Nonlinear dynamics of ion concentration polarization in porous media: The leaky membrane model

Abstract

The conductivity of highly charged membranes is nearly constant, due to counterions screening pore surfaces. Weakly charged porous media, or “leaky membranes,” also contain a significant concentration of coions, whose depletion at high current leads to ion concentration polarization and conductivity shock waves. To describe these nonlinear phenomena in the absence of electro-osmotic flow, a simple leaky membrane model is formulated, based on macroscopic electroneutrality and Nernst–Planck ionic fluxes. The model is solved in cases of unsupported binary electrolytes: steady conduction from a reservoir to a cation-selective surface, transient response to a current step, steady conduction to a flow-through porous electrode, and steady conduction between cation-selective surfaces in cross flow. The last problem is motivated by separations in leaky membranes, such as shock electrodialysis. The article begins with a tribute to Neal Amundson, whose pioneering work on shock waves in chromatography involved similar mathematics.