Transition to electrokinetic instability near imperfect charge-selective membranes

Abstract

The hydrodynamics and electrostatics of imperfect electric membranes are examined numerically. The investigation is based on the Nernst–Planck–Poisson–Stokes system of equations. A three-layer geometry, electrolyte–nanoporous membrane–electrolyte, is considered. The threshold of the electrokinetic instability of the one-dimensional quiescent state and the corresponding change of the current regime to the overlimiting one are studied. A map of the bifurcations, transitions, and regimes is constructed in the coordinates of the selectivity, the applied potential difference, and the Debye number. For good membrane selectivity (it corresponds to the dimensionless fixed charge in the membrane, N > 10), the hydrodynamics and electrostatics are quantitatively the same as for perfect membranes: the instability is monotonic and nonequilibrium and the voltage–current (VC) characteristic has all three portions: the underlimiting, limiting, and overlimiting regimes. For intermediate selectivity (1 < N < 10), the imperfect membrane behaves qualitatively as a perfect membrane. For poor membrane selectivity (N < 1), the nonequilibrium electro-osmosis turns to the equilibrium one and the monotonic instability is replaced by an oscillatory one. The concept of slip velocity loses its meaning, surface spike-like coherent structures disappear, and the equilibrium instability is caused by the bulk residual charge. For poor membrane selectivity, the VC characteristic dramatically changes: transition to the overlimiting currents occurs, bypassing the limiting current regime. There is a qualitative agreement between theoretical prediction and experimental observations of the microvortex expansion.