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We analyze the steady-state behavior of a general mathematical model for reversible galvanic cells, such as redox flow cells, reversible solid oxide fuel cells, and rechargeable batteries. We consider not only operation in the galvanic discharging mode, spontaneously generating a positive current against an external load, but also operation in two modes which require a net input of electrical energy: (i) the electrolytic charging mode, where a negative current is imposed to generate a voltage exceeding the open circuit voltage, and (ii) the "super-galvanic" discharging mode, where a positive current exceeding the short-circuit current is imposed to generate a negative voltage. Analysis of the various (dis-)charging modes of galvanic cells is important to predict the efficiency of electrical to chemical energy conversion and to provide sensitive tests for experimental validation of fuel cell models.Notably, we consider effects of diffuse charge on electrochemical charge transfer rates by combining a generalized Frumkin-Butler-Volmer model for reaction kinetics across the compact Stern layer with the full Poisson-Nernst-Planck transport theory, without assuming local electroneutrality. Since this approach is rare in the literature, we provide a brief historical review. To illustrate the general theory, we present results for a monovalent binary electrolyte, consisting of cations, which react at the electrodes, and non-reactive anions, which are either fixed in space (as in a solid electrolyte) or are mobile (as in a liquid electrolyte). The full model is solved numerically and compared to analytical results in the limit of thin diffuse layers, relative to the membrane thickness. The spatial profiles of the ion concentrations and electrostatic potential reveal a complex dependence on the kinetic parameters and the imposed current, in which the diffuse charge at each electrode and the total membrane charge can have either sign, contrary perhaps to intuition. For thin diffuse layers, simple analytical expressions are presented for galvanic cells valid in all three (dis-)charging modes in the two subsequent limits of the ratio δ of the effective thicknesses of the compact and diffuse layers: (i) the "Helmholtz limit" ( δ → ∞ ) where the compact layer carries the double-layer voltage as in standardButler-Volmer models, and (ii) the opposite "Gouy-Chapman limit" ( 0 δ → ) where the diffuse layer fully determines the charge-transfer kinetics. In these limits, the model predicts both reaction-limited and diffusion-limited currents, which can be surpassed for finite positive values of the compact-layer, diffuse-layer and membrane thicknesses.2

Possible mechanisms for over-limiting current (OLC) through aqueous ion-exchange membranes (exceeding diffusion limitation) have been debated for half a century. Flows consistent with electroosmotic instability (EOI) have recently been observed in microfluidic experiments, but the existing theory neglects chemical effects and remains to be quantitatively tested. Here, we show that charge regulation and water self-ionization can lead to OLC by "current-induced membrane discharge" (CIMD), even in the absence of fluid flow. Salt depletion leads to a large electric field which expels water co-ions, causing the membrane to discharge and lose its selectivity. Since salt co-ions and water ions contribute to OLC, CIMD interferes with electrodialysis (salt counter-ion removal) but could be exploited for current-assisted ion exchange and pH control. CIMD also suppresses the extended space charge that leads to EOI, so it should be reconsidered in both models and experiments on OLC.

We present theoretical models for the time-dependent voltage of an electrochemical cell in response to a current step, including effects of diffuse charge ͑or "space charge"͒ near the electrodes on Faradaic reaction kinetics. The full model is based on the classical Poisson-Nernst-Planck equations with generalized FrumkinButler-Volmer boundary conditions to describe electron-transfer reactions across the Stern layer at the electrode surface. In practical situations, diffuse charge is confined to thin diffuse layers ͑DLs͒, which poses numerical difficulties for the full model but allows simplification by asymptotic analysis. For a thin quasiequilibrium DL, we derive effective boundary conditions on the quasi-neutral bulk electrolyte at the diffusion time scale, valid up to the transition time, where the bulk concentration vanishes due to diffusion limitation. We integrate the thin-DL problem analytically to obtain a set of algebraic equations, whose ͑numerical͒ solution compares favorably to the full model. In the Gouy-Chapman and Helmholtz limits, where the Stern layer is thin or thick compared to the DL, respectively, we derive simple analytical formulas for the cell voltage versus time. The full model also describes the fast initial capacitive charging of the DLs and superlimiting currents beyond the transition time, where the DL expands to a transient non-equilibrium structure. We extend the well-known Sand equation for the transition time to include all values of the superlimiting current beyond the diffusion-limiting current.