Time-dependent modelling and asymptotic analysis of electrochemical cells |
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Authors: | G Richardson J R King |
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Affiliation: | (1) School of Mathematical Sci., University of Nottingham, University Park, Nottingham, UK |
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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. |
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Keywords: | Butler– Volmer equation Electrolyte Matched asymptotic expansions |
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