Long-time self-diffusion of rigid rods at low concentrations: A variational approach

No theoretical predictions exist for the concentration dependence of long-t ime self-diffusion coefficients of rod-shaped Brownian particles with a fin ite aspect ratio. The reason for this is that the relevant Smoluchowski equ ation is extremely complicated and cannot be solved explicitly, even on the two-particle level. We present an alternative approach where the Smoluchow ski equation is solved in approximation by a variational method. The variat ional principle is applied to calculate the dependence of the long-time tra nslational self-diffusion coefficient of spherocylinders with hard-core int eraction to leading order in concentration, with the neglect of hydrodynami c interactions, up to aspect ratios of 30. The first order in concentration coefficient alpha is found to depend on the aspect ratio p as alpha = 2 10/32(p - 1) + 1/53(p - 1)(2).