Electrodiffusional free boundary problem, in a bipolar membrane (semiconductor diode), at a reverse bias for constant current

Abstract

Abstract.A singular perturbation problem, modeling one-dimensional time-dependent electrodiffusion of ions (holes and electrons) in a bipolar membrane (semi-conductor diode) at a reverse bias is analyzed for galvanostatic (fixed electric current) conditions. It is shown that, as the perturbation parameter tends to zero, the solution of the perturbed problem tends to the solution of a limiting problem which is, depending on the input data, either a conventional bipolar electrodiffusion problem or a particular electrodiffusional time-dependent free boundary problem. In both cases, the properties of the limiting solution are analyzed, along with those of the respective boundary and transition layer solutions. Introduction.In our recent paper [1] we analyzed the electrodiffusional free boundary problem that arose asymptotically in the singularly perturbed model of electrodialysis for a vanishing perturbation parameter.This model concerned the passage of a specified direct electric current through a layer of univalent electrolyte adjacent to the wall (cathode, cation exchange membrane) selectively permeable to positive ions (cations) only. The simplest version of the governing equations was t > 0: pf = {p£x +pe4>%)x Vx€(0,l), Here p£(x,t), n£(x,t), cp£(x,t) are, respectively, the cation and anion (negative ions) concentrations and the electric potential. Equations (0.1) and (0.2) are the Nernst-Planck

Limiting current singularities in electro diffusion

“…Since a(x\I) is pointwise monotonic, we conclude that for the aforementioned N(x) the limiting current is characterized by the appearance of discontinuities in the first derivative of u(x\I) at x. The latter is accompanied by an unbounded growth of the electric field at x, which is the feature traditionally associated with the limiting current (see [4,1,2] -I -N' tj>' = (3.13) a and, generally, in the neutral symmetric case /hm ^ -N'(x), whereas cr(x) = 0 for / = /hm. Let us allow now for multiple roots of N(x), still, for a neutral symmetric case (dropping the latter assumption would require a reference to continuity of the solution with respect to the current, which has not been proved thus far).…”

“…As pointed out in previous studies [2,3], the limiting current may be viewed as a current value above which the straightforward use of boundary layer method fails in the perturbed problem, and, thus, has to be modified. Such a modification is provided in terms of the respective free boundary formulation.…”

“…The described singularities, developing at the limiting current, may be viewed, in a quasi-electro-neutral electrodiffusional system with an analytic distribution of fixed charges, as precursors of the free boundary and their related discontinuities of charge carrier concentrations which are expected to develop for currents above the limiting value. Dynamics of such free boundary have been described for systems with piecewise constant fixed charge distribution [1,2,3]. Such study for an analytic fixed charge distribution, to the best of our knowledge, remains to be done.…”

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 14: extension of the patch-adaptive strategy to time-dependent models involving migration–diffusion transport in one-dimensional space geometry, and its application to example transient experiments described by Nernst–Planck–Poisson equations

“…For this reason, we consider Example 4, which is a model of chronopotentiometry for a bipolar membrane (a sandwich formed by an anion exchange membrane adjacent to a cation exchange membrane). The model has been analysed by Primicerio et al [89], without performing numerical calculations for the time-dependent case. In this example, the application of an overlimiting current pulse creates a central zone of uncompensated spatial charges, outside which electroneutrality is obeyed well.…”

“…The interfaces are characterised by steep fronts of the concentrations. The widths of the fronts decrease with the decreasing value of the dimensionless dielectric permittivity e (which is also interpreted as a perturbation parameter [89]), because they are proportional to e 1=2 . Therefore, Example 4 exhibits a new category of difficult-to-solve moving fronts occurring in electrochemical systems, which has to be added to the collection of other kinds of moving fronts that have been already simulated by the PAS [6][7][8]12,13].…”