Effective diffusivity for a mixed-matrix membrane

The effective diffusivity or permeability of a mixed-matrix membrane composed of spherical fillers of one material distributed within a continuous matrix of another material is computed by a boundary-element method. The boundary value of the concentration of the diffusing species on the surface of the fillers is proportional to that of the adjoining matrix phase according to a linear sorption/desorption kinetics law that is responsible for an interfacial partition coefficient, while the diffusive flux is continuous across the interface. The solution of Laplace’s equation for the concentration field is constructed in terms of the single- and double-layer harmonic potentials involving the boundary values and boundary distribution of the normal derivative of the concentration and the free-space Green’s function of Laplace’s equation in three dimensions according to the standard boundary-integral formulation. The effective permeability of the membrane depends on the ratio of the filler-to-matrix permeabilities and not separately on the diffusivity ratio and partition coefficient. The results are in excellent agreement with the predictions of existing analytical models for random and ordered distributions of spherical particles within an infinite, continuous medium in the entire range of filler volume fractions considered. Extensions to systems governed by an interfacial contact resistance and nonlinear interfacial kinetics are discussed.