Elastic continuum theory

ELASTIC CONTINUUM THEORY 3.2.1. The Vector Field Director Axis n (f) 

In elastic continuum theory, introduced and refined over the last several decades by several workers,nematics are basically viewed as crystalline inform. An aligned sample may thus be regarded as a single crystal, in which the molecules are, on the average, aligned along the direction defined by the director axis n (r). 

The crystal is uniaxial and is characterized by a tensoiial order parameter  

Liquid Crystals, Second Edition By lam-Choon Khoo Copyright 2007 John Wiley Sons, Inc. 

The first principle of continuum theory therefore neglects the details of the molecular structures. Liquid crystal molecules are viewed as rigid rods their entire collective behavior may be described in terms of the director axis n (f), a vector field. In this picture the spatial variation of the order parameter is described by 

---

Next, let us compile some quantitative relations which concern the stress field and the energy of dislocations. Using elastic continuum theory and disregarding the dislocation core, the elastic energy, diS, of a screw dislocation per unit length for isotropic crystals is found to be... 

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 displacement field of an edge-dislocation in the SmA phase has been calculated by De Gennes in the framework of the elastic continuum theory... 

The lattice distortion energy, W E ), generated by a solute atom in the bulk can be calculated by using elastic continuum theory as " ... 

The elastic continuum theory is based on the assumption that at each point within the liquid crystal a preferential direction for the molecular orientation is given which is described by a unit vector L, and which varies continuously from place to place — except for a few singular lines or points. Any distortion of the undisturbed state requires a certain amount of energy since elastic torques attempt to maintain the original configuration. The elastic energy density of a deformed nematic liquid crystal is given by... 

Introduction to the Elastic Continuum Theory of Liquid Crystals... 

Eqs. [5] and [6] are the fundamental equations of the elastic continuum theory of nematic and cholesteric liquid crystals (for nematics Xo in Eq. [5] is set equal to ). In the following section we use the fundamental equations to solve four examples as illustrations for their applications. 

It must be stressed that, unlike our expansion of Eq. [30], which is valid only for vanishingly small values of the order parameter, Eqs. [38] and [39] are valid for arbitrary values of the order parameter. In fact, the magnetic and electric energy density expressions of the elastic continuum theory (Eqs. [3] and [4] of Chapter 8) are special cases of Eqs. [38] and [39]. For example, consider the magnetic energy density of the low temperature, anisotropic phase of a system in which only spatial variation of ri(r) is important. S f) in Eq. [38] can then be replaced by its equilibrium value , where < > denotes thermal averaging. The term proportional to H can be neglected because of its spatial invariance, and one obtains Eq. [3] of Chapter 8 directly. 

To characterize the diffraction properties of the LC grating, we estimated the director distribution in the cell based on the elastic continuum theory of nematic LCs. The elastic energy density of a deformed nematic LC, u, is given by... 

Generally speaking, we can divide liquid ciystalline phases into two distinctly different types the ordered and the disordered. For the ordered phase, the theoretical framework invoked for describing the physical properties of liquid crystals is closer in form to that pertaining to solids it is often called elastic continuum theory. In this case various terms and definitions typical of solid materials (e.g., elastic constant, distortion energy, torque, etc.) are commonly used. Nevertheless, the interesting fact about liquid crystals is that in such an ordered phase they still possess many properties typical of liquids. In particular, they flow like liquids and thus require hydrody-namical theories for their complete description. These are explained in further detail in the next chapter. 

---