"Concentrated capacity" model of ion-exchange funneling in a modified (thin film coated) heterogeneous electrodialysis membrane

Inexpensive highly permselective heterogeneous ion exchange membranes are prohibitively polarizable by a direct electric current for use in electrodialysis. According to recent experiments, polarizability of these membranes may be considerably reduced by casting on their surface a thin layer of cross-linked polyelectrolyte, weakly charged with the same sign as the membrane's charge. The present paper is concerned with this effect. In order to explain this feature, a simple limiting ion-exchange 'funnel' model of a modified membrane is derived from the original two-layer model. In this model, asymptotically valid for a thin coating, solution of the ionic transport equations in it is replaced, via a suitable averaging procedure, by a single nonlinear boundary condition for the membrane/solution interface, which itself has the same order as the bulk equation. Rigorous analysis of the 'funnel' model shows that the value of the limiting current through a modified membrane, which is the main quantitative characteristic of its polarizability, is equal to that through a homogeneous membrane for any positive value of the funneling parameter. The limiting problem for a vanishing funneling parameter is analyzed.