Analytical solution for the steady-state diffusion towards an inlaid disc microelectrode in a multi-layered medium

Abstract

The rigorous analytical expression for the steady-state diffusion-limited current produced by an electroactive species diffusing towards an inlaid disc electrode through a multi-layered medium (such as the membrane covered sensor) is presented. This theoretical approach reformulates the problem in terms of a dual integral equation and takes full account of the edge effect that one-dimensional formulations cannot include. The solution is found by solving a linear system of equations where the entries of the matrix are integrals involving Bessel functions, and can be easily and quickly computed with standard software such as Mathematica. Comparison of this analytical solution (corresponding to the axisymmetric geometry) with previous models (linear, spherical and mixed) for the membrane-covered gas sensor allows us to demonstrate the restricted ranges of applicability of these previous models. Through a variety of approximations, we show that it is possible to obtain accurate estimates of the current (also valid for liquid samples) using very simple expressions, which in turn allows procedures such as the fast estimation of a medium permeability. The model can be easily extended to any number of parallel layers with a variety of boundary conditions in the interface or layer furthest away from the insulator plane; thus some interesting limiting cases are analysed (such as the validity of ignoring a very thin layer next to the electrode or the unattainability of steady-state just by horizontal radial diffusion).