizer (adapted from reference [178e]).
rieties: ;1/2 and A/4 panels, the latter being of particular interest. How they function is explained in Fig. 115c. The A/;?cell uses two crossed polarizers on the two sides of the FLC, here called polarizer and analyser, and a diffuse reflector behind the analyzer. In the dark state (E=O) the directors in the bookshelf stack are aligned along the polarizer direction. Say that the polarizer direction is along (1) in Fig. 61, whereas the analyzer direction is perpendicular to it. The polarized light therefore traverses the FLC without being influenced and is extinguished by the analyzer. Now a switching pulse (with E >Ec) brings the directors to the bright state in which the directors have been switched away from the polarizer direction to (ideally) the direction which bisects the angle between polarizer and analyzer. If the A/2 condition is fulfilled (which cannot strictly be true for white light) all light would therefore pass the FLC film and further through the analyzer to the reflector. During its way through the cell, the light continuously changes its polarization state as depicted to the left of Fig. 115c: the vibration starts out parallel to the polarizer, then becoming elliptic (with the main axis still along the polarizer direction), then in the middle of
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9 FLC Devices
diffuse reflector ( ~ 0 . 8 5 )
diffuse reflector
Figure 115d. Estimation of the available contrast in the one-polarizer A/4 reflective FLCD. The polarizer is assumed to transmit in the extinction direction, 90% in the admittance direction, and to have a 40% efficiency of transforming unpolarized light into planepolarized. (The theoretical limit of 50% may in extreme cases be better approached and be about 43 or 44%.) (Adapted after reference [178c].)
the cell ( z = d/2) circular, after which it develops ellipticity with the main axis perpendicular to its initial direction and finally exiting as linear without loss through the admittance direction of the analyser. A smarter way to do the same thing is, however, to cut the cell in half and place a mirror at z =d/2. This transforms the cell to a A/4 cell with only a front polarizer. In the former dark state, the FLC, as before, does not influence the polarization state of the light which is reflected back and is now admitted by the entrance polarizer. This dark state thus has now become a bright state. In the switched state, however, the polarization has become circular as it encounters the mirror but is now propagated back through the cell, but now with opposite handedness. It therefore exits the A/4 cell being linearly polarized but with the vibration direction perpendicular to the original direction. That is, it will be extinguished by the front polarizer. An alternative way to describe this state of affairs is shown at the bottom right of Fig. 115c. The initial vibration along the polarizer is split in two components along the rt direction and perpendicular thereof. These two components propagate with different velocities and have a phase difference of A/4 when they meet the reflector where one of them suffers a phase shift of 180" or A/2. This means that the resulting vibration is perpendicular to the polarizer and is going to be extinguished. We can try to examine the achievable contrast of such a device in the following way, cf. Fig. 115d. Let us assume that the polarizer used is a common laboratory polaroid sheet. It has a transmission coefficient as a single polarizer of 40% for unpo-
9.4
The Stuttgart Technology
253
larized light, of 90% for polarized light in the admittance direction and lo-'% in the extinction direction. Now considering light traversing the display vertically in the figure, of the total incoming ambient light 0.9% is first reflected off the front polarizer which has a transmission coefficient o f t = 0.4, permitting 40% of the light, now linearly polarized, to penetrate the FLC. The FLC slab is not an ideal retarder, especially because of the chevron structure which means that the optic axis has some variations and is not entirely in the direction along the vibration of the light. Therefore 100% of the light is not transmitted but rather 86%. There remains 34% of the incoming light and after the reflector and the second passage 29% and 25%, respectively. The second passage of the polarizer then finally leaves 22.6% of the incoming light reflected back from the display in its bright state. In the dark state we have 25% reaching the polarizer as before but now in the extinction direction, leaving 25 . lop5or 0.002%. Thus in the dark state the effective light is the 0.9% reflected off the polarizer at the entrance. The contrast of the reflective display is therefore 22H0.9 or 25 : 1 , which is very close to what the Stuttgart group measured (24 : 1). For a reflective display this is a high value. From our crude estimation we can learn that the maximum reflective contrast is mainly limited by the reflection from the front polarizer. If we want to improve contrast and brightness we have to choose a polarizer with low reflectance and high transmission. The absolute limit in transmission of present sheet polarizers is considered to be about 44%. If we use this figure in our estimation above we arrive at 24.9% reflected light instead of 22.6% or a contrast raised to 24.9/0.9 = 28 : 1. The effect is not big enough to justify a change in polarizer. On the other hand, we see that if we could, by an antireflection coating, bring down the reflectance by a factor of three, the contrast would go up to 22H0.3 or 75 : 1. This is therefore the way to proceed. By going from the A/2 to the A/4 mode one can dispense with one polarizer and thus increase the brightness. This was measured to 32% of the standard barium sulfate white reflector, instead of 18% for the two-polarizer display [178e]. The A/4 mode requires a cell gap of 0.8 pm instead of 1.6 pm. This can conveniently be provided by etching a regular pattern of spacers in the 100 pm plastic foil. This spacer technology further increases the impact strength so that the display resists a point pressure of at least 125 N cm-*. Because of the very high pretilt, 32", the contrast has some dependence on viewing angle and is slightly butterfly-shaped. This shows that there are still improvements which could be made, in addition to lowering the reflectance of the front polarizer. New FLC materials, to be described in the next sections, will allow such improvements. These materials automatically go into quasi-bookshelf instead of chevron structure and can be used with a = 0. Furthermore, the birefringence is half of the value for conventional materials, whence the cell gap can be made wider in order to facilitate manufacturing. In addition to FLC, cholesteric materials also permit reflective displays showing a kind of bistability or rather quasi-bistability, because it is not at all symmetric (the
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9 FLC Devices
display cannot be switched back and forth between the two states) and has an entirely different origin. In the cholesteric state it is rather a question of being in the “written” or the “erased” state. Anyway this is a highly interesting and promising technology for many applications. Its main great advantage is that it works without polarizers. The cell gap is typically 4-5 pm which makes manufacturing a lot easier. The Liider group has developed both smectic and cholesteric bistable technologies and also makes a comparison of the two. The interesting thing is that the cholesteric cells also show mechanical fragility. In fact, they are more sensitive to both pressure and bending than FLC cells. This, together with the high driving voltage, about 30-35 V, makes them unsuitable for applications like smart cards, card calculators, and anything requiring CMOS compatibility.The k 2.5 V addressability together with the writing speed gives the FLC advantages in a number of areas. We may close this section by giving an illustration (from the Ph.D. thesis of R. Burkle) of the at first very surprising fact that the FLC displays become mechanically more resistant when thin plastic substrates are used rather than glass. The explanation is given in Fig. 115e. To the left is shown the case with glass substrates, which are always thick and rigid. When they are bent as shown, the two aligning surfaces next to the FLC will exert a powerful shear that will cause a flow across the smectic layers (which are originally standing perpendicular to the glass plates). This destroys the bookshelf order irreversibly. In order to illustrate the amount of shear corresponding to a certain bending curvature, the neutral lines (which keep their length) have been drawn in the middle of the plates. The shear deformation in the plates is considerable, due to the large distance D + d between these lines. In the plastic case, to the right, this distance is about one order of magnitude smaller. In addition, the plastic yields elastically to the bending action. Furthermore the sticky spacers are here helpful (which they are not in the glass case) to make the shear deformation as small as possible. The result is that the FLC layer is protected from excessive shear until the bend curvature grows very large. If the flow is thus prevented, the smectic layer ordering persists.
.D+d.
D+d
Figure 115e. Bending a display with glass substrates (left) immediately transfers a shear to the liquid crystal layer in between. Smectic layers in bookshelf geometry (perpendicular to the glass surface) are therefore ripped off from this order and the structure is irreversibly destroyed (has to be realigned).In the case of the thin plastic substrate, the confining material suffers a much smaller shear and also yields elastically, so that no shear is exerted on the FLC layer until the curvature attains a considerably higher value (after reference [178dl).
9.5 Material Problems in FLC Technology
255
9.5 Material Problems in FLC Technology FLC materials are complex and we are just at the beginning of exploring their physics and chemistry. As we have mentioned, they have nine independent elastic and 20 viscous coefficients, to which we may add that there are 15 different independent sources of local polarization (10 if we assume incompressible layers). One of them is related to chirality, whereas all the others are flexoelectric coefficients (thus equally relevant to the nonchiral SmC phase) describing local polarization resulting from the different deformations in the director field. With a local polarization vector and with complicated smectic layer structures, it is evident that we will also have an extremely complicated hydrodynamic coupling to an external electric field E. These hints may suffice to indicate the completely new level of complexity in these materials. However, already the fact that they are not three-dimensional liquids like nematics, but rather two-dimensional liquids with solid-like properties along one space dimension, makes them extremely fragile in their useful geometry - a pressure by the thumb is in practice sufficient to irreversible destroy the ordered bookshelf structure. On the other hand, the wealth of physical phenomena and the richness of potential effects to be exploited in future in these materials should be recognized. The molecular engineering of chiral tilted smectics is correspondingly complicated with a large number of parameters to optimize. In Table 1 we have listed the most important (so far recognized), in comparison with the standard nematic parameters. Beside phase sequence and sufficiently large phase range (not minor problems), we have to be able to control all the parameters with which we have already made acquaintance. The elastic constants can be taken as an example illustrating the relative progress on the way to full control of the materials in the nematic and in the smec-
Table 1. Parameters for nematics and smectics C*. Material requirements Phase range
Preferred phase sequence Birefringence Dielectric anisotropy Dielectric biaxiality Twist viscosity Elastic constants Tilt angle Spontaneous polarization Smectic C* pitch Cholesteric pitch Smectic layer thickness
Nematics
Smectics C*
(Cr) - N- (I)
( 0 ) - SmC - (SmA) (SmC) - SmA- (N) (SmA) -N - (I) N-SmA-SmC
An
An
A&
A&
a& P Z
@)
P
4T )
256
9 FLC Devices
0 0 T
~
T T <~T~~ ~
0 T < < T ~ ~
Figure 116. Two possible mechanisms for having a tilted smectic phase with a temperature-independent smectic layer thickness below the SmA-SmC transition. In (a) the end chains are first disordered but get more straight as the temperature is lowered; in (b) each individual molecule keeps its tilt constant relative to the layer normal, but the azimuthal direction of the tilt is unbiased in the SmAphase. At the SmA-SmC transition the average direction of the molecules begins to get biased.
tic case. Whereas in nematics we can balance and adjust the three Kij values to suit any particular application, the relevant elastic constants have hardly been identified yet in smectics, and even less measured - there have been much more serious problems to take care of first, for instance, the matching temperature behavior of 8 and 6 (layer tilt) mentioned earlier in the description of the Canon C1 technology. The most serious problem was initially recognized alongside the growing understanding of the relation between zigzag defects and chevrons and is connected with the parameter d(T) listed earlier, i.e., the smectic layer thickness as a function of temperature. And here is the challenge: to synthesize and mix materials to form a smectic C that keeps its layer thickness constant in spite of an increasing tilt angle 8 below the SmA-SmC transition. This would revolutionize the further development of FLC technology by avoiding all complications with chevrons and finally achieving the initially conceived structure of Fig. 6 1, with superior brightness, contrast, and viewing angle. That such materials could even be imaginated to exist is illustrated in Fig. 116. In this very simplistic picture (which may or may not be realistic), let us assume that (a), as the central aromatic part (essentially representing the optic axis of the molecule, and therefore 6) tilts more and more away from the layer normal on lowering the temperature, the aliphatic end chains get less disordered and stiffen, making them effectively longer. Various degrees of interdigitation (decreasing with higher tilt) of molecules in the layer could also be imagined for materials in which the layer corresponds to more than the length of one molecule. A different mechanism (b) goes back to early ideas of the crystallographer de Vries at Kent State University, who proposed that the molecules could already be tilted in the smectic A phase, but in an unbiased rotation making the phase uniaxial [191a]. At the SmA -SmC transition, a rotational bias appears, growing stronger on lowering the temperature, which is equivalent to a tilt of the optic axis. This tilt would then appear without being accompanied by a decrease of the layer thickness. Today we know that the majority of liquid crystals undergoing an SmA-SmC transition do not behave according to the de Vries model. This would seem to require a very low degree of interaction between the constituents from layer to layer, or even
9.6 Nonchevron Structures
257
within layers, i.e., a high degree of randomness, yet connected with smectic order. However, now and then, evidence has been reported indicating that some materials may behave like this. They would not only be of eminent interest for FLC applications, but just as much for use as electroclinic materials.
9.6 Nonchevron Structures At the Japan Display 1989 Conference in Kyoto, two contributions independently reported on new materials which seemed to almost automatically give a defect-free alignment, avoiding the zigzag defect. The first one, a collaboration between Fujitsu Laboratories, Mitsui Toatsu Chemicals, and Tokyo Institute of Technology [ 1791 presented mixtures based on the naphthalene structure
with n = 9 and 10 and m = 2 - 6. The second contribution [ 1801 was from the central research laboratories of the company 3M in St. Paul, Minnesota, U. S. A., and showed the result of fluorination on a number of more conventional liquid crystal structures. The naphthalene compounds have since been used in research as well as in some Fujitsu FLC display prototypes. They are highly interesting, but suffer from a rather high viscosity. In the 3M research the starting point was 4’-alkyloxyphenyl4-alkyloxybenzoates
or corresponding thiocompounds
which were partially fluorinated to structures like
258
9 FLCDevices
This partial fluorination has a number of interesting and even surprising effects [ 18 I]. First, it almost always totally suppresses the nematic phase and strongly en-
hances the smectic phases, in general both smectic A and smectic C. Moreover, it lowers the birefringence of the liquid crystal: An may go from typically 0.20 to 0.07 or 0.10. It also makes the smectic C* phase less hardtwisted: the helical pitch may increase from about 2 pm to 5 - 10 pm. Another important effect is that it makes the smectic C phase occur at a shorter chain length than in the unfluorinated analogs. This means that fluorinated SmC* is faster as an FLC material; it has remarkably low viscosity. But first and foremost: these materials do not make chevrons. This means that the smectic C layer thickness must be essentially temperature-independent. Partial fluorination leads to large terminal dipole effects. The fluorocarbon tail is also stiffer than the hydrocarbon tail. It is therefore relatively easy to understand the suppression of the nematic phase and strengthening of layered phases. However, it is less clear, as also in the naphthalene case, how to explain why the layer spacing should be temperature-independent. In both materials there seems to be a tendency for molecular interdigitation, and thus the smectic layer is thicker than the molecular length. This may be important, but we still have no precise molecular interpretation of the resulting effect. The 3M synthetic research in SmC* materials began in 1983 and has been pursued with a high degree of purposefulness. (It is thus not true that only Japanese companies invest in long-term research goals.) A number of pure compounds have been developed that exhibit almost continuous layer expansion on lowering the temperature in the smectic C phase, as well as in the smectic A phase. By mixing with other compounds, a practically temperature-independent spacing can be achieved. In addition to phenyl benzoate cores, 3M has largely used phenylpyrimidine cores, of which a general structure can be exemplified as [1821
As an example of how different materials behave, we compare their smectic layer spacing as a function of temperature in Fig. 117. For the Chisso-1013 mixture the decrease in the layer thickness is pronounced as soon as we enter the smectic C phase.
259
9.6 Nonchevron Structures
36 -
Figure 117. Smectic layer spacing for a number of different FLC materials as a function of temperature. The SmA-SmC transition is indicated by a vertical bar. Chisso CS- 1013 is a mixture showing conventional behavior, FA-006 is a Fujitsu naphthalene-based mixture, 3M-C is the pure pyrimidine compound with formula shown below, whereas 3M-A is a mixture (after [1831).
3
34-
00
.-
8
32-
8
3
300
28
-
Chisso CS-1013 261 0
,
I
20
,
I
40
,
I
60
,
,
80
I
100
,
0
T Wl
The naphthalene mixture FA-006 has a behavior that i s qualitatively similar, though much less dramatic. The fluorinated compound 3M-C, of formula
shows the opposite behavior, with a small, but characteristic negative layer expansion coefficient, and the mixture 3M-A i s qualitatively the same with a weaker temperature dependence. Although the 3M materials lack a nematic phase and thus have the phase sequence I-SmA-SmC, they align fairly well on nylon 6/6 to a bookshelf structure with a small pretilt at the surface. They switch symmetrically according to Fig. 5 1 and it has been confirmed that the switching angle corresponds to the full cone angle. In principle, therefore, with a 8 optimized to about 22" they should have memorized states corresponding to the ideal switching angle of 45" (in comparison see Fig. 109, illustrating the "forced" C2 switching states). Although it must be a particularly hard task to develop such materials with all other parameters optimized, it seems that they are now very close to being used in FLC panels. Figure 118 a shows a Canon prototype (not yet commercialized) which evidently is the first attempt to put chevronfree materials into a display. The lower birefringence of these materials has permitted the enlargement of the cell gap from 1.1 pm to 2.0 pm, which is an important advantage in manufacturing. The non-chevron structure has more than doubled the contrast, to about 100 : 1 and raised the brightness from 80 to 120 cd mP2, as compared with the first color panel shown in Fig. 87. The display has XGA resolution (1024 x 768) where every pixel is 300 pm x 300 pm and consists of three RGB stripes, each of which is subdivided into four dots. This gives 16 different color shades per pixel. Using two subframes, as described in more detail later (Sec. 10.2) this nominally gives about 16 million hues. This "digital full color" display is interesting be-
260
9 FLC Devices
cause it demonstrates that a binary technology (based on just two states, i.e., without any gray shades whatsoever) is capable, with this simple dither technique, of quite satisfactory rendition of full color pictures. The condition for this is that the spatial resolution is sufficiently high. Of course, in the limit of the highest resolution, not even temporal dither is needed, just as in newspapers and magazines. Compare also the limiting case of the silver grains in a photographic film. Hence the photographic film’s “analog” rendition of grey levels has “digital” character on a microscopic scale. The materials development is far from finished in the case of chiral smectic materials. The observation of highest importance for smectic C materials now relates to the property of a negative layer expansion coefficient 6 ., In order to significantly increase the useful temperature range for high performance FLC displays, especially the lower end (for shipping by air a SmC* phase stable down to -40 “C is desirable) we have to increase the available pool of molecular structures with this property of negative yd. This is an important but difficult task for synthetic chemists in cooperation with physicists, and may still demand a long time. With the most critical problems now being understood or even solved, several groups are attacking the problem of sufficient depth in the rendition of gray levels. At least this could relatively soon be approached by going to a higher resolution, of which FLC is eminently capable. Another approach is active matrix driving, which uses charge control, and which may obviate the use of the very expensive color filter by use of color sequential backlighting. Other methods have been proposed which use amplitude control in the same kind of symmetric driving as known from antiferroelectric displays [ 1841, eventually in combination with color sequential backlighting of high P, QBS structures [ 1851. Finally, special addressing methods are being worked out using a number of different physical principles. Some of the most interesting of these are represented by the so-called binary group addressing method [ 186, 1871, combining a time-dithered gray scale with a substantial extension of the illumination period to nearly over the entire frame. Once the FLC devices are on the market, we may expect a diversity of techniques and a rapid development into different directions.
9.7 The Sharp Video Prototype After about ten years of active involvement in FLC research, Sharp showed their first prototype in 1998. This is a 17 in., full color video screen, developed in a collaboration between Sharp Japan, Sharp Europe (Oxford), and DERA in Malvern, Great Britain. It uses the British electronic addressing scheme originally developed in the JOERS/Alvey program and already referred to in Sec. 6.8, together with a material brought into a uniform C2 chevron structure. A photograph of the panel is shown in Fig. 118 b. It has 720 x 9 16 picture elements and is operated at a duty ratio of 1 : 360,
9.7 The Sharp Video Prototype
26 1
i.e., the electronics are doubled to scan each of two halves of the screen simultaneously. At the operating temperature of 35 "C, the strobe or scanning pulse voltage is 30 V and the data pulse voltage 6 V across the FLC cell gap which is 1.3 pm. The two-slot addressing scheme then gives a line address time of only 12 ys. This high speed permits generation of a grey scale of sufficient depth mainly in the time domain. In fact, two bits of spatial dither are combined with four-bits of temporal dither [ 187al which gives (nominally) eight bits of grey per color or about 16 million colors. This is discussed in detail in the next chapter. Actually, all these hues cannot be updated at video speed because scanning 360 lines at 12 ys/line corresponds to about 4 ms frame time or 240 Hz which is sufficient for using eight frames (rather than 15) to write one picture. Thus, some of the hues are updated at a little slower rate. Nevertheless this combination of spatial and temporal dither gives an excellent color rendition. The Sharp FLCD is the first passively driven full scale screen showing full color video performance. To be sure, the Canon prototype described in the last section would do the same if it were driven from two sides so that only 384 lines were scanned instead of 768. (In the Canon 2 ym gap prototype the line address time is 36 ps. Scanning 768 lines and using two frames to write a picture gives 18 Hz update rate.) The main problem with the Sharp technology is certainly the high voltage required by the z(V)-min mode. Another serious problem is the quite pronounced temperature dependence of the z(V) characteristics, which can be seen from Fig. 118. A similar temperature dependence is a major problem also for the Canon screens and, generally, for all FLC displays which are passively driven.
PULSE WIDTH
20 10 5
1
l o 2o 30 40 50 PULSE VOLTAGE (V)
Figure 118. Memory pulse width (p) as a function of applied pulse voltage (V) for the mixture FDS-71 used in the Sharp FLCD. The material has a phase sequence N*-A*-C* with a P, value of 12 nC cm-' at 25 "C.At the operating temperature, 35 "C, V is equal to 30 V for r,,, equal to 6 ps. However, as can be seen, the r(V)-min characteristics has a very strong temperature dependence (from reference [ 187al).
262
9 FLC Devices
Figure 118a. Canon’s 1997 “digital full color” prototype is important in its demonstration that color pictures can be produced utilizing only two electrooptic states. The screen is here shown with two successive close-ups, the last of which with sufficient magnification to show the individual pixels. (Courtesy of B. Stebler.) The SSFLC structure in this panel is nonchevron and corresponds to Fig. 61.
Figure 118b. A video rate full color image picture of the Sharp 17 in. FLCD (courtesy of M. Koden, Sharp Corporation, Chiba, Japan).
10 Digital Grey and Color 10.1 Analog versus Digital Grey Although a ferroelectric intrinsically offers a binary technology, it is a necessity, for display applications, to generate, by some method, a grey scale of sufficient depth. In the limiting continuous case we talk about analog grey. However, much of what we normally conceive as analog is basically binary on a different scale. In the last section we made a comparison with the half-tones generated in a photographic film. Clearly, all these intermediate levels, which we perceive as continuous, are in reality discrete if we look at them under sufficient magnification, down to the silver grain size. The corresponding things in the FLC are the ferroelectric microdomains, which could give a quasi-continuous grey scale as we have discussed in Sec. 8.4.The size and shape of these domains sensitively depend on liquid crystal material, surface coating, rubbing conditions etc. An example is given in Fig. 119, showing two electrode stripes 100 pm wide, separated by equally wide areas of electrode-free glass. When activated by positive or negative pulses the liquid crystal in the electroded ar-
Figure 119. Microscopic observation of FLC switching in a cell with parallel1 stripes of conducting electrodes going horizontally in the picture and alternating with regions in which no field has been applied. In those, virgin, regions the spontaneous ferroelectric domains are visible as white and black microdomains of different size. The size distribution can be judged from the stripe width which is 100 pm. Sufficiently strong positive or negative pulses drive all microdomains into one uniformly white or black domain in the active areas. These homogeneous areas persist at zero field, with practically unchanged transmission, in the case of good bistability (after reference [96a]).
264
10 Digital Grey and Color
Figure 120. Illustration of the changed transmission state of an FLC pixel on applying voltage pulses of a sign such that they switch from bright to dark. Individual domains may typically be of the size 5 - 10 pm in cross section. From left to right the effective voltage level is 5 , 9 , 10, 11, and 15 V, which illustrates that the accessible window for brightness level variation is very narrow in passive addressing. In order to work, the method requires that all pixels, prior to the selection period, are reset to the same state by a sufficiently strong blanking pulse. This difficult method turns into advantage when each pixel is controlled by a transistor in a TIT matrix. By controlling which fraction of the pixel surface i s in the P up (+P) or P down (-P) state, the averaged transmission of the pixel can be very well controlled. This gives a quasi-continuous grey scale free from hysteresis effects (after reference [ZOS]).
eas goes into either of the two bistable states, seen as black and white. Between the electrodes, one can observe the same states in form of the spontaneous microdomains (virgin state of the liquid crystal). As in a photographic film, these domains can, in principle be used for a spatially generated grey scale. The first attempts [96a], [ 1741 were however unsuccessful. The idea is to use the smeared-out threshold due to the graininess of every pixel so that the level of the effectively applied voltage determines the average transmission state, as illustrated in Fig. 120. A photograph of the different transmission levels across a region of neighboring pixels at multiplex addressing is shown in Fig. 8 of reference [174]. Although the principle seems to work if every pixel is reset to the initial state between selections, it is limited in depth and too difficult to implement in practice. It shows both lack of reproducibility in time and local variation in space. However, if we could efficiently control these domains we would have more grey levels than anybody could use, i.e., we would have a perfect analog rendition of half-tones. The pioneering work of Philips [ 1651, [206], [207], partly continued by Sony [208] and by the Elston group [209], has shown that this might possibly be realized in the future. In Sony's method submicroscopic particles are mixed into the FLC material in order to serve as nucleation centers for controlled domain growth. A similar method consists of mixing in polymerizable material which after UV exposure gives submicroscopic dispersed polymer nucleation centers. However, at present, these methods have not been judged sufficiently practial to compete with other methods in passive driving. Philips has, however, convincingly shown that this control works very well in active driving, i.e., when each pixel is commanded by a transistor, which by charge control can precisely decide which area fraction of microdomains should be set in one of the two states, ovenuling both hysteresis effects and local variation of switching threshold over a large display.
10.2 Spatial and Temporal Dither
265
Consider a pixel which is fully switched into one state, say +P,, where P,, the spontaneous polarization density of the FLC is the dipole moment per unit volume, equivalent to charge per unit area. In this state, one of pixel electrodes has the surface charge density +o = P, whereas the opposite electrode has the charge -0.In order to reverse the state of the pixel we thus have to transfer the charge Q, = 20A = 2 f'$A between the electrodes, where A is the surface of the pixel and Q, here stands for the saturation charge. The important point is now that the transistor is current-controlled. Thus during the selection period of the addressing process we can transfer a predetermined fraction Q of Q, to the pixel and thereby switch the fraction SIA of the total area of the pixel. The transistor charge control is essentially insensitive to local variation of voltage threshold for switching (which may vary across the screen due, for instance, to small variations in cell gap). In passive driving the varying threshold would determine a grey level which is ill defined, as it is extremely sensitive to all variations in cell parameters. If we allow active driving there are, in addition, a number of ways in which ferroelectric or antiferroelectric materials could give an excellent analog rendition of grey. In principle, all director field configurations n(r) of these materials would do this, in which they do not show two stable states, i.e., in which they exhibit a dielectric response. The twisted smectic in Fig. 91 is one example to which we will return later. At this moment, however, we will limit our discussion to passive matrix displays. Antiferroelectric materials here offer a very convenient analog grey scale which will be discussed at some length in Chap. 13. As for passive FLC displays we have to resort to dither methods.
10.2 Spatial and Temporal Dither Dither methods use a subdivision of space or time to write a picture. Normally both space and time are subdivided, because it is most efficient, but we also encounter purely spatial or purely time dither, which might be simpler to implement, depending on the application at hand. Pure time dither gives the structurally simplest pixel, even without color filter mosaic if red, green, and blue light flashes are used for illumination, Fig. 121a. That time dither color may be permitted due to the very high speed of the FLC was first recognized by John White and convincingly demonstrated more than ten years ago in his group at Thorn EM1 [210]. As for spatial dither, the almost universally employed red-green-blue (RGB) filter is already a basic form of this category. The simplest structure is shown in Fig. 121 b. The pixel then has one horizontal connection for line scanning and three vertical connections for the independent data pulse control of the RGB subpixels. If each color pixel can just be on or off, we get 2' = 8 color combinations in this, the most primitive, form of color rendition. The designation
266
10 Digital Grey and Color
Figure 121. (a) If color hues are created hundred per cent temporally (color sequential illumination) no substructural features are needed in a picture element. (b) The conventional picture element comprises three subpixels, one for each color. (c) Canon’s pixel structure with four subpixels (from reference [2 111). (d) Subdivided RGB pixels as used by Sharp. (e) Spatial subdivision of a monochrome pixel in the area ratio 1:3.
“full color” is often used for the case that each pixel could be controlled to give eight bits or 28 = 256 individual shades of the color in question. This would give a total of 2563 color combinations, or 16 772 216 different colors, which is often written as 16M or 17M. In very many cases “16 million” different colors is just a fictitious figure often employed in the description of display performance but without any welldefined meaning. On the other hand, far less hues are, in reality, needed to give the impression of full color. In the following we will give some illustrations and simple estimations of the number of colors achievable in different dither techniques. In the first generation of the Canon FLC screen illustrated in Fig. 87, a slightly more sophisticated pixel structure than in Fig. 121b is used with white, red, green, and blue subpixels of different area weight. It is shown in Fig. 121c. This increases the number of vertical connections from three to four and gives 24 = 16 different color combinations. It is most surprising that such a limited gamut is able to give the excellent reproduction of artwork as demonstrated in Fig. 87. It is done by a sophisticated algorithmic method called error diffusion (ED) which is widely used in printing technology, for instance in color printers for producing professional quality photographic prints on paper. The method analyzes the content of a picture and calculates the optimum distribution of digital gradation steps to achieve a half-tone picture. By this method Canon achieves what is roughly equivalent to 32 000 different hues. It is, however, costly in time and electronics. Moreover, it requires a higher spatial resolution than any other technique in order to overcome a certain graininess in the picture. This resolution has not yet been achieved in the Canon screen and is unlikely to be achieved by any liquid crystal technology in the not-too-distant future. Dither methods would solve the problem in a way that is both simpler and more efficient.
10.2 Spatial and Temporal Dither
267
The simplest spatial dither technique uses an RGB-pixel according to Fig. 121d. For each color the element is subdivided into two unequal parts, which are independently controlled. This means that we have doubled the number of vertical electrode lines. Because a matrix display is scanned from top to bottom, one line at a time, doubling the number of horizontal lines would mean that it costs twice as much time to write a picture, i.e., it would bring down the speed of the screen, the update rate, by a factor of two. Doubling or tripling the vertical lines, however, has a much less dramatic effect; this is why RGB and further subpixel divisions are generally made along the vertical dimension. The subdivision of the pixel now means that it can produce four states of transmission in each color. Thus, without further tricks, we can generate 43 = 64 different colors. But we might also try to combine this with temporal dither. Suppose that we scan the matrix twice, so that each picture is produced by two individual subframes. As each pixel has 64 color states we might think that this would generate 64 x 64 = 4096 different colors. However, this is not quite true. The reason is that, as soon as we combine spatial and temporal dither, very many of the generated nominal states are redundant, i.e., they give the same transmission as do many other combinations. How much redundancy we have depends on how we subdivide the pixels. If the two areas in Fig. 121d are weighted as 1:2, the four transmission states of a pixel is, beginning with black: 0, 1,2, and 3, giving a linear scale of brightness. If instead we divide them as 1:3 we get the four brightness levels (always with the smallest area as unit) 0, l , 3,4, which are no longer equally distributed. Generally speaking, a binary division of a pixel gives a linear brightness scale but a higher redundance in combination with two, four ... subframes because of the factor two involved. This is easily illustrated. Consider one color pixel and its division in area ratio 1:3 as in Fig. 121e. We have here two picture elements (two subpixels) each capable of two states, black (B) and white (W). With two frames, each element can have three combinations, namely BB, BW and WW. Two independent elements then give 3 x 3 = 9 transmission states. These 9 states are given in Fig. 122. Without a color filter we would thus have 9 grey levels including black and white. The example is actually taken from one of the earliest FLC prototypes (black-andwhite) developed by LETUGrenoble, France, as early as 1988 [212]. This 6 in screen with 320x256 pixels is described in detail in [213] and [214]. It was operated at video rate and was the first to demonstrate the combination of the spatial and temporal dither technique. A variety of different aspects of the LETI technology, including electronic addressing for FLC and dither techniques, are discussed in [215]. In a color screen with RGB pixels, the LETI method would give 93 = 729 different colors. As we see this is much less than 642 but still quite interesting. If, on the other hand, the subdivision had been in the ratio 1:2 instead of 1:3, it is easily seen from Fig. 122 that the third and fourth levels would merge to one and the same (both giving “transmission 2”) and similarly the sixth and seventh levels would merge (giving “transmission 4”). Hence we would only get 7 levels and in the color version only 73 = 343 levels. Therefore, a careful choice of the subdivision ratio in space may reduce redundancy.
268
10 Digital Grey and Color
I. first frame second frame
O
1m
4
10 10
5 6
10
8
Irn
10 U U
Figure 122. The two bits spatial, one bit temporal dither first demonstrated by the LET1 Liquid Crystal Team in Grenoble, France in 1988. Their video screen was the first European breakthrough in passively multiplexed FLC technology, ahead of its time. It generates 8 levels of transmission, in addition to black. The subdivision of the pixel, shown larger at the top, is in the ratio 1 :3. If instead, it had been 1 :2, the third and fourth states, and similarly the sixth and seventh, would have fused toone.
So far we have achieved an insufficient number of grey levels, but continuing with the same principles we can raise the number relatively quickly. Let us in the following assume that we are always working with RGB color pixels which look identical and concentrate on how we may further subdivide the pixel, keeping the condition that we have two subframes at our disposal. The next step is then a binary subdivision according to Fig. 123a. Here we have been forced to introduce a second horizontal lead to every RGB pixel which means that we have lost a factor of two in the time domain. For a later comparison we note that at the same cost we could instead have kept the singly-divided pixel in combination with four subframes instead of two, i.e., 2 bits of temporal dither (4 = 2*). The four subpixels in Fig. 123a, each capable of two states, can be combined in 24 = 16 different ways. Generally, n subpixels give 2" levels or, equivalently expressed, n bits of grey. It is instructive to check these 16 states. With 0 meaning all black, we get the combinations 0, 1 , 2 , 4 , 8 ; 1 + 2, 1 + 4, 1 + 8; 2 + 4 , 2 + 8 , 4 + 8; I + 2 + 4, 1 + 2 + 8, 1 + 4 + 8 , 2 + 4 + 8, 1 + 2 + 4 + 8 or, rearranging the transmission states in order T = O , 1 , 2 , 3 , ... 15
(4 16)
The brightness scale is thus perfectly linear from black to white. This is quite an advantage compared with the distribution of the grey levels in a twisted nematic controlled by T m s , which on top of that are both far from color neutral and quite dependent on viewing angle. The first of these things means that as we change the twist-
10.2 Spatial and Temporal Dither
269
Figure 123. Four bits of spatial dither are illustrated in these two different ways of realizing a binary subdivision of each pixel. Compared with the subdivision in Fig. 121 d now two, instead of one, scanning line connections are required for every pixel; (a) a standard subdivision, (b) the actual subdivision in the Canon “full digital color” prototype (from reference [21 I]).
ed nematic state to give a different grey, the effective birefringence changes at the same time, even looking in a direction perpendicular to the screen. The second means that the viewing angle is not at all the same for levels near white as for levels near black, or in between. If we now were to dispense from temporal dither, i.e., only use one frame for writing an image, the subdivision of each RGB pixel according to Fig. 123a would allow 163= 4096 hues. This is interesting, as it is quite superior to the LET1 scheme of just the corresponding “cost”, namely 2 bits spatial dither combined with two frames which gave only 343 colors, or 729 colors if we accept unequal spacing, which is a cost in itself in that all levels are not really efficient and that the numbers are thus overestimated. With two frames the situation changes as follows. Every color pixel is capable of I6 states. Two frames then give 16* = 256 combinations for every color. As 256 is also equal to 2’ we might say that we have 8 bits of grey per color and this makes the “full color” 2563 = 16772216 = 16M colors possible. However, this is the nominal number of combinations including all empty permutations and redundant combinations. First of all we know that we have those redundant combinations from using the binary division in Fig. 123 a. We could get away from the redundance by making the area weight ratio different from I :2:4:8. Let us instead choose 1:2:9:27. Our corresponding transmission states would then be 0, 1, 3, 9, 27; 1 + 3, 1 + 9, 1 + 27; 3+9,3+27,9+27; 1+3+9, 1+3+27, 1+9+27,3+9+27;1+3+9+27or, ordered in brightness, the transmission levels are T=O, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39,40
(417)
These levels avoid redundancy when we apply two frames but instead they are distributed in four clusters, very far from making a linear grey scale. This characteristics is even more enhanced when we construct a picture out of two subframes. Let us nevertheless accept this inconvenience and calculate the correct number of different grey levels, i.e., avoiding empty permutations. It is easily done. Consider that each color pixel can produce N states. ( N in our present example is equal to 16.) In the sequence of two frames we can begin by combining the first level of frame 1 with
270
10 Digital Grey and Color
all N levels of frame 2. Then we can combine the second level with N - 1 levels in frame 2, the first level already being counted. Then the third level with N + 2 levels of frame 2 and so forth. The result is SN=
1 + 2 + 3 + ... . N - l + N
(418)
or
For N = 16, SN = 136. We see that this is about half the nominal number of levels (16 x 16 = 256) in a rough estimation ignoring empty permutations. Thus with the subdivision according to Fig. 123a, if we chose 1:3:9:27 (3°:31:32:33)instead of the binary 1:2:4:8 (2°:2’:22:23)onacolor(RGB) screen wecangenerate 13G3 =2.515.456 colors. Although this number is formally correct, it corresponds to a lower number of hues if these were instead equally spaced. This example just discussed applies to Canon’s 1997 prototype, shown at the FLC’97 conference and described in Sec. 9.6. However, Canon did not, and for good reason, choose the subdivision 1:3:9:27. The pixel structure adopted is shown in Fig. 123b and is certainly quite complex. It conserves the binary division in order to get the levels equally spaced. In practice this is just as efficient as formally avoiding redundance but having clusters of essentially coinciding grey levels. However, it is more complicated than the scheme of Fig. 123a because it avoids the effect that the optical center of the pixel shifts (vertically) as the brightness of the pixel increases. Such a shift leads to disturbing artefacts in the picture shown on the screen. The 16 different transmission states of a single color pixel in this screen are shown in Fig. 124. The redundance due to the binary choice of subdivision roughly sets down the efficient number of grey levels by a factor of two. Nevertheless this gives about one million equally spaced hues which is sufficient to generate a truly “full color” picture, the quality of which can be judged from Fig. 118. In the introduction to this example we mentioned that four bits of spatial dither combined with two frames (one bit of temporal dither) is similar in device “cost” to two bits of spatial dither combined with four frames (2 bits of temporal dither). Thus, for the comparison, let us evaluate the performance of the latter. In the present case of four subframes we have two individual subpixels which each could have the five combinations BBBB, WBBB, WWBB, WWWB, and WWWW. Each color pixel is therefore capable of 5 x5=25 combinations and we can generate 2253 = 15 625 colors, far less than in the division in four subpixels. Four spatial subpixels is therefore a very efficient way of rapidly reaching a high number of digital grey levels. Combined with three or more subframes such a division gives more hues than are useful in any device application.
10.2 Spatial and Temporal Dither
27 1
Figure 124. The 16 different transmission states in each color pixel from black to white, in the Canon 1997 full digital color prototype illustrated in Fig. 118 are here shown in increasing steps of brightness. The subpixel division is such that the optical center does not displace vertically. (This would be more complicated to achieve if the subdivision were not binary.) The RGB colors merge horizontally.
