Continuum Theory of Liquid Crystals

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. 

It is worth remarking that the controversial conflict between the swarm theory and the continuum theory of liquid crystals is illusory. The swarm theory was a particular hypothetical and approximative approach to the statistical mechanical problem of interpreting properties which can be well defined in terms of a continuum theory. This point is seen less clearly from Oseen s point of departure than from that of the present paper. 

CONTINUUM THEORY OF LIQUID CRYSTALS 1.5.1. Order parameter... 

According to the elastic continuum theory of liquid crystals which was introduced in Chapter 1, the three kinds of deformations can be described by three elastic constants, An(splay), / (twist) and / (bend). In the case of small molecular mass liquid crystals, the three constants are mainly determined by the chemical composition of the liquid crystalline molecules. Among them, K22 is the smallest while the other two are approximately close. All three elastic constants are of the order of 10 12 N. The elastic constants of some important liquid crystals are listed in Table 6.1. Each kind of liquid crystals is a mixture of R5-pentyl and R6-hexyl homologues in the ratio of 40 60. The data are obtained at the temperature of T = Tc — 10 °C where Tc is the clear temperature. 

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]. 

I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals, Taylor and Francis, London, 2004. 

The Static and Dynamic Continuum Theory of Liquid Crystals... 

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 above discussion of the continuum theory of liquid crystals is by no means complete. There exist many more effects that can be described by the continuum theory, either in its present or modified form. In view of the diverse applications of the theory, the selection... 

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... 

Stark H (2001) Physics of colloidal dispersions in nematic liquid crystals. Phys Rep 351 387-474 Stewart IW (2004) The static and dynamic continuum theory of liquid crystals a mathranatical introduction. Taylor Francis, London... 

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. 

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