Frank energy

At this stage, two moduli Ki and Kg remain finite, for a spontaneous splay and twist, respectively virtual polar cholesteric). 

However, when molecules have mirror symmetry (achiral molecules), the inversion center appears and g ist becomes invariant with respect to the following frame transformation x = —x, y = —y, z = —z. As a result, modulus Kg vanishes and we have a tensor corresponding to a polar nematic  

taking the head-to-tail symmetry into account we make the following transformation of the frame x = x, y = y, z = —z. Now coefficients K12 disappear. At this atage, only four different moduli are left, Kn, K22, 33, K24. 

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. 

we would like to write the same in a more compact vector form. To do this, consider vector forms for each of the three contributions. 

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If molecules are chiral, the coefficient K2 from tensor (8.12) becomes finite. Formally it is possible to add it to the Frank energy introducing a scalar quantity = K2IK22 and obtain the following expression ... 

From here and Eq. (8.17), the density dist and the total Frank energy are given by... 

Mathematically the molecular field vector h can be found using the Euler-Lagrange approach by a variation of the elastic and magnetic (or electric) parts of the free energy with respect to the director variable n(r) (with a constraint of = 1). For the elastic torque, in the absence of the external field, the splay, twist and bend terms of h are obtained [9] fi-om the Frank energy (8.16) ... 

Here, g is Frank energy density, F 2 are surface energies at opposite boundaries. Our task is to find the equilibrium alignment of the director everywhere between and at the solid surfaces. It is determined by minimization of the integral equation (10.19), i.e. by solution of the correspondent differential Euler equation for the bulk... 

In the case where the transverse dimension of the beam is smaller than or on the order of the cuvette thickness X, the Frank energy due to the transverse gradients of the director becomes dominant. Let us first estimate the order of magnitude of the threshold power of the Fredericks transition. The energy of the perturbed state is... 

The free energy of nematic gels under electric fields may be written as the sum of the Frank energy (Fp), the electrostatic energy (Fed, and the gel elasticity energy (Fg) ... 

Frank energy, is n curl n (=-1kIP in the example of a simple cholesteric helix). A natural extension of the Frank energy density to chiral nematics is [1, 59] ... 

In a SmC phase, the layer normal N no longer coincides with the director field n. Non-zero twist and bend (i.e. curl n 0) are permitted. The well known structure of the director helix is n(r) = (sin0cos 2 rz/P, sin d sin 2 KzIP, cos 6) for a layer normal along z and a SmC tilt 0(N n=cos 0). The twist and bend terms of the Frank energy are ... 

For small dielectric anisotropy ( 0) the free energy density includes only two terms the flexoelectric and the elastic terms derived from the Frank energy (Eq. (24))... 

We have seen that the soft elastic energy of a smectic C material can be derived from the Oseen-Frank energy of nemahc liquid crystals (see (4.56)). 

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