Theoretical analysis of the Frank-Oseen equation for nonpolar liquid crystals

The article compares two main theoretical approaches in deriving the Frank-Oseen equation for the elastic free energy density of a liquid crystal. The methods under consideration have its roots in the classical works of Frank, Oseen, Nehring and Saupe and are based on the use of the properties of rotational symmetry of non-polar crystal structures with a preferred symmetry axis. Despite the fact that both methods are based on the same symmetry principles, one of them is essentially a microscopic approach to describing the deformation of the director of fixed molecules. At the same time, another method is a macroscopic approach to analyzing the transformations of various components of the elasticity tensor. Nevertheless, it has shown that both approaches lead to consistent results in describing the elastic energy of liquid crystals, which consists in its dependence on four mechanisms of elastic deformations - transverse and longitudinal bending, torsion and saddle-splay deformation.