Derivation of Frank-Ericksen elastic coefficients for polydomain nematics from mean-field molecular theory for anisotropic particles.

The complete free energy density, including all eight Frank-Ericksen elastic coefficients and all anisotropic Ericksen-Leslie viscosities of nematic and discotic polydomain nematic liquid crystals are derived from the kinetic model of a spatially inhomogeneous system of uniaxial liquid crystal molecules with given shape. The authors take into account the known anisotropy of the translational diffusion tensor and its dependence on shape, rotational diffusion, and a macroscopic flow field for elongated particles (including disks). In this manuscript they release all of the previously made assumptions about closure relationships or the interrelationship between Frank elastic coefficients (such as a simple quadratic closure, or the one-constant approximation) in order to derive results which not only generalize or improve earlier results, but also apply to more general cases, and for arbitrary forms of the mean-field potential in terms of the scalar order parameter (or temperature). The kinetic model is shown to confirm all proposed inequalities between Frank-Ericksen-Leslie coefficients, i.e., satisfies the main result of the macroscopic approaches. They resolve quantitatively the effect of molecular shape, order parameters, and mean-field strength and form of the mean-field potential on all results, compare with experimental findings, theoretical predictions, and discuss some implications for various special cases of the general result derived in this work.