Four subpixels can also be considered the practical limit of spatial subdivision. In fact, it is most often considered too costly from an engineering point of view. In contrast, the relative ease of writing more subframes is more evident as FLC materials are developed which respond more quickly than before. We will take the Sharp prototype described in Sec. 9.7 as an example. The response here is fast enough, not only due to the material properties but also to the addressing method, (at the cost of very high voltage) that 15 frames can be used to write one picture. This means that we have four bits of temporal dither. Each pixel has a simple subdivision as in Fig. 121d. Each subpixel can then have 16 different states, including black, in the writing of one picture. These states are of course linearly distributed and we can name them as WBB.. ..B WWB .... B
(1 white time window) ( 2 white time windows)
w w w . . ..B WWW ....W
(15 white time windows)
Including black (zero white) m subframes thus generate m + 1 states. Two independent subpixels gives 16x 16 = 256 different shades and in an RGB matrix we have 256’ = 16M colors. With the pixel subdivision 1 :2 we have some re-
272
10 Digital Grey and Color
dundancy as we have discussed before. However, this (nominally) 8 bit greykolor is indeed fairly near the ideal full color performance. That ideal would be reached in the case of pure temporal dither, either using the RGB filter and 255 scans (eight bits) or even skipping the RGB filter altogether and 3 x255 = 765 scans to achieve 8 bits of grey per color. This is in principle the approach taken by Displaytech in Colorado. It requires active matrix driving but has the tremendous advantage that the expensive color filter can be disposed of. For video performance (60 Hz) one complete picture (one full color frame) has to be written within 16.7 ms. FLC materials, with response times in the microseconds thus allow hundreds of subframes to be written in order to exploit the digital method of producing full color. However, if we have to divide 16.7 ms in 765 time slots, only about 22 ps remain to write one individual frame. Even with active matrix driving this is not sufficient to scan 1024 lines. The limit is now set by the line address time for the transistors in the silicon backplane matrix, which is of the order of 3 0 4 0 ns, and by the fact that to refresh an SXGA panel (1280x 1024) in 22 ps a data rate of (1280 x 1024)/22x low6or 60 gigabits per second is required. To transfer information at this rate into the panel is itself a major problem (equally important for all display technologies). Disregarding this, the silicon backplane gives =: 1000x 40 ns = 40 ps frame time as the presently shortest slot time available. This would just be sufficient for 128 frames or 7 bits purely timegenerated grey if the matrix were to be driven at the extreme limits. For this and other considerations, Displaytech decided to use simple subpixellation of the silicon matrix (ratio 2: 1) and thus increase the number of columns to 2 x 1280 = 2560. By this they reach full color by a good margin. One drawback with subpixellation is that the effective area (aperture ratio) of a pixel is lowered. In this simple subdivision it is not serious, though, giving 84% instead of previously 90% but still a very high aperture ratio in comparison with regular TFT panels. This small SXGA microdisplay with pixel size of 12.5 pm square is now being used in HDTV projection device prototypes of one meter size diagonal. We have seen that if we subdivide space in n subareas we get 2" greys or n bits of grey. If we subdivide time in M = 2" subfrarnes we get M grey levels (in addition to black) or m bits of grey. We have also given several examples of combining the subdivision of space and time. In the Sharp case we saw that 2 bits spatial and 4 bits temporal dither gave (nominally) 2 . 4 = 8 bits of greykolor. However, it is not, in general, true that n bits spatial and m bits temporal dither combines to m x n bits of grey. At the beginning of this chapter we concluded that TFT driving has important advantages in controlling FLC panels. In fact it has many more than we could indicate there. We will return to those in the concluding chapter, but for now we would like to conclude our discussion about analog grey levels from Sec. 10.1. Suppose that every pixel is capable of N analog levels in TFT addressing, cf. Figs. 119 and 120. Using two frames, we then get the number of grey levels S , per color, given by Eq. (419). How many analog levels are then required to have 8 bits of grey per color?
10.2 Spatial and Temporal Dither
273
The relation (419) tells us that N would be of the order of 22, i.e., we would have to have 20 levels between black and white to achieve full analog color. This is certainly quite easy to achieve with the charge control in TFT driving of FLC. Therefore, as TFT now develops into a generally accessible and reasonably cheap technology, the advantage of combining FLC with TFT is obvious and will probably change the mainstream FLC from passive to active matrix displays.
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11 Elastic Properties of Smectics Almost every treatise on liquid crystals offers some exposition or survey of the Oseen-Frank theory of nematic elasticity. In Sec. 4.10 we have extended this kind of description to account for certain properties of the smectic C state and we have used this “nematic” description repeatedly in later sections. In contrast to the case of nematics, the case of smectic elasticity is relatively seldom treated, or only very briefly. Especially this is true for smectic C. We have so far essentially only used our description to look at the ground state, or undeformed state of the smectic C*. In order to include the energetic effect of deformations in SmC or SmC* we will have to develop a new continuum theory which replaces the Oseen-Frank description. This is what we will do in this chapter. But before going to the rather more complicated case of tilted smectics, let us look at the elastic characteristics of the simplest orthogonal smectic, in order to learn of the elastic peculiarities which are simply a consequence of the fact that the smectic is layered, i.e., that it has solid properties everywhere in directions which are perpendicular to the liquid-lke layers.
11.1 Continuum Description of the Smectic A Phase To study the consequences of the layering itself we may first look at the smectic in the approximation that the layers are incompressible, i.e., we look at deformations for which the smectic layer thickness d is a constant. Neither in smectic A nor in smectic C we can have a bend in the director field without violating this condition (cf. Fig. 41b), and the same is true for twist. Only splay would be allowed, as depicted in Fig. 40. We can look at this in a different way. If we focus on some local area across undisturbed layers, like the one in Fig. 125, we see that the value of the line integral over the director field n (r)between any two points between layers is equal to the thickness of the number of layers we cross,
In.dr = Nd
(420)
For instance, the value of this integral between the two points A and B in the figure is equal to 3d. If we take the integral over any closed loop Tits value must be zero.
Figure 125. For an incompressible smectic A with no singularities in the director field n (r)the line integral of n is independent of the path between two points and equal to zero for a closed loop.
276
11 Elastic Properties of Smectics
Using the Stokes theorem we then have
f n .dr = f ( V x n ) . dS = 0
r
S
which requires that
Vxn=O
(422)
We have in (421) assumed that n (r)is a vector field which is continuous and differentiable and that the derivatives of n are continuous over the surface S enclosing the loop r The condition (422) was recognized by Oseen who concluded that the only term left for the free energy, Eq. (96), of a smectic which is free of singularities is the splay term 1 G=-KII(V.n)2 2
(423)
If we induce a sufficiently strong bend in n (r)we will create defects: the material will respond by letting a pattern of singularities, in this case edge dislocations, invade the smectic in an attempt to keep the layer thickness constant, as in Fig. 126a. If we induce a corresponding twist, i.e., force n to change as we go perpendicular to n, we will break up the layers differently and create a pattern of screw dislocations, as in Fig. 126b. In chiral materials where the inherent twisting power is sufficiently strong such a state with periodically spaced families of parallel singularity lines twisting in space around the twist axis will even be the lowest energy state as shown by Renn and Lubensky [216] in 1988. These Twist Grain Boundary or TGB phases
Figure 126. (a) A layer bend is a soft deformation (corresponding to splay in the director field). Imposing a bend in the director field (equal to splay in the layers) results in creation of edge dislocations in order to try to keep the layer thickness locally constant. In order to reduce the deformation energy further the singularity lines do not lie on top of each other as in the figure but become alternately displaced, with the tendency to form a lattice. (b) A corresponding twist introduces screw dislocations with an elementary layer step such that following a layer on a circle around the singularity line (s) one ends one floor up after completing the circle, just like riding one floor up in a parking garage. The singularity lines turn in a helical fashion around the twist axis.
11.1 Continuum Description of the Smectic A Phase
277
Figure 127. A closed loop r enclosing two elementary edge dislocations in an area of slight bend in the n field. At top and bottom we move perpendicular to n, hence giving no contribution to the line integral of Eq. (424). Parallel to n we count six layers going down, eight going up, the difference being equal to two (after reference [217]).
exist as special thermodynamic subphases of both SmA* and SmC* and correspond to the Blue Phase manifestations of cholesteric order. They were experimentally found in the following year by Goodby et al. [216a]. If, in contrast to Eq. (421) the line integral is nonzero
we must inside the loop r h a v e singularities where V x n #0, i.e., we may have twist or bend deformations. But if the integral is nonzero it is also quantized. This is illustrated for the case of bend in Fig. 127, where the value is equal to 2d. In general it is equal to Nd if rencloses Ndislocations. As pointed out and developed by de Gennes [217], the relations (420) to (423) disclose interesting analogies between a smectic and a superconductor. A superconductor expels magnetic fields which are not allowed to penetrate into its interior. As the magnetic field B can be expressed by the vector potential A by
B=VxA,
(425)
we see that B = 0 corresponds to Eq. (422) with the director field n corresponding to A. Superconductors of type I1 do admit an exterior magnetic field to penetrate it in form of quantized vortex lines of magnetic flux, which have a tendency to form a two-dimensional periodic pattern sometimes called a Shubnikov phase or Abrikosov lattice. There is the same tendency for the dislocation lines to arrange in a two-dimensional pattern for a pure bend or twist deformation imposed on the n field. But in general terms, we may state that, just as superconductors expel magnetic fields, the Oseen condition (422) expresses that smectics expel bend and twist deformations. This is just a geometric consequence of the lamellar order of the smectics with the desire to keep the layer thickness constant. Our second approximation will now be to ease this hard condition d = const and admit a certain layer elasticity. If we consider a stack of layers which are originally flat and at equilibrium we can describe the deformed state by giving the distortion
278
11 Elastic Properties of Smectics
X \
,
-/ ----------
c1
Figure 128. A stack of smectic layers originally flat parallel to the xy plane. The deformation is described by the local deviation u (x, y) of a layer from its initial state.
u ( x , y ) relative to the original position along the z axis which is the original layer normal, cf. Fig. 128. For small distortions, au/& = tan a= -nx and similarly for auldy, so we can write
au-_ ax au- ny 12,
--
aY
Using the new variable u, the Eq. (423) for incompressible layers would read
The layers will be compressed or dilated if au/dz #O. We now assume that there is a linear stress-strain relationship across the layers, like in a solid (though this case is “softer”), i.e., with a stress proportional to au/az. The corresponding term in the elastic energy is therefore proportional to (dulaz)2 and we write the smectic A energy density as
This expression describes the mixed elasticity of a smectic. We have a first term expressing first-order elasticity, like in a solid, and a second term expressing the second-order elasticity (not containing first derivatives) characteristic of liquid crystals. A term (au/dx)2 does not appear because a constant au/ax only means a homogeneous rotation around the y axis and has no influence on the elastic energy, and correspondingly for (du/ay)2.On the other hand, this means that long wavelength undulation fluctuations will be inexpensive in energy and thus have large amplitudes in smectics. The coefficient B is about three orders of magnitude smaller than in normal solids. This mixed elasticity has peculiar consequences which we will now examine. To do this it is sufficient to look at the problem in two dimensions and let the deformations be independent of y in Fig. 128. We will now ask the question: specify-
11.1
Continuum Description of the h e c t i c A Phase
279
ing some boundary condition not compatible with undeformed layers, what will the equilibrium structure look like in a smectic sample. The structure will be the one that minimizes the total elastic energy, or the integral
taken over the whole volume. We therefore have to find the functionalfwhich minimizes this integral, where the functionalf, in a more convenient form of writing the derivatives, is identical to
+-
~ ( u , u ~x x, u. . ~ , , ~ , Z . . . ) 1= G = 2 - B1~K11uXI , 2 2 2
(430)
This is a problem in variational calculus, the solution of which is given by the Euler-Lagrange equations
afau aza au, af +..*---a af +--a2 af ax dux ax2 au,
+...= 0
(431)
The expressions in (430) and (431) emphasize the fact that here u, ux, u, etc. all have to be treated as independent variables. Because the functional does not depend on u, nor on itx, the first and third terms in Eq. (431) vanish, and we get
A a f - - -a2 af dz
au, ax2 au,
=O
(432)
Inserting f from (430) yields d Bu, - d2 Ki1 U, = 0 dz dx2
-
~
(433)
(434) This partial differential equation, which can also be written
contains the information on how a certain imposed deformation varies or is propagated along the x and z directions in a smectic A. We can solve this equation by sim-
280
11
Elastic Properties of Smectics
ply separating the variables, with a trial solution u =X(x) *
Z(z)
(436)
Inserted in (435) this gives
a2z BX---K11 az2
z a4x 7 = 0
ax
(437)
Introducing the parameter
we can now write the equation such that the left-hand is independent of x and the right-hand side independent of I, implying that both must equal a constant value,
Z” - A2 X”” = Const Z X
-
~
(438)
This constant we put equal to A2k4, after which we have split the partial differential equation in two ordinary differential equations, the first
with periodic solutions for X
and the second
with the solution
Z = e -+Ak2z
(442)
There are of course also nonperiodic solutions to (439) but they are of no interest to our problem, as we can always build up other solutions by superposition and because we will now adjust the solution to the specific boundary condition which we take from de Gennes, chosen to illustrate this problem and here reproduced in Fig. 129. This boundary condition is a sinusoidal undulation at the surface of the smectic. As u cannot grow with increasing z , we have to choose the minus sign in the exponent
11.1 Continuum Description of the Smectic A Phase
Figure 129. Layer distortions fall off very slowly in the direction across layers and therefore propagate over very long, often macroscopic distances in smectics (from reference [38]).
28 1
1 f
2x /k
of (442) and write the solution, according to (436). u = e k i k x . e- l k 2 z
(443)
The parameter A = V K ,,/B, formed from the two constants which determine the elastic properties, has the dimension of length and, being the only intrinsic characteristic length of the smectic, should be expected to be of the order of d, or about 20 A. The combination k2il has the dimension of inverse length, the size of which is influenced by the wave vector k of the undulation we impose at the boundary. We can then introduce a second characteristic length 1 = i/aP
(444)
and finally write the solution as = e k i k x . e-z/l
(445)
describing the fact that a distortion u characterized by the spatial period 2 d k along the smectic layers falls off exponentially across the layers with the distance z, in a way characterized by the length 1, which we therefore call penetration length. Equation (444) means that l is proportional to the square of the characteristic length of the distortion along the layers. Let us imagine a surface waviness of the order of 10 ym. With A = 20 A = 2 x ym we find 1 = 1 0 0 / 4 2 x 2 x lop3ym = 1200 ym equal to 1.2 mm. As pointed out by de Gennes, this distinguishes smectics from both ordinary crystals and nematic liquid crystals, in which distortions relax in the medium with relaxation lengths that have the same order of magnitude as the distortion itself. In smectics the layers transfer small distortions over almost macroscopic distances across the layers. This is perhaps the most striking general result of smectic elastic-
282
11 Elastic Properties of Smectics
ity. We can see several manifestations of this on smectic samples in which we have bookshelf geometry, i.e., where the layers are stackedperpendicular to the glass plates along the length of the sample. For one thing, this elasticity facilitates any chevron structure to be homogeneous, as the fold is a distortion which propagates across thousands of layers. On the other hand, defects in the layer structure will also have a tendency to develop preferentially over large distances across the smectic layers, which we also observe: defect lines across the layers tend to be long, defect lines along the layers are short, cf. Fig. 93. The one-dimensional positional order and the mixed elasticity in smectics have other far-reaching consequences, like the Peierls-Landau instability and other peculiarities which cannot be treated here. For these we refer the readers to references [38], [42], [218], [219], and [220].
11.2 Continuum Description of the Smectic C Phase To describe the elastic properties of smectic A we needed two elastic constants, Kl1 and B, from which we formed the first characteristic length being of the order of the layer distance. For incompressible layers we would need only one, but in skipping the compressional constant B, we also lose most of the interesting physics. For the smectic C phase we will need nine independent constants in addition to the compressional and in this case most of the interesting physics is contained in the description where the layers are considered incompressible. It turns out that we get four terms describing distortions of the director when the layers are undistorted, three terms for curvature distortions of the layers and two terms describing the coupling of these two types of distortions. The first family of four terms were introduced in 1969 by Saupe [221], the additional five by the Orsay group [222] in 1971. In the chiral (SmC*) case three further constants are needed, introduced by de Gennes [223] in 1975. This elastic theory, in its most general form, is not very easy to handle. We will therefore try to find tractable forms which can be used for the most important physical calculations, but begin by sketching the de Gennes approach [38]. In the last section we have seen that smectic layers cannot be splayed or twisted (corresponding to V x n #0, bend or twist in the director, forbidden at constant d ) without introducing defects (dislocations) which release the deformation in the surrounding parts of the smectic. We call deformations which are connected to layer compressions or layer breaking “hard deformations” because they cost a lot of energy. Local variations in the smectic C tilt angle 8 are hard deformations because they are coupled to variations in the smectic interlayer distance d. At constant temperature, 6 is a constant, and when we ignore hard deformations in the first approximation, we ignore variations in 8 and d.
vm,
11.2 Continuum Description of the Smectic C Phase
283
In the extension of smectic elasticity from SmA to SmC, de Gennes begins by introducing what he calls the c director, defined as the unit vector along the projection of n on the smectic plane, and he takes the x axis to be along this direction, cf. Fig. 130. We have repeatedly used this concept of “c director”, for instance in Secs. 4.10, 5.11, and 5.12, mostly in a slightly different way, as it is sometimes convenient to let c be the true projection of n on the smectic plane. In that case c does not only give the direction of tilt but also includes the magnitude of tilt, like in Fig. 75. In the following, however, we will treat c as a unit vector. In either case, c does not have “director symmetry”, i.e., c and - c are not equivalent. Apart from this, the two-dimensional vector c and the three-dimensional vector n have many character properties in common and smectics C therefore have many features in common with nematics. In de Gennes’s book the elastic energy of non-chiral smectic C is analyzed first. The deformations are described in the language of small rotations Qjaround the space axes. The rotation L2, around the z axis corresponds to a rotation of the c director, thus fiZcorresponds to cp in Fig. 130. It does not cost any energy (in the static continuum theory) if c is rotated by the same amount cp everywhere, but if cp depends on x , y or z we get the first family of elastic terms in the free energy
These are the terms introduced by Saupe. G, is quadratic in the spatial derivatives, including a coupling term. The terms are associated with distortions of the c director with the layers being fixed, that is with the cone motion of the director n, within and across the layers. Such deformations are “soft”. In the language of de Gennes the terms read
In addition to the variable aZ(or cp) describing c fluctuations, the Orsay group now introduces the variable u, which we have already used in the last section. Since a
284
11
Elastic Properties of Smectics
constant aulay amounts to a rotation around the x axis we can write
au -=a,
(448)
JY
and similarly
_ -au = szy
(449)
ax
Implying that
%+='O
asz
(450)
ax
ay Uniform rotations a (a,.
0,)do not change the free energy. Moreover, terms linear in the gradients or first derivatives of a cannot occur in the free energy for a nonchiral smectic C, which has local monoclinic symmetry including a center of symmetry. The required invariance under inversion in this symmetry cannot be fulfilled by V a because the V operator changes sign under inversion whereas a does not, being a pseudovector. Thus the terms of lowest order in VIRwill be proportional to ( VQ2. The group of terms describing layer distortions (strains) can then be written Qy,
G s = 21- B y2 + -21A i l
('2 ~
(4); ++ (2) 2
++ A12
Azl
(451)
The first term is the compressional term, with y= au/az. Using (450) we have also removed some redundant terms which can be transformed into each other by partial integration [222] which was also done in (446) and (447). We may compare this expression with the corresponding expression (428) for smectic A which in the new language would read 2
G , = -1B y 2 + -1K I , 2 2
(452)
The only torsional constant K,,in smectic A is replaced by three torsional constants A , , , A12, and A,, in the smectic C phase. (The elastic constant A , , is called A by de Gennes due to a misprint in the original paper by the Orsay group [222].) Finally there are the terms describing the coupling between the distortions of the smectic layers and the c director distortions in the layers
aa, aa, +c2----aa, aa, ax ax aY aY
G,, = C1 ___
~
(453)
1 I .2 Continuum Description of the Smectic C Phase
285
In the following we will neglect the layer compression. We are then left with nine independent elastic constants which are all torsional or liquid crystalline in the sense that they all describe first-order deformations in an angular variable 0 (corresponding to second-order deformations in a translational variable). Summing up all terms, the free energy can be written G = G, + G, + G,,, or
+-A1,[%) 1 2
2
2
2
This expression is not very transparent and it is not easy to grasp the physical meaning and importance of each term. It is not very well adapted for attacking specific calculations, for instance to find out equilibrium configurations in a number of technically important problems. Therefore this formalism has rarely been used. Instead a formalism of more transparence but somewhat limited in scope, introduced in 1979 by Pikin and Indenbom [224] has been used in a variety of problems. This formalism is only valid for undeformed layers and does not make any difference between soft and hard deformations. It is essentially convenient for describing small deformations around the ground state of smectic C*. Therefore, in 1984 Dahl and Lagerwall undertook to reformulate the more general approach of the Orsay group into a different and more applicable formalism using a local frame of reference which not only follows the director but at the same time the layer curvature everywhere in space [ 1271. This local frame of reference is shown in Fig. 131. The unit vector k is the local layer normal which together with the unit vector c describes the tilt plane. The third unit vector p has the direction k xc and will conform with the direction of the local polarization in the case that we have a chiral material with P > 0. The vectors c,p, and k thus form a right-handed system with the particularity that it follows the
k
f
Figure 131. In the local reference frame k describes the local smectic layer normal, c the local director tilt direction. In the local frame the directions of the axes are thus position-dependent.
286
11 Elastic Properties of Smectics
twists and bends of the layers through the medium. We might now say that we have two directors, k and c, which are varying continuously in three-dimensional space. The local relation between old and new directors is simply, from Fig. 131 n=kcos 8+csin 8
(455)
Our task is now to express the continuum description in the new variables k and c.
11.3 The Smectic C Continuum Theory in the Local Frame of Reference In reference [ 1271 the continuum theory of de Gennes is reformulated by deriving expressions relating the rotations Qi with the vectors k and c and, automatically,p, sincep is sterically coupled to k and c. To do this it is not sufficient to work in the limit of small rotations, which implies
+ f2,y - Q,Z p = - f2,x + y + axz k = OYx - f2,y + z c=x
(456)
but by a unitary transformation a higher-order expression is found that reduces to this correct small angle relation. The full expression for c, p , k is expanded to second order in 0,whereafter all derivatives in (454) are expressed as functions of k and c and their vector gradients. Of V k only the symmetric part Vsk need be considered in an uncompressional medium free of singularities. By finally demanding that all expressions should be invariant under the simultaneous transformations k +-k and c +-c, Dahl and Lagerwall arrive at the full expression G(k,c ) corresponding to G(f2) in (454). The result for the soft part is
G , = 1- B ~ ( ~ . V X C 1) ~ + - B ~ ( V . C ) ~ 2 2 1 +- B3 ( c . x c + c . k . k X C ) * 2 + Bl3 ( k . x c ) c . (V x c + vsk . k x c )
v v
vs
(457)
corresponding to (447). In this expression the symmetric part of the tensor Vk is given by (458)
11.3 The Smectic C Continuum Theory in the Local Frame of Reference
287
We will not derive the expression (457) here, but in the next section we will derive an expression suitable for calculations in the important applications which we will encounter. However, we will discuss the meaning of the different terms which appear in the free energy. The B, term can immediately be identified as splay of c within the smectic layers. The B, and B, terms may be identified as bend and twist of c (we will be able to see this more transparently below) although the B, term looks a little complicated. A layer bend in smectic A is a soft deformation [corresponding to a pure splay in n, cf. Eq. (423)l. In smectic C, layer bend (described by V s k )is also the only soft deformation of the layers, but in this phase the bending induces a twist in c between adjacent layers. The third term B, corresponds to twist of the c director between adjacent layers and it expresses just the fact that the layer bending induces a twist. Finally, the B , , term couples bend of c inside the layers to twist of c between the layers. Layer bend thus also induces bend of c inside the layers. The G,(Q terms of (45 1) in our local frame of reference (ignoring the compressional term which is the same in both languages) transform to 1 1 Gs=-All (c.VS~ . ~ X C ) * + - A I ~ ( C . V ~ ~ . C ) ~ 2 2 1 + - A*, ( k x c . V sk . k x c ) ~ 2
(459)
The interpretation of these terms which are all quadratic in the layer bend is somewhat complex but the deformations are illustrated in Fig. 42 of Sec. 4.10. In order to identify them we may write p instead of k x c in (459) and get the expressions (c - Vsk.p) and ( p . V s k . p )in the A , , and A,, terms, respectively. We then see that A , , corresponds to the deformation (3), A , , to (1) andA2, to (2) in Fig. 42. In order to be more explicit in distinguishing the last two cases we may draw the c-director field in the layer. This is illustrated in Fig. 132. Finally the coupling terms of (453) transform to
F,, = - C , (C F k . k x C ) (k . V X C ) - Cz (k x c . E k . k x C ) (V x C ) (460) *
Figure 132. The two layer bend deformations corresponding to the elastic constant A ,2 and A,, In the first case the c director lies in the plane of curvature, in the second it lies perpendicular to it. The two cases correspond to (1) and (2) from Fig. 42. ,
A12
A2 I
288
11 Elastic Properties of Smectics
These couple layer deformations to deformations inside the layers. For a description in more detail we refer to reference [ 1271. The advantage of the expressions (457) to (459) is that they are very applicable to specific important geometries, for which we easily gain more concise expressions. It is true that the derivation of the full expressions in [ 1271 is complex and lengthy. However, in the next section we will derive the corresponding expressions for the most important applications in a much simpler way, namely directly from our “nematic” description which we will transfer to our local frame of reference. The result is quite useful. Not only are the expressions concise enough to be easily interpreted and applied - in fact these expressions are practically the only ones to have been applied to SSFLC device problems - but they still contain the essential physics, also from a more philosophical point of view. The constants A , ..., B , . .., C, can be expected to be of the same order of magnitude as the Oseen constants Kii. We will show this for the soft deformation constants and we will be able to relate them to the Ki, values and deduce their dependence on the tilt angle 8.
,
11.4 The Case of Undistorted Layers The previous results are general but somewhat complex. All important applications, however, require undistorted smectic layers which we have, or try to get, in electrooptic devices. It is true that chevron structures, which are very special cases of bent layers, are very common. However they consist of flat parts except for the very tip of the chevron bend. In this area, the full expressions (457) to (460) have to be used together with the compressional term in calculating equilibrium configurations, eventually in slightly different formulations [225] and [228]. In the quasibookshelf (QBS) geometry we achieve the flat layers to a reasonably good approximation. It is instructive not to specialize the expressions (457) to (460) for the case of undisturbed layers - which would be easy enough - but to derive the expressions directly, making an immediate connection to our previous “nematic” description in Sec. 4.10 and extending it to our frame of reference c , p , k . In doing this we will also get a better understanding of how to interpret the terms in our general expressions (457) to (460). As the more general case of deformed but incompressible layers has been discussed above and is treated in [ 1271, for the most general case of deformed and compressible layers we refer to [ 1281 and [228]. Our starting expression is then the Oseen energy for a director field n (r),which we now repeat from Sec. 4.1
1 1 1 G = - K11 (V X n ) 2 + K22 (n . V x n)* + - K33 ( nx V x n ) 2 2 2 2 ~
(46 1)
289
I I .4 The Case of Undistorted Layers
This expression is to be transformed to the variables k and c, which is done using (455) which we write
n=ak+bc
(462)
To express splay, twist, and bend in the new variables is a simple formality.We have
V.n=aV.k+bV.c
(463)
Vxn=aVxk+bVxc
(464)
n - V x n= ( a k + b c ) ( a V x k + b V x c ) = a 2 k .V x k + b2 c .V x c + a b ( k . V x c + c . V x k )
(465)
In the same way (466)
n x V x n = a 2 k x V x k + b 2 cx V x c + a b ( k x V x c + c x V x k )
We will now limit ourselves to the case where we have not only incompressible but even undeformed smectic layers. This means that V k = 0, i. e., that
V-k=O and
Vxk=O everywhere. The first of these relations means that we have no layer bend (splay in k ) .The second corresponds to the Oseen condition (422) and means that we have no singularities in the k field. If our layered medium had been a smectic A, the vector k would be identical to n and the relations (467) and (468) would then have implied that we had no deformations in the medium whatsoever. However, the smectic C has a rich “inner life” and, notwithstanding (467) and (468) we will have twist, splay, and bend deformations in n as well as in c, governed by the elastic constants B,, B,, B,, and B 1 , . Because k is now a fixed direction, our reference frame is less “local” than before, and we could just as well replace k by z. However, we will keep the variable k so far, in order to facilitate comparisons and connections with the earlier expressions. With (467) and (468) the expressions simplify to
V . n= b V 7 . c
(469)
n . V x n = b2 c . V x c + a b k .V x c
(470)
n x V x n =b2cx V x c + a b k x V x c
(47 1)
290
11 Elastic Properties of Smectics
Figure 133. Pure splay in the two-dimensional c field.
The splay term in the free energy density hence transforms according to
In the new variables, this just expresses the in-plane splay, or the splay of the c vector, cf. Fig. 133. We will give it its own elastic constant, which we call B 2 , and write 1 1 - K , 1 ( v . n ) 2 = - 2B 2 ( v . C ) 2 2
(473)
Thus the relation between the new elastic constant and the Oseen constant is
B, = K , , sin2 o
(474)
The expression (472) means that a pure splay in n will appear as a pure splay in c . This is not so when we come to twist and bend. Pure twist and bend deformations in n will appear as mixed in c (and vice versa - a pure twist or bend in c is a compound twist-bend deformation in n, cf. Fig. 135) and we will have coupling terms. When we now continue with the other terms in the elastic energy, we will first look at the non-chiral case as in the previous sections and as represented by (461). For the twist and bend terms we have
v
(n. xn)2 = b4 ( c . v xc)2 + u 2 b2 ( k . v xc)2
+2b”(c.
(nx
(475)
v x c ) ( k .v x c )
v xn)2 = b4 (c x v xc)2 +a2 b2 ( k x v xc)2 +2 b b ( C x v x c ) ( k x v x c )
(476)
In order to identify the different terms, let us look at the geometry of twist and bend in the c field. For a twist [cf. Fig. 134al the curl, V x c, is directed along the c vector itself, i.e., c and V x c are collinear. Consequently the term (c . V x c ) gives ~ a contribution to the twist energy whereas the term (k . Vx c)* does not, since k is perpendicular to V x c , so their scalar product is zero. On the other hand, for a bend in the c field [Fig. 134bl the curl V x c is perpendicular to c , thus collinear with the layer normal k. This means that the term (k . V x c ) in ~ (473) gives a contribution for bend and that the term (c . V x c ) (k . V x c ) is a coupling term which is non-
29 1
1 1.4 The Case of Undistorted Layers
Figure 134. (a) The curl V x c is always antiparallel to c in a twist deformation. This means that c . V x c # 0, whereas k . V x c = 0. The unit vectors c and p are sterically connected and can be described by a scalar phase angle relative to an arbitrary direction in the smectic layer plane.
Figure 134. (b). The curl V x c is perpendicular to c in a bend deformation. It is parallel or antiparallel to k depending on the direction of curvature, as indicated. Hencek , V X c # 0.
k
Figure 135. Pure bend deformation of the vector field c
I
C
zero for a simultaneous twist and bend in c. A bend within a layer is thus coupled to a twist in the c vector between adjacent layers and vice versa. If the local twist vector (helical wave vector) in c is q, we have already seen [cf. Eq. (116d)l that c . V x c = - q . If we have a local bend with radius of curvature p. we find that k . V x c = Up. This is easily seen if we use cylindrical coordinates with the layer normal k in the center of curvature (Fig. 135). With c = (cp, ce, ck)= (0, 1, 0), the only component of V x c is along k with the value
(VXC>k= -
a
-(pC@)---p Pl i "aP ae
whencek. V x c = llp.
1
= -; -; p P = ;
1
(477)
292
11 Elastic Properties of Smectics
Investigating now the director bend expression (476) we notice that, because k and c are perpendicular unit vectors, we have equivalence between the forms
(CXVXC)*++(k.v x c y
(478)
. v xc y
(479)
and
(k x v x C)*
tj (c
The two geometries in Fig. 134 further illustrate that (c
x v x c ) . (k x
vx c ) =
-
(k . v x c ) (c . v x c)
Thus the expression (476) can now be rewritten
(n x v xn)2 = a 2 b 2 ( C . v x c ) 2+ b 4 ( k .v x c ) 2 - 2 b 3 a ( k . V x c ) (c . V x c) After regrouping, the twist and bend terms in the Oseen energy 1 K22 (n.V X 2
-
+-21 K33 (n x V ~
T Z ) ~
n
)
~
give rise to the three following terms
( k .V x c ) =~ 1B, ( k . V X 2
C ) ~
(K22 b 3 a - K33 b3 a ) ( k .V X C ) (C . V X C )
= - B13 ( k .v x c ) ( c.vx c )
(484)
With the new elastic constants the expression for the elastic deformation energy in a smectic C can then finally be written
1 1 1 G, = - B1 ( k . V x c ) +~ B2 (V . c ) +~ B3 (c.V X 2 2 2 - Bl3 ( k . v x c ) (c v x c) ’
C ) ~
(485)
This expression thus corresponds to the basic Oseen expression for the elastic energy in the nematic case. It is valid for soft deformations in a nonchiral tilted smectic,
1 I .4 The Case of Undistorted Layers
293
i.e., for all deformation allowed by the constraint that the director can move around the smectic cone. This corresponds to deformations in the phase angle cp. The layers are considered undeformable, represented by the layer normal k . The description has a certain two-dimensional character in that the vector c has two components only, like in a two-dimensional nematic with splay and bend deformations. This of course corresponds to the fact that each smectic layer is a two-dimensional fluid. However, the third dimension is represented by the twist through the layers and by the fact that this couples to the bend deformations in the layers. Splay deformations inside the layers do not couple at this level of approximation. Considering the different terms in the free energy expression (483) we notice that B , is the elastic constant for bend, B, for splay and B, for twist in the c vector field. The B,, term describes the coupling between in-plane bend and twist between layers. The values of theses elastic constants depend on the tilt angle 6 and they all, more or less rapidly, go to zero as we approach the smectic A state ( 6 = 0). From (472), (473), and (482), (483), (484) we see how they are related to the Oseen-Frank constants:
,
B = K,, sin2 6 cos28 + K33sin' 8
(c bend)
B2 = K , , sin2 6
(c splay)
B, = K2, sin4 6 + K3, sin2 8cos26
(c twist)
B,, = (K33- K,,) sin3 QCOS 8
(486)
These relations were first derived by Lagerwall and Dahl [46]. The minus sign in front of B,, in expression (484) has been chosen in order to give B,, a positive value (as K,, > K,,) like the other constants. Otherwise the order and numbering of the B:s has been made in conformity with the treatment by de Gennes in the non-local description of smectic C elasticity reviewed above. From (486) we see that in the one-constant approximation K , = K22= K,, = K , B , and B,, i.e. bend and twist, merge to one constant and their coupling term vanishes. In {he small angle approximation we have
,
This leads to the perhaps unexpected result that as 0 starts out to tilt from zero below the A + C transition, only the twist constant K2, contributes to the bend constant B, and only the bend constant K33contributes to the twist constant B,. This geometrical feature, illustrating how twist and bend are being mixed, is evident from
294
11 Elastic Properties of Smectics
Fig. 135. As 0 grows the bend and, respectively, twist contributions grow, however, and as we will see, the contributions are balanced at 0 = 45". It is instructive to insert some values in order to look at the character of the B : s depending on the tilt. In the A phase, of course, all constants vanish. For 0 equal to 22.5" we get the B :s roughly one order smaller than the K : s,
B , =: 0.125 K22 + 0.02 K33 B2 =: 0.15 K,, B,
^I
0.02 K22 + 0.125 K33
Bl, = 0.05 (K33 - K22) We see that the c bend constant ( B , ) is mostly composed of twist in n with only a small contribution from bend in n. For the c twist constant (B3)the reverse is true. The splay is, as always, just splay. If we increase 8 to 45", which can be regarded as the practical limit for FLC materials, we get
B1 = B3 = 0.25 (K22 + K33) B2 = 0.5 K , , 4 3
= 0.25 (K33- K22)
(489)
Thus B l = B3 are equally composed of twist and bend in n and merge to one single constant even if K33# K22.In the one-constant approximation we get, for 8 = 22.5" B , = 0.13 K B2 =: 0.15 K
B3 = 0.13 K
(490)
and for 0 = 45" B , = B2 = B3 = B = 0.5 K
(491)
In general, we may then say that the one-constant approximation in K corresponds to a similar one-constant approximation B,where B I (1/2) K. To be explicit, we introduce the one-constant approximation for small tilt angles in (486)which then takes the form
i3 = K sin2 6 with typical values being of the order of 1/10 of K.
(492)
11.5
The Elastic Energy Expression for Smectic C*
295
So far, few experimental measurements have been performed to find the values of the B constants. As in the case of measuring the nematic constants Kij,a kind of Frederiks transition may be used. Pelzl, Schiller, and Demus [230] in this way were able to measure B, and to compare its value to the K , , value measured in the nematic phase in different compounds. B, was always found to be smaller than K , ,. However, if they compared K , , with the bare value B,, as defined by B2 = B,,82
(493)
,
they found that B, was always higher than K , . The estimations gave typically B2 =: N with B, = lo-" N. Similar results were found by Dierking et al. [231]. Recently Carlsson, Stewart, and Leslie proposed a wedge cell configuration for a Frederiks transition geometry, which would permit the estimation of some A and B constants [232]. Measurements in this geometry have recently been performed by Findon and Gleeson which confirm the results on B, and also give an estimation for A,, + A , , [233]. Dunn et al. found B,= 5 . lop" N for a pyrimidine material [233a] and confirmed the validity of Eq. (486). Having arrived so far, we now only need a minor extension for the theory to be applicable to ferroelectric and antiferroelectric liquid crystals. This extension means that we have to include the case that the medium is chiral.
