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. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3539–3555, 2013

Concentration polarization, surface currents, and bulk advection in a microchannel

“…In microsystems, surface effects are comparatively important, and for this reason their behavior is fundamentally different from bulk systems. For instance, an entirely new mode of overlimiting current enabled by surface conduction, has been predicted by Dydek et al [26,27], for which the current exceeding the diffusion-limited current runs through the depletion region inside the diffuse double layers screening the surface charges. This gives rise to an overlimiting current depending linearly on the surface charge, the surface-to-bulk ratio, and the applied potential.…”

Analysis of electrolyte transport through charged nanopores

“…It is well known that interfaces between charged membranes or nanochannels and unsupported bulk electrolytes lead to ion concentration polarization outside the membrane, e.g., in classical electrodialysis [33][34][35], but complex nonequilibrium electrokinetic phenomena resulting from strong concentration polarization have recently been discovered inside membrane pores or microchannels, such as deionization shock waves [32,[36][37][38][39][40][41] and overlimiting current sustained by surface conduction (electromigration) and electro-osmotic flow [42][43][44][45] with applications to nanotemplated electrodeposition [46] and water desalination by "shock electrodialysis" [47]. In most situations for nanochannels, the ions remain in local quasiequilibrium, since electromigration FIG.…”

Experimental Verification of Overlimiting Current by Surface Conduction and Electro-Osmotic Flow in Microchannels

“…Finally, the electroconvection becomes very strong and unstable in case of the 22 μm deep microchannel, but still occurs behind the shock away from the membrane surface. Although this unstable flow resembles the bulk behavior [6,13,[28][29][30], the channel is not nearly thick enough to see the Rubinstein-Zaltzman electroosmotic instability, which originates on the membrane and is predicted to dominate only above 400 μm depth [31].…”

“…(6) The salt concentration tends to form very sharp gradients between the depleted and concentrated regions (on the anodic, depleted side of the membrane), perhaps first observed a decade ago [25]. In micronanofluidic device in which steady over-limiting current has been observed [26], salt gradients propagate as shock waves, [6,27] or "deionization shocks" [28][29][30] at constant current, due to the nonlinear effect of ion transport in the electric double layers of the sidewalls. These observations suggest that multiple transport mechanisms may be involved when overlimiting current occurs under strong confinement.…”

“…The details of the changes can be found in the videos in the Supplemental Material [39] for each depth. At 2 μm depth, a flat depletion zone propagates as a deionization shock wave from the nanojunction over the microchannel [6,25,[27][28][29][30], and the strong vortical motions are largely suppressed and hardly observed because of geometrical constrictions [19,26]. Transverse diffusion of the dye across the depth also eliminates concentration gradients [31,34].…”