Integral-equation analysis of transient diffusion-limited currents at disk electrodes: Asymptotic expansion and compact approximation

TL;DR

This paper develops an analytical framework using integral equations and asymptotic expansions to accurately model transient diffusion-limited currents at disk electrodes, aiding interpretation and parameter extraction in chronoamperometry.

Contribution

It introduces a compact analytical expression for transient currents that improves upon existing numerical methods, enhancing practical analysis of disk-electrode chronoamperometry.

Findings

The model recovers Saito's steady-state equation.

A Padé approximant accurately describes current over relevant times.

The approach captures both short-time Cottrell behavior and long-time steady state.

Abstract

The transient diffusion-limited current at a disk electrode following a change in interfacial ion concentration induced by a potential step is analyzed with direct relevance to chronoamperometric measurements. The mixed-boundary diffusion problem is formulated in the Laplace domain and reduced to a Fredholm integral equation that directly determines the Faradaic current. The steady-state limit recovers Saito's equation, while a systematic long-time asymptotic expansion quantifies the approach to steady state. A Pad\'{e} approximant yields a compact analytical expression in the time domain that accurately describes the current over experimentally relevant time ranges. In contrast to existing high-accuracy numerical procedures based on hybrid asymptotic and polynomial approximations, the present formulation provides an explicit and compact analytical representation that facilitates interpretation and practical implementation. The short-time response exhibits Cottrell's equation with edge effects characteristic of disk electrodes. Overall, the framework provides practical tools for analyzing transient currents, extracting diffusion parameters, and assessing the accuracy of widely used analytical approximations in disk-electrode chronoamperometry.