11.5 The Elastic Energy Expression for Smectic C* The elastic energy expression (485) is a translation of the Oseen expression (96) from 1928 for a nematic to the smectic C case. In 1958 Frank [ 11 11 extended the Oseen expression to the chiral nematic case, introducing the (n . V x n + q ) 2term to account for spontaneous twist (1 16). In 1984 Lagerwall and Dahl 1461, guided by Meyer's recognition that the helical smectic C* is a spacefilling twist-bend structure [43], finally introduced the corresponding term (1 17) (n x V xn - B)2to account for spontaneous bend to be able to treat the smectic C* phase in the "nematic description". We now want to translate this description into our local frame of reference. This means that we start from the free energy (494)
In addition to our previous treatment this gives two linear terms in the derivatives of n:
KZ2q n . V x n - K33 B . ( n x V x n )
(495)
296
11 Elastic Properties of Smectics
With B = - fik x n = - f i b k x c [cf. Eq. (118)], this translates to K22q (b2c . V x c + a b k . V x c ) + K33 (fib ( k x c)
. (b2c x V x c + a b k x V x c)
(496)
In the last term, the vectors c x V x c and k x V x c are parallel and antiparallel, respectively, to k x c, which gives
These terms, using a different sign convention for D, than in [38] and [128], can be written as Q(~.VXC)+D~(C.VXC)
(498)
with D , =K,,qsin0cos0+K,,~sin20 D , = K,, q sin20 - K3, p sin’ Ocos 0
(499)
Including these first-order D terms, the elastic energy for the smectic C* state can now be written
1 1 1 Gc* = - Bi ( k . V x c ) +~ B2 (V . c ) +~ B3 (C . V x c )2 2 2 2 - Bi3 ( k . V x C ) (C . V X c ) + Di k . V x c - QC .V x c
(500)
The new chiral terms are first order and therefore describe deformations which are spontaneous. The D ,term means that there is a tendency, in a chiral tilted smectic, for the c vector to bend in the plane of the layer, which means that the coupled local polarization will have a tendency to splay outwards, toward the edges of the smectic layers, or inwards, depending of the sign of the polarization P.This tendency may be suppressed by the boundary conditions, which will favor P pointing in or out of the surface, but for high P, also by the “bend stiffening effect”, cf. Sec. 13.9. The D, term is a spontaneous twist, whichjust describes the helical order of the SmC* ground state, as it is described by the expression (500). In this ground state there is a pure twist in c. This means that k . V x c = 0 and c . V x c = -q in (500),which reduces to
Hence
acc,iaq= B~
-
=
o
11.5
The Elastic Energy Expression for Smectic C*
297
which gives, for the helical wave vector in the ground state
for the smectic C* pitch. From (499) we see ( c o d being -- 1) that the wave vector will be zero (infinite pitch material) if the spontaneous twist and bend tendencies would compensate each other such that
The elastic energy expression (500) is simple and tractable, and gives a convenient base for most calculations involving the switching behavior of ferroelectric liquid crystals, which are always made under the assumption that the layers are fixed and undeformable and that any configuration of the director in the sample can be described by the c vector or, equivalently, by thep vector which is perpendicular to c. Alternatively, the c orp configuration in the layer can always be given in terms of the distribution of the scalar phase angle cp, cf. Figs. 130 and 131. Still, the expression (500) is perhaps not transparent enough. But if we notice that
where c x V x c is a vector in thep direction, we see more clearly that the B , and D , terms describe bend in the c director. B, and D, describe the twist. If we then incorporate D , and D, into the B , and B, terms and use (497) in the GI3term we can write G,, as
1 2
1 2
G p =Go + - B , ( c ~ v ~ c . p - D ~+/ BBz~ ( )V~. C ) ~
+1
-
2
B3 (c . V x c + D3/B3)2+ B, (c x V x c) . p (c . V x c)
(507)
where
These terms are all easy to interpret. The B , term describes a c director bend in the plane of the layer and expresses the fact that in a chiral material there is a spontuneous bend D , / B , in this plane. It also means that in a chiral material the local polarization has a tendency to point outwards toward the edges of the cell or inwards de-
29 8
11
Elastic Properties of Smectics
YYYY
>>>>
Figure 136. Spontaneous sideways bulging out of the c director in an SSFLC cell (for a material with an appreciable value of D,) at a boundary condition such that the polarization points out of the material at the confining surfaces. The boundary conditions will themselves have this kind of effect.
pending on the sign of D,. In a surface-stabilized cell the c director is fairly well anchored at the boundaries. Therefore there will be a tendency for the c director to bulge out in a direction along the cell plane coupled with the tendency for the polarization to be directed along the layers and across the length of the cell as illustrated in Fig. 136. We may also note that D , # 0 means that the flexoelectric effect from the deformation depicted as ( 5 ) in Fig. 42 is spontaneous in a chiral material. Whereas this spontaneous bend may be more or less suppressed -hard to judge - in SSFLC cells, it should be observable in experiments on freestanding smectic C* films. The different flexoelectric contributions to P in free films may possibly be distinguished due to their different dependence on the tilt angle 6, in combination with the fact that they are oriented differently, along k, c orp, see below. The second term in (507) is the only “simple” term, the splay. It does not couple to other deformations and there is no spontaneous splay. If, however, a splay in c is imposed by boundary conditions, like in Fig. 136 near the upper and lower surfaces, we will have two flexoelectric contributions to the local polarization, one along c and the other along k (i.e.. toward the observer). These correspond to the case (4)in Fig. 42. The third term describes the twist between c in successive layers and recalls the fact that in a chiral material there is a spontaneous twist equal to - D3/B3.Again this means a spontaneous flexoelectric effect for D, # 0 [deformation (6) in Fig. 421 as we have already discussed at length. While it is easy to get D, = 0 by doping with chiral materials, there is yet no real clue as how to suppress the intrinsic bend. Finally, the cross term B,, describes the coupling bend-twist. The forms c x V xc and c . V x c are completely analogous to n x V x n and n . V x n.
1 1.6 The Energy Expression in an Electric Field
299
11.6 The Energy Expression in an Electric Field If we apply an electric field to our SSFLC cell we have to add the term - P . E to our previous free energy expression. We deliberately ignore dielectric effects but want to emphasize that P should now also strictly include the contributions from the flexoelectric effects to the local polarization. Thus, if we write P = P , + P,, there are four contributions to P , in the case of unperturbed layers ( Vk = 0), corresponding to our three c deformations in Eq. (507). They correspond to the three geometries (4), ( 5 ) , and (6) in Fig. 42. If we use the same numbering as in reference [ 1271 (where the spontaneous polarization has the coefficient el”)we have the bend contribution
P,= e I 6(k . V . c ) p ,
with
e16-
O2
(509)
In Fig. 42 the vector is written (c . Vc . p ) p which is equal to - ( C X V X C = ) k~ . V X C . The splay contribution is
P , = e I 3( V . c ) c + eI9( V . c ) k , with e 1 6 - 8* , e I 9- 8
(510)
and finally the twist contribution
P , = e I 4(c . V X C ) ~with
eI4 -
8
(511)
This vector is written as (k . Vc . p ) p in Fig. 42, which is the same thing. Alternatively, these expressions may be written with k x c instead ofp. The total flexoelectric contribution is thus
P,=P,+P,+P,
( 5 12)
giving an additional contribution of -P, x E to the static energy and a torque contribution of P , x E to the dynamic switching equation. These contributions have been considered in some rare cases [229]. For a long time only the nematic state had a continuum theory giving a reasonably general still tractable expression for the elastic energy, Oseen’s description of the smectic A state being too rudimentary. De Gennes initiated the modern description of smectic elasticity using four independent constants in the smectic A case, (one of them solid-like) and ten independent constants in the smectic C case, augmented to thirteen in the chiral case SmC”. His approach is general but not very useful for applications and, as we pointed out, most of the theory of smectic C literature therefore has been dominated by the approach by Pikin and Indenbom which is less general but much easier to handle. Dahl and Lagerwall later translated the formalism of de Gennes to a local frame of reference. Finally, we have now simplified the treat-
300
11 Elastic Properties of Smectics
ment to the case of undeformed smectic layers which is the general condition for essentially all problems of practical importance, and which is also the domain of validity of the Pikin-Indenbom description. We did this here, however, not by translating de Gennes via Dahl-Lagerwall (which would only be necessary if we allow layer deformations), but by going directly from the extended Oseen formalism. This quasi-nematic description has shown to be very simple to apply and has been adopted in most quantitative simulations. We will now apply it to real device problems in the next chapter.
12 Smectic Elasticity Applied to SSFLC Cells 12.1 The P(q)-c (cp) Description For calculation of equilibrium structures as well as switching between different states, it is convenient to use a fixed reference frame xyz and let cp be the angle between the c director and one of the x or y axes or the angle between P and one of theses axes. The electric field is supposed to be applied along x or y . We will here make the choice that cp is the angular distance of c from the x axis with E along y , see Fig. 66. Let us start from our fairly general expression (500) and apply it to this case. For the two vectors k and c we write
k = (0,0, 1) c = (cos cp ,sin cp,
0)
(513)
Using the notation acp/ax = cpx etc., we then have
V .c = - sin cp cpx + cos cp cp,.
(5 14)
Further
=(-coscpcpp,,-sincpcp,,coscpcpx +sincpcp,) = - cpz c
+ (coscp cpx + sincp cp,)
k
(5 15)
Hence the curl of c has one component along c and one along k . Therefore c.
vxc=-cpz
k . V xc = coscp cpz+ sincp cp, Inserting (514), (516), and (517) in (500) gives the free energy expressed in cp and its spatial derivatives
302
12 Smectic Elasticity Applied to SSFLC Cells
If K , K,, and K,, are known, this relation can be used for computer calculations, via the connections (487). Presently, however, sufficient data are not really available to warrant this still complicated expression. We therefore go to the one-constant approximation and set B, = B, = B, = B. [As there is hardly any risk of mixing up B with the compressional elastic constant for smectic A discussed earlier, we now simply use B instead of as in (492).] Because B , , = 0 in this approximation, the expression simplifies to
This is quite convenient to use for calculations of static equilibrium in which the integral IGc,dV has to be minimized. In most calculations we can even assume that the orientation is uniform along the y direction, and as there is often no twist, except for short-pitch materials, also cp, vanishes and we get a one-dimensional problem
To repeat the same thing in the n director language, even in the one-constant approximation, is more complicated. Nevertheless, if we do, we gain a last bit of information which is quite valuable. The calculations are a little lengthy and tedious if done in a straightforward way. But with K, = K2, = K33 = K the basic expression (461) is written G = 1 K [(V . n ) 2+ ( n . V xn), +(n X V Xn)2] 2 1 = - K [ ( V . ~ Z+) (~V X ~ ) ~ ] 2 -
because the scalar and vectorial products take the components of V x n parallel and perpendicular, respectively, to n . If we develop the squares we get a sum of the particular derivatives squared plus mixed terms which, on integration, can be transferred to surface integrals. Hence, they do not contribute to the energy within the volume, and we may write
This is the expression for the free energy of distortions in a ferromagnetic medium (at cubic symmetry) according to Landau and Lifshitz [234] if we exchange the director components nj for those of the magnetization Mi. This gives a hint that a spin
12.1 The P ( y ) - c ( y )Description
303
system also behaves as an elastic medium with second-order elasticity. Although we only mention this in passing, it means that we have another domain of close analogies (disclinations, Bloch and NCel walls, etc.) between liquid crystals and magnetic systems. The difference between (46 I ) and the chiral expression (494) is that the latter (except for a trivial constant) contains the two additional terms K2,qn . V x n and - K33B . n x V x n . With
we may write (494) in the form
an. an. axi
G,, = - K 22+ K q n . V x n + K P ( k x n ) . ( n x V x n ) 2 ax;
(524)
which is easier to handle than (494). The first term in (524) means nine derivatives which reduce to six, since nZis constant. With
n = (sin cp cos cp, sin cp sin cp, cos cp) the sum of these derivatives is
Further, with
we get
(525)
304
12 Smectic Elasticity Applied to SSFLC Cells
Then, with K sin2 8 = B
In the last equation we have introduced the bend vector qb and the twist wave vector qt in the plane of the layers (qx,qy)and across layers (q,),thus describing c bend and twist. The bend vector qb like qt is an inverse length. Actually, it is the inverse radius of bend curvature. Comparison with ( 5 19) shows that
According to (530) we further have qb = p sin 8 + q cot 8 qt = p
case -
(533)
These relations again clearly show that the c vector bend and twist are both mixed from bend and twist contributions. If the FLC material is pitch-compensated (q = 0)
which confirms our previous finding that the bend in the IZ director (p) contributes equally to bend and twist of the c director for 8 = 45". To sum up, we have found transparent expressions for the elastic energy in the c director ( B ) as well as in the n director ( K ) descriptions, including the relations between the B and the K constants. In the one-constant approximation we have found that
B = K sin2 8
(535)
and the chiral bend and twist elastic constants D are given by B multiplied by the bend vector and the twist wave vector, respectively, for c deformations. The basic starting expressions for further calculations is Eq. (531). However, when we add the terms in the free energy representing the presence of an electric field, often only the first term of (531) is maintained. If the field is applied in the y direction, as in Fig. 66, its free energy contribution is
G, = - P . E = - P E cos ~p
(536)
12.1 The P(cp)-c(cp) Description
305
and the total free energy thus
1 G = - B ( V V )-~P E C O S ~ 2
(537)
The equilibrium configuration is the one which minimizes the integral JGdV, and in this case the corresponding Euler-Lagrange equation is
wherefis the functional G, i.e.,
This yields the equilibrium condition
B V 2 q - PE sinq = 0
(540)
{ V 2 q-si n 9 = O
(541)
or where we have introduced the characteristic length 5 = (B/PE)"2 corresponding to the expression 5 = (K/PE)'" which we used earlier in Eq. (296). In the meantime we have obtained a better understanding of the nature of the elastic constant Kin (296) as it is now identified as B. In a situation where we are not at equilibrium, q will approach the equilibrium state with the characteristic time constant T,according to
With the difference that we have now ignored the dielectric torque we thus recognize our earlier relation (294) from the discussion of the switching dynamics. This nonlinear partial differential equation has analytical solutions in a number of simple cases. If the sample can be considered uniform, i.e., if we disregard distortions of q which always exist near the surfaces, the elastic term in (542) vanishes and we simPlY get &plat
+ sinq
=0
which is the director equation of motion discussed in Sec. 6.5.
(543)
306
12 Smectic Elasticity Applied to SSFLC Cells
12.2 Helical States, Unwinding and Switching Equations corresponding to (531), (537),and (541) have been widely used, for studying both equilibrium states and switching dynamics in SSFLC cells. Most of this work has been pioneered by the Boulder group. It has become standard in these calculations [98], [235] to use a different convention for the angular variable than we have used above. Our cp is the angle between the c director and the x axis but also between the polarization P and the applied field E (cf. Fig. 66) which we have taken in the y direction. Furthermore the relation between z, c, and P corresponds to the right-handed system k , c , p which we have used above. This means that P precedes c which it does for a positive FLC material. However, most materials in use have turned out to be negative materials ( P < 0). Therefore, Handschy and Clark introduced the reference system of Fig. 137 which since then has been used in most of the SSFLC simulation work. Here P lags behind c by 90" and 4 is the angle between P and the field E which is supposed to be applied in the x direction. The connection between the two angles is cos cp = cos (90" + cp) = -sin 4, sin cp = cos $, and in the new variable 4, the c director is c = (-sin 4, cos $). Our expression (531) in the new variable reads 1 G p = - B (V$)2+ Bqb (sin4 $x 2
-C O S $ ~,)
+ Bq, $z
(544)
which is the same as Eq. (2) from reference [98] if we recognize that a different sign convention is used for qb. The equilibrium condition (541) reads the same,
50'4
-
sin4 = o
(545)
because of the additional effect that the field, so far, was in the y direction, while it is now in the x direction. With these observations it is quite easy to switch between the two equivalent descriptions.
t2
smectic layers
n
C
Figure 137. Reference system used in SSFLC simulations. The variable 4 is the angle between the polarization P and the positive x direction. The chosen P direction corresponds to the spontaneous polarization of a negative FLC material. P therefore lags behind c by 90". The field is assumed to be applied along the x direction. P i s then in the field direction for 4 = 0 (from reference [981).
12.2 Helical States, Unwinding and Switching
307
Figure 138. Brunet-Williams distortion in a smectic C* sample of medium thickness. The liquid crystal is confined between parallel bounding plates, shown as double lines, with the smectic layers perpendicular to the plates. The line segments between the layers show the director field, with a cross-bar denoting the end of the director projecting out of the page. The director is parallel to the page along the solid contours. In the core of the sample, the director forms a helix around the layer normal, which is parallel to the page and midway between the bounding plates. By favoring the orientation shown, the surfaces unwind the helix. The unwound regions near the surfaces are connected to the core of the sample by line defects, parallel to the surface and perpendicular to the page (from reference [237]).
Figure 139. Structures corresponding to Fig. 138 but showing the polarization vector field P,whereas the smectic layers and the director field are not shown for clarity. In the absence of applied field there are two JT walls per helix pitch 2, within which the regions of unfavorable direct01 orientation are confined (from reference [92]).
A survey of some simulation results is given in [236] but we may here resum6 the most important features. One of the first problems attacked [98] was the unwinding of the helical smectic C* structure by either boundaries or by the applied electric field. The first observation of unwinding phenomena was made by Brunet and Williams in 1977 [237] who analyzed the appearance of defect lines and were able to deduce that a partial unwinding of the helical structure takes place in a smectic C* sample with the layers perpendicular to the glass plates. Their proposed structure is shown in Fig. 138 and has been the starting point for many experimental and theoretical investigations of this subject. In Fig. 139the same structure is illustrated showing neither smectic layers nor the director field but only the connected spatial field of local polarization P (arrows). The boundary condition is that P should be perpendicular to the surface, without favoring any of the two possible directions. In (a) we see a periodic array of r walls between regions where the director makes a shift of n (1 80"). The core of the sample still has the helical structure. In (b) we see a Bru-
308
12
Smectic Elasticity Applied to SSFLC Cells
net-Williams distortion. The boundary condition now is supposed to favor the P UP state. The outer regions are unwound, adapting P to the admitted boundary condition. In Fig. 140 we see the case where instead we have the same polar surface condition at the boundaries, in this case chosen such (which is the common case) that the surface prefers the polarization to point away from it. Again, for a sample which is not too thin, there will be a helical structure in the bulk, but the disclination lines mediating the unwound surface regions with the helical regions are now displaced, both sets forming a periodic structure of period Z, the helix pitch. In (b) and (c) we see the helix unwinding taking place under application of an electric field, according to GlogarovB, Fousek, Lejcek, and Pave1 [238]. Regions of favored P direction are growing across the sample in the x direction but also narrowing the defect regions in the z direction, until the defect lines annihilate at or near a boundary. The calculation of the helical structure in the presence of boundaries and applied electric fields belongs, as is evident from Fig. 140, to the problems requiring two spatial variables when solving Eq. (540). Analytical solutions exist only for some special boundary conditions if E # 0. The calculation of the unwinding field is, however, simple and has been presented in a number of varieties [43], [240]-[242]. Various sophisticated structural problems related to unwinding have been discussed in references [238] and [243].
12.2 Helical States, Unwinding and Switching
309
As for the elastic unwinding by the boundaries, i.e., with no field, we know that it essentially takes place when the cell thickness is reduced to the intrinsic characteristic length, the pitch Z. In a thick sample the helix should unwind by an electric field when the characteristic length 4, describing the balance between electric and elastic torques, is about equal to the pitch. This gives
5 = (B/PE)'I2= Z
(546)
or, if we solve for the unwinding field E
As an example, take B = lo-'' N , P = 10 nC cm-2 and Z = I pm. This gives E = 1000 V cm-' or 0.1 V ym-'. We would have the same value for a short-pitch material with Z = 0.3 pm and with P = 100 nC cm-' (a rather typical value for short-pitch materials). This is the correct order of magnitude of the unwinding field, and we see that the voltage needed to unwind the helix in typical bookshelf cells with a gap of a few micrometers will be very low, whereas something of the order of 1 kV might be necessary to unwind the helix in a free-standing film where the electrodes cannot be placed very close to each other. Many features of the unwinding processes may naturally be similar to those which might be studied in switching processes with low or intermediate fields. Thus we can compare the process in Fig. 140 with the switching process outlined in Fig. 141, which is a description of simulation results from Handschy and Clark [98] based on Eq. (542) with appropriate boundary conditions. The bookshelf layer geometry is the same as in Fig. 140 but we will now more and more forget about the local directors and instead concentrate on their attached local dipole density P which is the one that couples to the applied field. The initial state is P homogeneously DOWN when the field at t = 0 is applied in the UP direction (a). After the characteristic time t = z = yq/PE the bulk has switched to be practically in the field direction whereas we have zones attached to the surfaces characterized by strong gradients in the P field, expressing the conflict between electric and elastic torques via the boundary condition. The thickness of these zones is characterized by our now familiar length {= ( K / P I ~ ) ' ' ~ . If the switching pulse is too short, i.e., if the field is shut off too early, the elastic torques from the boundaries will force P to switch back to the initial state. In order for the SSFLC cell to latch into the UP state, i.e., in order to reach the second bistable state, it is necessary that the suface states switch, a process illustrated in (c). Qualitatively, this makes it understandable why there is no threshold in the field as such, but rather in the voltage-time pulse area. Thus if we apply short pulses of 10 ps length, the required amplitude for switching might be, say, 30 V, while, if we can afford pulses of 1 ms length (which we can in active or TFT driving) only a frac-
310
12 Smectic Elasticity Applied to SSFLC Cells
. $
\-y
’-T
t
(c)
,:!
\
i
I
_
XV‘P
. .
-/-----___ ~. cone .-. .-\ ............... .._,_ ‘.,, ; +
.
-,
+
”
I
, --- -_--/+. . .;,/...\.......\.. t I + I
.
,
9
Figure 141. Switching sequence in SSFLC cells for intermediate applied electric fields, after simulations by Handschy and Clark. Note that in (a) the smectic layers are perpendicular to the paper, symbolized by the smectic tilt cone. Thus the vectorsc and P move perpendicularly to the paper (rotate around the z direction). In (b) and (c) on the other hand, we are in the plane of c and P,where the reorientation can be given in terms of the angle 0.The core of the cell reorients rapidly from P DOWN to P UP, leaving behind surface layers of thickness 5 which are still down. In (c) domains of UP polarization have nucleated at the surfaces. Bistable latching requires the surfaces to switch. The contours of constant P orientation correspond to 45” (dashed), 90” (dot-dashed), and 135” (dotted) (after reference [98]).
tion of a volt might be necessary for switching between the two states. Without surfaces there would be no latching at all. The simulations related in Fig. 141 were made before the discovery of the chevron layer structure. (In fact reference [98] is the first paper in which the suspicion is clearly expressed as to the general presence of tilted layers.) The chevron structure of course gives an additional complication both in computing and in interpretation. But the most important effect on the switching behavior of SSFLC cells can immediately be recognized: the chevron surface is a third surface, and switching the P state at this surface gives latching and bistability whether the top and bottom surfaces switch or not. Therefore we can have surface-stabilized cells which show good bistability although the surfaces may never switch, and we can produce such cells in a number of ways. The most popular way to do this is to have strong anchoring corresponding to the case shown in Fig. 90. On the other hand, this figure also demonstrates that the chevron per se is not a necessary requirement for achieving bistability. However, bistability is always a surface effect, mediated by bounding surfaces, chevron surface or both. In Fig. 141c is illustrated how the latching begins when “holes” of P UP state are created in the P DOWN state at the surface. We have discussed the switching in chevron structures to some extent in Secs. 8.3 and 8.7. The “boats” shown in Fig. 105 are such “holes” in a P UP or DOWN domain moving along the chevron surface. This
12.2 Helical States, Unwinding and Switching
Figure 142. Smectic C* bookshelf structure seen in cross sections cutting the layers (top) and along one layer (bottom). In the bottom pictures the polarization vectors connected with the directors are also shown. To the left is shown a homogeneous structure which can be switched between P UP and DOWN. To the right is shown a splayed structure where the P vector turns 180" or almost 180" (in case of some small director pretilt at the surface, indicated in the figure bottom right). To the extreme right is shown the corresponding deformation of the c director. Its bend vector may more or less satisfy the intrinsic tendency for bend, for which the bend vector qb (the inverse radius of bend curvature) is qb = D , / B .
UP
DOWN
+
311
SPLAYED
\
motion can be studied in the microscope at low or intermediate fields as the beginning of the latching process, after the directors between the chevron plane and top or bottom surfaces have already more or less adjusted to the new state. Further analysis of the elastic equations in the one-dimensional approximation has shown that the switching transitions, disregarding chevrons, are generally of two types: second order with a continuous evolution of Cp (x) and first order, with a discontinuous appearance of new lower energy states in the response to the applied field [244]. Thus even without chevrons we find transitions with a characteristic threshold and it was possible to show that the discontinuity of the transition as we change the applied field is due to polar surface conditions and a spontaneous splay in the P field, features which will be described in more detail below. In reality these transitions are mediated by domain walls which are at least two-dimensional. The description of switching processes in principle thus requires two spatial dimensions and even three in the case of chevron structures. The same is true for the calculation of equilibrium structures of domain walls. The elastic unwinding is also a very subtle process, the topology of which is quite involved due to the presence of chevrons. Observation and analysis of defect lines have here been very helpful to gain conclusive information of local layer structures. A review of this work with many new contributions has recently been given by Brunet and Martinot-Lagarde (2451. However, there is one extremely important aspect of the elastic unwinding that can be understood even as a one-dimensional problem. This is the calculation of the equilibrium structure in the limit that the helix is already spontaneously un-
312
12 Smectic Elasticity Applied to SSFLC Cells
wound (for instance there are no disclination lines), nevertheless the structure is not yet homogeneous. We will now consider this problem starting from the drawings in Fig. 142.
12.3 Splayed States In the top left part of Fig. 142 we see a bookshelf SmC* structure in a cross section parallel to the xz plane. The orientation is homogeneous with the director pointing out from the page toward the left everywhere. At top right we have the same orientation at the bottom cell surface while at the top surface the director tip lies on the opposite end of tilt cone diameter. In the vertical direction the director here makes a continuous transition in each layer from one of these orientations to the other. The three-dimensional curvature of the director field describes a twist-splay-bend deformation with a left-handed twist. In the midplane between the surfaces the director lies in the paper plane pointing upward to the observer at right. This observer looks at the director distribution in the layer plane, and what he sees is shown below. This is the xy or c director plane where the P vector turns by 180” (or almost so - in the bottom right figure we have admitted a certain pretilt at the surfaces) in a certain direction from top to bottom corresponding to a certain bend in the c director. The direction of the c bend vector, like the direction of splay in P,is connected to the twist sense. Actually, the c and P deformations are both combined splay-bend. However, this type of structure is commonly called “splayed’ because the splay in the P field is of special importance. Such splayed SSFLC cells are common. The reason is specific for chiral smectics as, in addition to a non-polar boundary condition for the director (a “nematic” interaction between n and the surface normal s) there is a polar boundary condition required forP. If we have two identical surfaces with strong polar anchoring the splayed state is favored. As this state has inferior optical properties (normally a bluish milky appearance between crossed polarizers, with bad extinction) one tries to avoid it by coating the surfaces with non-polar materials (like an inorganic, for instance SiO,) or with materials giving a weak polar interaction and facilitating the switching of the surface states. If the tendency to splayed states is strong it can, in principle, be overcome by going to thinner cells, as sooner or later the elastic torques due to the splaybend deformation will force one of the surface states to switch, releasing the elastic energy and giving one of the homogeneous states to the left in Fig. 142. This is a problem which can be treated as one-dimensional, because the c director field can be described in the angular variable (x). The surface contribution to the free energy has two terms, one corresponding to the nematic-like or quadrupolar interaction which we write $J
12.3 Splayed States
3 13
and the other polar one, which is related to the chirality of the medium
G, =
'/2 (P . s)
(549)
The surface normal s points out of the surface, thus in thex direction at the top, whereas the unit vector p points in the direction of local polarization P . Equation (548) means that the lowest surface energy state is the one in which the c director is parallel to the surface, forp . s = d .P is then along s with no discrimination between up or down state. With only this term present we would have a non-polar surface. For the polar interaction we choose the + sign which corresponds to the case at the right bottom of Fig. 142, with pretilt zero. (With a slightly more complex expression for G, we could easily have accounted also for a certain pretilt.) The lowest surface energy state is then the one where P points into the surface, p . s = -1. If P makes the angle Cp with the x direction, the total surface energy from the two contributions can be written G,=G,+Gp=-~l~~~2(P+~2~~~(P
(550)
Each surface has to be considered separately. The bulk free energy is given by (544) which now reads 1
G ~ =. - B ~p: 2
-
B4b sin ~p qix - PE cos Q
The general expression to be minimized is
in which Cpt and &, designate the Cp values at top and bottom surfaces (in our example already simplified to n and 0, respectively). The Euler-Lagrange equations then yield $xx -
( C U ~sin ) ~Q = o
(553)
and
Q~ + ( y , / ~d>sin2$ - (y 2 / + ~ qb)d sin @ = o
(554)
~p~ - 1 ;: sin ~p = 0
(555)
or and
qX + A , sinZ(P-jl, sin2$= 0
3 14
12 Smectic Elasticity Applied to SSFLC Cells
with the abbreviations
(557) For zero applied field, we have to solve this system of equations in bulk-surface self consistency. With E = 0, 4 = 0 and (555) reduces to
with the solution
@=ax+b
(559)
For a = 0 we get the two solutions UP (b = - IT)and DOWN (b = 0). For a # 0 we have a linear variation of @ across the sample. With x = 0 midways between the two plates we get two equivalent splayed states for b = f d 2 . The case depicted in Fig. 142 corresponds to b = + 7d2. Inserting these solutions back into (552) and minimizing the total energy we can check the stability of these solutions in terms of the parameters il,and 4 [235], [98]. The result is shown in Fig. 143. To interpret this diagram we first have to check the meaning of ill and 4.They are both proportional to the cell thickness d as we see from (557). The surface parameters A1 and 4 are surface anchoring energies with dimension of energy per unit area. The bulk elastic
Figure 143. Stability diagram showing the regions for the appearance of splayed and uniform states. For a sufficiently high value of the parameter which is proportional to the cell thickness d, the splayed state will always be of lower energy than the uniform states. Also, for a sufficiently large value of A,, (A, > d 2 ) , the splayed state is at least metastable (after reference [2351).
A,
10-2
10-1
lZ/2
10
lo2
non-polar surface strength (h,)
lo3
12.3 Splayed States
315
constant B has dimension energy per unit length. Hence y,/B is an inverse length and A, = (yl/B)da dimensionless parameter which is 9 1 if the surface torques (yJ dominate the elastic torques (B).A surface for which AI%l is called a strong surface, and we may give A itself the name surface strength. For the polar surface strength & the same thing is true but we now also have a contribution from bend. A strong spontaneous bend (large qb) will have the same effect as a small B value giving &=1. As we see from the diagram we have one region where the uniform state is unstable (has no minimum) and therefore the splayed state is the only state, another where the splayed state is unstable and where we thus only can have a uniform state, and finally two regions where one of the states can be metastable (both have local minima). The uniform state is always favored in a thin enough sample, but in general the stable state depends on the balance between non-polar and polar surface strength, the letter also reflecting the chiral strength of the material. On the one hand the uniform state is always stable for
0.2 B
d<
Y2 -t B q b
On the other hand it is always unstable or metastable for & > 214 = 2.5, i.e.,
2.5 B
d>
Y 2 +- B q b
If we ignore y2 relative to spontaneous bend we get as a reasonable estimation that the uniform state is achieved for d I l/qb or,
or using (532)
To summarize the results, with reference to Fig. 142, we first remind ourselves that in a smectic C* material there is an inherent tendency for a twist-bend which is thus not a distortion from the point of view of this smectic, but rather the lowest energy state for the medium. Nevertheless it is of course convenient to talk about it as a distortion when we discuss and compare it relative to uniform states. We can study the unwinding of the helix in such a material when we bring the smectic to the bookshelf structure and make the cells continuously thinner, and the unwinding can be studied by observing the squeezing out of the defect lines (unwinding lines) out of the sample. However, although the helix will now be suppressed by the surfaces for some value of the gap thickness d, a non-uniform state may then first appear, where
316
12 Smectic Elasticity Applied to SSFLC Cells
the director distorts in the natural way of the medium, connected with a splay-bend in the polarization field. Such splayed states may very well form even in the case of a compensated-pitch material, because there still is a spontaneous bend. The splayed P state is stabilized because of two effects, both of which are chiral in origin: (i) the polar surface interaction (y2 # 0) and (ii) the spontaneous bend in the c director (qb # 0). One of these effects would indeed be sufficient, as both contribute independently to the free energy. The spontaneous bend is illustrated to the lower extreme right of Fig. 142 but also more clearly in Fig. 136. From our previous analysis we have found that the value of its equilibrium bend vector q b is equal to D,/B which is always non-zero in a chiral material. Hence, the splay in P is an inherent, spontaneous property of the smectic C* phase. This explains why the uniform state is not stable in a thick or even relatively thin sample of a pitch-compensated material, as already commented on in Sec. 3.3 (cf. Fig. 22). Note from Eqs. (531), (512), and (532) that the bend vector for c, q b = D,/B, and the wave vector q = D,/B, for n or c depend on chiral elastic constants D 1and D, which as we have seen are independent. There is thus no simultaneous compensation of the bend and twist tendencies. This is also illustrated by the fact that in a pitch-compensated FLC material (q = 0), q b is equal to p sine according to Eq. (534). Thus not only is it necessary to unwind the helix in order for the macroscopic domains and the characteristic bistability to appear, but also to overcome the spontaneous bend, which requires a thinner sample. We have been able to illustrate only a few of the examples of applying the quasinematic description to the calculations of equilibrium structures and switching dynamics in the smectic C* phase. This description has also been shown to be invaluable for analyzing the propagation of light through these structures and calculating the optical transmission characteristics of SSFLC cells. Such computer simulations for chevron structures have been performed by Maclennan together with matching experimental measurements [246]. Some further discussion of chevron switching, domain walls, and optical properties can be found in Secs. 6.2-6.4 of reference [236]. Discussions in more detail are found in references [247]-[252]. Valuable discussions of both defect structures and optical properties are also found in [253].
12.4 Characteristic Lengths It is often convenient to describe the influence of external fields applied to a liquid crystal by considering the size of a penetration length, or a coherence length or, generally, a characteristic length describing the torques exerted by the field in relation to opposing torques from the liquid crystal or its boundaries. For instance, de Gennes in his book [38] introduces the magnetic coherence length of a nematic
5,
(H) = (K/x,H2)”2- 1/H
(564)
12.4 Characteristic Lengths
3 17
describing the fact that if the director is anchored somewhere, for instance at the boundaries, in a direction different from the aligning direction of the magnetic field, then in a transition layer with thickness of the order of 5, the orientation will change from the bulk orientation along the magnetic field, to that given by the surface. In (564) K is the elastic constant (one-constant approximation) and is the anisotropy of the magnetic susceptibility. The characteristic length therefore expresses the width of the layer in which the director is subject to considerable changes. Otherwise expressed, ( is the linear dimension of the region where the overall cell alignment is perturbed. The corresponding electric coherence length in a nematic is analogously
xa
(d
= (K/E,E2)112-1IE
(565)
where E, is the dielectric anisotropy. In the electric as in the magnetic case the characteristic length is inversely proportional to the applied field. In both cases the length is recognized as the scaling parameter when we write the equation expressing the balance between field and elastic torques in a dimensionless form. We have given examples of this in Secs. 6.3, Eq. (295) and 12.1, Eq. (541). The relations (564) and (565) are valid in gaussian units. The corresponding expressions in SI units are
5, = (KIP, H 2 )' I 2 and
where we have included and E~ in the anisotropies. For instance, E, is not dimensionless in (567) while it is in (565). The elastic constant K has the dimension WIL, energy per unit length. If we divide it by &E2which is energy density, WIL3,we get a length squared, L2.As the energy density in SI units is - q%oE2, we see that in (567) E, must contain G. In polar liquid crystals which have a local polarization P, the ferroelectric torque P x E , or dimensionwise PE will dominate the dielectric torque, except at very high fields. P E has dimension torque density which is the same as the dimension of energy density. Hence a characteristic length appears which can be written
te= (KlPE)'I2
(568)
td,
to be distinguished from which depends on the property of dielectric anisotropy. (, is the length introduced in Eq. (296). If we have no external field applied to a polar liquid crystal yet another characteristic length gains importance, at least for high values of the spontaneous polariza-
318
12 Smectic Elasticity Applied to SSFLC Cells
tion P. If, as for instance in the last section, we have polar boundary conditions such that the divergence V .P is not zero everywhere, then V . P is identical to a local polarization charge density
which is a measure of the non-uniformity of the polarization inside the liquid crystal. Identical means: as long as V P does not vanish, the individual volume elements appear to be charged. This is illustrated in Fig. 144. The polarization charge density is associated with an electrostatic field energy density ( V . P)2,hence with a field energy (SI units)
-
In order to lower this energy, the P field will try to be as uniform as possible, which leads to a stiffening of the P and c field. This is counteracted by the boundary conditions and within a region of linear dimension we will have a very non-uniform orientation of P and c. It is easy to estimate if we look at the balance of elastic and electrostatic energy in a region of, say, thickness L. This non-uniform region costs elastic and electrostatic energy. The elastic part is
tp
tP
per unit volume (we skip the factors 1/2), and per unit area, dimensionwise, (KlL2)L
(572)
Adding the electrostatic energy per unit volume according to (570) (note that V . P has dimension PlL)
(PIL)* (L3/&L )
(573)
and per unit area
(P2/&)L
(574)
gives the total energy per unit area
+
u = LP2/& KIL
(575)
12.5 The Electrostatic Self-Interaction
319
which we minimize. (576) giving
L2 = K&/P2
(577)
The length L is our tp,thus the polar coherence length is written
tP=(K&IP2)1/2
(578)
In the presence of an external field, the energy term PE adds to P2/&and the coherence length becomes
tp=[(KI(PE + P 2 / ~ ) ] ’ / 2
(579)
Because PE + P*/E = (PIE) ( E E + P ) = (PIE)D , where D is the dielectric displacement, can be written more compactly
cp cP=(KEl(PD)’/2
(580)
In gaussian units D = E + 4x P which gives the corresponding forms
tP= (KEIPD)’/2 +
= [KE/(PE 4 x P 2 ) ]
(58 1 )
or, for E = 0
tP=( K E I ~ Z P ~ ) ” ~ Small values of the polar coherence length lead to a kind of electrostatic stiffening which gives the appearance as if the bulk had a much higher elastic constant than the regions near edges or surfaces (represented by IQ. This effect was first noticed by Meyer [43] and later discussed by Nakagawa and Akahane [254] and by Dahl[255]. It can have a remarkable effect in FLC devices but is also observable in free-standing smectic films [43]. The meaning of E will be discussed below (Sec. 13.11).
12.5 The Electrostatic Self-Interaction Let us now return to the Eq. (545) giving the equilibrium configuration of P. This equation can easily be extended to a dynamic equation analogous to Eq. (293) with
320
12 Smectic Elasticity Applied to SSFLC Cells
or without inclusion of the dielectric torque. We will here disregard dielectric effects and therefore write it
B V 2 @ -PE sin@ = 0
(583)
E in this equation is the field acting on P. So far we have taken for granted that E is equal to the external field that we apply across the sample. But this will not in general be true for a non-homogeneous P field.A striking example of this is the case of a splayed SSFLC cell discussed in Sec. 12.3. In the reference system of Fig. 142 we can write P = (Px, P,,) = P (cos @, sin@)
(584)
whence
(585) We now assume that here is no y dependence, as in the case illustrated in Fig. 136.Then
and the polarization charge density in the bulk is
p = Psin@-d@ dx
(587)
Now we apply Gauss’ law
$ E = Q/E
(588)
to a surface enclosing the volume between the top plate and the parallel plane on the distance x from the top plate, see Fig. 144. Q is here the total charge, the sum of free charges and polarization charges. Let us assume that no ionic impurities are present - which might not necessarily correspond to reality - then the only charges are polarization charges and charges on the top plate. We further assume that the surface charge density CI = P . s from the FLC is just compensated by the plate charges from the external source. Then Q is simply the polarization charge density p integrated over the volume corresponding to the distance x from the top plate, and (588) gives
$(X)
= ( H E ) sin@d@= (P/E) [1 -cos@ (x)] 0
(589)
12.5
The Electrostatic Self-Interaction
* * * * f
f
f
f
f
* * f
f
32 1
++++++++++ ++++++++++
+++++I+
Figure 144. If we have a rigid boundary condition for the polarization field P,then in an SSFLC cell this leads to V . P # 0 in the volume of the cell and thereby to a polarization charge density p proportional to this divergence. To the left is shown an example of a cell for which P . s > 0 (s is the surface normal), and to the right the corresponding distribution of polarization charge. This leads to an internal electric field Epwhich is illustrated by the bold arrow. The function p (x)given by Eq. (587) and Ep(x) given by Eq. (589) are shown in the lower part. The positive charge density is compensated by negative surface charges. All illustrations correspond to a linear variation of $I across the sample.
