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. In the model, we consider effects of diffuse charge on electrochemical charge-transfer rates by combining a generalized Frumkin-Butler-Volmer equation for reaction kinetics across thecompact 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 ¿ of the effective thicknesses of the compact and diffuse layers: (i) the ¿Helmholtz limit¿ (¿¿¿) where the compact layer carries the double layer voltage as in standard Butler-Volmer models, and (ii) the opposite ¿Gouy-Chapman limit¿ (¿¿0) 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 thickness.@en

When high electrical fields are applied, and especially above the glass transition temperature, ion transport through the epoxy molding compounds that encapsulate the integrated circuit (die) strongly increases, leading to the accumulation of charge at the passivation-epoxy interface, and possibly to leakage currents because of the formation of parasitic transistors. Recovery from this failure mechanism is possible after annealing the package in the absence of the applied voltage, at sufficiently high temperature. The high electrical fields originate either from within the package due to high-voltage bondpads or are due to an external high-voltage source. We have developed a numerical transport model that is able to describe ion transport through IC packages including the formation of 'ionic diffuse layers', or 'polarisation layers', at the passivation-epoxy interface. In these layers the ion concentration strongly increases leading to a modification of the electrostatic boundary conditions (Gauss' law), as well as to a voltage step over the passivation-epoxy interface. The exact distribution of the ions in this diffuse layer is a very important quantity as it strongly influences the voltage profiles in the bulk of the material via the boundary conditions, and therefore the ion transport rates. We will briefly outline how the relation between charge and voltage (integral capacitance) can be obtained experimentally from the dielectric response when a DC voltage is applied across a thin polymer sample. The data that are obtained can be used to derive realistic microscopic models for the ion distribution in the diffuse layer formed within an epoxy resin next to electrodes and passivation layers.@en

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 Frumkin-Butler-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 (DLs), which poses numerical difficulties for the full model but allows simplication by asymptotic analysis. For a thin quasi-equilibrium 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 (numerical) 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.@en

When a monolayer of negatively charged surfactant molecules is brought in contact with an aqueous solution containing mixtures of counterions of different size and valency, very large deviations from Poisson-Boltzmann theory (PBT) develop at a high surface charge, with the smaller counterion outcompeting the larger one (even if divalent) near the interface, leading to counterion segregation [V.L. Shapovalov, G. Brezesinski, J. Phys. Chem. B 110 (2006) 10032]. We use a modified PBT that empirically includes an extended Carnahan-Starling equation-of-state to describe hard-sphere interactions in electrical double layers containing ions of different size and charge. Model calculations are made for ion concentration profiles, free energies, surface pressures, and differential capacities. At high surface charge, volume interactions become important, leading to significant deviations from PBT. In contrast to PBT, at high surface charge, contributions to energy and pressure are no longer mainly entropic, but instead volume and electrostatic field effects now dominate. When the hydrated size of the divalent ion is used as an adjustable parameter, the theory is in good agreement with the experimental data. Keywords: Poisson-Boltzmann theory; Electrostatic double layer; Surfactant monolayer; Volume effects@en

We present a numerical method to describe the transport of ions through polymeric electrolytes under the influence of an applied electrostatic field. The transport of ions results in an ion concentration profile with a large gradient near charged interfaces, the so-called diffuse layer. We show that the Poisson-Nernst-Planck theory, which generally describes the transport of ions, can be simplified significantly if the dimensions of the bulk material are much larger than the thickness of the diffuse layer. In this case, charge transport through the bulk of the material is driven by the electric field only (Ohm's law). At all interfaces an analytical mathematical relation between the surface charge and the voltage difference over the diffuse layer can be used. Several of these relations are used, including relations that account for ion volume effects. We use experimental data from literature [J.A. van der Pol, R.T.H. Rongen, H.J. Bruggers, Microelectron.Reliab.40 (2000) 1267] of charge accumulation at the surface of an integrated circuit (IC) resulting from the transport of ions through the polymeric material encapsulating the device. The data are very well described by a two-dimensional finite element model based on the simplified Poisson-Nernst-Planck theory. In addition, the theory predicts that the accumulated charge at the circuit interface will slowly drain away after having reached the experimentally observed maximum.@en