Long-Time Self-Diffusion of Rigid Rods at Low Concentrations:  A Variational Approach

No theoretical predictions exist for the concentration dependence of long-time self-diffusion coefficients of rod-shaped Brownian particles with a finite aspect ratio. The reason for this is that the relevant Smoluchowski equation is extremely complicated and cannot be solved explicitly, even on the two-particle level. We present an alternative approach where the Smoluchowski equation is solved in approximation by a variational method. The variational principle is applied to calculate the dependence of the long-time translational self-diffusion coefficient of spherocylinders with hard-core interaction to leading order in concentration, with the neglect of hydrodynamic interactions, up to aspect ratios of 30. The first order in concentration coefficient α is found to depend on the aspect ratio p as α = 2 + 10/32(p − 1) + 1/53(p − 1)2.