Diffuse-Charge Effects on the Transient Response of Electrochemical Cells

TL;DR

This paper develops theoretical models to analyze the transient voltage response of electrochemical cells considering diffuse charge effects, providing simplified analytical formulas and extending existing equations for super-limiting currents.

Contribution

It introduces a comprehensive model incorporating diffuse charge effects and derives simplified analytical expressions for cell voltage over time, extending the Sand equation for super-limiting currents.

Findings

Effective boundary conditions for thin diffuse layers derived

Analytical formulas for voltage in Gouy-Chapman and Helmholtz limits obtained

Extended Sand equation to include super-limiting current regimes

Abstract

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 monolayer 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 simplification 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 formulae for the cell voltage versus time. The full model also describes the fast initial capacitive charging of the DLs and super-limiting 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 super-limiting current beyond the diffusion-limiting current.