Modes of electrokinetic instability for imperfect electric membranes

Abstract

The direct transition to overlimiting current bypassing the stage of limiting currents is considered for imperfect membranes. Instability of the quiescent steady-state one-dimensional solution, which is the result of a balance of diffusion and electromigration, is investigated on the basis of the full Nernst-Planck-Poisson-Stokes system and a simplified quasielectroneutral system. A three-layer geometry, electrolyte-porous membrane-electrolyte, is considered. The usual assumption of a constant electrochemical potential along the membrane surface is removed from consideration. The effect of bulk and surface effects on the instability and transition to the overlimiting currents is evaluated for a different membrane selectivity. It becomes clear that for sufficiently small fixed charge concentration (large ion concentration in the electrolyte), the monotonic instability is replaced by an oscillatory one. The dependence of instability on the membrane porosity is found to be weak.

1D Mathematical Modelling of Non-Stationary Ion Transfer in the Diffusion Layer Adjacent to an Ion-Exchange Membrane in Galvanostatic Mode

“…The scheme in Figure 1 is based on numerical calculations of the steady-state ion concentration distribution in the diffusion layer at the cation-exchange membrane according to the (potentiostatic) model proposed by Rubinstein and Shtilman [ 2 ]. According to Reference [ 2 ] and other studies [ 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 ], the concentration of anions c 2 decreases monotonically when approaching the membrane to reach a relatively low value within the membrane. This low value is caused by the Donnan co-ion exclusion (the electrostatic repulsion of the ions, whose charge has the same sign as that of the fixed ions).…”

“…The value of c 1m is close to the concentration of the fixed ions in the membrane [ 2 ], which is of the order of 1 mol/L (=10 3 mol/m 3 ). The sharp increase of c 1 in the vicinity of the membrane (in the quasi-equilibrium part of the EDL) is due to the condition of continuity of concentrations when passing across the solution/membrane boundary [ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 ].…”

“…This approach allowed Rubinstein and Zaltzman to obtain a large number of physicochemical results concerning the mechanisms of mass transfer in membrane systems in potentiostatic mode [ 3 , 28 , 29 ]. Their results, in turn, stimulated other authors to develop productive models giving new insights into the theory of mass transfer in ion-exchange membrane and nano/microfluidic systems [ 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 ].…”

“…The boundary-value problems involving these equations are mathematically complicated, since a small parameter is present in the Poisson equation (in its dimensionless form) [ 1 ]. The most models are developed for the potentiostatic mode, where electric regime is set through the potential difference between two equipotential planes enveloping an electrode/membrane and parallel to it [ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 ]. The relationship between the local electric potential and ion concentrations is given by the Poisson equation.…”

“…The study was based on the numerical solution of the NPP equations with a boundary condition, which allowed setting the total current density as the parameter governing the electric regime in the system. Unlike potentiostatic (potentiodynamic) models [ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 ], where the potential difference was set, in Reference [ 27 ] the time derivative of the electric potential gradient at the outer edge of the diffusion layer was specified as an explicit function of the current density.…”