Memory function for collective diffusion of interacting Brownian particles

– The problem of memory contribution to the collective diffusion coefficient of interacting Brownian particles is considered. A well-defined theoretical expression for this contribution, free of divergent integrals, is derived. Its value is then estimated for hard sphere suspensions numerically by means of extensive computer simulations. A good physical model for a broad class of suspensions is the system of interacting Brownian particles. Despite a great number of important contributions to the field (see [1] for a review), there are still many areas in which theoretical results are scarce. One of these is the problem of memory contribution to the collective-diffusion coefficient in colloidal suspensions. We consider N identical spherical particles performing Brownian motion in an incompressible viscous fluid at temperature T . On the time scale characteristic for light scattering experiments the evolution of the configuration space distribution function P (X, t) is described by the generalized Smoluchowski equation [1] ∂ ∂t P (X, t) = D(X, t)P (X, t), D(X, t) ≡ N ∑ i,j=1 ∂ ∂Ri ·Dij(X) · [ ∂ ∂Rj + β ∂Φ(X) ∂Ri ] , (1) where X = (R1,R2 . . . ,RN ), Ri being the position of the i-th particle and β = 1/kBT . The potential Φ(X) incorporates both an external force field and direct pair interactions. Next, D(X) is the diffusion matrix connected with the mobility matrix μ by the generalized Einstein relation Dij = kBTμij . According to the definition of the mobility matrix, the contribution of force F j acting on particle j to the velocity of particle i is given by μijF j . In general, due to hydrodynamic interactions, μ depends on the configuration X and is non-diagonal in particle indices. Discussions of topics related to the mobility matrix can be found in the monograph [2].