For a linear variation @ = (dd) x across the sample, we find p = (dd) sin (dd)x according to (587) and the polarization field Ep = ( P / E )[l - cos ( ~ d x )according ] to (589). These functions are illustrated in Fig. 144. The polarization self-field obviously cannot be neglected for materials with sufficiently high P. The external field E in (583) then has to be replaced by the total field E,,, = E + Ep = E
+ (P/E)(1 -COS$)
(590)
and the equilibrium condition (583) changes to
BV’#= P [ E + ( P I E ) ( ~ - C O S # ) ] ~ ~ ~ # = ( P E + P 2 / & ) s i n @ - 1- ( P 2/&)sin2@ 2
(591)
When no external field is applied ( E = 0) the equilibrium condition is (592) or
(593)
322
12 Smectic Elasticity Applied to SSFLC Cells
with
tP = ( B E / P ~ ) " ~
(594)
The solution of (592) shows that the dipole distribution will organize itself to eliminate the electric field due to the polarization charge in all regions of space, except for sheets of width 5, as required by the boundary conditions [256]. If we apply an external field in the x direction across a cell of the kind depicted in Fig. 144 (corresponding to {p = d) the total field, which is the local field acting on the dipoles will initially be very non-homogeneous. Epacts together with the applied field in the lower half, against in the upper half. On the other hand, the counteracting torques are stronger in the lower part. The switching might therefore still be relatively homogeneous across the cell. To investigate the switching dynamics Zhuang [257] starts with Eq. (519) in the one-dimensional form and with D,= 0 for the elastic energy. After adding the electrostatic terms 1/2 €E2- (P. E ) , the total energy is minimized and a switching equation analogous to Eq. (542) is then solved numerically with the appropriate boundary conditions. The results are shown in Figs. 145 and 146. In the first of these the field-free state is assumed to be uniform with strong anchoring such that the surfaces never switch. The change in the P vector distribution in response to an increasing applied field is given as the angular variable Qt (x, E ) and also illustrated pictorially below. In Fig. 145a we have the case = d, and when we look at 145b for a comparison, where 4 d, we see the stiffening effect in a material with higher P:now in the bulk the P field rotates as an entity into the E direction. Only the thin surface sheets of linear dimension are opposing and appear to be electrostatically charged. In Fig. 146a and b we see the corresponding profiles for the case that we have nonswitching surfaces giving a splayed state. For 5, = d the P field is non-uniform (a) except at very high fields. The boundary condition for P is here reversed relative to Fig. 144, so the polarization charge distribution at zero external field corresponds to d, however, the one in Fig. 144, except that the charges now are negative. For ~$4 even the zero field state is uniform except for the thin surface layers and it follows the changes in E in a uniform way. The optical properties of such a cell therefore correspond to a homogeneous retarder with a field-controlled optic axis. Most of our discussion of simulation results has been conducted under the assump tion that we have chevron-free cells. The reason has of course been to avoid unnecessary complications and emphasize the basic physics, as the physics behind the chevrons themselves has been dealt with already in Secs. 8.3 to 8.8. It is also true that for high P, materials, the chevron structure readily is transformed to quasi-bookshelf, so that we may often, to a good approximation, disregard the chevron. On the other hand, many of the simulations, for instance those by Zhuang and by Maclennan, have indeed treated the chevron case. As a general rule the effect of the chevron structure is to split the cell in two halves, each half as thick as each other, and to introduce a new
tP
tP
tp
12.5 The Electrostatic Self-Interaction
OV
323
I
Figure 145. (a) Numerical simulations of the polarization field profile as a function of applied field for 5, = d and the case of uniform initial alignment (unequivalent and fixed boundary conditions corresponding to a monostable cell). The dipoles reorient giving strongly non-uniform n,c , and P fields except at very high applied voltages (after reference [255]).
0
1
0s xJd
Figure 145. (b) Numerical simulations as in (a) but for the case that &,9d. The bulk P is uniform and rotates as an entity in response to the field. The polarization charge density in the bulk can now be neglected but charges appear at the surface. Monostable cell. The corresponding chevron version of this cell could be bistable.
E
-increasing
5, << d
surface. The simulations can then be performed for each part and matched by the boundary condition at the chevron surface. This boundary condition is typically described by a surface energy density corresponding to the polar interaction (549) and written
324
12 Smectic Elasticity Applied to SSFLC Cells
Figure 146. (a) Simulations by Zhuang as in Fig, 145 but now corresponding to identical strong anchoring at both surfaces giving a splayed cell configuration. The case shown is for 5, = d .
+0,2v
1
0
Figure 146. (b) Simulations as in (a) but for the case of tP4d. Monostable cell with essentially homogeneous director fields for all applied voltages, including V = 0 (from reference [255]). The simulations in Figs. 144 and 145 were performed assuming anchoring surface energy coefficients fi = = 3 . lo-' erg cm-', B = lo-' erg cm-' and P = I nC cm-2, i n the case (a), and yz = 5 . lo-' erg em-' with P = 10 nC cm-2in the case (b). The higher P value in the latter case is supposed to increase the polar surface interaction
x
x.
x
where the angle is the angle of discontinuity of P at the chevron surface, cf. Fig. 96. The presence of a chevron may have a pronounced effect on the monostability or bistability of the cell under various surface alignment conditions. Normally a bistability which is weak or non-existing is promoted by the chevron. But one of the most interesting results of the simulations on chevron cells [257] is that the bistability may in fact get lost by the polarization charge interactions at high P,.
13 Antiferroelectric Liquid Crystals Although less well known, because of their much smaller application range so far in the solid state, materials with antiferroelectric properties are in fact more numerous than materials with ferroelectric properties among solid crystals. It would therefore seem natural to look for corresponding materials among liquid crystals. We are then of course not thinking of the very special case of the helielectric order already treated at length, which is a helicoidal antiferroelectric, but of the straightforward lund of antiferroelectric where the dipolar cancellation does not take place via a helix but directly from neighbor to neighbor. This kind of antiferroelectric liquid crystal (often AFLC or just AF in the following) would then correspond to the dipolar AF pattern in Fig. 5. This means that alternating smectic layers would constitute two sublattices of opposite polarization flsand therefore have opposite tilt, in the kind of herring-bone pattern illustrated in Fig. 147. If we ask why such a structure should exist, we have to remind of the principal difference between solid and liquid crystal polar materials. In a crystal the electric dipole-dipole interaction is responsible for the order and this may favor (most often) antiparallel adjacent dipoles (antipolar interaction) and in other cases parallel adjacent dipoles (synpolar interaction). In a proper ferroelectric this dipole-dipole interaction causes a symmetry change, P is so to speak responsible for the symmetry, whereas in a liquid crystal (improper) ferroelectric, P is the resulr of a certain symmetry. The spontaneous polarization in this case results from a symmetry constraint, a bias in molecular rotation. If this bias grows very large for molecular parts containing powerful dipoles, not only will the non-centrosymmetric monoclinic structure give rise to a local polarization along a precise direction in the layer, but
Figure 147. Herring-bone pattern in the hypothetical structure of antiferroelectric liquid crystal. Molecules, on average, have alternating tilt in alternating layers, corresponding to dipole moments pointing alternately into and out of the paper. The director n would now refer to a single layer like the c director, and the c directors of adjacent layers would be antiparallel. The structure would be biaxial with the layer normal bisecting the angle between the optic axes.
326
13 Antiferroelectric Liquid Crystals
the dipole-dipole interaction may itself be strong enough to change the lattice structure such that an alternating tilt results, or at least contribute to such a change. Intuitively, one has to expect the alternating tilt to be counteracted by the liquid crystal elasticity-adjacent molecules want to be parallel even across the layers - but other sterical factors may enter, for instance a tendency for dimerization of two molecules contained in adjacent layers with the effect that they rather act as one bent molecule. Anyhow, we may assume that both steric and polar factors are involved, the latter bound to chirality. We may also assume that, as far as the polar contribution is concerned, it cannot any longer be neglected when P , acquires very high values. It was, in fact, when synthesizing materials with the aim of reaching very high P, values, that antiferroelectric behavior was finally unambiguously detected, even if, in one of the two decisive cases, the recognition was made on a substance with unusually low P,, in this case by sophisticated arguments not linked to the electro-optic behavior in the bulk.
13.1 The Recognition of Antiferroelectricity in Liquid Crystals Antiferroelectric liquid crystal materials are very similar in most structural respects to the ferroelectric materials but their recognition dates about 15 years later. Their young history is quite interesting. In 1977, only two years after the paper by Meyer, Strzelecki and Keller [62], Michelson, Cabib and Benguigui developed a Landau description for the smectic A to C transition, [43b], with the particular aim to investigate which kinds of polar order would be permitted in liquid crystals. This and sequel papers [43e] and [43fl have already been referred to in sections 3.4, 5.9 and 5.12. Among many other things in that rich paper, the authors conclude that a transition from smectic A to polar antitilt, i.e. antiferroelectric, smectic C phase is only possible if the C phase is of the type CM (cf. Section 3.4). They also remark that such a transition, which would correspond to a doubling of the lattice period has never been observed experimentally. This is equally true today. In fact, the liquid crystal antiferroelectric phase we know today is a result of a transition, not from smectic A but from smectic C, C* +C,* on lowering the temperature. It means that the antiferroelectric phase in liquid crystals lies at lower temperature than the ferroelectric phase, which is opposite to the situation in the solid state, where the ferroelectric is the low temperature phase. This is another indication that the mechanism responsible for the antiferroelectric order is in essential parts different in the two cases. Four years later, in 1981, Beresnev and coworkers reported antiferroelectric order in a smectic liquid crystal [258], [259]. The antiferroelectric behavior was concluded from the pyroelectrically measured polarization reponse as a function of an ap-
13.1 The Recognition of Antiferroelectricity in Liquid Crystals
327
Figure 148. Two different two-ring cinnamates with certain similarities. (i)4-nonyloxybenzylidene4'-amino pentyl cinnamate. Doped with a chiral additive (*) it was reported in 1982 to have antiferroelectric properties. (ii) 4-( 1methyl pentyl oxycarbonyl) phenyl-4-nonyloxy cinnamate. It was shown in 1997 to have a monotropic SmCZ phase and two further antiferroelectric or ferrielectric phases. The same is true for the 1-methyl heptyl compound. If in the latter we replace a hydrogen in the benzene 2 position with a fluorine as in (iii) the Cf phase appears without being monotropic'.
plied electric field. A herring-bone packing of the molecules with tilt angles +f3and -8in adjacent layers was proposed to account for the non-polarized state at E = 0 together with the observed threshold and saturation behavior. This work was not further pursued and seems to have fallen into oblivion like reference [43b], though it was also discussed in the extensive review article by Blinov and Beresnev a couple of years later [45], in which the anti-tilt configuration with compensating local polarization vectors - exactly as for the phase of our Fig. 147 - was reproduced. The material used by the Moscow group in [259] was a tilted smectic two-ring cinnamate abbreviated NOBAPC, shown in Fig. 148(i), to which a chiral dopant had been added. In the Mol. Cryst. Liq. Cryst. Special Issue on Ferroelectric Liquid Crystals [47] Beresnev and coworkers describe that adding HOBACPC to NOBACPC gives rise to new phases not existing in the pure non-chiral NOBACPC, at least one of which ought to be antiferroelectric. With the knowledge of today one would be tempted to say that the electric response reported in these papers was probably misinterpreted. This is based on the empirical fact that up to now the only essential difference between materials showing ferroelectric and antiferroelectric behavior has been that in the latter case at least three rings have been required. However, recent unambiguous results from the Heppke and Goodby groups [260]show that two-ring antiferroelectrics exist, for instance in the second and third cinnamate compounds shown in Fig. 148. Therefore the 1981 experiments should be taken seriously and ought to be repeated.
328
13 Antiferroelectric Liquid Crystals
MHTAC
cryst 95 SmO* 130 SmQ* 133 is0
/"C
cryst31 SmI,* 63 SmC,* 118.5 SmCy* 119,2 SmC* 120.9 SmC,* 122" SmA* 148" is0
IT
Figure 149. Some antiferroelectric or anticlinic materials referred to in the text. The structures (iv) and (v) can be considered as "prototype" AF compounds. MHPOBC belongs to this category (ii). The compound (iii) is non-chiral anticlinic but its chiral version, with unequal branching parts like in (iv) and (v), can also be considered an AF prototype.
In 1976 Keller, Liebert, and Strzelecki [ 1081 had described the synthesis of a variety of aminocinnamates, most of them compounds with two benzene rings. There was also a three-ring compound with two stereogenic groups, today abbreviated MHTAC and shown as (i) in Fig. 149, the phases of which were reported [ 108al by Levelut, Germain, Keller, Liebert, and Billard in 1983. By x-rays, miscibility and texture studies they found two new mesophases between crystal and isotropic for which they propose the names smectic 0 and smectic Q. The latter is a hard-twisted smectic existing only in a narrow temperature interval below the clearing point and reminiscent of a blue phase (the phase is still not identified). The smectic layer thickness was determined but - as an alternating tilt cannot be observed in a conventional x-ray experiment - it could only be concluded that the molecules were tilted. The smectic 0 was therefore described as "disordered lamellar, similar to smectic C but distinct from both smectic A and smectic C". In the chiral forms (S,S- or R,R-)the SmO* exists from 95 "C to a couple of degrees below the transition to iso-
13.1 The Recognition of Antiferroelectricity in Liquid Crystals
329
tropic at 133 “C, whereas in the racemic mixture SmO exists in the broader range from 112 “C to 158 “C and the SmQ is absent. Such drastic displacement of both melting point and clearing point is unusual. The authors also report the most puzzling observation by G. Sigaud in Bordeaux that a great number of 1/2 disclination lines are visible in free drops of the racemate. Such half-integer defects are typical for nematics but not allowed in SmC. The study of this material was only to be continued about five years later, by Galerne and Liebert. At this time the efforts in synthetic chemistry of ferroelectric materials had shifted mainly from Europe to Japan (with a number of new players involved, in addition to Chisso Corporation, industrial groups like Mitsui Petrochemical, Showa Shell Sekiyu, Mitsubishi Gas Chemical, Nippon Denso and others) and an impressive amount of new materials with very high values of spontaneous polarization generally around 100 nC cm112 and more were synthesized. One such material was MHPOBC, shown as (ii) in Fig. 149, first made by Chisso in 1985 [261]. Avery high value of the electroclinic coefficient was measured in the A* phase of this compound with an induced tilt saturating at about 15” [262]. Below the C* phase was a new phase identified by Chisso as distinctly different with a high polarization value (measured at high field E> but a low dielectric permittivity (measured at low E>. At the First International Symposium on Ferroelectric Liquid Crystals in Arcachon (Bordeaux, France) in 1987 Furukawa reported a DC threshold for switching and designated the phase as SmY* [261]. At the same conference Hiji reported that in addition to the states identified as ferroelectric there was a non-polar state which was called the “third state” [263]. In hindsight it is surprising that even in spite of the fact that the characteristic double hysteresis loop was found several times - perfectly corresponding to Fig. 5 - it took further effort to identify the third state as antiferroelectric. Two examples of such double loops given in Figs. 150 and 151 referred instead to “tristable switching” in SSFLC and “smectic layer switching”, respectively. However, at the Second International Symposium on Ferroelectric Liquid Crystals in Goteborg, 1989, the time was finally ripe for clear evidence of antiferroelectric order in smectic liquid crystals. As the chairman of that conference I had the pleasure to place the contribution from Galerne and Liebert [266] (“The Antiferroelectric Smectic 0 Liquid Crystal Phase”) and that from Takezoe, Chandani, Lee, Gorecka, Ouchi, Fuku-
Figure 150. Hysteresis observed in an “apparent tilt angle” as a function of the applied voltage (from Tristable Switching in Sugace Stabilized Ferroelectric Liquid Crystals wirh a Large Spontaneous Polarization by Chandani et al., reference [264], May 1988). The material is MHPOBC.
7
Applied Voltage.( V
I
330
13 Antiferroelectric Liquid Crystals
Applied
Voltage
/
V
Figure 151. Hysteresis in an “apparent tilt angle” as a function of an applied DC voltage observed at 50 ”C (in MHPOBC) (from Smecric Layer Swilching by an Electric Field in Ferroelectric Liquid Crystal Cells by Johno et al., reference [265],January 1989).
da, Terashima, Furukawa, and Kishi [267] (“What is the Tristable State?”) on facing pages in the abstract book; cf. also references [268] and [269]. Just before the conference a paper by the Tokyo group entitled “Antiferroelectric Chiral Smectic Phases Responsible for the Tristable Switching in MHPOBC” appeared in the Japanese Journal ofApplied Physics, thus answering the question asked in the abstract. In this paper [270] Chandani et al. introduce the designations SmCi and SmIx for the two identified antiferroelectric phases. Referring to the conjecture by Beresnev et al. in reference [47], pp. 73-75, they conclude that the electro-optic as well as the optic properties of MHPOBC might be well explained by the herring-bone packing and also supported by selective reflection experiments at oblique incidence relative to the layer normal. The experimental evidence obtained in an ingenious way by Galerne and Liebert is rather more direct. They study thin films of MHTAC, which can be grown epitaxially in a layer-by-layer fashion at the free surface of the same material in its racemic form and in isotropic state, when the system is slowly cooled down. The number of layers can be directly counted and electric fields can be applied along the film at the same time as the birefringence can be measured in different directions against the background of the isotropic drop. Galerne and Liebert find that the physical properties depend on the parity of N, the number of layers, such that the interaction with the field depends on whether it consists of an even or an odd number of layers. They further show that the average molecular direction is along the normal to the layers. The experiments are explained by the proposed herring-bone arrangement and the alternating tilt makes the in-plane polarization alter in opposite directions from layer to layer. The spontaneous polarization of MHTAC was measured to be of the order of only 1 nC cm-*. These experiments [268] have been followed up in the references [27 I], [272], [273]. An equally convincing and direct argument to confirm the alternating tilt in adjacent layers was later found by the Tokyo group. It will be discussed in the next section. The ten years that have followed since the recognition of antiferroelectricity in 1989 have yielded an enormous amount of new AFLC structures. Today more than
13.2 Half-Integral Disclinations
33 1
1000 different compounds have been synthesized which show antiferroelectric behavior. With some exceptions they are all relatively similar. They often show a succession of phases from ferroelectric to antiferroelectric order, which are distinctly different and to date not fully identified. However, we still do not understand the decisive factors that control the stability of the different states. The discussion in Sec. 13.3 will describe the present situation.
13.2 Half-Integral Disclinations Let us consider a non-chiral smectic having the phases SmA and SmC. If we cool a sample down below the A + C transition a vector field is created along the smectic layers, which does not exist in the A phase. This is the “c director” field illustrated in Fig. 152. As already pointed out the “c director” is not really a director, i.e., it has no director symmetry, c and -c do not describe the same state. Instead c is a true vector and c and - c describe opposite tilt directions. Otherwise c has many similarities to the nematic director n. If we have a sample oriented homeotropically in the SmA phase, we will observe a schlieren texture showing singularities in the C phase, quite similar to a nematic schlieren texture, but with the important difference that disclinations appear only with integral (s = 2 1) strength. This difference is easily understood if we look at the nematic case. In Fig. 153a we have drawn the director field around a - 112, + 112 dipole of singularities. The line between these half-integral disclinations looks as if it were a line of discontinuity, as n is directed to the left above it but to the right below. However, it is not, because t and + designate the same thing. We can always change sign of n everywhere in a nematic without changing anything physically (except at surfaces, but the director n itself is to be regarded as a bulk property). Thus disclinations of half-integral value are allowed in nematics. But if the field lines in Fig. 153a are instead considered to represent the c vector field, we cannot cross the line between the singularities because this would mean that the tilt abruptly changes direction to its opposite and we would have a surface of discontinuity going across the layers.
tic A to smectic C. At any constant temperature, c can be considered a unit vector describing the tilt direction, including its sign. If we align a sample homeotropically in the A phase, it will be quasi-homeotropically in the C phase, which means that the smectic cone axis will be perpendicular to the surface.
Z
332
13 Antiferroelectnc Liquid Crystals
+++ + t i - /
/+++
/
t t t
/+++
/
Figure 153. (a) Half-integral singularities exist in the director field of a nematic because of the inherent director symmetry (n + --A is a symmetry operation). They cannot exist in a smectic C where c -+ - e is not a symmetry operation. (b) Anticlinic or alternating tilt structure represented to the right by the c field in successive layers. (c) A local displacement amounting to one layer between the two singular points allows a discontinuity along the dotted line in an alternating tilt structure to be avoided. This displacement constitutes a screw dislocation dipole. The line singularities (point singularities in the plane of the paper) now each consists of an inseparable pair of wedge disclination and screw dislocation.
Now let us look at the case of the anticlinic structure illustrated in (b), to the left represented by the alternating tilt and to the right in the corresponding c field representation. From the latter we see immediately that if we displace the layer between the singularities in (a) by one unit - this is done in (c) - there is no discontinuity any more along the dotted line. Thus the combination of a unit screw dislocation and a half-unit disclination is a singularity which would be allowed in an anticlinic structure. Such a defect is called a dispirution. In our illustration we have the combination of an Is1 = 1/2 (or 7c) disclination with a b = 1 dislocation ( b is the value of the Burgers vector, in this case amounting to one layer), but it is immediately clear that we could have other combinations of a screw dislocation and a wedge disclination, like b = 3 plus J s J= 112 or b = 1 plus J s J= 3/2, etc. For energetic reasons (deformation energy b2,s2) only small values will occur. Whereas we cannot see the screw dislocation in the microscope, the disclination affects a very large area and is easily seen. By counting the extinction lines the strength of the defect is obtained. Hence, this explains the early observation of 1/2 disclinations by Sigaud as reported in [981] and referred to above. The displaced lip in the layer shown in Fig. 153c is actually a screw dislocation dipole. Thus we have a pair of singularity lines being at the same time the core of a disclination and a screw dislocation. Hence the configuration also represents a dispiration dipole. But dispirations are of course also allowed as separate singularity lines. The c vector field around such a line with s = +1/2, followed from layer to
-
13.2 Half-Integral Disclinations
333
Figure 154. Half-integral disclination observable in an anticlinic smectic phase. The vertical singular line traversing the layers is singular both with respect to the c field and the layer ordering and is the core of a dispiration consisting of an s = + 1/2 disclination and a b = 1 screw dislocation. On passing from one layer to the next the tilt direction changes without any discontinuities except along these singular lines. Thus in (non-chiral) anticlinic materials s = 1/2 defects can be observed in addition to the s = & 1 defects, which are the only ones observable in smectic C materials (from reference [274]).
*
layer, is shown in Fig. 154. The screw dislocation allows for the molecules to have alternating tilt from layer to layer but the defect as such is a wedge dispiration of strength +1/2. Of course, the screw dislocation can as well be combined with an s = +1 disclination. This gives an integral strength dispiration which is going to be observed as if it were an integral strength disclination. The discovery that the half-integral disclinations observed in non-chiral, racemic, phases (corresponding to the chiral phases with antiferroelectric properties) could be interpreted as dispirations was made by the Tokyo group [274], [275] in 1992. Their analysis is an original and beautiful contribution to our understanding of smectics and brings striking evidence for the existence of anticlinic structures. In a substance which had a conventional SmC phase above the supposedly anticlinic phase, they could observe N 2 defects in addition to the +1 defects in the latter phase. On raising the temperature to the C phase all 1/2 defects vanished. These studies were later extended to other dispirations than corresponding to b = 1, s = 1/2 [276]. The concept of dispiration was introduced by W. F. Harris [277]. The reason for the name is that this specific kind of defect is characteristic of materials which have a helical or spiralling periodicity in addition to translation order. In unstructured materials dislocations and disclinations are the only possible line defects. If we introduce translational order the dislocation will be the characteristic defect, and if we introduce directional order, the disclination will play the corresponding role. But if we have both structural elements, a new distinct kind of defect, the dispiration, will be possible. A dispiration has a translational and a rotational component but must not be regarded as a superposition of a dislocation and a disclination, because neither the translation nor the rotation is a symmetry operation of the structure. This is a situation known in crystallography when we go from local symmetry (i.e., point group symmetry) to space group symmetry, which is characterized by the appearance of the new symmetry elements called screw axes and glide planes. The alternating tilt structure in the SmC, phase is the simplest illustration, cf. Fig. 155. The symmetry element can be considered to be a screw axis which twists as well as translates. In
334
13 Antiferroelectric Liquid Crystals
Figure 155. Smectic liquid crystal with alternating tilt (anticlinic order). This simple structure has a glide plane as symmetry element (reflection combined with a translation by half the periodicity; left) but can equally well be considered to have a screw axis, in this case a screw dyad (rotation 180" combined with a translation by half the periodicity; right). If, in addition, we assume that the molecules are chiral, the glide plane is no longer a symmetry element, only the screw axis remains a valid description. I I
Figure 156. Tilted smectic structure with a four-fold screw axis. Such a structure is helielectric. From reference [324].
this case it is a combination of a 180" rotation with a translation by half the lattice spacing along the two-fold axis. In the special case of 180" rotation the twist is not apparent but it appears more clearly in Fig. 156 where the rotation combined with a one-layer translation (1/4 of the lattice period) is only 90". In this case we have a 4fold screw axis. In such a structure one should, in principle, be able to observe dispirations of strength 114. While dispirations were introduced to account for defects in helical structures, such as found in many biologic materials, the example in Fig. 155 shows that ckality is not a prerequisite for the appearance of dispirations. The basic condition is that only a combined translation-rotation is a symmetry operation in the medium. This makes those two essential elements inseparable. Thus, for example it is not possible to separate the dislocation from the disclination in Fig. 153 - both have to appear and to vanish simultaneously. For an introduction to the study of dispirations, the reader is recommended the article by Harris in Scientific American [278] from 1977.
13.3 Antiferroelectric and Ferrielectric Phases
335
13.3 Antiferroelectric and Ferrielectric Phases From the two preceding sections we can safely consider the existence of alternating tilt in smectics as an established fact. It has also now been checked by the pyroelectric technique [278a] as well as been supported by direct microscopic evidence given by deuteron NMR [279]. If we believe that this structure is preserved in the nonracemic bulk form of these materials (which is far from obvious) then the antiferroelectric can be depicted as in Fig. 157 where the effect of an applied electric field is compared with the corresponding effect on a ferroelectric liquid crystal. Both materials are considered being in their surface-stabilized states, i.e., we disregard eventual helical superstructures. Under suitable conditions the ferroelectric is symmetrically bistable, the hysteresis is then centered around E = 0, giving two stable states for P in the absence of an electric field, Fig. 157a. Because this phenomenon is here a surface effect, under different conditions the hysteresis might be asymmetric even to the point that the cell becomes monostable, with the curve shifted to the right or left of E = 0. The antiferroelectric material, on the other hand, is always monostable, i.e., only the anticlinic state is stable in the absence of an external field. In this state the local dipoles from adjacent layers cancel each others influence, which explains the small value of E measured at very low fields. If we increase E slowly from zero, the polarization (it is here the question of inducedpolarization) will increase in a linear fashion up to moderate fields. The optical transmission corresponding to this dielectric mode constitutes a powerful linear electro-optic mode with a cut-off at about 1 MHz. It has the same symmetry as the electroclinic mode in the S m A * phase, but the underlying molecular mechanism is entirely different. It is not a soft mode but generally a two-layer cone mode [280], [281]. In order to obtain high contrast in an AFLC display, however, this mode is a nuisance since it leads to light leakage and a bad dark state, cf. Sec. 13.6. If the molecules would be aligned as in the figure (which may be rather unlikely at least at the surface) a field applied perpendicularly to the paper would essentially try to turn around the molecules in every second layer into the direction of the field. This corresponds to inversion of a sub-lattice polarization and as the electric torque P x E in this situation is zero or at least very small, the effect is small at the beginning and the turn-over is associated with a considerable threshold indicated in the figure. Ultimately all molecules will however be aligned along the field (for E > Eth).We have then arrived at what is called the ferroelectric state of the antiferroelectric. The word state is here essential as opposed to the often heard confusing statement that this is a transition to a ferroelectric phase. However, we do not mean that the material changes its thermodynamic phase when a field is applied. (If we did there would be thousands of “phases” and “phase transitions” in liquid crystals -the switching of a twisted nematic of the unwinding or a helix would be a phase transition etc.) At this transition to the ferroelectric state we can now measure the value of the spontaneous polarization P,, by measuring the cur-
336
13 Antiferroelectric Liquid Crystals
E
monostable
\ B bistable
Figure 157. Simple planar structure for the antiferroelectric phase to the right (b), as compared with the ferroelectric, left. The direction of the dipoles is indicated in the middle of the molecule. The two states in (a) are spontaneous and exist in the absence of a field. The corresponding states are achieved in the antiferroelectric on applying a field IEl > Eth.
rent corresponding to the change from macroscopic polarization zero, to macroscopic polarization P,. We underline that this view of the meaning of P, is in conformity with the operational definition of P, in Sec. 3.4: the spontaneous polarization is the intrinsic polarization attached to one layer, free from any flexoelectric effects and is what we measure when we have aligned all the dipoles homogeneously in an electric field. If the material has both an antiferroelectric phase and a ferroelectric phase, P, thus naturally is the same quantity for both, and measured in the same way. The only difference is the inevitable temperature dependence, which is already present within each phase. In contrast to this, the mesoscopic polarization P,, is the effective po-
13.3 Antiferroelectric and Ferrielectric Phases
337
larization of a few layers (in practice from 2 to lo), which is not a characteristic of the material as such (MHPOBC for example) but sensitively depends on the tilt ordering from layer to layer and therefore is quite different for different thennodynamic phases of the material. Thus in the AF phase of Fig. 157b P,, = 0 while P , # 0. In Fig. 158 we have depicted a corresponding tentative structure for aferrielectric liquid crystal, according to the interpretation by the Tokyo group [282]. In this case there are two stable states ( E = 0) showing a macroscopic polarization, in which two successive layers are synclinic and the third anticlinic. The lattice period is thus three layers and the mesoscopic polarization P,, is equal to 1/3 of P, according to this model. Ferrielectric materials constitute a subclass of ferroelectrics, for which P,, c P, but distinct from zero. The ferroelectric structure of Fig. 158 is not tetrastable as often stated but bistable like all ferroelectrics. Likewise the antiferroelectric in Fig. 157b is not tristable but monostable like all antiferroelectrics.If we want to keep one meaning to the word “bistable” then the obsolete expressions “tristable” and “tristable switching” should be avoided. Any dielectric is “tristable” by the same logic because it stays in the same polarization state +_Pif we apply a holding voltage ?V. Furthermore, the monostability of the AFLC is really a key point in its application in display devices, giving the very important grey scale capability of these materials. The anticlinic order in the AF configuration of Fig. 157b could in a way be looked upon as the ultimate limit for a helielectric, i.e., a helicoidal antiferroelectric (helical SmC*) when the helical pitch shortens to finally equal the length of two layers. The field-induced transition to the ferroelectric state accordingly can be looked upon as the helix unwinding in an electric field. This analogy tends, however, to give much to high values for the threshold field in the AF case, of the order of hundreds of volts per pm instead of the typically observed, of the order of 20 V pm-’ . At the beginning of this chapter we concluded that high values of P, would promote the occurrence of antiferroelectric order as a result of dipolar interaction. This is certainly true. As the dipole-dipole interaction energy is proportional to the polarization squared
G,-P,2-e4
(596)
we can expect the antiferroelectric phase to appear for high values of the tilt. Thus the expected phase sequence is SmA* - SmC* - SmC:. ..
(597)
on decreasing temperature. This is also what is found: the C,* phase always lies below the C* phase and has a higher value of the tilt angle. However, one must not overrate the importance of P, if different molecular structures are compared. The most strilung illustration of this is the discovery by Nishiyama and Goodby [283] in 1992 that non-chiral compounds may also exhibit alternating tilt phases. One of these compounds is shown as (iii) in Fig. 149. It belongs to the so-called swallow-tailed
338
13
Antiferroelectric Liquid Crystals
bistable
Figure 158. Simple planar structure of a possible ferrielectric phase. The two stable states are indicated on the hysteresis curve and correspond to the two inner configurations, which thus are spontaneous, with a macroscopic polarization of (113) P,.
compounds, some of which had been known to give smectic phases far earlier [284]. Nishiyama and Goodby found that these structures showed complete miscibility with known standard antiferroelectric compounds. They could also be made antiferroelectric if doped with chiral additives according to
although this “reaction” does not work in several of the few cases where it can be tested. (AFLC materials for displays, unlike FLC materials, are not prepared this way, first of all because of the scarcity of C , compounds.) These non-chiral anticlinic compounds were initially designated Calt;we will just call them C, where a stands for “alternating” or “anticlinic”. The chiral version is then accordingly C?. This is the antiferroelectric which is also commonly written Cg. We will however save the capital letters for the main phase designation like N,
339
13.3 Antiferroelectric and Ferrielectric Phases
SmA*
SmC,*
+p
I SdA
I
SmA*
*+
SmC,*
A, C, I ... and use small letters as indices, also in accordance with C2, Cp*,C; for antiferroelectric or ferrielectric phases. Use of capitals would also be in conflict with symbols like TGB(A), TGB,, TGB, etc. where A and C denote smectic A and smectic C order. The anticlinic order exhibited by compound (iii) is obviously not due to chiralityrelated polarity but to steric action, and it was found that increasingthe extent of branching of the terminal alkyl chain promoted the appearance of anticlinic order and counteracted the synclinic (smectic C) order. Measurement and comparison of the layer thckness in these and non-branched compounds supported the interpretation that there is some interdigitation of the molecules between the layers in the first case. If the nonchiral structure (iv) is transformed to a chiral counterpart by making one of the branching tails different, like in (iv) or (v) it acquires antiferroelectricproperties. In reality the situation turned out to be still more complex. Subsequent investigations by the Tokyo and Hull groups showed that the transition SmA* + SmC* in these compounds often takes place via another distinct phase, called SmCz. However the SmC, phase does not exist, that is, the corresponding transition in the racemate is S m A + SmC. Likewise, the transition SmC* + SmCz takes places via a distinct phase denoted SmC;, but SmC, does not exist; the racemic transition is SmC + SmC,. The high-temperature part of the phase diagram of MHPOBC therefore looks as shown in Fig. 159. Sometimes C 2 and C; are called subphases because they only appear in optically very pure compounds, but they have to be regarded as phases in their own right. Possibly more such phases with very limited range exist as indicated by the thermogram obtained by adiabatic calorimetry in Fig. 160.These phases have been detected by a number of experimental methods in addition to adiabatic or differential scanning calorimetry. The electro-optic behavior indicates that besides Cz there seems to exist a three-layer period ferrielectric phase which could be identified with C; and having P,, = P,/3. The distinction of this phase was also confirmed by conoscopy [282]. All these phases have turned up in dielectric spectroscopy, at least now and then, and in addition a new antiferroelectric phase with a tentative four-layer period was reported to occur in thick samples. This phase was called "AF". Unfortunately the terminology is, as we see here, not very consistent.
340 1350
13 Antiferroelectric Liquid Crystals I
1
--
z
7 ’
Y
1250
2.
4 1150
400
390 temperature (K)
Figure 160. Heat capacity measurement by adiabatic calorimetry showing the “subphase” C*, between A* and C* as well as C: (and possibly a second phase) between C,* and C*. The “subphases” do not exist in the racemate. From reference [282].
Figure 161. Possible helical structures in the antiferroelectric SmC: phase. The structure in (b) means that the phase angle increment is constant from one layer to the next. This means that there is only partial cancellation between local polarizations from one layer to the next.
Selective reflection studies then revealed that the phases seemed to be helical in bulk. It was found that the optical periodicity is equal to half the helical pitch in the C,* phase, as distinct from the periodicity in the C* phase, which is a full period at normal incidence, but similar to the case of the N* phase, which at normal incidence also shows a half period. But in the C z phase the optical periodicity showed to be half the pitch even at ablique incidence. This indicated that the anticlinic C,* in reality has a superstructure consisting of a double helix. A simple and straight-forward way to think of this helical structure would be to imagine that the molecules, keeping their anticlinic configuration from one layer to the next, would spiral as a pair, cf. Fig. 161a. However, a second consideration shows that the helix, being the result of the chiral interaction between two adjacent layers, might tend to distribute the twist equal-
13.3 Antiferroelectric and Ferrielectric Phases
34 1
Figure 162. The alternating tilt model of the SmC; phase showing the double helical superstructure (from Blinc, reference [ass]).
ly from layer to layer - which is also natural from an elasticity point of view - leading to the alternate structure in Fig. 161b, which has a longer pitch relative to the picture in (a). The interesting thing with this second, more realistic structure in (b) is that, in contrast to the structure in (a), its mesoscopic polarization Pmsis not zero. In fact, keeping in mind that P, is typically one order of magnitude higher in conventional antiferroelectric liquid crystals than in ferroelectric liquid crystals, the P,,value in the former case may not be essentially different from P, in the latter case and thus the unwinding of the helix is not only qualitatively the same process but also quantitatively quite similar. In Fig. 161 the twist is highly exaggerated. Normally the pitch is found to be similar to that in the C* phase, typically lying in the range 0.1-1 pm. It is incommensurate as in the C* phase. A somewhat less exaggerated picture of the twist is shown in Fig. 162, clearly illustrating the double helix structure. The designations C;, CJ, and C ; stem from the first Japanese studies of the compound MHPOBC where a /3, y was used in order of descending temperature. Later it turned out that CB is simply identical to the normal C* phase. Therefore the designation C$ does not appear any more. As for the remaining notations, C,* corresponds to the “official” antiferroelectric phase. Whereas the racemate, C,, is planar and biaxial, corresponding to Fig. 157b (showing no helix), the chiral (2: seems to correspond to Fig. 161b or Fig. 162, with a helix often somewhat smaller than the C* phase helix of the same compound. As for the C g phase, it remained mysterious, but the experiments indicated an extremely short pitch. The C ; phase on the other hand, quoted as ferrielectric, seemed to have a pitch typically five times longer than
342
13 Antiferroelectric Liquid Crystals
in the C* phase. It was also reported that the helical structure showed a discontinuous jump at the C* + C$ transition accompanied by a handedness change. Thus, the phase situation is quite complex. How can the experiments be interpreted? Before we can try to answer this question, however, we have to bring up another relevant matter, which will make the situation even more complex.
