Concentration Polarization and Nonequilibrium Electroosmotic Instability At An Ion-Selective Surface Admitting Transverse Flow

The passage of ionic current through a charge-selective surface is a classic problem that has witnessed a recent resurgence of interest, motivated by experiments on electrokinetic transport in micro- and nano-fluidic devices. In certain cases, a charge-selective surface can also accommodate a fluid flow across itself. Here, we analyze a model system that elucidates the influence of such "transverse flows" on concentration polarization (i.e. ionic concentration gradients) at an ion-selective surface. Specifically, we consider a cation-selective surface, or membrane, that admits a uniform transverse flow. The membrane contacts an electrolyte, whose concentration is uniform in a "well-mixed" region at a prescribed distance from the membrane. A potential difference across the system drives an ionic current, leading to concentration polarization in the "unstirred layer" between the membrane and well-mixed bulk. The concentration polarization reflects a balance between advection of ions with the transverse flow and diffusion. A Peclet number, Pe, parameterizes the relative importance of these effects; notably, Pe is signed, as the flow can be toward or away from the membrane. An asymptotic analysis for thin Debye layers reveals a nonlinear concentration profile for non-zero Pe, in contrast to the familiar linear (diffusive) profile at Pe=0. Next, we explore the structure of the unstirred layer at over-limiting currents, wherein a non-equilibrium space-charge layer emerges near the membrane surface. Finally, we examine the instability of the quiescent concentration polarization due to second-kind electro-osmotic flow in the space-charge layer. A linear stability analysis shows that transverse flow can enhance or retard the instability, depending on its direction.