Continuum Theory for Liquid Crystals

Leslie F M 1998 Continuum theory for liquid crystals Handbook of Liquid Crystals Vol 1. Fundamentals ed D Demus, J Goodby, G W Gray, Fl-W Speiss and V Vill (New York Wiley-VCFI)... 

The presently accepted continuum theory for liquid crystals has its origins going back to at least the work of Oseen [214, 215], from 1925 onwards, and Zocher [286] in 1927. Oseen derived a static version of the continuum theory for nematics which was to be of instrumental importance, especially when the static theory was further developed and formulated more directly by Frank [91] in 1958. This static theory, introduced in Chapter 2, is based upon the director n and its possible distortions. 

The above comments represent only some particularly chosen topics and events in the mathematical development of the continuum theory for liquid crystals, focusing especially on nematic and smectic C materials. In a book of this scope it is inevitable that some major topics have not been included, such as soliton-like behaviour in nematics or the flexoelectric effect, and many important contributions to the field have been omitted for the sake of brevity. Nevertheless, readers should have no difficulty in accessing these topics in the current literature if they have been armed with the material presented in subsequent Chapters. Interested readers can find more extensive details on the development of liquid crystals in the historical review by Kelker [143] or the forthcoming volume in this book series by Sluckin. Dunmur and Stegemeyer [253]. 

F.M. Leslie, Continuum Theory for Liquid Crystals, in Handbook of Liquid Crystals, Volume 1, D. Demus, J. Goodby, G.W. Gray, H.-W. Spiess and V. Vill (Eds.), 25-39, Wiley-VCH, Weinheim, Germany, 1998. 

Of all these contributions, many seminal, that by Sir Charles Frank clearly stands out. This unique paper was presented at the 1958 Discussion, entitled Configurations and Interactions of Macromolecules and Liquid Crystals, and it earned him his enviable international reputation in the field. The paper describes what is known as the continuum theory of liquid crystals and is now widely used to design and optimise display devices for the LCD industry through the prediction of the macroscopic organisation of the liquid crystal when subject to competing external constraints, such as surface interactions and electric fields. 

The static continuum theory of elasticity for nematic liquid crystals has been developed by Oseen, Ericksen, Frank and others [4]. It was Oseen who introduced the concept of the vector field of the director into the physics of liquid crystals and found that a nematic is completely described by four moduli of elasticity Kn, K22, K33, and K24 [4,5] that will be discussed below. Ericksen was the first who understood the importance of asymmetry of the stress tensor for the hydrostatics of nematic liquid crystals [6] and developed the theoretical basis for the general continuum theory of liquid crystals based on conservation equations for mass, linear and angular momentum. Later the dynamic approach was further developed by Leslie (Chapter 9) and nowadays the continuum theory of liquid crystal is called Ericksen-Leslie theory. As to Frank, he presented a very clear description of the hydrostatic part of the problem and made a great contribution to the theory of defects. In this Chapter we shall discuss elastic properties of nematics based on the most popular version of Frank [7]. 

One optical feature of helicoidal structures is the ability to rotate the plane of incident polarized light. Since most of the characteristic optical properties of chiral liquid crystals result from the helicoidal structure, it is necessary to understand the origin of the chiral interactions responsible for the twisted structures. The continuum theory of liquid crystals is based on the Frank-Oseen approach to curvature elasticity in anisotropic fluids. It is assumed that the free energy is a quadratic function of curvature elastic strain, and for positive elastic constants the equilibrium state in the absence of surface or external forces is one of zero deformation with a uniform, parallel director. If a term linear in the twist strain is permitted, then spontaneously twisted structures can result, characterized by a pitch p, or wave-vector q=27tp i, where i is the axis of the helicoidal structure. For the simplest case of a nematic, the twist elastic free energy density can be written as ... 

In the continuum theory of liquid crystals, the free energy density (per unit volume) is derived for infinitesimal elastic deformations of the continuum and characterized by changes in the director. To do this we introduce a local right-handed coordinate x, y, z) system with (z) at the origin parallel to the unit vector n (r) and x and y at right angles to each other in a plane perpendicular to z. We may then expand n (r) in a Taylor series in powers of x, y, z, such that the infinitesimal change in n (r) varies only slowly with position. In which case... 