13.4 A Complicated Surface Condition One consequence of the alternating tilt structure is that AFLC materials would require a non-homogeneousboundary condition in order to fit the boundary layers with the bulk. As far as is known there is no recipe on how to make such a boundary: it would require a precoating of the surface with an alignment layer which itself had a periodic structure with a period matching the smectic pair of layers. Such alignment materials may be developed in the future but are presently not available. At least one would have to require a two-fold degenerate tilt alignment. Some surfaces with limited capacity in this direction have been achieved and studied [286], [287], [288], but the tilt is generally not stable against temperature changes and these methods, therefore, have never found industrial application. With standard aligning materials it is almost certain that there is a transition layer from the boundary to the bulk, which is mediated by defects. This is a situation that is also valid for a short-pitch SmC* material in bookshelf geometry and indeed also for a cholesteric in the geometry that the helix is parallel to the surface. But in the case of SmC: or short pitch SmC* the situation is even more severe than for the non-chiral SmC, or the cholesteric due to the additional polar part of the surface condition. This is illustrated in Fig. 163. Very probably, the fact that it is much more difficult to achieve a good alignment (and good dark state) in C z or short-pitch C* materials can be traced back to these phenomena. Recently Lee et al. [289] have also reported that crossrubbing considerably improves the dark state in AFLC cells. Crossrubbing had already been used by Canon in the FLC case, in order to improve the bistability. In the AFLC case the split in two degenerate tilt states is an even more severe affair because the tilt angle is twice as large (Canon used a tilt angle 8 less than 14”). Whether the normal AFLC case corresponds to the left or the right of the bottom pictures in Fig. 163 or to anything else is hard to judge based on available experimental data. The left picture corresponds to Fig. 157b whereas the right picture would correspond to an unwinding in an applied electric field as shown in Fig. 164. Surfaces normally have a certain polar preference and they normally also have only one “easy axis” for the director. This means that with a chiral tilted smectic the surface is most comfortable with a pure SmC* state, i.e., even with only one of the possible cone states. For a material with a transition C* + C: at a certain temperature To the surface layers can therefore be expected to stay in the C* state well be-
13.4 A Complicated Surface Condition
343
Figure 163. Hypothetical surface conditions for SmC: materials (bottom) in bookshelf geometry. In the first, non-chiral, case we envisage two situations, one with the tilt plane lying in the surface (left), the other with the tilt plane perpendicular to the surface. Both might roughly correspond to rubbing in the direction (r)shown. In the chiral case (bottom) the situation is further complicated by the local polarization connected with every layer. Thus the surface is now asked to accept not only a rapidly alternating tilt but also a rapidly alternating polarization. This would require a polarity-neutral surface, which might be achieved by so-called cross-rubbing [ 1741. For instance, if the surface prefers the P direction to be into the liquid crystal (up in the picture) it could, at least partly, be neutralized by rubbing in the direction corresponding to the alternate tilt, as indicated.
low To,while the bulk is in the C: state. In addition there is a coexistence in the bulk itself because the transition is first order. If we make the sample very thin an appreciable part of it will then be in the C* state, to the point that the C,* phase might be squeezed out altogether. That surfaces in general tend to stabilize the C* phase and destabilize the C$ phase, and that the apparent phase transition temperature depends on sample thickness is an experimentally established fact. That the appearance of other, more subtle antiferroelectric and ferrielectric phases should be surface and thickness dependent, must be expected afortiori. This is also what is frequently observed [290], [291] in samples with planar orientation. Indirect and direct evidence for the coexistence of the C* and C: phases over large temperature intervals have been reported in [290]-[294] and in [294a]. The most direct evidence is found as striped domains of alternating C* and C$ bands [290] and in the simultaneous peaks from the two phases in the dielectric spectra [294a], [260], in both cases with striking thickness dependence. The striped bands were observed equally well [290] in the racemic case (C,C,). Sometimes the dielectric data can be very eloquent in showing how the less stable phases vanish. Fiitterer [260] has traced the detected temperature for the transition C* -+ AF as a function of cell thickness. This temperature is constant, equal to 26.4 "C, in thick cells of about 20 pm to 50 pm, but decreases rapidly beyond 10 pm and the transition never occurs below 2 pm, cf. Fig. 165. The C* phase has then taken over; AF is completely squeezed out.
344
Ee
13 Antiferroelectric Liquid Crystals
E=O
E@
Figure 164. Unwinding of the antiferroelectric “helix” in an external electric field E > Eth,i.e., transition from the antiferroelectric to the ferroelectric state. The AF state for E = 0 corresponds to the local polarization vectors lying along the plane of the surface (after reference [295]).
Cell thickness d (Wm)
Figure 165. Phase transition temperature To for the transition C* + AF (evaluated from dielectric spectroscopy) as a function of cell thickness L. The material is compound (ii) from Fig. 148 (from reference [260]).
If, on top of all this, we add that chevrons form in the Cz phase as well as in the C* phase, it is even harder to find a surface condition which might cope with the bulk structure. In this case there is no possible solution with a planar director as to the left in Fig. 163 but only some compromise between the left and right version. The most natural one is in Fig. 166a corresponding to alternating directors but with constant pretilt at the surface, whereas (b) seems entirely unrealistic. Both have been drawn with the layer leaning angle 6 equal to the director tilt angle 0, corresponding to the finding by the Bordeaux group [249] that 6tends to equal Bfor C$ phase chevrons while 6 < 8 for C* phase chevrons. However, even in (a) the pretilt is extremely high (equal to 0) and furthermore the condition at the surface S is not a possible condition in the chevron plane. Therefore it is more realistic to imagine that a zero
13.4 A Complicated Surface Condition
345
(d)
Figure 166. Hypothetical director conditions for SmC: (SmC,) materials in a chevron structure.
8
pretilt state is present at the surface (which corresponds to synclinic smectic C* order) and that the same horizontal state is present in the chevron plane. This chevron region (c) would correspond to a discontinuous jump of the P vector between two opposite horizontal directions at the chevron plane. However, the chevron plane cannot be infinitely sharp but has to be rounded off over at least a small region. In this region the tilted smectic becomes an orthogonal smectic (SmA* state) and thus P goes to zero on reaching the chevron plane from both sides. This would be the proposed structure (d) for the antiferroelectric chevron, taking into account the experimental evidence that 6 = 8 and having the feature that P goes to zero in the wall region between two domains of opposite polarization, which is the well-known behavior in walls in solid ferroelectrics. This structural feature is of course also to be assumed in the FLC case.
346
13 Antiferroelectric Liquid Crystals
13.5 Landau Descriptions of Antiferroelectric and Ferrielectric Phases Soon after the recognition of antiferroelectricity in 1989 Orihara and Ishibashi [296] presented the first theoretical model to account for the antiferroelectric behavior. This was a natural extension of the Landau expansion for the ferroelectric case using the Pikin-Indenbom two-component order parameter. This description was then developed further by &kS, Blinc and CepiE [297] and by i e k s and CepiE [298]. In this phenomenological, continuous, description the unit cell consists of two smectic layers. In each of the two layers a Pikin-Indenbom vector
ti
is introduced, and the pair is assumed to be able to change continuously when we move along the layer normal ( z direction). The directors niare the directors in odd and even layers and will then tilt in the same direction for the ferroelectric state, in opposite direction for the antiferroelectric state and with an angle different from 0
L
Figure 167. (a) Geometrical relation between the c director and the tilt vector 4, which shows the direction of the polarization in each layer. 4 is the Pinkin-Indenbom vector order parameter, which can also be regarded as an axial tilt vector. (b) Relation between polarization directions (t,,&) of alternate layers in the ferrielectric case (y2*O), in the middle shown for four consecutive layers in the z direction. The cones also viewed along this direction although in the figure for convenience spread out vertically in the paper plane. This state has a macroscopic polarization (directed towards the right in the figure). To the outermost right the helical superstructure has been added (double helix drawn with 16 layers period (2 x 8) in this example, although it is generally incommensurate. The structure is thus helielectric just like the helical SmC*, but the surface-stabilized state, represented by the middle figure, is femelectric. (c) Hypothetical ferrielectric structure with a difference in tilt angle (and therefore polarization) in alternate layers. This structure seems unrealistic and also lacks experimental support.
13.5 Landau Descriptions of Antiferroelectric and Ferrielectric Phases
347
or Kcorresponding to a ferrielectric state. At the beginning the magnitude of the tilt, 8, and 8,, was also allowed to be different. As order parameters in the expansion are chosen
5,
where represents synclinic order and 5, anticlinic order. To get a feeling for the meaning of t,,C2, tS, let us make the tilt small (niz-- 1). The c director c, = (njx, n-) is then preceded by {i which is d 2 in advance according to Fig. 167a. Thus 5, and represent the directions of the local polarization in odd and even layers. For the ferroelectric case is equal to zero and we have, from Eq. (600)
5,
t2
5,
whereas for the antiferroelectric case 5, equals zero and thus
The case that 5, and 5, are both nonzero corresponds to possible ferrielectric states. The expansion in 5,and 5,takes the form
It will also contain Lifshitz invariants in both 5, and t,, but we will here limit the discussion to the two bilinear coupling terms with the coefficients y, and E. They are nonzero only in the ferrielectric case. The first corresponds to coupling between the order parameters as expressed by their magnitudes, the second between their directions. For y2 > 0 this second term gives a positive contribution to the free energy except when 5, and 5,are perpendicular. For sufficiently large y2 > 0 the term will thus tend to lock and 5, to the situation {a I which also corresponds to 6, I cf. Fig. 167b. For y2< 0 5, and 5, tend to be parallel. This gives a hypothetical phase in which I I # 1 C2 1, i.e., with different angles of tilt in alternating layers, Fig. 167c. It would correspond to a smectic with two different periodicities along z , which has never been observed. The only realistic proposal for a ferrielectric offered by the model is thus the one given in (b) where the phase difference A 9 between the c director of alternate layers is locked to be ad2.
5,
5,
t2,
348
13 Antiferroelectric Liquid Crystals
0
112
113
114
115
215
(4
(b)
(c)
(4
(4
(f)
1
(9)
(h)
Figure 168. Long range order tilt configurations as a result of applying the ANN1 (Axial Nearest Neighbor Ising) model with long-range repulsive interaction of Bak and Bruinsma to the smectic case. The wave vector q is given on top. The presumed antiferroelectric state "AF" (g) is not contained in the model and therefore has no q value. The structures (a) and (h) represent the two stable ferroelectric states. For the other states the effective tilt angle 0 can roughly be considered proportional to the macroscopic polarization which is from (a) to (h): - P s , 0, f (113) P,, 2 112 P,, 315 P,, ? 115 P,, 0, +P,.
The continuous model by Orihara-Ishibashi and by ZekH and collaborators thus gives, in addition to the ferroelectric and antiferroelectric phase, a possible fenielectric phase which might tentatively be identified with the C; phase. However, as concluded from conoscopy studies by the Tokyo group [299], when an increasing electric field was applied in this phase it behaved in a way incompatible with the structure in the middle of Fig. 167b. As a result of these studies Fukuda and his collaborators instead proposed [299] a planar structure for the C; phase shown as Fig. 168 (f). This structure has a period of five layers and a mesoscopic polarization P,, = (1/5)P,. A little later the same year they proposed an alternative structure [300] with a period of five layers (e) but with P,, = (3/5)P,, and in the following year a three-layers (c) structure [301] with P,, = (1/3)Ps. These structures were postulated partly with the support of the fact that there are discrete models which are able to account for such fractional values of the polarization. Furthermore, such discrete theories could give many more phases - in principle an infinity of phases - between C,* and C*. While %kH and CepiE [298] could account for more than 20 possible phase sequences, among them CZ-C;-C*-A*, their model could not reproduce the sequence CZ- C;-C*- C i - A*. In addition, the Fukuda group had now detected still more subphases, so that the most complete experimental sequence observed was
Here FIL is a ferrielectric phase (often called FI,) to the lower side of C; and FI, (often called FI,) one on the higher side. Thus there were three ferrielectric phases between the two antiferroelectric Cz and AF, the latter postulated to have the structure of Fig. 168(g). In fact, near the ferrielectric phases FI, and FIH further ferrielec-
13.6 Ising Models
349
tric phases were reported [282]. As the continuous models do not seem capable of giving enough subphases, we will now have to turn to a discussion of the discrete models. In an attempt to provide the desired subphases, and espcially to account for the C ; phase, Orihara and Ishibashi were soon able to present discrete models [302], [303] as an alternative to their previous ones. The prototype for such a description is the so-called Ising model, which goes back to an idea by Lenz from 1920 for a very simple description of ferromagnetism [304], an idea that he proposed to his doctoral student Ising [305].
13.6 Ising Models In order to make the phenomenon of magnetism accessible to simple calculation, the idea by Lenz was to assume that the magnetic moments of the atoms could only take one of two opposite directions and that there is an interaction between nearest neighbors (but only between nearest neighbors), such that the spin sitends to be parallel to the spin of the neighbor sj. Hence a state is conceivable where all spins are in the same direction but this state is counteracted by thermal disorder, represented by kT. The interaction between nearest neighbors can be described by a hamiltonian
where the summation is taken to be over all pairs, and where s can take only the discrete values + 1 or -1. J , is the interaction strength which for J , > 0 promotes ferromagnetic order. For Ju < 0 the hamiltonian (605)gives an equally simple description for the antiferromagnetic state. The hamiltonian H corresponds to our previous free energy except that there is no entropy term in H. This term has to be added in order to find the equilibrium state. For finite temperature, T > 0, there can only be partial order and the order parameter, which is the magnetization
M = -1E
si
Ni here taken to be the fraction of up (or down) spins, (-1 < M < l), is generally a function of temperature. We have here normalized M by dividing by the total number of spins, N . Ising showed in his thesis [305] that for a one-dimensional problem there can be no magnetic order, i.e., that M = 0 for T > 0. The problem was then taken up by others but not until 1944 was the corresponding problem in two dimensions solved analytically (showing magnetic order), by Onsager in a paper which was a breakthrough in the theory of phase transitions and critical phenomena [306]. When we deal with smectics we would seem to have a one-dimensional king problem along
350
13 Antiferroelectric Liquid Crystals
L
+ + + + + t i ++ + + + + + + + + ++++++j-
+““.
- - - +++ - - + -
1 1 1 1 1 1!0 0 0 0 1 1 1 0 0 1 0
..... ” ” ’
c c c c c c 0 0 0 0 c c c 0 0 c 0”’” s s s s s s a a a a s s s a a s a.....
Figure 169. One-dimensional king chain. A spin interacts only with its neighbor and wants to be in parallel direction. A homogeneous spin up state is therefore conceivable, but any spin flip that breaks the state at some point also breaks the connectivity between the spins which can no longer influence each other on either side of the line L. The chain is therefore macroscopically disordered for any temperature T > 0. An king variable can only take one of two states. Some other representations are shown below the spins (c, o closed-open, s, a synclinic-anticlinic, etc).
the L direction (layer normal), but this is not quite true. To have some order in the Ising variable we would therefore have to add some long-range interaction in addition to the hamiltonian in Eq. (605). We can see why when we realize that the reason for the disorder in an Ising chain, cf. Fig. 169, is the lack of connectivity: once the parallel order has been locally broken there is no connectivity and no interaction between the separated parts, as long as we are confined to one dimension. Figure 169 also illustrates how the Ising model could be applied to smectic liquid crystals or, in fact, to a lot of systems which have nothing to do with magnetism whatsoever. The prerequisite is that there is a two-state variable, +1 and -1,O and 1, + and -, blue and red, absent or present, active and passive, and J, etc., and the tendency (expressed by the interaction) that atoms, people, etc. do “what their neighbors do” (or contrary to what their neighbors do if we want analogs to the antiferromagnetic state). This explains the extremely wide field of application for this model beyond phase transitions, spin-glasses and also far beyond physics, for instance in modelling of neural networks, or social and economic phenomena. Still it is not apriori clear that it should work for smectics. In the presence of an external magnetic field H , even the spins in a 1D Ising chain would line up, and to the previous hamiltonian we would have to add a field term proportional to -MH, or
g=-ZH i
Jii si s j
sj -
0
If we want to introduce a temperature, the hamiltonian could in principle be turned into a free energy by subtracting TS. In order to describe long-range periodic or incommensurate structures in solids, Bak and Bruinsma [307] had, in 1982, introduced
13.6 Ising Models
35 1
a hamiltonian of a similar form with the interaction J , = -J, J > 0, representing an antiferromagnetic ordering tendency between adjacent spins. Their expression can be written
The interaction H wants to have adjacent spins parallel while the interaction J wants not only these but all spins to be antiparallel. The outcome of this conflict depends on the relative sizes of H and J where His short range (nearest neighbors) and J long range and supposed to fall off like
J ( i - j)--
1
li- j I 2
with the number I i-j( of lattice planes between the two spins. As J is opposed to parallel spins in general we might say that it is a long-range repulsive potential. Now, it turns out [307] that this model gives an infinity of states with various degrees of magnetic order M between 0 and 1. Each state has a certain stability, which can be calculated and expressed in AHH. In fact, every rational fraction q = m/n < 1 corresponds to a possible state where q is the fraction of UP states and where the ordered appearance of UP states (+) along the chain is given as a sequence
xi = Integ (ilq)
(610)
which we will discuss in a moment. Thus q = 1 corresponds to all spins up or M = 1 and q = 1/2 to half the spins up or M = 0. The first is a ferromagnetic state, the second an antiferromagnetic state. For 1/2 < q < 1 we have an infinity offerrimagnetic states. The system is symmetric around q = 1/2 such that for q < 1/2 instead a majority of spins point down and M = - 1 for q = 0 which is the equivalent ferromagnetic state of opposite direction to the state represented by q = 1. The curve representing q as a function of H/J has a very peculiar shape and is shown in Fig. 170. It is known by the name devil’s staircase and is the brainchild of the German mathematician Georg Cantor. It is continuous, has a finite length but has an infinite number of points where it is not differentiable. It proceeds from zero to one by an infinite number of steps. It may superficially look as if there were only a finite number of steps in Fig. 167 but between any two steps there are infinitely many steps (between any two rational numbers there are infinitely many rational numbers) and soforth in infinity. Consequently every step itself is infinitely small though it may look as if there were big jumps here and there in the figure. The devil’s staircase is a fractal object (its fractal dimension is here about 0.87, i.e., it is “less dense” than a normal line which has dimension I). As a fractal object it is scale invariant, which is demonstrated in the figure: if you look at a small inter-
352
13 Antiferroelectric Liquid Crystals
0 0
I
I
1
2
I
3
1 4
I 5
I
6
MAGNETIC FIELD H/J(l)
Figure 170. Density of up spins q as a function of the strength of the magnetic field H in the ANN1 model. The spin density is described by a fractal curve going from 0 to 1 called the devil's staircase. The length of each step shows the stability range (AH) of the corresponding state. In addition to the two equivalent stable ferromagnetic states q = 0 and q = 1, the antiferromagnetic state q = 1/2 has a wide range of stability. The other states represent ferrimagnetic order, and q appears also as a characteristic wave vector for this long-range spatial order (from reference [308]).
Val of the curve under magnification, it looks like the whole interval. If, so far, we have been referring to magnetic systems, it is clear that the model by Bak and Bruinsma can equally well be tried on a number of other systems, like solid ferroelectrics and polar smectics. (Whether it correctly describes any system in nature can only be decided by experiment.) Thus we ought to test what it would tell about the observed polar smectic subphases. However, this bizarre function does not mean that in reality we should expect to observe an infinity of states between ferroelectric and antiferroelectric because, as Fig. 170 shows, the stability range would be far to small for most of them. If we consider the case 4 = m/n < 1/2 then the stability AH falls off rapidly with increasing n (roughly like l/n3). We should therefore investigate what the solutions 4 = 1/2, 1/3, 1/4, 1/5, and 2/5 mean in the first place. The meaning is contained in Eq. (610) which says that xi is equal to the integer part of the fraction i/4.For 4 = 1 we have x, = i which gives a dipole up for every i, thus the configuration ?"???T ...or + + + + +... . This is the ferroelectric state corresponding to + P,. For q = l / n we get xi = Integ (ni)= ni, i.e., x1 = n, x2 = 2n, x3 = 3n. .. . Thus 4 = 1/2 gives an UP state for
13.6 king Models
353
x = 2 , 4 , 6, 8,... or -
+ - + - + - +...
L R L R L R L R... for polarization and tilt direction (left, right), respectively.This is the antiferroelectric state (C:) of Fig. 168(b) with the period 2 and P = 0. In the same way q = 1/3 corresponds to up states at
x = 3 , 6 , 9,... o r - - + - - + - - +
...
(612)
with the period 3 and P = - (1/3) P,. The two equivalent ferrielectric states for P = +(1/3) P, are given in Fig. 168(c). We see here that for q = l/n the period is generally equal to n, that is to llq, and we could therefore say that q is the wave vector or the wave number for the long-range order. For q = 1/4 and 1/5 the configurations are
and
corresponding to the tilt representations of Fig. 168(d) and (e). In (613) and (614) we have given the two equivalent configurations representing the two presumably bistable ferrielectric states. The given P values are the macroscopic polarizations. If there were to be a long-wavelength helical superstructure (not accounted for by the model) superposed on these structures, P would be equal to P,,, the mesoscopic polarization. If the structure is flat, this mesoscopic polarization is also equal to the macroscopic polarization. The state q = 0 corresponds to q = I/n when n + a)i.e., when the period gets infinitely long. Then we have only down states, thus
corresponding to q=l:
++++++
If we now turn to the case q = 2/5 we get
xi= Integ (ilq)= Integ (5i/2)
P=+P,
(6 16)
354
13 Antiferroelectric Liquid Crystals
Figure 171. Macroscopic polarization P of the spatially modulated structure as a function of the fractional number q of local polarization UP states (tilt-to-the-right states in Fig. 168). The only antiferroelectric state is represented by q = 112.
As the sequence 5i/2 is 2.5, 5, 7.5, 10, 12.5, . . the corresponding sequence for Integ(W2) gives UP states for x i = 2, 5,7, 10, 12, 15. ... - + - - + - + - - + - + shown as tilt in Fig. 168(f). Like the configuration (614) for q = 1/5 this has the period 5 but is otherwise different with a ferrielectric polarization equalling only 1/5 of P,. As a final example we choose a value q > 1/2, let us say q = 3/4. With the sequence 4i/3 being 1.33,2.67,4, 5.33,6.67, 8,9.33 ... we get UP states for xi=1,2,4,5,6,8,9,10,12
.......
+ + - + + + - + + + -+...
P = (1/2) P,
(619)
If we consider all sequences as infinitely long this one is identical to the lower sequence of (613). They are thus describing the same state and illustrate the fact that the solutions are symmetrical around q = 1/2. Thus q = 1/4 and q = 3/4 describe the same ferrielectric state with P = +(1/2) P,. In the same way the reader might persuade himself that q = 2/3 describes the same state (+ - + + - + + - + ...) as q = 1/3 with P = f (1/3) P,, and so forth. Fig. 171 shows the P value as a function of q. The apparent tilt angle of the optic axis in the ferrielectric states should increase roughly in the same way. Summing up, we can see that this axial nearest neighbor Ising (ANNI) model by Bak and Bruinsma if applied to the chiral smectic case would give, in addition to the ferroelectric phase C* ( q = 0, q = 1) and the antiferroelectric phase C,* ( q = 1/2), a number of ferrielectric phases, among them q = 1/3 (or q = 2/3) which could be identified as the C; phase. This phase would have a macroscopic polarization (1/3) P, where P, is the spontaneous polarization and it would have an apparent tilt angle 6, which is a third of the tilt angle in the C* phase. This seems to be born out by experiment [309]. On the other hand, the model gives only one antiferroelectric phase ( q = 1/2),whereas experiments show that several antiferroelectric phases exist [218].
13.6 k i n g Models
355
In 1978 Bak and von Boehm constructed another model with staircase behavior [310] which is in fact better suited to be applied to the smectic case. In this model which was made with the purpose of accounting for the important role of temperature in determining the sequence of periodic and incommensurate layered magnetic structures, there is no long-range interaction. In addition to the interaction in the layer, which wants the spins to be parallel, and the interaction between nearest layers which promote the same parallel order, there is an interaction between next nearest layers which wants the spins to be antiparallel. This description is therefore often called the ANNNI (axial next nearest neighbor Ising) model. This ANNNI model indeed gives the antiferroelectric state in Fig. 168(g) if applied to our case, but it gives neither the Cz state ( q = 1/2) nor the C T state ( q = 1/3) among the infinite number of possible states that it predicts. Yamashita and Miyazima [ 3 111 tried to remedy this by adding a third neighbor interaction to the ANNNI model, and they were then able to reproduce the phase sequence Cz - C; - AF - C ” . In addition some ferrielectric phases were found which could be candidates for FI, and FI,. So far the Ising variable has been the tilt itself. Because the tilt occurs in both chiral and non-chiral systems the Ising molecular field H must contain both steric and polar interactions. The steric interactions are clearly very strong within a layer and also force the dipoles into parallel order. Between nearest layers it is not unreasonable to assume that the dominant steric interactions favor synclinic tilt while the dominant polar interactions favor antiparallel dipoles just like in the nematic case. This latter interaction is in principle long-range. However, Bruinsma and Prost have shown that from the next nearest layer on this dipole-dipole electrostatic interaction is zero [312]. In the Ising model we therefore have to find a separate expression for the long-range interaction or at least an interpretation of where it comes from. As Bruinsma and Prost found out, the c director fluctuations may be responsible for the long range repulsive force needed. This is because these fluctuations are coupled to fluctuations in the P field and thus to fluctuating polarization charge p - - V . P.The coulomb interaction - ( V . P,) ( V . 4) is then with certainty long-range and repulsive. Because the simple Ising model discussed so far failed to provide any other antiferroelectric phase besides c:, the Tokyo group embarked on a different interpretation of the model. They argued that the subphases result from a competition between two different interactions contained in H , one favoring ferroelectric order, H,, the other favoring antiferroelectric order, H A , such that H = HF- H A . H, and H A are supposed to act on a pair of neighboring layers such that “F orderings” (layers having the same tilt) and “A orderings” (layers having opposite tilt) occur. The q value is now assumed to control, via Eq. (610), not the occurrence of UP states, but the occurrence of F states. It is claimed (reference [282], pp. 1006-1007) that the F states repel each other and in fact represent the J interaction, while it is not certain whether the A states repel one another. These ideas that there could be repulsive interaction between layer pairs if they represent change in layer tilt or not are somewhat ar-
356
Antiferroelectric Liquid Crystals
q =I
2a
5 5 b
B c’
C;
(115)q
9 =a5
9 =3/4
a -
P=O
AF
P=O
Figure 172. Smectic ordenng corresponding to the same wave numbers discussed before but now with this wave number (qT)referring to the character of the layer interface instead of tilt. The phase is antiferroelectric if either m or n is even in qr = mln, otherwise it is ferrielectric with P = (Iln)P,. A relation between q and qT often stated is q = 112 (1 - qT),but is not generally valid. For instance, there is no q value corresponding to qT = 112, or to other higher-periodic phases with P = 0.
tificial. The existence of HF and HA are said to have support in miscibility experiments, but the reasoning is fairly circular, because the only indication for a strong HF is a wide occurrence of the C* phase (for instance, it is said that racemization increases HF and decreases HA) at the same time as HF and H A are used to explain the appearance of C*, C$ and so on in other diagrams. Further the concept of a “temperature-induced staircase” is introduced related to the new Ising variable with a new q value designated qT,in contrast to the old staircase which is called field-controlled. The reinterpretation of the variable and the distinction between the q:s remain obscure. However, if we accept this interpretation of the model, then the ordered sequences corresponding to the same most stable q values as before would look as in Fig. 172. Here we have used the symbols s (synclinic) and a (anticlinic) for the layer surface couplings or “bonds”, instead of F and A. In addition to the C*, C$, and C; phases we now get the AF phase with a period of 4 and further three antiferroelectric phases ( P = 0). The C; phase has a period of 3 and is the same as before (even the q number happens to be the same in this case). The other ferrielectric phase has a period 5 , the first antiferroelectric beyond AF a period of 8 (qT = 1/4) and so on with
13.6 Ising Models
357
higher values of the period. If we go on to less stable states we would get a number of new ferrielectric and antiferroelectric states. A final new feature of the change to a new Ising variable is that the Cz phase comes out to have an infinite stability range AH just like the C* phase, because it now corresponds to qT = 0. Thereafter the AF phase with 4-layer period occurs as the most stable phase, even considerably more stable than the CT phase. With these new states the Tokyo group attempts a better fit to their experimentally found phase sequence (604), especially now with at least two new antiferroelectric phases besides Cz. In their review article (reference [282], p. 1011) they also state with certainty that the C$ phase is antiferroelectric, shown by its electro-optic behavior. They also assign qT = 1/11, 1/9, 1/7, 1/5, and 3/11 to additional ferrielectric phases found in the neighborhood of FI, and qT = 3/7,5/11, and further states to additional ferrielectric phases found in the FI, region. At least in part these results seem to be confirmed by observations on free-standing films although the Cz phase is shown to be complex. Bahr et al. [313] have reported to observe a sequence of alternating antiferroelectric and ferrielectric states in the narrow Cg region as a function of temperature in elliptometric studies on thick films. The number of observable phases increases with increasing film thickness, which is a reasonable result. The compound studied is 12FlM7 which in bulk samples shows the sequence Cz -C;-C* -A* [314]. In freely suspended films there is also a temperature range of some degrees between C* and A* which might be attributed to C;. Already at 10 layers several subphases can be distinguished in this region, because sequences of zero polarization and finite net polarization can be detected. In a 122-layer film Bahr et al. find, presumably, not less than four antiferroelectric and three ferrielectric states between C* and A*. The sequence is a- f - a- f -a-f-a, where a stands for antiferro and f for ferri. Results on freely suspended films are interesting because the surface effects from substrates are absent. If the films are too thin, however, they cannot represent an ideal bulk state because helicoidal modulations cannot fully develop. The Tokyo group has recently described an experiment [3151 in which a free-standing film, about 100 pm thick, is kept in a temperature gradient, in order to be able to observe the subphases simultaneously, also under application of electric fields. They are able to confirm the whole sequence (604) but now conclude that the modified ANNNI model of reference [311] offers the best interpretation. They also confirm by x-ray synchrotron scattering that the ferrielectric C; phase has the expected three-layer spacing. However, looking at the presented evidence, one must conclude that the confirmation of one or the other of all proposed theoretical models is still at most conjectural. There is no doubt that a large number of subphases seem to be able to exist under certain conditions, but what these conditions are, is still extremely confusing. Regarding the Ising models, they all have the drawback of being “flat”, incapable of including any chiral interactions. On the other hand we know that the C z and C ; phases only occur in almost pure enantiomers. It therefore seems likely that chiral effects are essential and it is unlike-
358
13 Antiferroelectric Liquid Crystals
ly that these phases can really be accounted for by non-chiral models. The same is true for all other subphases. Ising models should therefore be regarded with some scepticism in this domain even if they seem to predict the appearance of certain subphases. After all, experiments guided by specific models have a tendency to c o n f m the prediction of the model. At the other end, if some predictions are confirmed while others not, theories are often very adaptable to adjust to these discrepancies, also when they are basically wrong or unphysical.
13.7 Helix Models As pointed out by Lorman [3 161 models with one-dimensional tilt order parameters are not consistent with the symmetry properties of the system. All Ising models thus suffer from a limitation that is basic. Sun, Orihara, and Ishibashi, as already referred to above [302], [303] introduced discrete models as an extension of their previous Landau approach. These models use a two-dimensional order parameter corresponding to the xy model in ferromagnetism. Such models can of course easily include chiral effects and are often called “clock models” because they describe how the tilt vector or the c vector rotates one way or the other around the cone. We will refer to them as helix models because some kind of helix is always permitted and generally involved in the chiral case. A particularly attractive description belonging to this category, but really of a new kind, was published in 1996 by Wang and Taylor [3 171. It is a breakthrough not only because of its simplicity but also because it, finally, takes the effect of the bounding surfaces into the consideration. We will therefore discuss it in some detail. The sample is to be of thickness L and consists of N layers in bookshelf geometry bounded by surfaces at x = 0 and x = L, cf. Fig. 173. If the layer normal is considered to be along the z direction as usual, the hamiltonian of Wang and Taylor can be written N !7i
=Zjdxdy i=l
K
[($y +(2,’j.
U cos ( @ j - Qii-,) + bsin(@j- @ j - l ) (620)
In a way this is a smart hybrid of a continuous and a discrete representation. The first four terms correspond to the first three terms in Eq. (446) which were
13.7 Helix Models
359
L2222Vl ‘
I ‘ I ‘ I ‘/ I ‘ I ‘ I ‘// I \ I I \ / / /
“t
i Figure 173. A stack of N smectic layers in bookshelf geometry bounded by two surfaces at x = 0 and x = L. The azimuthal angle $ ( z ) = @idescribes the local tilt direction through the sample. The tilt angle
0 is considered constant in the problem (after reference [317]). (b) The cosine term gives the herringbone antitilt structure ($, -$;-, = f x) depicted in (a). The sine term changes this to a helical structure (chooses the handedness) by “repelling” deviations toward +7d2 while “attracting” deviations toward -7d2. Alone, it would induce a jump in the tilt direction by -7d2 from layer to layer. (c) Assumed surface anchoring energy.
with the one-constant approximation ( B , = B , = K ) applied within the layer, and with the third and fourth terms replacing what would have been the helical term in (621) in the chiral case, namely
expressing the fact that there is a helical wave vector k along the z direction. To see what the discrete terms
( b 4 U ) mean we note, cf. Fig. 173b, that the cosine term has a maximum for (& - qi-,)=O and a minimum for (@i - @i-l) = k r. This means that it describes a preference for anticlinic order. The more the angular increment differs from +ITor - r in
360
13 Antiferroelectric Liquid Crystals
the direction + d 2 or 4 2 the higher is the energy. However, this degeneracy of the krc states is broken by the smaller sine term which has a maximum for ($i - Qti-,) = + d 2 but a minimum if ( k - $ii-l) equals - d 2 . If alone, this term would induce a displacement in $iby - d 2 from layer to layer, but the effect is now to give the lowest energy for a shift in direction which is biased toward the negative side. Thus, together the two t e m s describe a deviation from the herring-bone antiferroelectric state assumed to be present at the surfaces, such that lA$l = I $ii-,l < rc from layer to layer and giving the structure a handedness. The next two terms are the field energy - P . E and the dielectric anisotropy term, which we are already familiar with. The W,= W (x) sin2$i represents the surface anchoring energy. W, is equal to zero everywhere except at the boundaries where it has the value
ei-
This function has a minimum for qi= 0 or x,i.e., when the c director (as well as the director n) lies in the y z plane. This is the case of so-called planar anchoring. Note that this boundary condition treats synclinic and anticlinic order in the same way, which probably is the only unrealistic feature of the model. The last term expresses the inertia, which gets involved in dynamic problems. In this model 8 is considered constant. The only order parameters are therefore and There are only nearest neighbor interactions, expressed by the discrete form (623).This interlayer interaction is assumed to favor the herringbone structure having antiparallel orientation of adjacent dipoles, by the predominance of U. The small perturbation b can be interpreted as a steric hindrance which acts to create a small chiral deviation from a perfectly antiparallel order. But how could we have order in this system with only nearest neighbor interactions, considering the argument in Fig. 169? The answer is that while a smectic has a 1D translational order it is not a 1D system. The in-plane order in the layers guarantees the connectivity in this case. The smectic is not really an king system. The two discrete terms can be set to work directly. If we put $i-$i-l = a and minimize H with respect to a, we get -
U s i n a + b cos a= 0
(625)
or
a = arctan b/U
(626)
This is thus the equilibrium value of the angular deviation from layer to layer. We have here looked upon the hamiltonian as an effective free energy because several parameters are temperature dependent. A temperature dependent value of b gives a temperature dependence of a. If we take the derivative of % with respect to we
ei
13.7 Helix Models
get the restoring torque if nal field
@j
36 1
is brought out of equilibrium, for instance by an exter-
The motion induced by this torque is counteracted by the viscous torque ry=
-
ya@jiat
(628)
The dynamic balance T E+ T y =0 would then give us the dynamic equation for qj in the presence of an external field. Wang and Taylor continue along these lines but we will not study these solutions as we are here only interested in the static equilibrium properties at zero field. We will further assume that there are no spatial variations of #j within the layers, i.e., that and &$ilay are equal to zero. With E = 0 and &at = 0 the expression (620) then simplifies to
a@j/&
where we have performed a simple integration along y, the transverse dimension of the sample. When we take the derivative of this expression four terms in the discrete part get involved in the summation over i. These are
U cos (@i- @ji-l)+ b sin (@;- @ji-l) + U cos (@j+l
- @J+ h
sin
(630)
- @J
After differentiation with respect to @; we get - U sin
(qj - @ j p , ) + b cos (4; - $i-,) + U sin (@j+l - @J- b cos(@j+l- 4;)
=- UsinA+bcosA+UsinB-bcosB
(63 1)
Because blU = tan a we can put b = C sin a and U = C cos a with C2 = U2 + b2.Using the relation sin (A - a) = sin A cos a - cosA sin a
(632)
the result is C sin (A - a) + C sin ( B - a) = U [- sin (9; + sin ( @ j + l - @ j - a>l
-
a) (633)
because C = \ U 2 + b 2=U
(634)
362
13 Antiferroelectric Liquid Crystals
Hence, d%/d& = 0 gives N
YC i=l
{U 1- sin ($i - +ii-l -a> + sin ($i+l
- $i
-
a>]
+ W ( x ) ~ i n 2 4=~0)
(635)
or, after integration with respect to x N
Y C { L U[- sin ($i i=l
-
4i-1 - a ) + sin (&+I
- $ j - a11
+ 2 Wosin 2 &} = 0
The equilibrium condition then is 2 r s i n 2 $i = sin (&-
-
a) - sin
-
$i-
a)
(637)
with
The solutions of this set of coupled non-linear difference equations are helices whose pitch form an incomplete devil’s staircase as had been shown earlier by Banerjea and Taylor [318]. If we plot a as a function of Twe find domains within which the change of tilt from one layer to the next has a constant value. This is shown in Fig. 174. If for a fixed C that is a fixed layer thickness, we change the temperature and thereby a, we go through domains where the pitch of the helix is constant. Because all these structures have the same symmetry the transitions between them, at constant pressure, must be first order. From one to the other there is a discontinuous jump in helicity. The pure bulk behavior is obtained if we set I-= 0 in Eq. (637). This means that either the surfaces are very far apart (I-+ 0 when L + -) or the surface anchoring energy W, is negligible. In this case the difference equation reads sin (qi - $i-l
-
a) - sin
-
ei
-
a) = 0
(639)
As can be immediately checked, the solutions to this equation are given by
& = i ( n + a ) , i = 1 , 2 , 3 ,... For a = 0 we get $i= i n corresponding to the azimuthal tilt angle
o,n,o,n)...
(640)
363
13.7 Helix Models
r
Srn C
0
I
1/5 1/4
1/3
2/5
Srn c*
I
1/2
3/5 2/3
3/44/5
1 dll
Figure 174. Stability domains with constant helical twist according to the model of Wang and Taylor. Each domain is assigned by the number of layers corresponding to the helical pitch. The broad domains to the right (-) and left (2) are the ferroelectric (C*) and antiferroelectri (C:) phases. The intermediate domains come down on the line r=0 like leaves on a flower touchmg this line. Between two leaves there are infinitely many other leaves. If, at small and constant I-, we move to the right (lower the temperature) we would jump from one twist value to the other on a devil’s staircase from 0 to 1, corresponding to the many modulated states between C: and C*. At large values of the surface strength (thin samples) only three modulated phases are predicted to exist between C: and C* according to this model, with helical periods corresponding to three, four and six layers.
Figure 175. Variation of the azimuthal tilt angle 4; according to Eq. (639) for a equal to rd2 and 2 rd3 corresponding to twists with four and six layer period, respectively.
