Diffusion of interacting Brownian particles in a fluid with fixed macroparticles

Abstract

The diffusion of interacting Brownian particles in a fluid with an added macroparticle species is discussed. The added particles are fixed on the time scale of the Brownian motion. The effective (collective) diffusion coefficient De of the Brownian particles, characterizing relaxation of Brownian number density fluctuations, is calculated to linear order in Brownian concentration. We construct the friction tensor ζ(r) describing Brownian pair interactions by considering the reflections of the fluid from the Brownian and fixed particles, and averaging over fixed particle configurations. We find that a naive expansion in fixed particle concentration na leads to divergent integrals. Introducing a diagrammatic representation, we sum to all orders in na and obtain convergent results, with ζ(r) depending on na in a nonanalytic fashion. This ζ(r) is inverted and used in a Smoluchowski equation to find De. To lowest order, we find that the modification to De from the fixed particles enters as √na. Measurement of De by quasielastic light scattering techniques should confirm this behavior.