The second order concentration corrections to the mutual diffusion coefficient of Brownian macroparticles

Abstract

From the initial time dependence of the dynamic structure factor S(k,t), we obtain a general form, valid through second order in the concentration, for the mutual diffusion coefficient Dm. All effects of direct and hydrodynamic interactions (other than dynamic friction) are taken into account, including the three-point ’’0seen’’ hydrodynamic tensor Tijl, whose analytic form is obtained here for the first time. The concept of the reference frame correction is re-examined. The usual factor (1−φ) for transition from the fundamental hydrodynamic frame to the volume-fixed frame is argued to be a low-concentration approximation. The general form for Dm is evaluated for a model system of hard spheres. With the use of the auxiliary assumption (relaxed herein) that all hydrodynamic interaction tensors satisfy ∇ ⋅ t = 0, use of our new general method [J. Stat. Phys. 28, 673 (1982)] for reducing N-particle cluster integrals to (N−1)-dimensional integrals shows Dm = D0 (1−0.875φ−19.53φ2).