Torsional elasticity

In nematic liquid crystals we see a novel feamre. There is no shear modulus as in isotropic liquids, but the orientational, for example, torsional elasticity appears. Such elasticity is also characteristic of crystals but, in that case, the corresponding moduli are much smaller than the other moduli. The orientational elasticity determines almost all fascinating properties and applications of nematics. 

The presence in the system of inductive (rotational kinetic) energy and of capacitive (torsion) energy is made effective by the existence of the two systan constitutive properties, rotational inertia (or moment of inertia, also notated I) and torsion elastance (or more classically torsion constant ) ... 

Elasticity is a macroscopic property of matter defined as the ratio of an applied static stress (force per unit area) to the strain or deformation produced in the material the dynamic response of a material to stress is determined by its viscosity. In this section we give a simplified formulation of the theory of torsional elasticity and how it applies to liquid crystals. The elastic properties of liquid crystals are perhaps their most characteristic feature, since the response to torsional stress is directly related to the orientational anisotropy of the material. An important aspect of elastic properties is that they depend on intermolecular interactions, and for liquid crystals the elastic constants depend on the two fundamental structural features of these mesophases anisotropy and orientational order. The dependence of torsional elastic constants on intermolecular interactions is explained, and some models which enable elastic constants to be related to molecular properties are described. The important area of field-induced elastic deformations is introduced, since these are the basis for most electro-optic liquid crystal display devices. 

The condensed notation for the elements of the torsional elasticity tensor is normally used, and the torsional strain elements are written as a column vector with the compo-... 

The strain tensor must conform to the symmetry of the liquid crystal phase, and as a result, for nonpolar, nonchiral uniaxial phases there are ten nonzero components of kij, of which four are independent ( i i, 22> A 33 and 24)- These material constants are known as torsional elastic constants for splay (k, 1), twist ( 22) bend ( 33) and saddle-splay ( 24) terms in 24 do not contribute to the free energy for configurations in which the director is constant within a plane, or parallel to a plane. The simplest torsional strains considered for liquid crystals are one dimensional, and so neglect of 24 is reasonable, but for more complex director configurations and at surfaces, k24 can contribute to the free energy [7]. In particular 24 is important for curved interfaces of liquid crystals, and so must be included in the description of lyotropic and membrane liquid crystals [8]. Evaluation of Eq. (16) making the stated assumptions, leads to [9] ... 

Torsional distortions can now be written in terms of derivatives of a and c, and it is found [10] that nine torsional elastic constants are required for the smectic C phase. Mention should be made of the biaxial smectic C phase, which has a twist axis along the normal to the smectic layers. This helix is associated with a twist in the c-di-rector, and so elastic strain energy associated with this can be described by terms similar to those evaluated for the chiral nematic phase. 

In two dimensions and setting all the torsional elastic constants equal, the free energy density expression Eq. (17) can be written as ... 

Because the torsional elastic constants are small, the terms in the expression for the free energy density are treated classically so that... 

Torsional elasticity is of special interest from a microscopic viewpoint since it is a property characteristic of liquid crystals, which distinguishes them from ordinary liquids. The elastic properties contribute to many physical phenomena observed for liquid crystals, and a molecular theory of torsional elasticity should enable the identification of particular molecular properties responsible for many aspects of liquid crystalline behavior. 

The two terms in the square brackets of Eq. (94) can be identified as a temperature independent internal energy term, and a temperature dependent entropy term resulting from the hard particle pair distribution function. From this equation it can be seen that the calculation of the principal elastic constants of a nematic liquid crystal depends on the first and second derivatives with respect to the angle 9 of the single particle orientational distribution function. Any appropriate angular function may be used for/(f2i, 9(R)), but the usual approach is to use an expansion in terms of spherical harmonics. The necessary mathematical manipulations are complicated, but give relatively compact results. Thus the ingredients of a molecular calculation of torsional elastic constants within the van der Waals... 

For the simplest distribution function, only the term involving the second derivative in Eq. (94) is nonzero, and the torsional elastic constants are given by an average over the square of the intermolecular distances x, y and z. Since macroscopic uniaxiality is assumed, the averages over x and y, perpendicular to the undisturbed director, will be equal, with the result [43] ... 

Various authors [46, 47] have reported calculations of torsional elastic constants for hard spherocylinders. These neglect any attractive interactions, and so only give en-tropic contributions to the elastic free energy their results can be summarized as ... 

---