Time-dependent modelling and asymptotic analysis of electrochemical cells

Abstract

A (time-dependent) model for an electrochemical cell, comprising a dilute binary electrolytic solution between two flat electrodes, is formulated. The method of matched asymptotic expansions (taking the ratio of the Debye length to the cell width as the small asymptotic parameter) is used to derive simplified models of the cell in two distinguished limits and to systematically derive the Butler-Volmer boundary conditions. The first limit corresponds to a diffusion-limited reaction and the second to a capacitance-limited reaction. Additionally, for sufficiently small current flow/large diffusion, a simplified (lumped-parameter) model is derived which describes the long-time behaviour of the cell as the electrolyte is depleted. The limitations of the dilute model are identified, namely that for sufficiently large half-electrode potentials it predicts unfeasibly large concentrations of the ion species in the immediate vicinity of the electrodes. This motivates the formulation of a second model, for a concentrated electrolyte. Matched asymptotic analyses of this new model are conducted, in distinguished limits corresponding to a diffusion-limited reaction and a capacitance-limited reaction. These lead to simplified models in both of which a system of PDEs, in the outer region (the bulk of the electrolyte), matches to systems of ODEs, in inner regions about the electrodes. Example (steady-state) numerical solutions of the inner equations are presented.

Theory of linear sweep voltammetry with diffuse charge: Unsupported electrolytes, thin films, and leaky membranes

“…Bazant and co-workers first used matched asymptotic expansions to treat Faradaic reactions using the PNP-FBV framework, applied to steady conduction through electrochemical thin films [12,13]. Richardson and King [93] extended this approach to derive effective boundary conditions for time-dependent problems, which provides rigorous justification for subsequent studies of transient electrochemical response using the thin double layer approximation, including ours.…”

Multiscale modelling and analysis of lithium-ion battery charge and discharge

“…In all but extremely narrow Debye layers (typical width 1 nm) about the surface of the electrodes, charge neutrality is almost exactly satisfied-that is c * p ≈ c * n (see, e.g. [26] for further details). This motivates us to write c * p ≈ c * and c * n ≈ c * and write down the following (approximate) conservation equations, in the standard fashion, for the two ion species in the bulk of the electrolyte away from the Debye layers: Fig.…”

“…The parameters D p , D n , F, R g and T are the diffusion coefficients of the positive and negative ions, Faraday's constant, the universal gas constant and absolute temperature, respectively. More details of the physical significance of these equations can be found, for example, in [26]. It is also useful to write down an equation for the current density j * in terms of the ion fluxes…”

“…In general the Butler-Volmer condition is given as an empirical law. However, in [26], the three continuity conditions, together with a version of the Butler-Volmer condition, are systematically derived from an analysis of the underlying Poisson-Nernst-Planck (PNP) equations (see, e.g. Newman [1, pp.…”

“…The solution to this system is, where I (and hence by (4.22) μ also) is constant, 26) where the degree of discharge D d is defined by…”