A stochastic theory of non-ideal behaviour

Abstract

Short range interactions in multicomponent systems are studied with the help of a Glauber-like stochastic model of intermolecular exchanges. The transition probability of a molecule depends on its local state (i.e. the instantaneous shell of interacting neighbours) and except in the simplest case, the process has to be evaluated with the help of computer simulation. In order to arrive at an analytical formulation we make use of the quasi-chemical (QC) approximation, which permits us to reproduce the QC estimation of the chemical potential difference μ_AB in a binary [A, B] system at equilibrium, as a function of the composition and pair interaction parameter θ. To simplify the stochastic model we introduce an additional approximation: the various local states are combined into microenvironments, one for each molecular species. We show that the QC estimation of μ_AB is not materially affected by the (additional) microenvironment approximation.

Since our calculation is based on a stochastic exchange it is applied to diffusion in an isothermal condensed medium. The diffusion is not attributed to a precise mechanism of molecular exchange between specified locations, but, to exchanges with a kind of semi-macroscopic reservoir. A proper definition of the transition probabilities ensures that the gain of molecules at one place implies their loss at another. A straightforward calculation gives the species' flow as a function of gradients of chemical potential, permitting one to derive the dependence of the mutual diffusion coefficient on the composition and on θ. Previous models of diffusion which do specify a precise mechanism of exchange between pairs of neighbour lattice sites are considered. Their results agree well with ours, showing the relative unimportance of the exchange mechanism in treating the effect of interactions.