(116f) An Electrokinetic Vortex Instability at the Interface of a Permselective Membrane: Beyond Limiting Currents

We examine the important limiting-current phenomenon, when a DC current penetrates a permselective nanoporous membrane or nanochannel, and showed that it breaks down beyond a critical voltage to produce an over-limiting current because of a nonlinear hydrodynamic instability. Unlike zero-flux Poisson-Boltzmann equilibrium distributions, a constant current flux through the membrane produces an electroneutral diffusion layer with a non-equilibrium linear concentration distribution that extends from the membrane interface to the electrode. The diffusion-limited current corresponds to the limit when the interfacial ion concentration approaches zero. This is, however, a singular limit as the local chemical potentials of the ions would approach negative infinity in a logarithm manner. Instead, an extended polarized layer occurs between the electroneutral diffusion layer and the thin electric double layer (EDL) at the interface with a thickness that exceeds the Debye layer. Rubinstein et al (Desalination, 69, 101 (1988)) first predicted that over-limiting currents beyond the limiting current occur because of a vortex instability associated within this extended polarized layer. We recently verified this instability experimentally with a nanoslot by rejuvenating the interfacial vortices repeatedly with a low-frequency AC field, such that an ionic fluorescent dye trace the vortices under confocal microscopy before complete mixing occurs (PRL, 101, 254501 (2009)). The diffusion layer is found to be established by a self-similar constant-flux diffusion front whose position advances diffusively as sqrt(Dt). The vortex instability arrests the front growth once it has advanced to a critical position, which is a function of the applied voltage. This position corresponds to the thickness of the bifurcated vortex layer and a new diffusion layer thickness which offers a good prediction of the over-limiting current (PRE, 79, 046305 (2009)). We examine this length scale selection mechanism of the vortex instability here by examining the stability of the diffusion front. The instability is attributed to the interaction between the diffusion front and the polarized layer to produce the classical coupled dynamics of two interfaces. The positive feedback mechanism is due to the enhanced Maxwell pressure, when the separation between the two layers decreases, which drives an electro-osmotic vortex that further compresses the separation. Using a spectral theory for transient base states (Phys. Fluid, 14, 999 (2002)), we find that the dominant wavelength increases as sqrt(Dt), as in miscible fingering, in quantitative agreement with the experiments. The critical time for the destruction of the diffusion front to establish the steady vortex array is found by matching the disturbance concentration to the base-state concentration. It produces a voltage-dependent vortex layer thickness that will be compared to the experimental data.