An analytical model of tracer diffusivity in colloid suspensions

We present an analytical model of the tracer diffusivity of a colloid particle applicable to a suspension at a density low enough that only pair interactions of the particles need be considered. We assume a pair potential consisting of an infinitely repulsive hard core together with an attractive short range tail as well as hydrodynamic pair interactions which are described by the general mobility series of Schmitz and Felderhof which allow differing hydrodynamic boundary conditions. Using our earlier low density expansion of the generalized Smoluchowski equation we obtain an expression for the tracer diffusivity identical to that obtained by Batchelor from a different derivation. For a class of simple analytic potential functions we show how the tracer diffusivity may be evaluated by quadratures. We present explicit high order mobility series for both stick and slip hydrodynamic boundary conditions and we use these to evaluate the tracer diffusivity numerically for a range of hard core radii. We distinguish short time and long time components of the tracer diffusivity and show that the relative magnitude of these components is quite sensitive to the hydrodynamic boundary conditions.