2
eoz4 3@o,6
1 ad2
a=2n/3
in consecutive layers. Thus a = 0 represents the antiferroelectric state ( C z ) ,as expected. For a = z we get & = 0 in all layers, representing the ferroelectric state (nonhelical C*). For a = z/2, the $ivalues are 0, 3 d 2 , 3 z, 5 d 2 , 4z ...giving a period of four layers, for a = 2 d 3 we get &= 0, 5 d3, 10 d 3 , 15 z/3,20 d 3 , 2 5 7d3, 10 z ... i.e., a period of six layers, cf. Fig. 175, and so forth. Within each domain we have written the helical twist period in number of layers in Fig. 174. In a bulk sample the helix could, in principle, lock in to an infinity of states, though of quite varying stability, between period 2 ( C z )and 00 (C*- infinite period means that the tilt never changes). When we increase T w e notice the influence from the surfaces as the less stable phases vanish, one after the other. Finally, we have only modulated structures with period 3, 4, and 6, in addition to C z and C*. We note that the figure is symmetric
364
13 Antiferroelectric Liquid Crystals
around a/z = 1/2. The reason for this is of course that we have applied a surface condition at x = 0 and x = L which is symmetric with respect to the phases C* and C z . With a more realistic surface condition put into the model -one which favors the C* phase at the expense of Cz - the phase map would have looked asymmetric, and at high value of which might be called the “surface strength”, we could expect the phases with period 3 and 6, then 4, and eventually even C$ to vanish. The strength of the model by Wang and Taylor is that it is physically transparent and uses very few ad hoc assumptions. The fact that it treats 8 as constant between C* and Cz must be looked upon as a very acceptable simplification. It should be mentioned that Wang and Taylor also treat the field-induced transition in the C,* phase from the antiferroelectric to the ferroelectric state, although we will not discuss this feature here. But what actually are the phases reproduced by this model? A consideration and a look at Fig. 174 reveals that there are no ferrielectric phases. There is also no antiferroelectric phase except C:. What the model predicts are that some helielectric phases with commensurate periods (3,4, 6) should exist in bookshelf geometry. It does not consider the fact that for any realistic boundary condition, no helical state can exist without being accompanied by a periodic lattice of defect lines near the boundary. While it may not be too serious that the model predicts no ferrielectric and antiferroelectric phases (considering how shaky the experimental evidence for these phases still are), something would have to be added to reproduce the helical C* states and the double-helix Cz states (both with incommensurate helices), because these are very well established. Although the model brings in fresh considerations, it is therefore certainly not the last word. A model with several similarities was independently and almost simultaneously worked out by Roy and Madhusudana [319]. Their free energy contains, in addition to the 82 and @ terms in a conventional Landau expansion (which we here can omit, because 8 is considered constant, not an order parameter in the temperature interval treated), the following discrete expression
r,
N
= i=l
11 e2C O S ( $ ~-+$~i )
I 2
- - 52 Q4 C O S ~($;+I -
+ e2sin (q++l - ei)+ y2 o4cos ($i+l - $; + Y, e2sin ($i+2 - 9)
9;) + 53 cos ($;+2 - 4)
sin
- $i
(642)
It is not as transparent as the Wang-Taylor expression and we will just comment on some of the terms. The first one favors antiferroelectric order as just discussed above, as long as J , > 0. For J , = Jo (T,-7‘) this is the case for T < T,, where T, is supposed to be the temperature at which the C z phase appears. At higher temperatures ferroelectric order prevails which corresponds to the fact that the synclinic order is favored entropically as it allows an easier translation of the molecules between adjacent layers at higher temperature. The J2 term tends to confine the tilt to a plane, fa-
365
13.7 Helix Models I
t 105
I
1
1
1
1
I
1
1
1
-
SrnA
I
I
SmA
I
-
t-GC---l-3@l
t
2
0
10
5 J3
Figure 176. Phase regions in temperature versus interaction strength parameters J , and J3. J , promotes synclinic or anticlinic order, but in a neutral way. J 3 promotes anticlinic order and represents the long range “repulsive” interaction between layers. The sequence of azimuthal tilt angles ( I , 2 , 3 , 4 ...) in adjacent layers is depicted for the corresponding phase. C& C* and FI, are ferroelectric, FI, and FI, ferrielectric and C,* antiferroelectric in the surface-stabilized state (unwound helix). After ref. [3191.
voring the F and AF states equally. The J3 term is a repulsive next nearest neighbor interaction (i.e., promoting AF) inspired by the Bruinsma-Prost interaction [ 3 121. It is the interaction fromc director fluctuations, which is proportional to P2/Kand hence independent of the tilt, as P2and K are both proportional to 8.This is a long-range interaction but here represented by J3 in a truncated form. The Y-terms make the interactions chiral and also go to next nearest neighbors (Y3).The model is thus an ANNNXY model. After numerical solution, with the parameters fitted to the data for C8-tolane [320] the phase diagram comes out as in Fig. 176, where the azimuthal angle difference between successive layers are also indicated for the different phases. These structures are in fact quite realistic candidates for the observed phases. They are all helielectric with C,* having a double helix. In this helical state it has a mesoscopic P f 0 which makes it effectively similar to the helical C* state as regards unwinding in an electric field. In the surface-stabilized state the C,* would be a planar antitilt structure and two of the phases (FI, and FI,) would be ferrielectric when surface-stabilized, i.e., have a macroscopic polarization in the absence of a field. The phase between A* and C* has a very hard-twisted helix which makes it natural to
366
13 Antiferroelectric Liquid Crystals
interpret it as the C z phase. Finally, the twist sense in the C z phase comes out to be opposite to that of the C* phase which is what is actually observed. Thus the model by Roy and Madhusudan seems to describe the experimental facts quite well. In their continuous Landau model, Lorman, Bulbitch and Toledano [330] start with the same Pihn-Indenbom order parameter as used by Orihara and Ishibashi, Eq. (599) and they also, in the beginning, consider a unit cell containing two smectic layers. They further construct the same order parameters 6, = 5, + and = used in the same Landau expansion which thus contains the invariants and (& . <J2 in addition to the second and fourth order terms in 6, and &, cf. Eq. 603). Their result is naturally identical to that of Orihara and Ishibashi: of the five phases found, three have a constant tilt angle and are the planar C*, the planar Cz and a helicoidal phase presumed to be C;. Allowing then and <, to vary with z to describe a modulated phase, they add four more second order invariants, two of which are Lifshitz [321], [316] invariants. Later Lorman even added the sixth order terms {,: {,6, {,’(<, . &J2 and {,’({,, .{,)’ and extended the model beyond two layers. For instance, in the case of four layers, four axial tilt vectors have to be introduced and the four basic order parameters constructed from their combinations. The physics is not very transparent but essentially this expansion gives a number of new helicoidal phases (described as ferro-, ferri- and antiferroelectric, the word “ferri” meaning that there is a non-zero mesoscopic polarization) with different pitch and handedness. The sixth order terms secure that the transitions C* + Cy* + C,* become first order. The model also predicts that the helical structures should be commensurate, a = 2 d n . This type of phenomenological Landau description can evidently be made sufficiently complex to accomodate all possible experimental observations. On the other hand, it then more and more gets the character of a posteriori adjustment, which phenomenological descriptions often have, but to a varying degree. For instance, if we let our unit consist of three layers, there will always be a solution corresponding to structure with a 3-layer helix. If we take the unit to be four layers, one solution will be a 4-layer helix, etc. The predictive value obviously degrades with the complexity of the chosen order parameter. While some king models predict several antiferroelectric states, and thus in principle could account for the AF state, but fail to give helicoidal states, none of the helical or clock models described so far give any other antiferroelectric state than C,*. Recently, however, &k5 and CepiE [322] modified their previous continuous Landau description [298] to a discrete model. We will finally consider the assumptions and implications of this model. In the discrete iekS-CepiE description interactions between nearest layers and next nearest layers are taken into account. These interactions are of steric, van der Waals and electrostatic (dipolar and quadrupolar) origin. The steric and van der Waals interactions between nearest layers are actually competing - the steric ones promote synclinic order, the van der Waals interactions anticlinic order (this is independent of
<
t2 <,
<:
<,
<:,
t2,C2,t3
367
13.7 Helix Models
chirality) - and could even eventually cancel each other. Therefore it is necessary to take the next nearest layer interactions into account. There are no steric interactions between next nearest layers, only electrostatic interactions. As there is no dipolar interaction between such layers, according to Bruinsma and Prost [3 121 they have to be of quadrupolar character or, more probably due to the Bruinsma-Prost fluctuation interaction. With the usual tilt order parameter the &kS-Cepii- free energy expansion has the form
ti
Here, as usual, only a is assumed to be temperature dependent, with a = a(T-T,) and b > 0. A negative a, favors synclinic order between adjacent layers, a positive value favors anticlinic order. Similarly a2 < 0 favors synclinic, u2 > 0 anticlinic order in next nearest layers. The coefficient b , represents a quadrupolar andfa chiral interaction. The coefficients a ; and a a; give the bilinear tilt interactions between nearest and next nearest layers, respectively. These terms are somewhat nebulous as for their interpretation. After expressing the tilt vector by
ti= (8cos qj,6 sin qj)
(644)
and assuming that 8 is constant, minimization of the free energy leads to the stable solutions. To get a connection to the Wang-Taylor model, we note that if a= 'pi+,- q, the phase difference of 5 between adjacent layers, . can be exand x pressed as cosa and sina, respectively. It turns out that there are stable (equilibrium) states in two distinct cases. The first means that a is constant, equal to q,for all layers, including zero (SmC) and n (SmC,) or very close to zero (SmC*) or n (SmC:), or between zero and n.The last case could be a candidate for SmCz and is
ti
ti
layer
Figure 177. The helical SmCZ phase according to the "clock" model of ZekS and CepiE. The left part (a) shows the successive director projections on the xz plane, the right part the corresponding projections in the xy plane. The phase shift in c director between adjacent layers is approximately 62". From reference [322].
layer 2
layer
layer layer
4
layer 5
layer
/
\
368 layer 7 layer 6 layer 5
layer 4 layer 3
layer 2 layer 1 b)
a)
Figure 178. The double-helix structure proposed for the SmC; phase, in the same projections as in the previous figure. From layer 1 to layer 2 there is a shift q,from 2 to 3 a shift Po in opposite direction, etc. In the figure q = 66" and Po = -60", which gives a pitch of 120 layers. Layers 1,3,5.. . and 2,4, 6.. . each represent a helicaI director configuration. From reference [322].
3fj9, 5
.........
3 @................ 1
@,.......
3,4 @ ............... (2
@ ....... , 3
d
e
1
4
3
4
a
b
C
Figure 179. Molecular configurations followed through four layers, given as stable structures according to the discrete phenomenological ("clock") model by CepiE, RovSek and keks [323]. These might account for C; (a), FI,(b), FI,(c), AF(d) and C;.
illustrated in Fig. 177. The periodic order is incommensurate. Experimentally this phase would only be distinguished as a separate phase if the C;-C* transition is first order, otherwise both phases would be recognized as one single C* phase with an anomalous temperature dependence of the pitch near the A* phase. The other kind of solution, which is an interesting new feature, is that a alternates between two values, a = a,and a= -Po for alternating layers. In non-chiral systems %=Po and only one intermediate solution exist between a,= Po = 0 and a,= Po = z (SmC,).In chiral systems cr, and Po are slightly different and a slow (hundreds of layers) helicoidal modulation occurs. A mesoscopic polarization results and this phase will be recognized as ferrielectric in its surface-stabilized state. Its structure is shown in Fig. 178. This structure could be a candidate for the observed Cy*.The period would in general be incommensurate.
13.8 The Present Understanding of the Antiferroelectric Phases
369
The model of iekS and CepiC is attractive because of its relatively transparent physics. It gives reasonable structures for some of the subphases between SmA* and SmC:. But it also gives one hypothetical ferroelectric phase and another antiferroelectric phase which are not observed. It is noteworthy that it does not, like most models, account for the claimed AF phase. In a later work [323] Cepii., RovSek and ZekS have simiplified the model further by discarding the a; and a; terms. This makes their discrete phenomenological ("clock") model still more attractive, as it also turns out that it describes the empirical data even better than the more complex model. If the coefficient al is allowed to change sign (and therefore can be regarded as small) in the region, the model now, in addition to the previous (incommensurate) phases C z and Cy*,accounts for one three-layer and one four-layer (i.e.commensurate) helical phase as well as the planar four-layer AF phase, cf. Fig. 179.
13.8 The Present Understanding of the Antiferroelectric Phases Many antiferroelectic liquid crystal materials have a range of polar phases between SmC* with synclinic tilt at high temperature and SmC: with anticlinic tilt at lower temperature. The nature of these phases and their stability is still by and large an open question. According to certain of the models which we have reviewed above there could be a series of ferrielectric phases with different mesoscopic polarization, while other models in addition predict a number of antiferroelectric phases different from SmC? but likewise with zero mesoscopic polarization. In any case, the appearance of these phases sensitively depends on the surface condition and the thickness of the sample. In thin samples of bookshelf geometry most of these phases will be squeezed out when we go to pm thickness. Most phases transitions vanish, and the SmC,* phase appears at continuously lower temperature when the thickness d decreases. Finally we may be left with only SmCz and SmC*, eventually with only SmC*. This fact, and its explanation, is of vital importance for the interpretation of electro-optic phenomena in thin cells of these materials. But equally important might be the fact that the coexistence of the two phases SmC* and SmC,* mimics the existence offerrielectric phases which do not really exist. Simulations by P. Rudquist show that the great variety of hysteresis loops which surround the V-shaped switching and change shape as function of frequency and temperature, can simply be accounted for by the coexistence of only two phases, ferroelectric (C*) and antiferroelectric (C:). While it is likely that a number of subphases have been proposed as a result of misinterpreted electro-optic experiments, it is likewise certain that some subphases do exist. This is evident from freestanding film experiments. Some electrooptic experiments may also be judged as reliable, for instance the observation of a triple hysteresis loop
370
13 Antiferroelectric Liquid Crystals
0) Smla' 71,6" SmCa* 95,l"FI1 96" Fl2 97" SmC' 104" SmCa' 105,5" SmA' 135,3"Is0
cryst 110" SmCa* 112" Fl1 114" Fl2 119" SmC' 120" SmCa* 124" SmA" 153" Is0
(iii) cryst64" SmCa* 80" FI 83" SmC* 93" SmA' 98" Is0
Figure 180. Three compounds used for the first x-ray tests of the nature of the subphases between SmC: and SmA*. The tolane compound (i) was investigated by conventional small angle x-ray scattering, the thiobenzoate (ii) 10-OTBBB 1M7 and the thiophene (iii) MHDDOPTCOB compounds were investigated by using the anoinalous (fluorescent) x-ray scattering resonant at the sulfur K edge of the incorporated S atom.
which indicates that the Cy*phase, in its surface-stabilized state, is femelectric. But the identification of the really existing subphases has not yet been performed. This is one of the outstanding problems and a great challenge, the outcome of which will be important both for the basic understanding of the ordering principles in smectics and for the technical applications of these materials. However, the work has at least begun and we would now like to discuss the results obtained so far. The first attempt to attack the problem came from the Bordeaux group and is quite recent [324]. It was done on the substance (i) of Fig. 180 which in the bulk in addition to SmCz shows two presumed ferrielectric phases, between SmC: and SmC*. This tolane compound can be compared to the compounds (ii) and (iii) in Fig. 149. Cluseau, Barois, Nguyen and Destrade performed small angle x-ray scattering experiments on oriented bookshelf structures of this material confined between glass substrates of 50 pm thickness. As the subphases are known to be easily suppressed in thin samples, they worked with a thickness of 20 pm. By following the layer spacing as a function of temperature they could first of all conclude that all subphases must be tilted because of their smaller period relative to the one in the S m A * phase. Further, the only periodicity found corresponded to one single smectic layer. While any periodicity corresponding to electron density variation can be
13.8 The Present Understanding of the Antiferroelectnc Phases
37 1
observed by conventional x-ray technique, it is not possible to conclude how the tilt direction varies along the layer normal. Hence, screw axes, corresponding to helix models cannot be seen. On the other hand, if there had been a periodicity according to some of the ferrielectric states in the Ising model, as in Fig. 168 or 172, this would immediately have been revealed. For instance, in Fig. 168 the C$ phase ( q = 1/2), like the C* phase would only give reflexions corresponding to a monolayer because in each case the layer interfaces are equal. In contrast the ferrielectric with q = 113 would give reflexions corresponding to three layers (and one third of the single-layer wave vector) because each third layer interface differs in electron density. No such reflexions exist however, which means that the Ising models can be ruled out, at least in this case. Instead, the layer interfaces must be identical, as far as their scalar properties (electron density) are concerned, which they are in the clock models. From these experiments the Ising models have to be dismissed. It is difficult to be surprised by this result. After all, it seemed unrealistic from the very beginning to try to describe the C ; , Cy* ... phases which only exist in chiral systems, by a model which is intrinsically non-chiral. But there is still no direct evidence for the competing models. Furthermore, the FI, and FI, phases cannot be resolved but merge in the narrow temperature region (= 2 K) in which they are observed. If the subphases are all helical, conventional x-ray experiments cannot discriminate between them, nor between them and C* or C z . In order to do this, a more powerful tool has to be used. Such a toool is the resonant x-ray diffraction technique. The first studies used x-rays of 2.5 keV, corresponding to a wavelength of about 5& resonant at the K edge of sulfur which was incorporated instead of oxygen in one of the ring linkages (COOCOS), making the thiobenzoate molecule 10-OTBBBlM7 of Fig. 180 (ii). These soft x-rays are absorbed by glass and even air, so the measurements have to be performed on free-standing films (about 250 layers thick) in a helium atmosphere [235]. Future studies may also use Br-containing compounds, permitting slightly harder x-rays with A= 1A [326]. The results of these resonant scattering studies are quite conclusive and reveal distinct superlattice periodicities. By working with x-rays corresponding to the K absorption edge the structure factor becomes a tensor instead of a scalar [327]. This means that any periodicity in the c director along z can be sensed. The structure factor Fijis a second order tensor analogous to the dielectric tensor. It relates the amplitude of a scattered j polarization to the incident i polarized wave [329]. The elements Fijrelated to the S atom are non-zero only in a narrow energy band of about 30 eV around the resonant energy. Only in this region the fluorescent anomalous scattering from the sulfur atom can be measured. The off-diagonal elements depend on the orientation of the bonds around the S atom with respect to the polarization of the incident beam. Thus by polarization-analyzed x-ray technique the scattered intensity gives direct information of the molecular orientation within the layer. For instance, in the region corresponding to the FI, phase, satellite peaks are now found at 1.2 Qo, 1.50 Qo, 1.75 Qo and 2.25 Qo, where Qo designates the wave vector corresponding to a single smectic layer. This corresponds to a four-layer super-
372
I3 Antiferroelectric Liquid Crystals
structure. In the same way a three-layer superstructure is found for the FI, phase. In the C; region a strongly temperature-dependent incommensurate periodicity appears with the period varying between about 5 layers at lower temperature and about 8 layers at higher temperature (only about 4 degrees higher, cf. Fig. 180). For the C z phase, finally, a periodicity is found which only slightly differs from two layers, 2% corresponding to the superposed double-helix lying in the optical wavelength range, at about 0.5 ym. The same measurements performed on the racemate, which has the phase sequence cryst-SmC,-SmC-SmA-iso, showed a period of exactly two layers in the C, phase, thus confirming the non-chiral planar anti-tilt structure. To sum up, the results for the (I?)- 10- OTBBB lM7 can be stated in the following phase sequence cryst 110" C: 112" CTI3 114" CT/4 119" C* 120" Cz 124"A* 153". Here we have discarded the unrational designation FI, and F12 (first of all because these structures are not ferrielectric) and instead used CI/3 and C,, related to the wavevector of their periodicity relative to the single layer. Instead of C$ we could of course also have used CTI2(and correspondingly CLI2in the racemate). A designation of type Ca seems rational for phases which have a commensurate period, while C& Cy*etc are appropriate for phases with an incommensurate period in the x-ray wavelength range. The subphases C& CTI4 and CT/3 correspond to the structures a, b and c of Fig. 179. In the continued study of the 10-OTBBBlM7 compound [328] it turned out that the ordinary SmC* phase could not be detected in the film experiments, although it is observed in the bulk, albeit over a range of only 1K. If we, somewhat arbitrarily, would attach the two limiting values 5 and 8 of the approximate periods to the Ca* phase, the observed phase sequence could then also be written
with the wave vectors related to the C* layer periodicity. A long wave incommensurate superstructure may be added in the case of the commensurate lattice structures. So far such a superstructure was found in CT/4but not reported in CrI3.The periods could then be stated, keeping the bulk structure C*, C" 2+&
3
4 k ~ I+&
5-8
The symbol E here stands for a small deviation, correspondingto an optical pitch which typically is in the 0.3-1 pm range or roughly corresponding to 100 layers. In this case the optical pitch is about 0.5 ym for both C$ and C*, and about 2.5 pm for CT/4. In a second series of measurements the tiophene compound MHDDOPTCOB was investigated in the same way [328]. This compound has the sulfur atom incorporat-
13.8 The Present Understanding of the Antiferroelectric Phases
373
ed in the ring system and the subphase investigated (FI) lies between SmCz and SmC*. The results show that this phase is also a C1,4 helielectric with a four-layer lattice superstructure. The fine splitting of the satellite peaks further indicates a superposed long wavelength optical helix of pitch 1.2 pm which is again about 5 times longer than in the C* phase of the same material. This means that the four-layer clock model confirmed already for 10-OTBBB1M7 is not a compound-specific feature. It is not unique to one material and will probably be found as a characteristic structure in many materials, although further work is required to establish such a fact. Assuming that these results, which constitute the first direct determination of the subphase structures, though admittedly collected only on two compounds, will turn out to be representative, the consequence is first of all that the king models can be dismissed (already concluded from the conventional x-ray results) as inconsistent with the position of the observed scattering peaks as well as with their polarization states, whereas the clock models [3171- [323] are supported, at least in principle. How many of the predicted clock model structures that will actually show up in experiments is something that only future can tell. We can also observe that in these first results all subphases are helielectric. They are in principle not different from the C* phase except that their pitch is very short and sometimes commensurate. As the SmCz phase can be regarded as the extreme example of such a phase, with a period of only two layers, we can conclude that all polar smectic phases observed so far are helielectric. This is even more true in consideration of the additional long helix in the optical wavelength range which is superposed on these elementary structure. Inversely, because the helielectric structure is a special case of antiferroelectric, it would be equally correct to say that, so far, all observed polar phases are antiferroelectric. This seems to be a natural consequence of their “fluid” lattice. But how about the observed ferrielectric state? Perhaps they correspond to some structure like (e) in Fig. 179 (not yet confirmed). But these observations could probably also be due to hardtwisted structures like the one with four-layer or three-layer period in (b) or (c). First of all, the electro-optic measurements are generally performed in thin bookshelf structures, thus in the surface-stabilized state. Hence, the long wavelength helix is unwound from the start. A hysteresis experiment amounts to unwinding the hard-twisted helix. The conditions for this cannot be too different from those for unwinding the anti-tilt state of C z . It would be interesting to find out if, for instance, the unwinding of the three-layer helix is a continuous or rather a quasicontinuous process, in which separate steps could be disinguished (which possibly could be taken for a triple hysteresis). A different consideration is that these short pitch helices, if present, very probably would be distorted by the surface interactions, which could make their unwinding non-uniform. A third consideration is that the interpretation of electro-optic processes reported so far ought to be reconsidered in the light of what we now know about coexisting (C: and C*) phases. Throughout this chapter we have given several examples indicating a subtle balance between synclinic and anticlinic order in polar smectics, especially if the sam-
374
13 Antifen-oelectricLiquid Crystals
Figure 181. (a) The chiral end chain of MHPOBC in a lowenergy conformation (cf. Fig. 149); (h) out- of-layer-tluctuations in SrnC*; (c) typical SmC* case with both end chains less tilted than the core; (d) MHPOBC case, where the chiral chain is almost orthogonal to the core. It is drawn as perpendicular to the paper plane but has a high degree of mobility within the interlayer region; (e) layer pair, in which the cores and the non-chiral tails repel each other.
7 ‘ I ......
(e)
ple is confined between surfaces, which always will have a low acceptance for any kind of inhomogeneous condition near the boundary. If it turns out that all subphases are helical they will also tend to unwind by the surface action in the bookshelf geometry just like, ultimately, the antiferroelectric C: phase itself. In order to better understand this complex domain, one would like to have, in the first instance, a reasonably clear picture of what factors promote synclinic and what other factors promote anticlinic order in the bulk. However, the competition between these two antagonistic states of order is nothing that could be considered well understood. Nevertheless we would like to conclude with some thoughts regarding this issue. The first thing to note is the similarity of almost all molecular structures presenting the C: phase. We have already said that essentially all of them have (at least) three rings. These could be fairly different but what is almost consistently the same is the chiral end chain and its connection, normally by a carbonyloxy group to the core. Thus in the prototype AF compound MHPOBC we have the tail -COO-C*H(CH,)-C,H,, connected to the ring (Fig. 18 l), and in the corresponding trifluorinated prototype compound TFMHPOBC we have -COO-C*H(CF3)-C6H,,. The latter species of end chain, C*H(CF3)-C,H2,+1 is now the most common chiral chain in AF molecules. Although this chain at first sight looks very “normal”, it is obviously the decisive part of the molecule. In order to investigate the molecular con-
13.8 The Present Understanding of the Antiferroelectric Phases
375
formations in compounds of this type, the Fukuda group in 1996 deuterated the chiral and achiral chains separately in MHPOBC and then performed polarized IR spectroscopy [331]. The results are very illuminating. It turns out that, in contrast to the achiral chain, the chiral tail adopts a strongly bent conformation relative to the core: surprisingly it almost makes an angle of 90 degrees with the rest of the molecule (exaggerated in Fig. 181a). Recent molecular dynamics simulations on MHPOBC confirm the Japanese results [332]. A naive and simplified picture illustrating the competition between the two ordered states represented by C* and C,* could then look as follows. In Fig. 181b are shown fluctuations in the smectic layer ordering involving the partial permeation of molecules from one layer to the next. Such fluctuations are permitted by the synclinic order, they are facilitated if the molecular conformation corresponds to a reasonably rodlike shape (c). At the same time, the fluctuations themselves stabilize this order. In the case that the molecule is strongly bent, the interpenetration of the layers must be much less frequent. The typical AF molecule seems to behave like in (d) where the chiral end chain spends a large part of the time in the interlayer plane, within which it has a great deal of mobility. A common situation is therefore that it is more or less perpendicular to both the director and the tilt plane. This preference for the interlayer plane sort of separates the layers from each other; the less frequent interpenetration gives the layers a sharper definition than in the SmC* case. It has been reported [383] that x-ray scattering spectra show more of the higher harmonics in the case of SmCz than in the case of SmC* [334], [335]. It is also a common observation when doing microscopy on SmC: materials that the layers are much more conspicuous than they are in SmC*: transitions and other phenomena very often propagate along layers which seem to have a lot of individuality and it is seldom hard to find the layer direction. This does not yet explain the tendency for antitilt. However if the permeation is suppressed then the dipole-dipole interaction gets more effective in making the dipoles antiparallel, provided in the first place that P, is high. That such a dipole-dipole mechanism must be involved is evident from the fact that (with very few exceptions) no antiferroelectric order is observed for materials with low P,. So, generally, the electrostatic interactions promote antiferroelectric order. According to Zeks and Cepii. [322] this is also true for the van der Waals attraction. But still this does not explain why synclinic order is rather common and anticlinic order relatively rare. But the fact that the latter is extremely rare in non-chiral non-racemic systems unquestionably must be interpreted such that the steric factors are by far the most important. It also means that the out-oflayer fluctuations give a thermodynamic (entropic) driving force for synclinic order. In contrast, anticlinic stacking suppresses the out-of-layer fluctuations and thus carries an entropic penalty. This is consistent with the fact that the ferroelectric phase is the high-temperature phase in liquid crystals, and the antiferroelectric phase the low-temperature phase. One issue brought up by Fig. 181 d is whether the invariance under n +- n is valid for a single layer in the C z case, or only for a pair of layers (e). If it is not valid
376
13 Antiferroelechic Liquid Crystals
for a single layer then we would have one kind of interlayer where the core parts come close together and another kind where the non-chiral alkyl or alkyloxy chains interact. Such a situation could for instance appear if we increase the degree of fluorination on one part of the molecule, to the point where it will almost behave in an amphiphilic way such that the unlike ends repel each other. This would have a number of consequences, one being to further stabilize the antiferroelectric order and further pronounce the anisotropic effects of the layering. If the discussion above has been primitive and superficial it should at least indicate that small but characteristic features might decide whether a chiral smectic phase of a certain molecular species will be ferroelectric or antiferroelectric. Even more evident is that if we mix different molecules the balance between the two competing polar ordered states might be very fragile. We will return to this issue in Chapter 15.
13.9 Freely Suspended Smectic Films Experiments on free-standing smectic films have been very important for clarifying the nature of the polar order in ferroelectric and antiferroelectric liquid crystals and they still play an important role in ongoing research. We have already made frequent references to them in this book. A comprehensive review of work done up to 1993, with a particular emphasis on dimensionality effects in phase transitions, has been given by Bahr [353]. The art of suspending a smectic film on a frame, allowing its observation without a substrate, was introduced around 1920 by Friedel [336], although he essentially limited himself to the interesting observation that it is possible to spread a piece of smectic A material such that it is stretched like a thin soap film across a hole. This may typically be about 5 mm in diameter. The film is then supported only at its edges and except in the edge region it tends to be of very uniform thickness. In modern liquid crystal research this technique was introduced by the Harvard group in the seventies. The first presentations were made at the international liquid conference in Kent, Ohio, in 1976 [337]. These experiments permitted some very illuminating observations on ferroeiectric smectic C* films. Among other things they contained the first measurement of the rotational viscosity y, in the C* phase, together with determinations of elastic constants and polarization, confirming previous results. In the absence of applied electric field there are strong fluctuations in the director field and the related local polarization field. These thermal fluctuations can be studied by light scattering which, in fact, allows the director splay and bend modes to be followed separately. The first account of these experiments is found in Meyer’s review [43] from 1977. Meyer there makes the interesting observation that the splay and bend modes do not seem to have the same curvature elasticity and explains the reason. Since the c director and the polarization P are perpendicular to
13.9 Freely Suspended Smectic Films
377
each other, the splay mode of the director corresponds to a bend mode for P. In Sec. 4.9 we showed that V . P for such a mode in the twodimensional case. On the other hand the bend mode in c is a splay mode in P, which means that the fluctuations in this mode entail a corresponding fluctuating charge density . This means that such fluctuations cost an additional electrostatic energy P2 originating from the coulomb interaction between all involved volume elements. The effect is that bend in the c director appears to have a higher elastic constant K,,, or rather B, (see Sec. l l .4) as we are here dealing with the two- dimensional elastic constants. The coulomb interaction gives an additional term proportional to P2 in the expression for B , in Eq. (486) in the chiral case. We have already mentioned this ”stiffening”, which can have quite remarkable effects, at least for materials with high P,, which will be discussed in Sec. 13.11. The free films are normally drawn in the A* phase, but they can also be produced directly in the C* phase. In the case of antiferroelectric materials they are normally drawn in the C$ phase. In all these cases they are stable from two layers up to hundreds and thousands of layers. Thick films are of interest mainly for studying the nature and temperature dependence of helical superstructures in different phases, measuring the pitch by Bragg reflection. In the other extreme limit, one-layer films can be made of SmC* but are not stable and crack in a short time. Therefore, few studies have been devoted to their physical properties so far [339]. Anyway, as the number of layers N in general can be well controlled, free films have a general interest also because they permit a system to be studied going from 2D physics to 3D physics. The layer number N is normally calculated from the optical thickness which is inferred from measuring the film’sreflectivity R, which increases accordingto R sin2N[345]. For the preparation, in a typical case, a rectangular hole of 3 mm x 10 mm is made in a microscope coverglass, about 0.15 mm thick. Then metal electrodes are evaporated on the glass on each side of the 3 mm gap. Usually a field of about 1 V/mm is sufficient for aligning the director to a homogenous state along the gap. This is about three orders of magnitude lower than a typical threshold field for switching in an SSFLC sample and is characteristic of the fact that there is no anchoring to any substrate. By measuring the depolarization of light reflected from the film, it is possible to deduce the direction of the c director, c (x, y), as will be discussed in more detail below. In a non-chiral smectic C, the c director behaves very much like a two-dimensiona1 nematic. This means that there is no characteristic length scale within the layer. The correlation function for the phase variable fluctuations falls off algebraically (instead of exponentially); the system is scale invariant and behaves as if we were at a critical point. In chiral smectic C* the fluctuations will be somewhat influenced by the presence of a local polarization (cf. the discussion of coulomb effects above) but the picture is essentially the same. The spatial correlations in both cases have been directly measured by van Winkle and Clark [352]. But in addition to the director fluctuations we also have layer fluctuations. In principle, these fluctuations lead to a corresponding behavior across the layers, with the same kind of algebraic correlation
-
-
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13 Antiferroelectric Liquid Crystals
function, which also means that the mean square deviation of a layer position from the strict position corresponding to long range translational order diverges (though only logarithmically)when we move from any one layer to another thousands of layers away. This is the Peierls-Landau instability describing a basic difficulty in having condensed states with only one-dimensional order. In this case it means that a smectic in very large dimensions (but they have to be very large!) will have a tendency to lose its layering order and look more like a nematic. It is not clear whether this instability will actually have many interesting physical implications, though the difference between bulk smectics and free-standing smectic films may turn out to be one case. But before continuing this line, let us look at the experimental situation. First of all it was observed [342] that the transition temperature T, for the transition A* to C* is not the same in a thin film, let us say of 2 to 15 layers, as in the bulk (and certainly the same thing is valid for A to C, since this is not a chiral phenomenon). We have to go to a higher temperature in the film to break the tilted smectic order. This is because at least a few surface layers stay tilted up to this temperature. Contrary to the case of a solid, the surface is here more ordered than the bulk. The size of the effect is remarkable. Measured as the average tilt as a function of N, the tilt angle 8 shows a classical 2.order behavior for N 2 20. For DOBAMBC the bulk T, is at 95°C. For N = 2 Heinekamp et al. [342] find that the C* phase extends as much as 30°C into the region in which the bulk shows the A* phase; T, now lies at 125°C and the transition undergoes some changes in character within this region for intermediate N . The surface tilt has been confirmed in ellipsometry studies by Bahr and Fliegner [344] who were also able to conclude that the size of this tilt does not depend on N. Important refinements of the technique were introduced by Pieranski and coworkers, who used a frame with a mobile side by which the tension of the film can be controlled [354] and by Stegemeyer and coworkers [346], [347] with, in addition, a special electrode geometry.A second observation regarding the A-C transition was made by Kraus, Pieranski and Demikhov, using the movable frame. By a stepwise increase of the film area, the tension can be increased (momentarily) driving the transition in the direction A --+ C. If the tension is instead decreased the transition goes in the other direction. This illustrates the coupling between tension and tilt. The Stegemeyer arrangement is shown in Fig 182.The films were drawn in a frame with one movable edge which is integrated in a hot stage. Furthermore two heatable microelectrodes were inserted directly into the film in order to eliminate the influence of excess material on the edges of the film-holding frame as shown in the figure. In this set-up the polarization of various materials was measured as a function of N by the triangular wave method. The results are surprising. For one material the measured P, was more than doubled relative to the bulk value when going to a film with N = 6 layers. For some other materials (both ferroelectric and antiferroelectric) the thin film P, values were even more than one order of magnitude higher than in the bulk. Such enormous differences indicate that the surface tilt alone cannot be suf-
13.9 Freely Suspended Smectic Films
Liquid crystal
Figure 182. The Stegemeyer arrangement for drawing freely suspended films with one movable edge. Below is shown how thin electrodes can be inserted, penetrating the film, which can be of thickness down to below 100 A, without cracking the film. From reference [347].
379
Free standing film
freesfandl
ficient to account for the effects. We might here see an additional ordering in the smectic layers which goes far beyond the limited order permitted in the bulk due to the Peierls-Landau instability. In fact it is known that any present field, even the weak gravitational field, will stabilize the smectic system such that the fluctuations are not any longer diverging [349]. Holyst has considered the case of free-standing smectic films and finds 13431 that the amplitude of layer fluctuations is strongly quenched by the surface tension in the film. This means that the layers themselves become better defined when we go to thinner films. Thus the increase of smectic order is much more general and important than if it were only due to the increased tilt angle in the surface layers. Evidently this strongly increases the rotational bias which leads to a much higher spontaneous polarization. According to this argument, one of the reasons for the very low rotational bias (of the order of percent) in the normal bulk SmC*, as inferred from measured P , values, would therefore be the Peierls-Landau instability. Sometimes it is stated that even if a smectic C* is not truly ferroelectric, a single layer or a couple of layers would be, in particular because we have now got rid of the helix, which is often thought to make the difference. Because one can isolate two layers as a free-standing film, this statement can be directly tested. The result is that the film is very far from ferroelectric behavior. There is no question of stable states or macroscopic polarization, nor domains in a ferroelectric sense. Instead the strong fluctuations in the c-P field show how sensitive the system is to external influences. In fact, the electric susceptibility tends to become infinite and the film gives a characteristic example of superparaelectric behavior.
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13 Antifenoelectric Liquid Crystals
n
Oblique Incidence
P
Decrossed Polarizen
Figure 183. The four-fold symmetry lobes seen in reflectance when the c director rotates through 360" is transformed to essentially two-fold symmetry on decrossing polarizer and analyzer.
The first free-film experiments in the antiferroelectric phase were performed by Galerne and Liebert before the antiferroelectric order was recognized, and led to its recognition. In these investigations, already referred to [268], [2711, they discovered the parity effect in the number of layers when these are evedodd and from this concluded the anti-tilt arrangement in succeeding layers. This allowed to identify the SmO* phase in MHTAC as a SmC: . Three years later Bahr and Fliegner were able to bring direct evidence for the coupling of tilt and polarization by studying the phase transition C* -C$ in two- and three-layer films of MHPOBC: they found that this transition took place by just reversing the tilt in one single layer [350].More recently some very illuminating experiments were performed on the same material by Link, Maclennan and Clark [35 11. These experiments generically are a direct continuation of the earlier free film experiments [337]-[341] but now on antiferroelectirc materials. Link et al. applied an electric field to influence and control the tilt direction while they were able to observe the twists and turns of the c director across the film by depolarized reflected light microscopy. When looking at specularly reflected polarized light through an analyzer, one finds that the reflection coefficient for incident light polarized parallel to c is larger than for perpendicular polarization. Further, the polarization plane is rotated slightly toward the slow axis, i.e. toward c.Thereby c (x,y ) can be visualized across the film. If the phase angle q o f c is rotated by 27c one would find four equal maxima in the reflected light. By decrossing polarizer and analyzer this four-fold degenerate pattern is transformed into one of two-fold symmetry depicted in Fig. 183. The film can be aligned by applying a field of about 10 V / m . This is about one order of magnitude higher than for films being in the SmC* phase, due to the lower mesoscopic polarization of the film, but the field is still two orders
381
Figure 184. Regions of even and odd layers on a film viewed with polarizer setting as shown to the left. The applied field aligns the c director (spikes) in perpendicular directions for the two sets of regions, such that the polarization (arrows) is along the field. After reference [351].
