Extended Poisson–Nernst–Planck modeling of membrane blockage via insoluble reaction products

A generalized time-dependent mathematical model is developed for a diffusion–migration–reaction system incorporating a pore blockage effect due to generation of insoluble precipitates in a porous membrane. The system behavior is investigated via direct numerical solution of an extended, highly non-linear equation set based on the classical Poisson–Nernst–Planck equations for ion transport. In order to treat the buildup of solid reaction products in the membrane, this novel formulation incorporates both a reaction term and a space- and time-dependent diffusivity expression based on a simple precipitation model. The model is demonstrated for a generalized case and then extended to cover the well-known reaction of silver and chloride ions to form insoluble AgCl. Time-dependent concentration profiles of all ions in the membrane are obtained and the effects of precipitate buildup in the pore space are investigated. The role of counterions in the transient behavior of the system is also clarified.

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