Van Hove function of colloidal mixtures: Exact results

A general theory for the collective diffusion of a polydisperse colloidal suspension is developed in the framework of the generalized Langevin equation formalism of time-dependent fluctuations. The time-evolution of the intermediate scattering functions, Fαβ(k,t)Fαβ(k,t), is derived as a contraction of the description involving the instantaneous particle number concentration, the particle current, and the kinetic and the configurational components of the stress tensor of the Brownian species as state variables. Analogous results also follow for the self-intermediate scattering functions, Fαs(k,t). We show that neglecting the non-markovian part of the configurational stress tensors memory, one obtains the multicomponent generalization of the single exponential memory approximation (SEXP), based on sum rules derived from the Smoluchowski equation, for both, Fαs(k,t) and Fαβ(k,t)Fαβ(k,t). As an illustrative example, the SEXP is applied to a simple model binary mixture of colloidal particles interacting through repulsive Yukawa pair potentials. The results are compared with Brownian dynamics simulations.