Statistical–mechanical theory of membrane transport for multicomponent systems: Passive transport through open membranes

Abstract

Equations for transport through membranes are derived from basic principles of statistical mechanics, with the classical–mechanical Liouville equation as the starting point. General fluid-dynamic equations are first obtained, following the methods of Bearman and Kirkwood and of Snell, Aranow, and Spangler; a quasicontinuum or coarse-graining assumption then allows these equations to be inverted to a Stefan–Maxwell form in which the diffusion and thermal diffusion coefficients can be given an experimental interpretation. The remaining assumptions pertain specifically to membranes. The membrane is taken as one component of the mixture, held fixed in space, and the structure of the membrane is assumed to lead only to a geometrical space-filling role. Finally, the assumption of nonseparative viscous flow allows the transport equations to be decoupled from the details of membrane structure. The results have the same final form as those obtained earlier by heuristic generalization of gas transport equations. Comparison is made with previously proposed phenomenological equations for membrane transport, including the Onsager (Kedem–Katchalsky) linear laws, the frictional model, the Stefan–Maxwell diffusion model, a hydrodynamic model, and Nernst–Planck equations. It is shown in what sense these different formulations are equivalent, where terms have been omitted, and how such omission can be compensated by reinterpretation of the transport coefficients. Effects of membrane heterogeneity and heteroporosity are briefly discussed.