Statistical mechanics of viscous flow in nematic fluids

Abstract

We derive Green–Kubo (GK) relations for the viscosity coefficients of nematic liquid crystals. These GK relations are similar to, but considerably more complicated than, those of an isotropic fluid. In addition to shear viscosities there are also twist viscosities and cross couplings between the symmetric strain rate and the antisymmetric pressure tensor and vice versa. We show that the twist viscosity is inversely proportional to the mean square displacement of the director. Using the so-called SLLOD equations of motion we construct nonequilibrium molecular dynamics (NEMD) algorithms that can be used to efficiently calculate the viscosity coefficients of nematic liquid crystals from atomistic computer simulations. We also devise an additional NEMD algorithm for controlling the angular velocity of the director in a nematic fluid. We derive a fluctuation relation for the alignment angle between the director and the streamlines in planar Couette flow and also for the shear induced molecular angular velocity. In an isotropic fluid, close to equilibrium, this angular velocity is equal to half the vorticity. In a nematic liquid crystal it is nearly zero because of cross couplings with the symmetric part of the strain rate tensor. We test the Green–Kubo relations and the NEMD algorithms in a nematic liquid crystal modeled using a modified version of the Gay–Berne potential. In general, the Green–Kubo and NEMD results agree very well.