Cluster expansion for concentration dependence of self-friction coefficients for suspensions of interacting spheres

We present a general cluster theory for the concentration dependence of the translational and rotational self- (tracer) friction coefficients for interacting colloidal spheres in a suspension at vanishing shear rate in the steady state limit. The sphere–sphere interactions enter through the Mayer cluster expansion of the various sphere distribution functions, and the calculation at any order of concentration is reduced to a series of quadratures. The contributions at each order of concentration involve integrals of these equilibrium distribution functions multplied by hydrodynamic functions. The latter are represented in an infinite multiple scattering series which is shown to involve an expansion in higher inverse powers of the interparticle separations and, therefore, to be convergent at low concentrations. This represents the first systematic derivation of the cluster expansion beyond the leading concentration dependence. The general theory is tested by the evaluation of the leading concentration dependent terms which are found to be in good agreement with earlier calculations for translational friction. The concentration dependent rotational friction has not been given previously. Explicit formulas are also presented for the calculation of the leading three-body hydrodynamic contributions.