Nematic liquid crystal distortion free energy

Fgi is the elastic Frank free energy which describes the slowly varying spatial distortions of the director the free energy density f i is a function of the elastic modes of deformation of a nematic liquid crystal and is given by [19,23]... 

The free energy of the distorted nematic liquid crystal can generally be written as a sum of two terms ... 

First let us go back to the same particular case with a constraint llz, and discuss the free energy of a conventional (uniaxial, nonpolar) nematic liquid crystal. We combine elementary distortions corresponding to splay (ai + as), bend ( 3 + ae) and twist ( 2 + ad and present the free energy as a sum of these combinations squared. 

The static free energy F of a nematic liquid crystal undergoing long wavelength distortions in a magnetic field is... 

As in the case of the Frederiks transition (discussed in Section 9.3.1.1 of this Chapter), the theoretical interpretation of the twist effect is based on the minimization of the free energy of the system. In this case, however, the problem is two dimensional, since both the azimuthal angle 0(z) and the tilt angle G(z) are considered to be dependent on the z coordinate. In the case of the infinitely strong anchoring, the threshold for the distortion includes all three elastic moduli of a nematic liquid crystal [111] ... 

In the case of completely asymmetric hypothetical fluid nematic liquid crystals, this means that "only" 45 elastic constants are needed. However, as has been shovm by Frank (see Appendix C.l), in the case of nonchiral and nonpolar nematic liquid crystals of uniaxial symmetry, the distortion free energy, in general, can be written as... 

This alignment displays a certain elasticity and it is known that an initial uniform alignment of a nematic liquid crystal commonly returns after the removal of any disturbing influences. It is therefore assumed that there is a free energy density, also called the free energy integrand, associated with distortions of the anisotropic axis of the form... 

M.G. Clark, Algebraic derivation of the free-energy of a distorted nematic liquid crystal. Molecular Physics, 31, 1287-1289 (1976). 

For each in a uniaxial phase there are two normal modes corresponding to a splay-bend distortion n q) and a twist-bend distortion ri2(q) biaxial liquid crystal phases have five normal modes for each value of q. The free energy density can be written in terms of the normal coordinates for torsional displacement in a uniaxial nematic as ... 

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