Automatic solution of integral equations describing electrochemical transients under conditions of internal cylindrical diffusion

Abstract

Diffusion of reactants inside cylindrical spatial domains, uncomplicated by homogeneous reactions, is sometimes considered in electrochemistry, for example in the theoretical modelling of porous electrodes, certain micro-electrode arrays, or ion insertion processes. Simulation of transient experiments, under conditions of such internal cylindrical diffusion, can be performed by the classical integral equation method [see, for example, R.S. Nicholson, I. Shain, Anal. Chem. 36 (1964) 706]. This requires an accurate computation of a specific kernel function and its moment integrals. By combining a formerly known series expansion for the kernel, with another expansion proposed in this work, highly accurate (16 digits) and cost-optimised procedures serving for this purpose have been devised. The procedures have been incorporated into the adaptive Huber method developed by the present author. The resulting simulation technique has been tested on examples of integral equations, including models of potential step chronoamperometry and cyclic voltammetry. The method is shown to provide automatic solutions, with a user-selected target accuracy. Errors corresponding to the range from about 10−2 of the maximum solution value, down to about 10−7 or even smaller, can be easily obtained at a modest computational cost.