Concentration polarization and second-kind electrokinetic instability at an ion-selective surface admitting normal flow

The passage of ionic current across a charge-selective surface has been studied for over a century and is relevant to well-established processes such as electrodialysis, electrodeposition, and electrochromatography. Recent years have witnessed a resurgence of interest in this subject, motivated by experiments demonstrating charge-selective transport of ions and solutes in nanofluidic devices. In this paper, we revisit and build upon the prototypical problem of one-dimensional ion transport across a flat ideally ion-selective surface, by examining the influence of imposed fluid flows on concentration polarization, over-limiting current, and second-kind (non-equilibrium) electro-osmotic instability at the surface. Specifically, we consider a simple model system of a cation-selective surface or membrane that admits a uniform fluid flow across itself. The membrane resides against a binary symmetric 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 profile reflects a balance between advection of ions with the imposed "normal flow" and diffusion. The relative importance of these effects is parameterized by a Peclet number Pe; notably, Pe is a signed quantity as the flow can be imposed toward or away from the membrane. An asymptotic analysis in the thin-Debye-layer limit reveals a significant impact of normal flow on concentration polarization and the advection-diffusion limiting current across the membrane. In particular, there exists a nonlinear concentration profile in the unstirred layer for non-zero Pe, in contrast to the familiar linear (diffusive) concentration polarization at Pe = 0. Next, we use matched asymptotic expansions to explore the structure of the unstirred layer at over-limiting currents, wherein a non-equilibrium space-charge layer develops near the membrane surface. A key step in this process is the derivation of a "generalized master equation" for the electric field across the unstirred layer. Finally, we examine the instability of the quiescent concentration polarization resulting from second-kind electro-osmotic slip in the space-charge layer. A linear stability analysis shows that normal flow can either enhance or retard the instability, depending on the flow direction. (C) 2011 American Institute of Physics. [doi:10.1063/1.3605693]