Flux Theorems for Linear Multicomponent Diffusion
Alan McNabb, L. Bass · IMA Journal of Applied Mathematics · 1990
Sten-Knudsen & Ussing (1981) and Bass et al . (1986) have shown that the flux of matter through membranes is proportional at all corresponding times to the flux in the opposite direction when appropriate boundary conditions are reversed, even if the diffusion-migration parameters and the trapping characteristics of the material vary arbitrarily with distance normal to the membrane surface. In this paper, we extend this flux-ratio theorem to some cases where the diffusing component is interacting with other diffusing species in the membrane. When the equations are assumed to be linear, as for tracer experiments, the amount of any component transmitted though any part of the boundary over the duration of an experiment is determined by the time integral of the given boundary conditions and is independent of pulse shape. If the interaction matrix A coupling the transport equations of the components is quasi-symmetric in the sense that it is the product of a symmetric and a diagonal matrix, the flux-ratio theorem remains valid. Interactions dominated by a single chemical reaction give rise to quasi-symmetric matrices of this form. An example involving three components illustrates the structure. It is shown that interaction parameters such as absolute reaction rates can be deduced for experiments based on these results.