Non-Equilibrium Statistical Mechanics of Nematic Liquids

Abstract

The rotational diffusion of a general-shape object (a molecule) in a flow of uniaxial nematic liquid crystal is considered in the molecular field approximation. The full corresponding Fokker-Planck equation is derived, and then reduced to the limit of diffusion of orientational coordinates in a field of uniaxial nematic potential and the flow gradient. The spectrum of orientational relaxation times follows from this analysis. As a second main theme of this work, we derive a complete form of microscopic stress tensor for this molecule from the first principles of momentum balance. Averaging this microscopic stress with the non-equilibrium probability distribution of orientational coordinates produces the anisotropic part of the continuum Leslie-Ericksen viscous stress tensor and the set of viscous coefficients, expressed in terms of molecular parameters, nematic order and temperature. The axially-symmetric limits of long-rod and thin-disk molecular shapes allow comparisons with existing theories and experiments on discotic viscosity. The article concludes with more complicated aspects of non-linear constitutive equations, microscopic theory of rotational friction and the case of non-uniform flow and director gradients.

This work has been partially funded by the Institute for Mathematics and its Applications (IMA), University of Minnesota, and the Cambridge Commonwealth Trust.