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Zbl 1090.34009
Chu, Kevin T.; Bazant, Martin Z.
Electrochemical thin films at and above the classical limiting current. (English)
[J] SIAM J. Appl. Math. 65, No. 5, 1485-1505 (2005). ISSN 0036-1399; ISSN 1095-712X/e

Summary: We study a model electrochemical thin film at DC currents exceeding the classical diffusion-limited value. The mathematical problem involves the steady Poisson-Nernst-Planck equation for a binary electrolyte with nonlinear boundary conditions for reaction kinetics and Stern-layer capacitance, as well as an integral constraint on the number of anions. At the limiting current, we find a nested boundary-layer structure at the cathode, which is required by the reaction boundary condition. Above the limiting current, a depletion of anions generally characterizes the cathode side of the cell. In this regime, we derive leading-order asymptotic approximations for the (i) classical bulk space-charge layer and (ii) another nested highly charged boundary layer at the cathode. The former involves an exact solution to the Nernst--Planck equations for a single, unscreened ionic species, which may apply more generally to Faradaic conduction through very thin insulating films. By matching expansions, we derive current-voltage relations well into the space-charge regime. Throughout our analysis, we emphasize the strong influence of the Stern-layer capacitance on cell behavior.

MSC 2000:: *34B08 Multi-parameter boundary value problems
34B16 Singular nonlinear boundary value problems
34B60 Applications of theory of BVP
35E05 Fundamental solutions (PDE with constant coefficients)

Keywords: Poisson-Nernst-Planck equation; electrochemical systems; limiting current; reaction boundary conditions; double-layer capacitance; polarographic curves

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