Figure 185. Transversal and longitudinal components of the polarization across films for the cases N = -, N = 4 and N = 3. The first case corresponds to the bulk. In each layer the molecules have been drawn according to their head-to-tail invariance in the bulk, corresponding to the fact that n +-n is a symmetry operation. The only non-vanishing polarization (P,) in each layer is due to the transverse component permitted by the C, axis. In a thin film, however, the situation is different: n +-n is not a symmetry operation for the surface layers, which not any longer possess a C, axis. As a result, there will be a net longitudinal polarization (along the c director) for thin films consisting of an even number N of smectic layers. After reference [35 11.
of magnitude lower than the critical field (- 1 kV/mm) necessary to entail the transition of the SmCz into its ferroelectric state. The film is normally inhomogeneous and shows different areas of constant but different number of smectic layers. Parts with N alternating between odd and even can often be seen as islands of alternating brightness as sketched in Fig. 184. Where N is odd the film has a net polarization perpendicular to c, therefore the electric field will orient c perpendicular to the field
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13 Antiferroelectric Liquid Crystals
and these regions will thus look less bright, whereas regions with N even will look bright. The explanation to this is given in Fig. 185. To the left we are looking at a cross section of a thick film, corresponding to N = M , i.e. the bulk. In the bulk we have director symmetry, hence the system is invariant under n +-n. As we have already repeatedly pointed out, this invariance is not valid at surfaces, which already from a geometrical point of view have a polar character, expressed by the surface normal s, a vector. If we take s to point out of the liquid crystal, that is in this case into the air, there is no reason to believe, with n regarded as fixed along the molecule, that n . s>O and n . s c0 should represent the same energy. In the middle part of Fig. 184 we have taken this lack of director symmetry into account in such a way that n # - n in the two surface layers and such that the liquid crystal has a dipolar component P into the medium at the surface ( P s < 0). (If it were the other way around it would not change our argument.) P is here a longitudinal dipole, PI,,along the molecular axis, which is now permitted to be nonzero in the surface layers due to the local violation of the n +-n invariance at the surface. The transverse dipole P', is the normal bulk spontaneous polarization. Assuming, for simplicity, that the next nearest surface layers already obey director symmetry we see that the smectic C,* film with N = 4 in the figure has to have (P') = 0, but (PI,)# 0 with an average layer in-plane polarization (PII)= PII/N.Correspondingly, in the right part of the figure, a film with N = 3 is shown to have = 0 but (P') = 2 P'IN. For the compounds in question we may expect that PIIQ 4 ,but this does not change the qualitatively important fact that antiferroelectric films with N odd have a nonzero tranverse polarization ( P Ic), whereas films with N even have a nonzero longitudinal or in-plane polarization (P 11 c). As all surface effects, this difference will vanish when N -+ m, but for N I 2 0 it is easy to observe, like in Fig. 184. It is also interesting to note that Link et al. in the absence of applied field observe large director fluctuations in the (bright) N even regions, smaller fluctuations in the (darker) N odd regions, but much smaller fluctuations in both regions when the film is in the C* phase. It is now easy to interprete this result. It is of course due to the stiffening phenomenon already encountered, increasing the resistivity toward bend fluctuations (connected to V . P # 0). The electrostatic contribution to the bend elastic constant increases as Pi/r where r is the radius of curvature of the bend fluctuation. MHPOBC is a high polarization material (P, = 72 nC cm2 10°C below the A*-C* transition) so the quenching effect is very strong in the C* phase. When cooled into the C$ phase, N-odd regions have a P reduced to (P') Q P, and the N-even regions have an even smaller P, reduced to (PI,)Q (p') , as PIIQ P,. Link et al. find that T, for the C*-Cz transition decreases with decreasing N, i.e. that the C* phase is stabilized relative to Cz in thin films. If this is true we would have a second independent mechanism giving a preference for synclinic order in addition to the surface mechanism operating in bookshelf geometry between substrates. Bahr et al. find, on the contrary, that the AF phase is stabilized in thin films, similar to what was found for the A*-C* transition in [342] in the sense that the more or+
13.10 Antiferroelectric Liquid Crystal Displays
383
dered phase is stabilized. They also observe a larger tilt in the outer layers [350]. CepiE, RovSek and %kS, applying their phenomenologicaltheory to the thin film case, find that the outer layers ought to have a smaller tilt angle than in the bulk and that the synclinic tilt is favored in the thin film, i.e. that the C*-A* transition should be lowered in temperature [356].This is still an open issue. A last striking observation by Link et al. is the characteristic feature that the layers tend to stick to each other in pairs, and that the layers with time segregate into larger domains consisting of only N-even regions and only N-odd regions. For instance, one domain may show a development from N = 4,to 6, to 8 layers and then transit to an adjacent large domain of N = 11, 13, 15.. layers. Steps with AN = 1 , or generally AN odd, seem to be energetically unstable and vanish until the whole sample is dominated by AN = 2 steps.
13.10 Antiferroelectric Liquid Crystal Displays The antiferroelectric order found in the C,* phase is extremely interesting for applications in large-size high-resolution displays. The basic features of the electro-optic switching used in such AFLC displays are illustrated in Figures 186 and 187. The smectic is arranged, just as in the FLC, in the quasi-bookshelf geometry and the molecules are switched back and forth between the two ferroelectric states on field reversal. The cell gap is about 2 pm. The superposed helical order on a pm scale is absent due to the surface-stabilization in the thin cell. When no field is applied, the smectic immediately returns to the antiferroelectric state which has its optic axis perpendicular to the layers. In the AFLC case, however, the polarizers are set parallel and perpendicular to the layers. This means that E = 0 is the dark (extinction) state, whereas +E and -E both give the same bright state (Fig. 186). Therefore, we can redraw the "optical" hysteresis curve, like the one of Fig. 151, into the symmetric shape of Fig. 187, where the transmission is along the vertical axis instead of the tilt, and where the horizontal axis represents the applied voltage V. Figure 186. Antiferroelectric displays use the switching between the two equally bright ferroelectric states symmetrically situated around the dark antiferroelectric state. This allows a very simple electronic addressing with dc auto-compensation and an amplitude-controlled grey scale. In operation a bright pixel is actively switched between the two ferroelectric states, while a dark pixel is - in principle - allowed to relax to the stable state. By use of a holding voltage a pixel can be kept in a level between these two limits, permitting a simple realization of a grey scale.
QE @ P
\\\\ BRIGHT
DARK
BRIGHT
384
13 Antiferroelectric Liquid Crystals
Figure 187. Transmission-voltage character-
0
V
istics corresponding to the optical hysteresis in Fig. 15I . The shape of the threshold is very sensitive to a number of parameters: AFLC material, layer thickness, surface treatment, temperature, pulse length, pulse shape and, in particular, the frequency of the applied wave form. The shaded areas indicate the light leakage giving a bad dark state: a dark pixel is brought to a non-zero transmission state due to the applied holding voltage i V,. The holding voltage holds the optical transmission in a state determined by the additional pulse amplitude in the select period. It is itself incapable of switching a pixel into a bright state, but its presence all the time (except in a reset slot) in every frame entails a disturbing average transmission which has to be minimized in order to obtain high extinction in the dark state. The quality of this dark state is the essential limiting factor for the achievable contrast in an AFLC display.
There are a number of interesting advantages with the AFLC subclass of materials, relative to FLC, for making displays, but also some disadvantages. We will now discuss these in turn. First of all a certain bright state can be written by application of a voltage +V over a certain pixel. In the next frame the same state can be written by application of -V, simply by inverting the line and column voltages. This means that we can utilize very simple drive schemes which have an automatic dc-compensation by simply reversing all voltages from one frame to the next. This also allows relatively conventional drivers, except for the high voltage (and about twice as many row voltage levels than for FLC). The price to pay is that we have to use two frames to write one image in these addressing schemes, as discussed below. A requirement for the AFLC display mode to work is that the transmission-voltage characteristics of Fig. 187 is perfectly symmetric, which requires careful control of alignment. The exact shape of the hysteresis loop is of course extremely important for the display performance and as this shape sensitively depends on temperature, this also has to be very well controlled over the whole display area. In that case we can use the hysteresis curve to generate a grey scale simply controlled by the pulse amplitude. An advantage of AFLC over FLC is that we do not have to reset the pixel to the same starting condition for every frame - this is done automatically, by the pixel itself, since the AFLC has no memory. In other words, because the ferroelectric state of the SmC: phase is not stable it does not have to be continually erased by a blanking pulse. On the other hand a zero voltage period, most often consisting of at least two time slots is normally required to reset the AF state. Alternatively, to gain speed, it
13.10 Antiferroelectric Liquid Crystal Displays
385
may actively be reset to one of the F states. But the characteristic hysteresis shape now allows us to apply a holding voltage +V, in each frame which, in a way that we will discuss below, together with the column voltage will determine the level of the bright state. Due to the high P, value of the AF material the applied voltage is sufficient to keep the chevron structure upright, “updating” it in this position every frame. This is another great advantage; as soon as electrically addressed, the AFLC display automatically goes into the QBS structure, which gives it the superior combination of high viewing angle and brightness typical of QBS. The AFLC (typically three ring) materials have higher viscosity than FLC (typically two ring) materials. But this is compensated for by the much higher P values that can be used in AFLC materials without running into the image retention (“sticking”) problems typical in FLC and also known from solid state ferroelectrics. There are these problems in AFLC as well, but much less pronounced. The mechanical fragility is a problem in AFLC displays as it is in FLC displays. On the other hand a high P, material in a QBS structure is less sensitive to small distorsions as the applied field will reset the structure. However, in principle, every chevron or quasi-bookshelf structure is sensitive to direct pressure even if the chevron angle is small. If the deformation is not too large, the structure will be regenerated when the electronic driving is turned on. If the deformation exceeds a certain threshold (lower for FLC) the structure will be irreversible damaged. The particular sensitivity of the FLC is perhaps enhanced by the fact that it is piezoelectric (in the SSFLC state) which means that a thumb pressure (via secondary strains) may set free local charges in certain areas of the display. Relatively harmless for small displays, there is probably only one radical way in the long run to overcome these problems of fragility common in AFLC and FLC: to utilize a display structure which cannot be deformed by thumb pressure. Such structures can easily be produced just by choosing the correct density (which is not high) and distribution of hard spacers [357].If combined with sticky spacers to withstand twist of the substrate, this technology would allow flat screens for FLC as well as for AFLC to be manufactured without a protective front glass which always impairs the optical properties, in particular reduces the viewing angle. The electro-optic response of an antiferroelectric liquid crystal has a quasi-linear part - it is dielectric at low voltages - and a strongly nonlinear part - it abruptly changes slope and becomes hysteretic at higher voltages. It is in this way similar to both nematics and ferroelectrics. It has no memory. But it does not trace the same curve when going back, once we have switched it into the ferroelectric state. Let us now see how we could take advantage of the characteristic AF hysteresis loop and the inherent monostability in order to realize grey levels. The idea is simple. If we apply a sufficiently strong vottage pulse to go all the way along branch 1 to the saturated bright state (we denote this level as 100 per cent transmission) and then lower the voltage, the transmission will decrease according to branch 1’ of the hysteresis loop. If we stop at V, on the way down, a corresponding transmission level T will result. Evidently T depends on V, but, in particular also on the value V that
386
13 Antiferroelectric Liquid Crystals I
Transmission
b
I
vo
Voltage
R
Voltage
Figure 188. The achievable grey scale and contrast depend sensitively on hysteresis shape. Without a holding voltage the response on an applied pulse relaxes back into the monostable (AF) state. However, the hysteresis curve here is somewhat academic, as it applies only to very low frequencies. It is drawn to demonstrate certain features but also to stress the frequency dependence on the electro-optic behavior.
we apply to the pixel in the select period before we stop at the holding voltage V,. For instance, let us take Vo to be 15 volts and in the short select period have a pulse of saturation amplitude, let us say 30 volts. This will bring the pixel to the bright state and after partial relaxation at 15 volts we get a transmission state T I ,cf. Fig. 188. For an off pixel holding and data voltage should not sum up to more than 20 volts, leaving the pixel in the transmission state T, after partial relaxation at V,. Complete relaxation to the AF state is prevented by this holding voltage. By varying the amplitude we see that we can slide along the curves 1 and 1’ but the grey scale is quite limited with such a hysteresis shape and the achievable contrast is quite low also because we cannot obtain a lower transmission than about T3which is given by the holding voltage and the quasi-linear response corresponding to the branch 1 at low voltage. Even if pixels are allowed to relax during a certain ”reset” period, dark pixels will never be darker than corresponding to some state in the shaded areas at the bottom of the figure, since the holding voltage is applied during the rest of the frame time. This provokes a light leakage which is one of the problems with AFLC displays as it limits the available contrast. Luckily, however, the hysteresis shape can
387
13.10 Antiferroelectric Liquid Crystal Displays transmission
Figure 189. As the frequency increases the hysteresis loop grows broader, first to a butterfly shape. Then the 1’ branch will cross the V = 0 line before reaching zero transmission. At higher but still modest frequency the switching between the outermost ferroelectric states becomes direct, the AF character gets lost (top). The hysteresis then has the shape typical of a ferroelectric liquid crystal with polarizer setting parallel and perpendicular to the layers. The horizontal arrows in the top figure indicate that the 1’ branch now crosses the V = 0 axis before the transmission goes down at all. Modulating the data pulse amplitude in the case of this broad hysteresis gives an analog grey scale with considerable depth (bottom). The hysteresis loop is different for every different value of V = V,+V, and a very large number of intermediate transmission states can be controlled.
V
vo
I_
I
vo
V
be conditioned to be very different from what one would believe from the Fig. 151, 187 and 188. In particular it is also by itself very dependent on the frequency of the applied switching pulses: in fact, this type of narrow hysteresis would be useless for applications but is found in academic papers; curves with the shape in Fig. 187 and 188 are only obtained if we use a triangular ramp voltage with frequency of typically O.1Hz or lower. As can be expected, the hysteretic behavior is much more pronounced at higher frequencies and the loop then takes a butterfly shape like in Fig. 189 where more realistic loops are shown. Then we see that if we take the holding voltage to be in the middle of the wing, a considerable span (T,, T,, T3...) in transmission and therefore grey levels can be achieved. Furthermore the linear effect can be made very small by good alignment techniques like crossrubbing, such that the light leakage caused by data pulses around V, is considerably reduced. Still very little is published about how AFLC hysteresis loops depend on, for instance, alignment, P , value and pulse shape. Most hysteresis curves in the literature have been taken with triangular voltage which is of course a completely unrealistic pulse form to be applied in displays. A simple waveform used in some of the very first (1989) AFLC display prototypes [358] is shown in Fig. 190. This waveform does only give black and white states but it is instructive to see how easily it can be modified to give a greyscale. The pixels
388
13 Antifenoelectric Liquid Crystals Pixel 1
Pixel 2
TH 1st frame
TH’ 2nd frame
J
Figure 190. Simplified addressing scheme for an AFLC display. The waveform is reversed every second frame. Two frames are required to write one picture. Like here a zero voltage slot (reset time) is often inserted before each frame in order to allow the pixels to relax (more or less) to the AF state. This relaxation will limit the available speed in high-resolution screens and is therefore normally replaced by a field-assisted return (counterpulse) to the AF state or, alternatively, actively reset to one of the F states. Modified after reference [358].
1 and 2 which are supposed to be in the same row are in the ON and OFF state, respectively. The driving voltage is +30 volts, the holding voltage 218 volts and the data pulses +13volts. The waveforms are inverted every second frame like in the case of driving a nematic display. The used material had a P, value of 225 nC cm2. This gives a very high power consumption taking into account that Pis reversed with twice the image frequency. However, modern materials have now been developed with more reasonable P, values. Simultaneously the driving techniques are now much more sophisticated which also has increased the dynamic performance of the displays. For instance the rather slow relaxation back to the antiferroelectric state (the historic name ”third state” for this is now obsolete) which most often was allowed during a “reset time” in the driving cycle, is now speeded up by applying a short push-pulse of appropriate sign. In the scheme of Fig. 190 there is such a reset time of length one slot (usually it is longer) immediately prior to the select time slot. If we denote the amplitude of the data pulses applied on the columns by +V, and the selection voltage applied to the addressed (select) row by V,, the ON and OFF pixels receive the amplitudes V,+V, and V,-V,, respectively, during the select period. In the example these amount to 30 volts and 24 volts, of which the first should
13.10 Antiferroelectric Liquid Crystal Displays
389
bring the pixel to the bright state and the second to the black. This requires quite a steep slope to the right flank of the bottom curve in Fig. 189.After this select slot both pixels are subject to the signal V,kV,, composed of the holding V, = 18 volts and V, = 3 volts. For both pixels dc compensation is achieved in the next frame where all voltages are reversed. This means that the picture is written in two frames. These two frames are always needed to dc compensate the holding voltage. But if we designed the other pulses, select and data, as bipolar pulses such that they were already dc compensated within each frame then we could equally well write a different picture in the next frame (without gaining anything in time). The immediate compensation has the advantage that the average holding voltage is not subject to any variation, it is independent on the incoming data information. In the scheme so far, we have considered the data pulses to be of constant amplitude +3 volts. In order to write a grey scale we now only have to vary V,, let us say from 0 to 5 volt. This can again be made in two ways, either with one-slot pulses and dc compensation over two frames, or with directly compensated bipolar pulses of varying amplitude. During operation many aspects of AFLC driving are fairly similar to the FLC case. For instance, when a bright pixel switches between the two equivalent ferroelectric states in continuous operation, it does so without, of course, passing any antiferroelectric state. Otherwise we would have a severe flicker problem. However, few details have been published on driving and switching dynamics. For instance, little is known about AFLC thresholds. It is clear from Fig. 190 that for AFLC switching there is no dynamic threshold corresponding to the voltage-time area (VZ), in the FLC case: if there were, the holding voltage would just switch every pixel back and forth by virtue of its enormous voltage-time area. But the shape of the select pulse, and not only the amplitude, probably has an influence on the switching dynamics. The pioneer in AFLC technology has been the Japanese Denso Corporation. Their first full color panel with analog grey scale was presented in 1993 and had a 6 inch diagonal with 220 lines. It had those good properties which can be expected from the quasi-bookshelf geometry: a hemispheric viewing angle and a very high brightness. The line writing time was 63.5 ps corresponding to a 72 Hz frame time and giving 36 Hz picture frame rate (2 framedpicture). The impression of this display is extremely pleasant in spite of the fact that the contrast ratio is only 20: 1 which, according to “general rules” is completely insufficient for colour reproduction. The fact is, however, that if the brightness is high enough, which is the case in the QBS structure, this compensatesfor a modest contrastratio. Recently,considerableprogress has been made, both in materials, alignment and electronic addressing. In addition to a much developed version of the 6 inch display for navigation instruments there is now a 17.4 inch AFLC full color flat screen ready for manufacturing. It is aimed for office use and multimedia and is shown in Fig. 191.The resolution is 1280x 1024 and the screen presents a very high brightness, (200 cd/m2) at 60 W power and with a contrast of 50 : 1 . (As a comparison the Canon FLC prototype in Fig. 118 a has a contrast of 100 : 1 and a brightness of 120 cd/m2 at the same power in image-changing mode but only 10 W
390
13 Antiferroelectric Liquid Crystals
Figure 191. Antiferroelectric liquid crystal displays from Denso Corporation, Japan. The 17.4 inch diagonal prototype at the top was demonstrated in 1998 and has 1280x 1024 picture elements of 0.27 mm x 0.27 mm size, each capable of 16 million colors (3 times 8 bits) in analog gradation. The pixel response time is 30ms. At bottom an earlier desk top prototype is seen together with the 6 inch display in a road navigation system. Courtesy of N. Yamamoto, Denso Corporation.
in sleeping mode.) Several other companies, e.g., Citizen and Casio, are developing similar but smaller size AFLC panels. These all use the same domain mode analog grey scale like Denso. It may be pointed out that it is also quite conceivable to use this mode in a dynamic way for writing a grey scale in AFLCDs, just as it is in FLCDs, by taking account of the relaxation process itself. In the FLC case this powerful mode was developed by Seiko [359] for spatial light modulators, and consisted in driving the FLC only so far that it did not latch. Thereby it looses memory and relaxes back to a monostable state. So far this principle has, surprisingly not been used for displays.
13.11 Thresholdless Smectic Modes In 1995 Fukuda [360] and slightly later Inui et al. [361] reported that in certain mixtures of antifen-oelectricmaterials the double hysteresis loop vanishes and is replaced
13.11 Thresholdless Smectic Modes
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€= 0
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Electric field alignment
@ Random from layer to layer
+E
0 Electric field alignment
Figure 192. V-shaped switching (left) observed by the Fukuda group on the three-component mixture shown (ratio 40:40:20) together with a simulated electric-field response (light transmittance) assuming a 2D Langevin-function.To the right the proposed microscopic model to account for this behavior. From reference [361].
by a continuous, V-shaped electro-optic characteristics of the kind shown in Fig. 192. The crossed polarizers are arranged parallel and perpendicular to the layers as in the conventional AFLC case. This characteristics is much smoother than any corresponding electro-optic effect in a nematic and can be almost linear from zero to 100%transmission. It appears in bookself geometry and the effective optic axis moving in the plane of the cell (in-plane switching) as is the conventional case. Since this electrooptic mode combines these properties with the very high speed of smectic materials it immediately attracted interest from industry and prototypes with outstanding performance have already been demonstrated. In fact, these materials have constituted the hottest topic in the field of antiferroelectric materials, until recently almost exclusively studied in Japan where they have been available [362]. The effect was dubbed “thresholdless antiferroelectricity” in [361] and this name was used in later papers. However, this nomenclature is not a particularly happy one as it is in conflict with the present meaning of the word (the threshold is an inherent part in the
392
13 Antiferroelectric Liquid Crystals
0 0
20
40
60
80
100
Mixing ratio (%)
Q-;wcF3GM,,w,
Figure 193. Phase diagram of the binary mixture of two compounds shown below, together with the AF threshold field vs. mixing ratio. From reference [361].
universally accepted definition). Furthermore, no findings have been reported so far which would really support the name (and a redefinition of the old concept). The Tokyo group interpreted the thresholdless switching by a model also shown in Fig. 192. According to this model the phase is a tilted smectic with the molecules making a constant tilt angle with the layer normal but with completely random tilt direction or with random tilt from layer to layer. The phase has then the symmetry of a smectic A (or A*, in the chiral case) and essentially corresponds to the hypothetic phase proposed by de Vries [ 191a] previously mentioned in Sec. 9.5. Such a phase is perfectly possible but not yet demonstrated. As we already pointed out it would be the optimum material for the electroclinic effect because it would allow very large induced tilts without any accompanying layer shrinkage. Since the molecules are chiral they would have a dipole perpendicular to each individual tilt direction and, under an applied electric field they would align continuously into the field direction like
13.11
Thresholdless Smectic Modes
C* phase
393
C*, phase
Figure 194. The Tokyo mixture studied in the C* phase and in the C,* phase in a geometry with the layers parallel to the glass plates. The field is applied along the layers. It is found that in the C* phase the field can swing around the c director continuously, without any threshold. (No surface-stabilization in this geometry.) This is the cone mode. In the C,* phase on the contrary there is a distinct threshold field typical for an antiferroelectric. Only above this field the director may switch between either of two opposite positions.
the molecules in a Langevin paramagnet do in an external magnetic field. The designation Smectic C, was proposed as a phase symbol, where R stands for “random”. The Fukuda group found that the V-shaped switching appeared when AF compounds were mixed in a certain ratio but only thin cells. The last condition was interpreted as a “randomization” effect caused by the surfaces, a feature which was also necessary to explain the apparent absence of thresholdless behavior in thick freely suspended films [363]. The first condition prompted them to study how the threshold field for turning the C z phase into its ferroelectric state depended on the mixing ratio between compounds. Thus, the threshold Ethwas measured as a function of this mixing ratio. One example [361] is shown in Fig. 193. Thresholdless switching is then supposed to occur when Ethgoes to zero. However, if we extrapolate the curve to E t h = 0 in that figure, we will find that we are not any longer in the Cz phase but in the C* phase. Furthermore, Eth= 0 has physically nothing to do with thresholdless switching but rather with the equilibrium between Cz and C*. If Eth= 0 it is confirmation that we are in the C* phase. This brings serious doubts as to the above interpretation. However, many more ad hoc models based on this first one were presented in order to account for the various features of thresholdless switching and many experiments were interpreted in these terms between 1995 and 1999 by several groups. As a result the Boulder chemists Walba, Wand and Chen decided to synthesize the three compounds depicted in Fig. 192 and mix them in the specified ratio. This mixture was then investigated in a collaborative effort between the Boulder and Goteborg groups. The results were first presented [364] at the 17th International Liquid Crystal Conference in Strasbourg (France), July 1998, and published in [365] and [366]. The bulk was studied by dielectric spectroscopy, the surface by depolarized total internal reflection. This was complemented by observations on free-standing films. Finally, as electro-optic experiments on bookshelf cells have turned out to be very ambiguous in this case, the electro- optics was instead studied with the layers parallel to the glass plates. In this geometry it was easy to see how the c director swings around without threshold in the C* phase, controlled by the applied field, whereas it only switches to the opposite tilt above a distinct threshold in the C z phase, cf. Fig.
394
13 Antiferroelectric Liquid Crystals
5 0
30
40
50 60 Temperature / "C
70
107 106
N
?
16
z? 104 103
I@
30
40
50 60 Temperature / "C
t---------, Thresholdless mode
7o
Figure 195. Dielectric spectroscopy data taken on the Tokyo mixture (Fig. 192) in a 2.3 pm cell with QBS geometry. The divergence-like rise of the dielectric constant in the A* phase, on approaching the C* phase is the typical behavior for a soft mode which is a precursor of a phase with strongly polar order. Both E and the relaxation frequency f show that there is considerable coexistence of C* and C,*.
194. Thus in this geometry it is easy to distinguish between the C* phase and the C,* phase and, as expected, the C* phase has no threshold whereas the C,* is thresholded. The results of dielectric spectroscopy in Fig. 195, unambiguously show that in the region of thresholdless switching, which primarily is between about 45°C and 65°C the material is in the SmC* phase, not in SmC,*. The thresholdless switching is clearly identified with a collective mode, the Goldstone mode. The collective behavior was also confirmed by the fact that in thicker samples we could observe that a helix exists - thus the phase cannot be random. New similar mixtures exhibiting V-shaped or thresholdless switching were then prepared by the Boulder chemists. The constituents belong to the class of partly fluorinated compounds with the non-chiral tail C4F9CnHZn,represented by the n = 6 compound in Fig. 196. In a 5.6 pm thick cell dielectric spectroscopy again clearly differentiates between A*, C* and C,* phases, with a 2 K range between C* and C,* normally assignated to the ferrielectric C;. This is shown to the left of Fig. 196. On the other hand the C; region in a 5.6 pm thick cell could just as well be interpreted as a pure coexistence between C* and Cz. If the measurements now are repeated on a 1.5 pm thick cell we see that not only is the Cf squeezed out but even Cz! The C* phase has taken over and persists all the way down to crystallization.
395
13.11 Thresholdless Smectic Modes
102
ER
10’
100
Id
fRW, 10‘
102 10’
1,
60
, , , ,
, 65
, 70
,I,
sample thickness =5.6p , ,
, 75
, ,
,
,
{
80
Temperature (“C)
103
102 50
55
60
65
70
75
80
Temperature (“C) F
MFA
Figure 196. Spectroscopic data taken on the n = 6 species of a series of partly fluorinated compounds synthesized in Boulder (shown at bottom) in a 5.6 pm thick cell and in a 1.5 pm thick cell. The coexistence of C* and C,* in the first cell is evident to the point that the existence of the C*yphase must be doubted. In the 1.5 pm thick cell not only the questioned C? phase has vanished but the whole C,* phase has been squeezed out, and the C* phase is stable down to crystallization.
Having now established that we are dealing with the ferroelectric SmC* phase and not with an antiferroelectric phase, we have to explain which features give rise to the thresholdless switching. The internal reflection studies show first of all that the surface states are monostable and synclinic. This should not surprise us too much after the discussion in Sec. 13.4. If both surfaces are equivalent, then we have the situation already met in Sec. 12.3, cf. Fig. 12.6, where we discussed the splayed states. The corresponding state is shown in the left part of Fig. 197. We have here arbitrarily assumed that the surfaces prefer the local C* dipoles to point out of the liquid crystal. This (like the other way round) gives rise to a splay-bend-twist deformation of the director, corresponding to a splay-bend deformation in the two-dimensional P field. The splay in P gives rise to a polarization density pp = - V . P as we have now discussed several times. For high P , materials this will lead to a stiffening of the director: the bulk becomes almost homogeneous, thus lowering the electrostatic energy and leaving only two thin regions of distance 6 = ( K d P 1)1’2 to be strongly nonhomogeneous [256]. We have already met this characteristic length in the discussion
396
13 Antiferroelectric Liquid Crystals
.
*
*
.
u- (V P)* '
Lowered
Figure 197. Twist-splay-bend states in a low-P, and in a high-P, material in the case of strong anchoring with the dipoles pointing out of the liquid crystal at the boundaries. To the left (low P,)we have a uniform twisted smectic C* structure formed when polarization charge effects are negligible. To the right the high P, case with stretched or stiffened directors.
of splayed states and electrostatic self-interaction in Sec. 12.3, 12.4 and 12.5. The dielectric constant E is here that part which does not contain the contributions from local polarization, i.e. we can consider it to be E&~ where E~ corresponds to the bare dielectric constant of the racemate. If 6 9 d, where d is the cell thickness, then the cell is going to be essentially homogeneous with the polarization P aligned along the glass plates as depicted in Fig. 198. This drawing explains the essential features of the thresholdless or V-shaped switching. The crossed polarizer directions are both horizontal, in the plane and perpendicular to the plane of the paper. Then we have a very good extinction state for E = 0. The thin non-homogeneous edge regions of the director are too thin to influence the transmission. When a downward increasing field E is applied as shown to the right the polarization follows E continuously and the c director swings out in the plane, and oppositely for reversed field. The transmission then grows in an essentially linear fashion as depicted at the bottom. The surfaces may finally switch or not without making much of a difference. In both cases optical saturation is achieved at high field. The described smectic C* state belongs to the category of SSFLC - it is a surfacestabilized state. It has a macroscopic polarization, P # 0 (otherwise it would not belong to that category). But how come then, that it is not bistable? This is because, although it is a ferroelectric state, it is employed in a dielectric mode. This is due to the fact that the field in Fig. 198 is applied along x (vertical in the figure, cf. the reference frame in Fig. 142), but P, = 0. In the conventional ideal case it is just the other way around: P has there only an x component. But now, when we apply the field, we immediately get a torque, P x E . This means that the electro-optic effect cannot have a threshold. (For bistable modes we would have to apply the field along the y z plane.) The described state also, more specifically, belongs to a category which is called twisted smectic [367]. The electro-optics of twisted smectics has been studied both theoretically and experimentally by Pate1 [ 1621, and we have already discussed the
13.11 Thresholdless Smectic Modes
397
Applied E
Figure 198. Proposed twisted C* structure shown as a cross section of the cell as a function of applied field, together with the corresponding transmission (grossly simplified). The P direction is along the glass plates which explains the SHG signal found from light traversing the cell. (A random order could not explain this). After Rudquist [368].
case briefly in Sec. 8.2. However, this is a special and much more useful case than the general one. We might consider it the “stretched” case, which is a result of the characteristic length 5 being much smaller than the cell gap. The physics of 5 has been discussed in Sec. 12.4. If 5 = d we get the conventional case which is not as useful because it has inferior extinction and contrast as an electro-optic device, in comparison with the polarization-stabilized twisted smectic C*. In particular, the first case does not give extinction at E = 0 between crossed polarizers.
398
13 Antiferroelectric Liquid Crystals
Figure 199. The full color Toshiba 15 inch FIT prototype showing an unmatched independence of viewing angle in both contrast and color hue for an LCD. Equally important, it works in video speed without showing any inertia in rapidly moving film sequences. FLT is a combination of FLC and TFT. In this case the FLC is a special surface-stabilized structure working in a dielectric mode (thresholdless smectic C*). The screen has XGA resolution (1024x763 color pixels). While TN-TFT (nematic) are excellent in PC’s for office use, their ability to display moving TV images has been found to be inadequate with respect to both grey level response time and contrast ratio. The combination of fast smectic materials and TFT’s offers the solution to the problem of combining required resolution for television with the required speed. Courtesy of K. Takatoh, Toshiba corporation.
To sum up: two effects dominate the intrinsic order of the polar smectic. These are polarization charge effects and surface effects. The latter may also be strongly polar and can give a surface twist of the director which may be interpreted as a surface electroclinic effect. Thus, polar surface interactions induce ferroelectric order, not randomness, at the surface. In the bulk we will then have a twist-splay-bend configuration of the director which is connected with a splay-bend configuration of the local polarization P.The splay of P leads to a non-zero polarization charge density such that every volume element seems to be charged. For high values of spontaneous polarization the whole material appears to be “stiffened” with an almost homogeneous direction of P in the whole sample, thereby reducing the electrostatic
13.11 Thresholdless Smectic Modes
399
energy -( V . P)’. Only near the surfaces, in thin layers given by a characteristic length 5 = ( K E / P ~ ) ’ does ’ ~ , the polarization have a rapid spatial rate of change, in the rest of the cell it is parallel to the cell plates (as confirmed by NLO measurements). If the sample contains a lot of free ionic charges, which is probably the normal case, a very high P, value will be required in order to keep small enough to get a quasi-homogeneous director structure, because the ionic charges tend to stabilize the non-desired continuous twist to the left‘in Fig. 197. In such a case the high contrast will only be a dynamic phenomenon. This means that the display works well when being repeatedly switched, at a frequency sufficiently high to prevent the slow ion clouds to respond to the changing internal field. In purified materials of very high resistivity it should, however, be possible to achieve the same kind of electro-optic performance at much lower P, values, which would reduce power and relieve some of the load on the transistors. Thresholdless antiferroelectricity is not a useful concept but misleading and selfcontradictory. The fact that the applied field E is perpendicular to P,means that the electro-optic effect can have no threshold and is purely dielectric, even if the structure belongs to the category SSFLC. The switching is a collective motion and not random as in the so far widespread interpretation, which also required that the surfaces have a disordering effect (surface-induced randomization). However, the opposite is true: the surfaces have a strong ordering effect. This explains a number of other observations which can be made on these materials. For instance it turns out that in the virgin state the Cz phase really behaves as an antiferroelectric both at the surface and in the bulk. In the beginning there is a clear switching threshold. But an increasing field will switch the sample to the ferroelectric state and then the surfaces only slowly relax back. This in turn promotes the ferroelectric state even in the bulk and the thresholdless (V-shaped) switching may continue way down in the C z phase (or what in the bulk would correspond to such a phase). Thus we can have a switching of a non-equilibrium twisted SmC* state that totally or partially persists during repeated switching due to the strong influence of the surfaces. The observations have also been strengthened by simulations. For instance, Maclennan has calculated the propagation of light through the twisted c * structures and finds good agreement with experiments. It is also clear that the coexistence of the two phases SmC* and SmCz mimics the existence of ferrielectric phases which may not really exist. Simulations by P. Rudquist show that the great variety of hysteresis loops which surround the V-shaped switching and change shape as function of frequency and temperature, can simply be accounted for by the coexistence of only two phases, ferroelectric (C*) and antiferroelectric (C:). The thresholdless materials thus bring up several fundamental questions concerning the existing subphases and their stability. For a discussion in more detail the reader is referred to the first published accounts of the properties of the polarization-stabilized twisted smectics [365],[366].
<
400
13 Antifemoelectric Liquid Crystals
In 1997 Toshiba demonstrated a full color 15 inch XGA TFT-LCD employing the thresholdless smectics. It is shown in Fig. 199 seen from different angles. With 1024x 763 x RGB picture elements it works in video speed showing 6 bit x 3 colors and a contrast >loo: 1. Its performance [369] is outstanding in at least two ways, in viewing angle and in the grey level response time allowing rapid sequences to be shown without being blurred. A full grey level image is written using only 2-3 frames which is far better than any nematic-TFT performance.
14 Current Trends and Outlook Before considering trends it should be fair to remind that many important topics have only been treated superficially or not are all in this text. One example is chemistry in general, which is however well covered in the chapters by Goodby and by Kelly in the Handbook of Liquid Crystals. Arecent review has also been given by Bezborodov and Dabrowski [370]. The polymer aspects are also well covered in the Handbook in the articles by Zentel (Vol. 3, 1.3) and by Dubois, Le Barny, Mauzac and Noel (Vol. 3, IV) including also non-linear optics. Non-display applications are treated at length in the Handbook by Crossland and Wilkinson (Vol. 1 , IX.2). Clearly in these areas there are important and exciting developments which we will not be able to comment on. Further topics that, regrettably, had to be left out, are the intriguing and interesting electromechanical effects (pioneering work has here been done by the Budapest group) and the field-induced rotational instabilities affecting the smectic layer ordering. Many other fundamental aspects would certainly also had merited a deeper treatment. In this outlook we will, however, go back to some more practical device aspects. The driving force for academic research is not only the generation of new knowledge, but also the development of industrial applications, even if these applications may lie far in the future. In liquid crystals it is evident that the industrial aspects are dominated by displays. The fact that displays are the most severe bottleneck in the present-day information technology motivates the enormous investments in this industry. The liquid crystal display market is today entirely dominated by nematic materials, mainly in form of twisted nematics for simple displays, supertwisted nematics, and, in particular, twisted nematics in combination with thin film transistor arrays for very sophisticated panels. During the last two decades, where practically everything has happened in the development of smectic display technologies, the nematic key technologies have constantly become better. The ferroelectric demonstrators have also inspired new developments in nematic technologies with the object that nematics should be able to do the same thing. Examples of such developments are those trying to mimic both the switching geometry and go to similar small cell gaps, for instance in the so-called In-Plane Switching nematic displays, and in the fast bistable nematic displays also using the in-plane switching mode in combination with two-fold degenerate surface conditions. In particular, huge industrial investments have now been made which lock the development into the direction of TFT panels, at least for a very long time. Especially for those who extrapolate present trends, it is therefore doubtful whether there is any future for smectic materials at all, in spite of the enormous potential in the form of much higher speed, brightness, color neutrality, ideal viewing angle, memory, and so on, which these materials have already
402
14 Current Trends and Outlook
demonstrated at the same time as they have shown to be very difficult to handle. The situation has some similarity to the one in the semiconductor industry where gallium arsenide and similar materials, in spite of their much higher speed (in particular), have a hard time entering a marked completely dominated by classic and much simpler silicon technology, which by huge industrial investments is constantly becoming better and, in fact, always seems to be good enough for the present demands. As always in history, the few pioneers, especially in Japan, have found a number of unexpected problems and difficulties in smectic materials that set back the development and which just had to be solved in order for the development to proceed. Regarding the serious nature and complexity of these problems, it is indeed remarkable what has been achieved, and the advent of the first FLC panels and also of AFLC prototypes in the last few years were important events. The development of FLC technology using flexible polymer substrates is quite extraordinary. Moreover, the all-polymer FLC displays simply have no counterparts in the nematic domain. But equally amazing, because of the substantial upshift they demonstrate in optical quality, resolution and speed, are the microdisplays using FLC on a silicon wafer substrate. At the same time, it is clear that the chemistry that has been so instrumental for the perfection of the nematic technology has not yet, by far, reached the corresponding level of refinement for smectic materials. First of all this has to do with the simple fact that the materials development started much later for smectics, at the same time as it was much less intense. However, the task is also more complicated by the very rich polymorphism found in smectics, and it is harder to implement the requirements set by industry. For instance, it is generally harder to phase-stabilize any smectic such that it meets the requirements of storage and transport (-40°C to 90 "C) and not only for the operation temperature interval, and this is still, to some extent, a limiting factor. On the other hand, there are yet many exciting discoveries to be made by synthetic chemists working with smectic materials, considering the richness in interactions when we go to the more complicated molecules that organize in smectic layers. There is no doubt that, in particular, Japanese industrial chemists have been very successful in this more speculative chemistry in the last decade, and their innovative activity will be very important for the future use of smectic materials. American chemists have also been quite successful for instance in developing interesting high performance materials for the electroclinic effect. Another recent example of high performance are some new Displaytech FLC mixtures with switching time of 1 ms at -2O"C, unsurpassed among liquid crystals in general. Smectic materials have already demonstrated their potential use in very large [15-24 in. (37.5-60 cm)] flat panels with very high resolution, a potential that nematic materials do not have if not used in combination with thin film transistors. All smectic materials essentially use the same in-plane switching mechanisms, the cone mode or Goldstone mode, based on the very fragile bookshelf geometry. This has long been considered a serious obstacle, but today well-known methods exist in spacer technology for making such displays mechanically rigid. With this and some sim-
14 Current Trends and Outlook
403
ilar problems eliminated, what we are probably going to see is not only powerful smectic displays working in passive matrix mode, but also a merging of the TFT technology and the smectic technology in panels of yet unachieved quality and sophistication. Ahint of this can be found in the Toshiba thresholdless smectic C* panel described in section 13.11. The rapid progress in the development of active matrix (TFT) screens with twisted nematics has considerably brought down the cost of TFT technology. Moreover, there is now the insight that while the nematic TFT screens are more than adequate for office purposes, they are not good enough for TV which is of course the main future product for the consumer market. Even aided by the TFTs, the nematics are by far not fast enough to write grey levels during one frame time. Instead quite a number of successive frames have to be used, which means that fast movement sequences become blurred on the screen. It is true that the TFT matrix secures that the charge is loaded very rapidly to every pixel. However, thereafter it takes 30 ms for the twisted nematic to switch. With the nematic In-PlaneSwitching (IPS) mode it takes even 60-70 ms. In contrast, with the combination of chiral smectics and TFT, the response time of the liquid crystal itself is now 0.5 ms, which is fast enough not only to write the grey levels but even allows to generate the color by sequential RGB backlighting instead of using a color (RGB) filter mosaic. Even if TN-TFTs are produced in high volume, one can now see a trend to combine TFT with smectics for future high resolution television screens of 15 inch and larger, as for instance demonstrated by Toshiba using the thresholdless chiral smectics. Toshiba and Canon have also demonstrated other combinations of ferroelectrics and TFT. Clearly the TFT-SmC* is therefore the most important current trend in the field. Let us then consider some of the merits of this combination, in addition to the high speed. As the active matrix offers direct drive, full advantage can be taken of the fact that the SSFLC has a dynamic threshold, i.e. a threshold not in the voltage V but in voltage-time pulse area, (Vz),. In direct drive, z is not any longer the pulse width limited by the maximum acceptable line addressing time but is the time for writing a subframe. Hence the necessary pulse voltage is considerably reduced, to the order of one volt, or at most a few volts for conventional addressing schemes. It may be noted that this advantage is not possible for the z (V)-min addressing schemes, since these use the balance between ferroelectric and dielectric torques, which intrincically requires high voltage to work. Thus, FLC displays based on this special driving will have to stay passive. Neither will it, presumably, be favorable to combine AFLC with TFT, because the holding voltages is determined by the AF + F threshold, which again is intrinsically high. Even if this holding voltage could be brought down to CMOS levels, it is not obvious that the combination would be particularly useful. On the other hand, it is a strength of the AFLC displays that they do no need the active matrix. Another great advantage of the FLC-TFT combination is that the temperature dependence of the accessible addressing window is considerably reduced. It may also
404
14 Current Trends and Outlook
be noticed that in combination with TFT, the property of bistability is not really used. This gives a great freedom of device structures and allows the use of much simpler surface conditions. For instance, a diversity of monostable configurations can be used. Canon has recently demonstrated such a mode in which the switching characteristics has practically no temperature dependence at all. A number of methods for producing analog or digital grey scale also become available, among other the already mentioned Seiko method of not driving the pixel to latching. Active driving would also allow going to a cell gap of for instance, 3 pm instead of 2 pm, which would facilitate manufacturing, not the least by simplifying the tedious filling procedure. On the other hand vacuum filling might be replaced by printing techniques, which would be a decisive step in manufacturing progress. Other fascinating possibilities with great consequences for applications would be light-induced FLC switching [3711 as well as command surfaces of thin ferroelectric polymer alignment layers which can be switched and, in turn, induces switching in almost any kind of liquid crystal used to produce various electro-optic effects between the substrates. The possibilities are diverse and general and will, no doubt, stimulate fundamental research along with the future device physics and chemistry.
