Ornstein–Zernike formalism for reaction rates in random media

Abstract

A useful, general model for the study of diffusion-controlled reactions in random media consists of spherical inclusions of the reactive phase dispersed randomly in a background matrix through which the reagents diffuse. We develop Ornstein–Zernike equations for the material correlation functions of such media. These functions are used both to characterize random media and to determine bounds on the rate of chemical reactions occurring in them. The Ornstein–Zernike equations are solved readily by using standard closures for any degree of correlation among elements of the reactive phase. This allows us to obtain bounds on reaction rates in a large class of random morphologies. In particular, we show that the hypernetted-chain (HNC) closure gives the exact material correlation functions when elements of the reactive phase have uncorrelated positions. On the other hand, the mean-spherical approximation (MSA) gives exact material correlation functions when the reactive phase is dispersed in nonoverlapping inclusions. Our formalism gives the first general method for calculating specific surface and volume fraction in an arbitrarily correlated two-phase medium. The approximations that we have developed for these quantities prove to be essentially exact in the cases for which one has simulation data for comparison and should be highly accurate in the general case.