Properties of coplanar periodic electrodes in confined spaces: Case of two-dimensional diffusion

TL;DR

This paper develops a general theoretical framework for analyzing the diffusion and concentration profiles in coplanar periodic microelectrode configurations within confined spaces, applicable to various electrode geometries and current conditions.

Contribution

It derives a comprehensive theory for two-dimensional diffusion in periodic electrode cells, independent of reaction reversibility, and provides properties to predict electrochemical behavior.

Findings

Properties relate concentration averages to net current and accumulation.

Predictions of collection efficiency and limiting currents.

Estimates of steady-state times in potential-controlled experiments.

Abstract

Periodic configurations of electrodes, in particular of microelectrodes, have been of interest since the advent of microfabrication. In this report, theory which is common to any periodic cell (or any cell that can be extended periodically) with finite height and two-dimensional symmentry was derived. The diffusion equation in this cell was solved and the concentration profile was obtained in terms of its Fourier coefficients and as a function of an arbitrary current density. From this base result, a set of properties were derived which are fairly general, since they don't assume restrictions such as reversible electrode reactions (Nernst equation valid when current circulates). These properties involve: horizontal averages and (weighted) sum of concentrations, both with a close connection to the net current and accumulation of species in the cell. The derived properties allow: to explain qualitative aspects of collection efficiency and limiting currents, to predict the concentration on counter electrodes and non-linearities caused by depletion of species at extremely polarized electrodes, and to estimate the time required by the current to reach steady state in potential controlled experiments. The theoretical results are illustrated analytically and numerically for the concrete case of interdigitated array of electrodes.