The committee set up a liquid crystal symposium in 1975 as an annual meeting to provide a forum for exchanges among young liquid crystal researchers. The 1st Liquid Crystal Symposium was held at Kyushu Uruversity in the aummn of 1975, with the Chemical Society of Japan as a cosponsor. There were three special lectures Liquid Crystal Research for Chemistry, by Professor Narikazu Kusabayashi of Yamaguchi University Physical Phenomenon of the Liquid Crystal for the Application Side, by Professor Masanobu Wada of Tohoku University and Continuum Theory of Liquid Crystal, by Professor Kohji Okano of... 

The aim of this book is to present a mathematical introduction to the static and dynamic continuum theory of liquid crystals. Before doing so, we outline some points on the discovery and basic description of liquid crystals in Sections 1.1 and 1.2. This is followed by a short summary of the development of the continuum theory of liquid crystals in Section 1.3. The Chapter closes in Section 1.4 with some basic comments on the notation and conventions employed in later Chapters and refers to some sources for those who may require further background on some of these conventions used throughout this book. 

This book grew out of a perceived need for a text specifically aimed at applied mathematics, theoretical physics and engineering graduates who wish to obtain some basic grounding in the static and dynamic continuum theory of liquid crystals. It is hoped that beginners and more seasoned readers will benefit from the topics to be raised and discussed throughout the book. 

In microscale models the explicit chain nature has generally been integrated out completely. Polymers are often described by variants of models, which were primarily developed for small molecular weight materials. Examples include the Avrami model of crystallization,- and the director model for liquid crystal polymer texture. Polymeric characteristics appear via the values of certain constants, i.e. different Frank elastic constant for liquid crystal polymers rather than via explicit chain simulations. While models such as the liquid crystal director model are based on continuum theory, they typically capture spatiotemporal interactions, which demand modelling on a very fine scale to capture the essential effects. It is not always clearly defined over which range of scales this approach can be applied. 

In most practical circumstances, however, this ideal conformation is not compatible with the constraints imposed by the walls of the container and by external forces such as electric and magnetic fields. These constraints cause some modifications of both n and S at each point. The modifications of n take place over macroscopic distances (typically a few microns) and are, thus, easily observed optically. From a more theoretical point of view, these macroscopic distortions can be described by a continuum theory (the analogue for liquid crystals of classical elasticity for solids). The modifications of S do not persist over long distance and cannot be detected optically [2,13]. 

An important aspect of the macroscopic structure of liquid crystals is their mechanical stability, which is described in terms of elastic properties. In the absence of flow, ordinary liquids cannot support a shear stress, while solids will support compressional, shear and torsional stresses. As might be expected the elastic properties of liquid crystals are intermediate between those of liquids and solids, and depend on the symmetry and phase type. Thus smectic phases with translational order in one direction will have elastic properties similar to those of a solid along that direction, and as the translational order of mesophases increases, so their mechanical properties become more solid-like. The development of the so-called continuum theory for nematic liquid crystals is recorded in a number of publications by Oseen [ 1 ], Frank [2], de Gennes and Frost [3] and Vertogen and de Jeu [4] extensions of the theory to smectic [5] and columnar phases [6] have also been developed. In this section it is intended to give an introduction to elasticity that we hope will make more detailed accounts accessible the importance of elastic properties in determining the... 

Continuum theory for nematic liquid crystals has its origins in the 1920s in the work of Oseen [1] and Zocher [2], who largely developed the static theory. The first to attempt the formulation of a dynamic theory was Anzelius [3], who was a student of Oseen, but an acceptable version had to await developments in non-linear continuum mechanics many years later, as well as further experimental studies by Zwetkoff [4] and Miesowicz [5]. A full account of the early development of dynamic theory for nematics can be found in a paper by Carlsson and Leslie [6]. 

G. Barbero and L.R. Evangelista, An Elementary Course on the Continuum Theory for Nematic Liquid Crystals, World Scientific, Singapore, 2001. 

F.M. Leslie, Continuum theory for nematic liquid crystals, Continuum Mech, Thermodyn., 4, 167-175 (1992). 

F.M. Leslie, J.S. Laverty and T. Carlsson, Continuum Theory for Biaxial Nematic Liquid Crystals, Q. Jl. Mech. Appl. Math. 45, 595-606 (1992). 

F.M. Leslie, Some flow effects in continuum theory for smectic liquid crystals, Liq. Cryst, 14, 121-130 (1993). 

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