Study of phase transition in homogeneous, rigid extended nematics and magnetic suspensions using an order-reduction method

Abstract

We study the phase transition in rigid extended nematics and magnetic suspensions by solving the Smoluchowski equation for magnetically polarized rigid nematic polymers and suspensions in equilibrium, in which the molecular interaction is modeled by a dipolar and excluded volume potential. The equilibrium solution (or the probability distribution of the molecular distribution) is given by a Boltzmann distribution parametrized by the (first-order) polarity vector and the (second-order) nematic order tensor along with material parameters. We show that the polarity vector coincides with one of the principal axes of the nematic order tensor so that the equilibrium distribution can be reduced to a Boltzmann distribution parametrized by three scalar order parameters, i.e., a polar order parameter and two nematic order parameters, governed by three nonlinear algebraic-integral equations. This reduction in the degree of freedom from 8 (3 in the polarity vector and 5 in the nematic order tensor) to 3 significantly simplifies the solution procedure and allows one to conduct a complete analysis on bifurcation diagrams of the order parameters with respect to the material parameters. The stability of the equilibria is inferred from the second variation of the free energy density.