15 References [ I ] J. Valasek, Phys. Rev. 1920, 15,537; 1920,17, 475. [2] K. Deguchi in Landolt-Bornstein, New Series, Vol IIU28, Springer, Berlin 1990. [3] J. Fousek, Ferroelectrics 1991, 113, 3-20. [4] The New Encyclopedia Britannia, Vol. 9, 15th ed., Micropedia, 1995, p. 825. [S] R. Williams, J. Chern. Phys. 1963,39,384. [6] A. P. Kapustin, L. K. Vistin, Kristallogrujiya 1965,10, 118. [7] R. Williams, G. H. Heilmeier, J. Chem. Phys. 1966,44,638. [8] P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, Clarendon, Oxford 1993. [9] L. M. Blinov, V. G . Chigrinov, Electroopic Effects in Liquid Crystal Materials. Springer, New York 1994. [lo] M. E. Lines, A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials Clarendon, Oxford 1977. [ 11) M. E. Huth, Dissertation, University of Halle 1909; D. Vorlander, M. E. Huth, 2. Phys. Chem. A 1911, 75,641. [ 121 K. Hermann, Trans. Faraday Soc. 1933,29,972. [13] A. Saupe, Mol. Cryst. Liq. Cryst. 1969, 7,59-74. [14] G. W. Gray, Mol. Cryst. Liq. Cryst. 1969. 7, 127. [I51 M. Leclercq, J. Billard, J. Jacques, C. R. C 1968,266,654. [16] W. Helfrich, C. S. Oh, Mol. Cryst. Liq. Cryst. 1971, 14,289-292. [ 171 W. L. McMillan, Phys. Rev. A 1973,8,192 1 - 1929. [I81 R. B. Meyer, Phys. Rev. Lett. 1969,22,918-921. [ 191 P. G. de Gennes, The Physics ofLiquid Clysta1.r. Clarendon, Oxford 1974, p 97. [20] S. Garoff, R. B. Meyer in 6th Int. Liquid Crystal Conference, Kent, OH, August 1976, Conference Abstracts, 1976, Paper (Bl-1 I). [21] S. Garoff, R. B. Meyer, Phys. Rev. Lett. 1977,38,848-851; S . Garoff, R. B. Meyer, Phys. Rev. A 1979,19,338-347. 1221 W. G. Cady, Piezoelectricity, McGraw-Hill, New York 1946; reprinted by Dover Publications, New York 1964. [23] C. Kittel, Introduction to Solid State Physics, Wiley, New York 1956 and later editions, Chap. 13. [24] J. E Nye, Physicul Properties of Crystals, Oxford University Press, Oxford 1957. [25] A. J. Dekker, Solid State Physics, Prentice-Hall, Englewood Cliffs, NJ 1957, Chap. 8. 1261 W. Kanzig, Solid State Physics Vol4,Academic, New York 1957, pp. 1- 197. [27] F. Jona, G. Shirane, Ferroelectric Crystals, Pergamon, New York 1962; reprinted by Dover Publications, New York 1993. 1281 G. S. Zhdanov, Crystal Physics, Academic, New York 1965. 1291 I. S. Zheludev, Physics of Crystalline Dielectrics, Plenum, New York 1971. [30] R. Blinc, B. ZekS, SoftModes in Ferroelectrics artd Ant~erroelectrics,NorthHolland, Amsterdam 1974. [3 I J S. B. Lang, Sourcebook of Pyroelectrici~.Gordon and Breach, New York 1974. [32] T. Mitsui, I. Tatsuzaki, E. Nakamura, An Introduction to the Physics ofFerroelecrrics, Gordon and Breach, New York 1976. [33] J. C. Burfoot, G. W. Taylor, Polar Dielectrics arid their Applications, Macmillan, London 1979. [34] G. W. Taylor, J. J. Gagnepain, T. R. Meeker, T. Nakamura, L. A. Shuvalov (Eds.), Piezoelecrricity, Gordon and Breach, New York 1981. 1351 G. A. Smolenskii, V. A. Bokov, V. A. Isupov, N. N. Krainik, R. E. Pasynkov, A. 1. Sokolov, Ferroelectrics and Related Materials, Gordon and Breach, New York 1984. [36] L. A. Shuvalov (Ed.), Modern Crystalbgruphj: Volume IV, Springer, Heidelberg 1988. [37] T. Ikeda, Fundamental of Piezoelectricity, Clarendon, Oxford 1990. [38] P. G. de Gennes, The Physics ofliquid Crystals, Clarendon, Oxford 1974, second printing 1975. 1391 G. W. Gray, J. W. G. Goodby, Smectic Liquid Crystals, Leonard Hill, Glasgow 1984. [40] S. A. Pikin, Structural Transformations in Liquid Crystals, Gordon and Breach, New York 1991, Chap. V.
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Index A-C transitions, Landau model 37 AC stabilization 189 Abrikosov lattice 277 active areas, digital grey 263 active driving - digitalgrey 265 - Displaytech devices 244 activity, optical 74; 83 f adressing, AFLC displays 388 AFLC displays 383 f alkoxy chains 376 alkyl chains 339,376
4-alkyloxyphenyl4-alkyloxybenzoates 257 alternating smectic layers 325 alternating tilt phases 337 amino acids, chirality 73 amplitude controlled grey scale 383 amyloxy terephthal cinnamate, smectic C* 4 analog grey levels 226 f angular distributions, SmA*-SmC* transitions 124 anisotropy - dielectric 255 - optical I85 f anticlinic order - antiferroelectric liquid crystals 339 - polarsmectics 373 anticlinic structures, AFLC 332, 369 antiferrielectric phases 335 ff antiferroelectric liquid crystals (AFLC) 3 2 5 4 0 0 antiferroelectric phases 335 ff antiferroelectricity 2 f - dielectrics 14 - SmA*-SmC* transitions 13.5 f antipolar interactions, AFLC 325 applications 2 15 ff Arrhenius law I99 axial nearest neighbor [sing model (ANNI) 348,355
bend deformations flexoelectric polarization 93 - singularities 99 - smectic C* phase 299 bend modes, AFLC films 376 bend stiffening, smectic C* phase 296 Bessel differential equation 60 biaxiality - dielectric 255 - flexoelectric polarization 105 -
nonchiralSmC 212 optical 185 f birefringence 3 - deformed helix mode 154 - digitalgrey 269 - electrooptics 182 - nernatics/smectics 255 - nonchevron structures 258 - optical activity 83 bistability - antiferroelectric liquid crystals 336 - chevrons 234 - digitalgrey 263 - smectic elasticity 310 boat domains, chevrons 225,234 Boltzmann distribution, mean field theory 24 bonds, Ising model 356 bookshelf geometry - antiferroelectric liquid crystals 342,369 - biaxiality 188 - devices 215 - future trends 402 - quasisee: quasi-bookshelf structures - smectic elasticity 3 11 Born mean field theory 22 ff Bose-Einstein condensation 3 Bouligand plane 101 boundary conditions - antiferroelectric liquid crystals 342 - SSFLCcell 298 - switching dynamics I77 Bragg reflection 377 branching, AFLC 328,339 Brand-Cladis model 70 Bravais lattices 59 breakdown, dielectric 8, 12 Brewster model 18 brick model, SmA*-SmC* transitions 120 brightness - antiferroelectric liquid crystal displays 383 - surface-stabilized ferroelectric liquid crystals 226 Bruinma-Prost interactions 365 Brunet-Williams distortions 307 building blocks, chirality 73 -
-
CUC2 chevrons 234 f Canon technology 241 ff - digitalgrey 266 - screen prototype 21.5
418
Index
Cantor function, AFLC 3 5 1 cell gap - antiferroelectric liquid crystals 397 - Luderdevices 254 - short pitch materials 154 characteristic length - antiferroelectric liquid crystals 399 - cell thickness 309 - smectic elasticity 3 16 charge annihilation, singularities 101 charge density, polarization 320 charge fluctuation, AFLC films 377 charged singularities 99 chemical properties, nematics/smectics 255 chevrons - nematicslsmectics 256 - antiferroelectric liquid crystals 343 - biaxiality 187 - C1/C2 234f - ferroelectric liquidcrystals 221 f - Sharpdevices 260 - smectic elasticity 3 10 - tilted smected layers 21 I chiral dopants 327 chiral end chains 374 chiral phases, dielectric spectroscopy 199 chirality 2 f, 72 ff - antiferroelectric liquid crystals 327 ff, 367 - electrooptics 169 f - flexoelectric effect 109 - helical smectic C* states 132 - Neumann principle 62 - SmA*-SmC* transitions 118 Chisso 1013 mixture 258 cholesteric materials 253 cholesteric phases, symmetries 8 I cholesteric pitch 255 cholesteric structures, flexooptic effect 101 chronocolor display 245 cinnamates, AFLC 327 Cladis model 70 clamped crystal condition, piezoelectric phases 52 clock models AFLC 358 CMOS VLSI chip 244 coating, IT0 249 coercive fields 14,47 coherence length, smectic elasticity 3 16 Cole-Cole relaxation, dielectric spectroscopy 200 collective modes - antiferroelectric liquid crystals 394 - viscosities 199 color, digital 263-274 color contrast Jee: contrast color filter mosaic 265
color illumination 172 color rendition 265 conductivity, electrooptic 175 cone mode, AFLC 335 cone mode viscosity 178 f - experiments 197 conjugated bonds, polar effects 22 continuous point groups, Curie principle 77 continuum theory - smectic A phase 275 f - smectic C phase 282 contrast - Canondevices 241 - chevrons 234 - Displaytech devices 244 - surface-stabilized ferroelectric liquid crystals 226 contrast enhancement 189 converse piezoeffect 12 coulomb interactions, helical smectic C* states I33 coupling, king model 356 coupling constants - dielectric spectroscopy 200 - Landau expansions 142 - Pikin-Indenbom 166 critical exponents 45 critical points, Landau model 29 crossrubbing, AFLC 342 crystallographic groups, Curie principle 77 Curie law 9,57 ff - macroscopic polarization 76 ff - SmA*-SmC* transitions 118,144 Curie temperature 3 ff, 14 f Curie-Weiss law 17,41 cut off parameters, dielectric spectroscopy 201 cyanobiphenyl compound 8CB, isotropic-nematic transitions 10
dark pixels, AFLC displays 383 dark states, AFLC 342 DC autocompensation, AFLC displays 383 de Gennes model 3 1,93 - smectic A phases 277 de Vries model - antiferroelectric liquid crystals 392 - electroclinic effect 153 - nematicshnectics 256 Debye model 20,40 f Debye viscosity 202 defect free surface, Canon devices 242 defects - antiferroelectric liquid crystals 332 - chevrons 225
Index
- nematicskrnectics 256 - smectic elasticity 3 1 I - surface-stabilized ferroelectric liquid crystals 22 1 deformations - antiferroelectric liquid crystals 395 - chevrons 238 - electroclinic effect 152 - Liiderdevices 254 - nematic states 93 f - piezoelectric phases 52 - SmA*-SmC* transitions 118, 136 - smectic A phase 276 f - smecticcphase 282 defonned helix modes (DHM) - electrooptics 183 - short pitch materials 154 f Denso AFLC display 390 deuteron NMR 335 device structures 215-239 devices 24 1-262 devil’s staircase 35 1 dextro handed helices 73 diamagnetic effect 17 dielectric anisotropy 255 dielectric biaxiality 187 f - nematics/smectics 255 - nonchiralSmC 212 dielectric displacement, electrooptics 172 dielectric effect, models 17 f dielectric permittivity, Landau model 32 dielectric permittivity tensor 206 dielectric spectroscopy 199-21 4 dielectric susceptibility, SmA*-SmC* transitions 143 dielectric tensor, biaxiality 186 dielectrics 7 ff diffusion coefficients, electrooptics 175 digital color 263-274 digital full color display 259 digital grey 263-274 dirnerization, AFLC 325 dipolar cancellation, AFLC 32.5 dipoledipole interactions 5 - antiferroelectric liquid crystals 325,375 - Tsingmodel 35.5 dipoles 2 f - meanfieldtheory 22f direct driving, Displaytech devices 244 director alignement, devices 21 5 director motion equation, solutions 179 director polarization couple, SSFLC 64 directors - 8CB 10 - antiferroelectric liquid crystals 33 1,342
419
chevrons 235 continuum theory 275 - electrooptics 169 f - films 381 - flexoelectric polarization 93 - helical smectic C* states 132 - nonchiralSmC 212 - Pikin-Indenbom order parameters 164 - SmA*-SmC* transitions 123 - smecticC* 70 - smectic elasticity 302 disclinations - antiferroelectric liquid crystals 329,333 - flexoelectric effect 107 - singularities 99 dispersion - biaxiality 187 - rotatory 74 dispiration, AFLC 332 displacements - antiferroelectric liquid crystals 332 - dielectric 7, 172 - smectic Aphase 276 displays - antiferroelectric liquid crystals 335, 383 f - future trends 401 Displaytech screen 215,244 f distortions - electroclinic effect 152 - flexoelectric effect 107 - layers 281 dither methods, digital grey 265 ff DOBAMBC - antiferroelectric liquid crystals films 378 - devices 215 - Meyercompound 75 - SmA*-SmC* transitions 128 - smectic C* states 131 domain growth. controlled 264 domain patterns 4 dopants - antiferroelectric liquid crystals 327 - SmA*-SmC* transitions 126 double helical superstructures, SmC: phases 341 driving modes, Displaytech devices 244 -
-
edge dislocation, smectic A phase 277 Einstein model 59 elastic constants, nematics/smectics 255 elastic deformation 238 elastic energy, sniectic C” phase 295 elastic properties, smectics 275-300 elastic torque, dielectric spectroscopy 205 elastic unwinding, smectic C* 66
420
Index
elasticity, smectic 301-324 electrets 21 electric field behavior, dielectrics 12 f electric field energy, smectic C* phase 299 electroclinic effect 105 - antiferroelectnc liquid crystals 329,392 - measurements 182 - SmA*-SmC* transitions 118,147 f electrooptic measurements 180 ff - viscosities 205 electrooptic response, AFLC displays 385 electrooptical properties - antiferroelectric liquid crystals 339 - chirality 75 - surface-stabilized states 169-198 - thin films 369 electrostatic self interaction, smectic elasticity 3 19 electrostriction 2 f, 12 enantiomers, helical Sm C* states I33 f enantiornorphism 83 f end chains, AFLC 374 energy equation, electric field 299 error diffusion, digital grey 266 Euler angles, dielectric spectroscopy 199 Euler buckling 218 Euler-Lagrange equation 60 - smectic Aphase 279 - smectic elasticity 305
Faraday induction 59 Fermirnodel 60 fermions 3 ferrielectricity 2 f. 14 ferroelectric materials 2 ferroelectric models 40 ff ferroelectricity 1 ff, 14 ferromagnetism 1 ff films - antiferroelectric liquid crystals 369,376 f - Isingmodel 356 flexoelectriccoefficients 94 flexoelectric effects, nonpolarity 12 flexoelectric polarization 93-1 14 flexoelectricity 2 f tlexooptic effect 101 ff fluctuation behavior, gauge invariance 117 fluorination, nonchevron structures 257 forced memorized states, chevrons 239 fragility - antiferroelectric liquid crystal displays 385 - future trends 402 - Luderdevices 247 Frank model 73 - flexoelectric effect 93, 108
Frederiks transition 294 free crystal condition, piezoelectric phases freestanding films - antiferroelectric liquid crystals 369 - Isingmodel 356 freesuspended films, AFLC 376 f Friedel model 18 full color, pixel control 266 full color panel, Toshiba 398 future trends 401
52
gas-liquid-solid phase diagram, Landau model 29 gauge invariance, SmC states 117 geometries - dielectric spectroscopy 209 f see also: bookshelf etc. Ginzburg model 28 glide planes, AFLC 333 Goldstone mode - antiferroelectric liquid crystals 394 - dielectric spectroscopy 199 - future trends 402 - viscosity I78 f, 197 grain size, silver 263 Grandjean model 28 grey, digital 263-274 grey levels - devices 226f - Displaytech devices 244 grey scale - antiferroelectric liquid crystal displays 383 - surface-stabilized ferroelectric liquid crystals 219 grey scale materials, electrooptics 183
half integral disclinations. AFLC 329 ff Hamilton formalism 60 handedness 84 - antiferroelectric liquid crystals 360 - flexoelectric effect I 10 - helical states 136 - SmC: phases 341 hard deformations, smectic C phase 282 hard distortions, flexoelectric effect 107, 1 13 hard twisted helix unwinding 373 Hams dispiration 333 heat capacity, AFLC 340 heat conductivity, electrooptics 175 Heisenberg model 28 helical antiferroelectricity see: helielectricity helical C* states I 15-262 - Landau expansion 156 f
Index helical modes, deformed 105 helical N* phases 81 helical periodicity 154 helical pitch, AFLC 337,363 helical smectic C* phase 13I , 295 - dielectric spectroscopy 206 f helical states, smectic elasticity 306 helices, surface-stabilized states 63 helielectric structures 334 helielectric suhphases 373 helielectricity - dielectrics 14 - surface-stabilized states 65 helix modes - antiferroelectric liquid crystals 358 - deformed 154 herring bone pattern, AFLC 325 Henmann-Mauguin notation 80 Herrmann theorem 79 ff - SmA*-SmC* transitions 122 Hertz induction 59 HF stabilization, hiaxiality I89 high-definition television 245 HOBACPC - antiferroelectric liquid crystals 327 - devices 215 - Kellercompound 75 HOBAMBC, SmA*-SmC* transitions 128 holes, chevrons 234 homeotropic geometry, dielectric spectroscopy 210 Hooke law - electrooptics 194 - piezoelectric phases 52 host materials, SmA*-SmC* transitions 127 hysteresis loop - antiferroelectric liquid crystals 369 - dielectrics 14 - digitalgrey 264 - displays 387
Idemitsu polymer FLC 245 Idemitsu screen 215 improper ferroelectrics 48 ff in-plane switching - antiferroelectnc liquid crystals 391 - electrooptics 170 - future trends 403 inclinated layers - tiltedsmected 211 - wall dimensions 230 Indenbom model 161 f indicatrix, optical 186 infinite point groups, Curie principle 77
interlayer regions, AFLC 374 inverse flexoelectric effect 96 inversion centers, Curie principle king model 349 f, 37 1 f isotropic liquids 2 IT0 coating 249
Jones long pitch method
77
213
Kedge, AFLC 370 Keller compound, HOBACPC 75 Kelvin model 18,62 Khachaturyan state, SmA*-SmC* transitions 139 kmetic coefficients, dielectric spectroscopy 200 kink angle 229
Lagerwall reference frame 286 Lagrange potential 59 lambda-half condition 182 lambda-halfpanels, Liider devices 25 1 Landau expansion 2 ff electrooptics 182 - helical C* states 156 f - polar nematic phases 140 - SmC states I15 f Landau-Khalatnikov equation, dielectric spectroscopy 200 Landau-Lifshitz theory 60 - smectic elasticity 302 Landau model 2 ff, 21,28 ff - antiferroelectric liquid crystals 346 f Landau viscosities 204 Langevin function 8, 17 ff - antiferroelectric liquid crystals 391 - mean field theory 24 latching, smectic elasticity 3 10 lattice breakdown 16 lattice distortions 9 layer bend 276 layer breaking 282 layer components, AFLC films 38 1 layer distortions 281 layer fluctuations 379 layer geometry 21 5-239 layer lunk angle 230 layer shrinkage 23 1 - antiferroelectric liquid crystals 392 - surface-stabilized ferroelectric liquid crystals 220 - antiferroelectric liquid crystals 329 ~
42 1
422
Index
layer thickness nematicslsmectics 255 - surface-stabilized ferroelectric liquid crystals 220 layers - macroscopic polarization 137 - tiltedsmectic 211 LEDs, Displaytech devices 244 Legendre polynomials 23 lengtwwidth ratio, dielectric spectroscopy 202 LET1 technology, digital grey 267 levo handed helices 73 Lifshtz invariants - antiferroelectric liquid crystals 347 - helical C* states 156 - Landaumodel 38 Lifshitz point, surface-stabilized states 65 light intensity measurements 18 1 light transmittance, AFLC 391 limiting factors, future trends 402 line defects, AFLC 332 f linear electrooptic effect 169 ff local layer geometry, FLC 215-239 long pitch method, dielectric spectroscopy 213 long range order, AFLC 348 Lorentz field 27 lozenge loops, chevrons 223,229 -
3-M materials, nonchevron structures 258 Machmodel 77 macroscopic polarization 57-9 1 magnetic liquids 3 magnetic susceptibility - dielectrics 16 - Landaumodel 32 magnetization, order parameters 30 Maier-Saupe theory 28 ff material problems 255 Matsushita screen 215 Maxwell theory 59 - piezoelectric phases 54 McMillan model 5 mean field theory, Born 22 ff measurements - electrooptic 180ff - limitations 213 melting point 3 memory, AFLC displays 384 memory pulse width 262 Merck cholesteric mixture TI827, flexooptic effect 103 f mesoscopic polarization, AFLC 336,369 4-( 1methyl pentyl oxycarbonyl)phenyI-4-nonyloxy cinnamate 327
Meyer compound, DOBAMBC 75 Meyer model 5,63,95 MHDDOPTCOB 370 MHPOBC - antiferroelectric liquid crystals 328 - films 380 MHTAC - antiferroelectric liquidcrystals 330 - films 380 - optical activity 85 Michelson-Morley experiments 59 microscopic models, SmA*-SmC* transitions 125 Miller indices 58 modulations, electrooptic 184 f molecular dimensions, dielectric spectroscopy 202 molecular model, Meyer 95 molecular shapes, SmA*-SmC* transitions 121 monochrome displays 241 monomer FLC mixture 247 monostability, AFLC 336,395 multi color panels 215 myelin sheath, flexoelectric effect 106
N* phases, symmetries 81 N*-C* transitions 5 N-AP-C transitions, Landau model 37 naphthalene 257 nearest neighbor interactions, AFLC 360 Nehring Saupe model, flexoelectric polarization 98 nematic isotropis transitions 27 nematic orders I 1 nematic phases - Landau expansions 140 - Neumann principle 61 nematic smectic A transitions 221 nematic states 93 f nematic viscosity 192 nematics, physical properties 255 Neumann principle 57 ff - biaxiality 186 nonchevron structures, material parameters 257 nonctural helielectrics 135 f nonchiral polar liquid crystals 88 f nonchiral smectic C, dielectric spectroscopy 212 noncollective modes, viscosity 202 nonlinear optics (NLO) - active liquid crystals 88 f - dielectrics 16 nonpolarity 10 ff - dielectrics 7 ff
Index
4-nonyloxybenzylidene-4'-aminopen ty 1 cinnamate (NOBACPC) 327 nucleation centers - chevrons 234 - digital grey 264 Oersted experiments 77 optical activity 83 f - chirality 74 optical anisotropy I85 f optical biaxiality 185f optical efficiency, SSFLC 219 optical indicatrix 186 optical purity, helical smectic C* states 133 optical response 181 optical transmission - antiferroelectric liquid crystals 335 - measurements 181 - SmC*phases 185 optical transmittivity, surface-stabilized states order parameters - Born mean field theory 23 - improper ferroelectrics 48 - Landaumodel 29ff - Pikin-Indenbom 346 - SmC states 1 15 ff Oseen constants - electrooptics 194 - smecticcphase 290 Oseen-Frank theory 275 ff Oseen model 28,40 - flexoelectriceffect 108 - helical C* states 156 10-OTBBB 1M7 370 out-of-layer fluctuations, SmC* states 374
rt electrons, Born mean field theory 22 P-Ecurve, dielectrics 13 f packing effect, molecular model 96 paraelectric materials 17 paraelectric phases 51 paraelectricity 2 f, 14 paramagnetic materials 17 Peierl-Landau instabilities - antiferroelectric liquid crystals films 378 ff smectic Aphase 282 penetration length, smectic A phase 281 periodicity, layer shifts 368 permittivity - biaxiality 187 - dielectric 7ff,32 - tilted smected layers 21 1 permittivity tensor 206 ff ~
69
423
phase angle, surface-stabilized states 63 phase diagram - binary AFLC mixtures 392 - Landaumodel 29 - MHPOBC 339 phase sequences, nematicsismectics 255 phase transition temperatures see; temperatures, Curie temperature, etc. 26 phase transitions 2 ff - antiferroelectric liquid crystals 326 - helical C* states 161 - SmA*-SmC* 115-262 phases 4ff phenylpyrimidine cores 258 Philips texture method 227 physical properties - nematics/smectics 255 - smectics 275-300 piezoeffect, dielectrics 12 piezoelectric analogy, electroclinic effect 152 piezoelectric effect - flexoelectric analogy 97 - smectic A phase 5 piezoelectric phases 5 1 piezoelectricity 2 f - macroscopic polarization 76 ff Pikin-Indenbom description, smectic C* phase 299 Pikin-Indenbom order parameter - antiferroelectric liquid crystals 346 161 f - Sm A*-Sm C* transitions pitch - antiferroelectric liquid clystals 337,363,372 - helical periodicity 154 - nematics/smectics 255 - smectic elasticity 309 pitch cell thicknes ratio, dielectric spectroscopy 213 pixels - antiferroelectric liquid crystal displays 383 - devices 217 - digitalgrey 264 - Displaytech devices 244 planar alignement, dielectric spectroscopy 209 plastic substrate layers. Luder devices 247 Pockels modulators 16 point singularities, AFLC 332 point symmetry. Curie principle 77 polar dielectrics 7 ff polar effects, (historical) models 16 ff polar materials 2,7-55 polar phases, AFLC 369 polar tensors, Curie principle 78 polarization 2 f, 7 f - antiferroelectric liquid crystal films 381
424
Index
- antiferroelectric liquid crystals 325 ff - flexoelectric 93-1 14 - macroscopic 57-91 - nematics/smectics 255 - surface-stabilized ferroelectric liquid crystals 220 polarization reversal current 180 f polarization temperatures 68 polyarylate 249 polyestersulfone 249 poly(ethy1ene terephthalyte) 249 polyimides 249 primary order parameters 48 see also: order parameters pyroelectric materials 9 pyroelectricity 2 f quadratic torques I72 f quadupolar symmetry, SmA*-SmC* transitions 121 quartz 72 quasi-bookshelf structures (QSS) - antiferroelectric liquid crystal displays 385 - Canondevices 242 - chevrons 225 - dielectric spectroscopy 210 - smectic constituents 182 quasi-homeotropic structures, dielectric spectroscopy 2 10
racemic mixtures antiferroelectric liquid crystals 329 - chirality 72 randomization effect, AFLC 393 randomness 3 reciprocal indices, Miller 59 red-green-blue (RGB) color pixels 268 red-green-blue (RGB) filters 265 red-green-blue filter triads 172 redundancy, digital grey 267 reflective displays 250 reference frames, smectic C phase 286 refractive index 185 relaxation - electroclinic effect 151 - walls 231 relaxation spectroscopy, dielectric 199 f reset time, AFLC displays 388 response, optical 181 response time - biaxiality 189 - electroclinic effect 151 - quadratic torques 174 -
Rochelle salt I , 20 rotational modes, viscosity 191 f, 199 f rotator dispersion, optical 74 rubbing direction, Canon devices 243
saturation - antiferroelectric liquid crystal displays 329,385 - dielectriceffect 12 - electroclinic effect 149 f saturation polarization, ferrielectrics 15 Saupemodel 4 - flexoelectric polarization 98 - smectic C phase 282 scaling law, cone mode viscosity 178 f Scherrer model 20 Schiff base 85 Schoenfliess notation 20,84 Schrodinger model 20 screens, FLC applications 215 f screw dislocations - antiferroelectric liquid crystals 333 - smectic A phase 276 second harmonic generation 82 secondary order parameters 48 see ulso: order parameters Sharp video prototype 260 shear 128 shear deformation 254 shear stress, Curie principle 78 short pitch materials - deformed helix mode I54 f - dielectric spectroscopy 213 shrinkage, layers 220,231,392 Shubnikov phases 277 sign conventions, flexoelectric effect 97 singularities - charged 99 - halfintegral 332 - smectic Aphase 276 Sm4-SmC transitions - antiferroelectric liquid crystals 326 ff - biaxiality 187 SmA*-SmC* transitions 1 15-262 - Idemitsu polymer FLC 245 SmC*-SmC: transitions, AFLC 339 smectic A phases - axiality 185 - dielectric spectroscopy 199 - flexoelectric effect 106 - Neumann principle 61 - viscosities 203 smectic C phases - biaxiality 187
Index dielectric spectroscopy I99 - flexoelectric effect 107 - Neumann principle 61 - order parameters I15 ff - quadratic torques 172 f - reference frame 286 - surface-stabilized ferroelectric liquid crystals 220 - viscosity 191 f, 203 smectic C* materials 242 smectic C* phases 4 - helielectric 66 - optical transmission 185 - order parameters 129 f - physical properties 255 smectic elasticity, SSFLC cell 301-324 smectic layer switching 329 smectic 0 phases 328 smectic phases 4 ff smectics, elastic properties 275-300 SmI:, antiferroelectric liquid crystals 330 SmY*, antiferroelectric liquid crystals 329 sodium chlorate, chirality 72 soft deformations, smectic C phase 283 soft distortions, flexoelectric effect 107, 113 soft mode - antiferroelectric liquid crystals 335 - SmA*-SmC* transitions 118 soft mode viscosity - dielectric spectroscopy 199 - electroclinic effect 150 smectic C phases 192 solid state ferroelectrics 1 space-time behavior, electrooptics 176 spacefilling twist-bend structures 136 - smectic C* phase 295 spacers, Canon devices 243 spatial dither, digital grey 265 f splay-bend deformations - antiferroelectric liquid crystals 395 - singularities 99 splay-bend structures - flexoopticeffect 101, 113 - optical activity 86 splay deformations - flexoelectric polarization 93 - singularities 99f - smecticcphase 293 splay modes - antiferroelectric liquid crystals films 376 - smectic C* phase 299 - smectic elasticity 3 11 splay-twist states, chevrons 237 spontaneous domains, surface-stabilized states 67 -
~
425
spontaneous polarization - antiferroelectric liquid crystals 325 - dopants 127 - nematicshmectics 255 stability - chevrons 238 - splay states 314 stabilization, AC/HF 189 stacking, AFLC 375 Stegemeyer model 126 stereospecific centers 86 steric effect, molecular model 96 steric factors, AFLC 375 steric hindrance 360 steric polarity, SmA*-SmC* transitions 138 Stokes law - continuum theory 275 - dielectric spectroscopy 202 strain - electrostrictive 13 - piezoelectric phases 52 stress, Curie principle 77 stress fields, piezoelectric phases 5 1 stress-strain relationship, smectic A phase 278 Stuttgart technology 247 subphases, AFLC 340,369,373 subpixels 217,265 f sugars, chirality 73 sulfur K edge 370 superconductivity 17 superconductors, smectic A phase 277 superstructural chirality 72 superstructures, SmC; phases 341 surface conditions, AFLC 342 surface defects, chevrons 222 surfaceeffects, AFLC 335,398 surface layers, AFLC films 381 surface-stabilized ferroelectric liquid crystal (SSFLC) 67f - boundary conditions 298 - short pitch materials 154 - symmetries 82 surface-stabilized states 63 - antiferroelectric liquid crystals 396 - biaxiality 188 - devices 215,218f - electrooptics 169-198 - nonchiralSmC 212 surface strength, handedness 364 susceptibility - dielectric 7f, 143 - magnetic 16,32 suspended films - antiferroelectric liquid crystals 376 f - Isingmodel 356
426
Index
switching antiferroelectric liquid crystal displays 383 - chevrons 225,234 - electroclinic effect 148, 152 - electrooptics 175f - future trends 402 - in-plane 170 - SmC*phases 206 - smectic elasticity 306 - thresholdless smectic modes 391 switching angle - biaxiality 188 - Canondevices 241 SXGA panels 272 symmetry 2 f - antiferroelectric liquid crystals 392 - biaxiality 186 - Curieprinciple 77 - electrooptics 169f - Neumann principle 61 - Pikin-Indenbom model 162 - surface-stabilized states 68 symmetry change - antiferroelectric liquid crystals 32.5 - Landaumodel 30 - Neumann principles 57 f symmetry groups, SmC states 115 symmetry loss, SmA*-SmC* transitions 122 symmetry paradox, Oersted 77 synclinic effect, AFLC 395 synclinic order 339,373 synclinic tilt 369 synpolar interactions, AFLC 325 -
Taylor series expansion 42 temperature behavior - nematicslsmectics 256 - order parameters 3 1 temperatures - antiferroelectric liquid crystals 344 - Luderdevices 250 - phase transitions 26 - polarization 68 - SmA*-SmC* transitions 146 - SmC states 115 ff - surface-stabilized fenoelectric liquid crystals 220 temporal dither, digital grey 265 f tensile stress, Curie principle 78 tensor components, dielectric spectroscopy 199-2 14 tensor formalism - Curieprinciple 78 - Landau model 33 f
terminal alkyl chains 339 texture method, grey levels 227 TFMHPOBC, antiferroelectric prototype molecule 138 TfT technology 403 thermal fluctuations, AFLC films 376 f thick walls, devices 228 f thickness dependence, AFLC 343,369 thin walls, devices 228 f thiobenzoate 370 thiocompounds 257 thiopene 370 thresholdless smectic modes 390 tilt angles - antiferroelectric liquid crystals 392 - electrooptics 169f - measurements 181 - nematicslsmectics 255 - SmA*-SmC* transitions 130 - smecticcphase 293 - surface-stabilized ferroelectric liquid crystals 220 tilt deformations, SmA*-SmC* transitions 118 tilt fluctuations, dielectric spectroscopy 199 tilt planes, SmA*-SmC* transitions 121 tilt representations, AFLC 3.53 tilted smectic layers - dielectric spectroscopy 21 1 - elasticity 310 tilted smectic phases - antiferroelectric liquid crystals 334 - dielectric spectroscopy 206 - flexoelectric effect 111 - Landaumodel 37 tilted smectic polymers 91 time constants, electroclinic effect 150 time modulation, Displaytech devices 244 time scales, electrooptics 176 tolane compounds 370 torque - antiferroelectric liquid crystals 335,361 - biaxiality 187 - elastic 205 - electroclinic effect 149 - electrooptics 170 - smectic elasticity 309 Toshiba full color panel 398 transition orders, Landau model 36 f transition temperatures, SmA*-SmC* 146 transitions, field induced 15 transmission - antiferroelectric liquid crystal displays 384 - chevrons 234 - digitalgrey 264ff - optical 335
Index surface-stabilized ferroelectric liquid crystals 226 transmissive displays, Luder devices 250 transmittance - antiferroelectric liquid crystals 391 - surface-stabilized ferroelectric liquid crystals 219 transmitting states, electrooptics 170 transmittivity, optical 69 triangular director profiles, nonchiral SmC 2 12 tristable switching - antiferroelectric liquid crystals 337 - surface-stabilized ferroelectric liquid crystals 329 twist-bend structures - helical states 134 - smectic C* phase 295 twist deformations antiferroelectric liquid crystals 395 - flexoelectric polarization 93 - smectic Aphase 277 - smectic C* phase 299 twist grain boundary (TGB) phases 277 twist-splay-bend deformations, smectic elasticity 312 twist states, chevrons 237 twist vectors, SmA*-SmC" transitions 126 twist viscosity - electrooptics 173 - nematicslsmectics 255 twisted smectic states 396 -
~
unwinding aritiferroelectric liquid crystals - smectic elasticity 306 - three layer helix 373 -
344
V shaped switching, AFLC 39 I Valasek model 21 van der Waals isotherms 47 van der Waals model 1 1, 17, 22 f VGA resolution 2 17 viscosity 199-2 14 - cone mode 178 f - nematicskmectics 255 - rotational modes 191 f - soft mode 150 viscous torque antiferroelectric liquid crystals 361 - electrooptics 173 Voigt model 19 ~
wall dimensions, devices 228 f Wang-Taylor model 358 Weiss model 17,24
X-ray scattering
370 f
Zeks-Cepik model 346,366 zero dipole moment, dielectrics 9 zigzag defects - nematicslsmectics 256 chevrons 222f ~
427
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