# Steady-state theory of interdigitated array of electrodes in confined spaces: Case of pure diffusion and reversible electrode reactions

## TL;DR

This paper derives analytical equations for steady-state current in interdigitated electrodes within confined spaces, considering reversible reactions and pure diffusion, revealing how cell dimensions influence current behavior.

## Contribution

It introduces a conformal transformation approach using Jacobian elliptic functions to solve the diffusion equation in confined geometries, providing new analytical expressions for steady-state current.

## Key findings

- Steady-state current depends on cell dimensions and elliptic integrals.
- Shallower cells exhibit smaller steady-state currents.
- Tall cells' currents approach semi-infinite geometry values.

## Abstract

Analytical equations were found for interdigitated electrodes, which considered reversible electrode reactions and pure diffusion within confined spaces. A conformal transformation, obtained by the use of Jacobian elliptic functions, was applied to solve the diffusion equation in steady state. The obtained steady-state current depends on the ratio of elliptic integrals of the first kind, in which their moduli are functions of the relative dimensions of the cell. The current is smaller for shallower cells, but approaches similiar values to those of semi-infinite geometries when the cell is sufficiently tall. Approximations using trigonometric and hyperbolic expressions were also found for the steady-state current in the cases of shallow and tall cells respectively.