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.

A Limiting Case of Constant Counterion Electrochemical Potentials in the Membrane for Examining Ion Transfer at Ion-Exchange Membranes and Patches

“…Figure a illustrates the system. A number of papers employed the limiting case of constant electrochemical potential of counterions within idealized ion-exchange media in the analysis of electro-convection under overlimiting-current conditions − and in the modeling of the “electrokinetics of the second kind”, which always occurs under non-1D conditions. However, these models are complicated (for example, they included deviations from local electroneutrality and convection) so the results are far from analytical, and even the numerical procedures are intricate and not very straightforward.…”

Potentiodynamic and Galvanodynamic Regimes of Mass Transfer in Flow-Through Electrodialysis Membrane Systems: Numerical Simulation of Electroconvection and Current-Voltage Curve

“…Models based on the Nernst-Planck-Poisson and Navier-Stokes equations that directly take into account the formation of the extended SCR were considered in the works of E.A. Demekhin et al [15,28,29], S.V. Pham et al [16,30], and K. Druzgalski, E. Karatay et al [18,31], P. Magnico [32,33].…”

Transitions and Instabilities in Imperfect Ion-Selective Membranes

“…The properties of an imperfect membrane can be very different from its perfect analog. A mathematical formulation for the imperfect ion-selective membrane was first put forth in [ 21 ] and exploited in the works in [ 22 , 23 , 24 , 25 , 26 ]. In all the mentioned works only the electrostatic properties of an electric membrane were taken into account and the fluid inside the porous membrane is considered immobile.…”