A singular perturbation problem, modeling one-dimensional time-dependent electrodiffusion in an electrolyte layer flanked by charge-selective walls (electrodes, ion-exchange membranes), 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 certain limiting problem which is, depending on the input data, either a conventional diffusion problem or a diffusional free boundary problem equivalent to the one-phase Stefan problem with superheating. Spatial boundary layers in the perturbed problem are analyzed in both cases, together with the extended space charge zone which develops for electric currents above a certain critical (“limiting”) value. In this framework, the relaxational, vanishing at steady state, components of the ionic fluxes are being introduced and evaluated along with the respective parts of the electrochemical potentials of the ions. The analysis is constructive and yields, in particular, the explicit ionic concentration and electric potential profiles in the typical regions in the system.
Some remarks on stability of cones for the one-phase free boundary problem
