  The conductivity of highly charged membranes is nearly constant, due to
counter-ions screening pore surfaces. Weakly charged porous media, or "leaky
membranes", also contain a significant concentration of co-ions, whose
depletion at high current leads to ion concentration polarization and
conductivity shock waves. To describe these nonlinear phenomena 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.
