A transformation for the treatment of diffusion and migration. Application to stationary disk and hemisphere electrodes

Abstract

A method is presented for solving coupled, nonlinear ionic transport equations governing steady-state transport by diffusion and migration. The technique can be used on any system in which only one charged ion reacts and the Nernst-Planck equation describes the ionic transport. It is shown that the concentration of each of the ionic species can be written as a function of the potential in solution, and the transport equation for the reactant species becomes a nonlinear equation in this unknown potential. A transformation of the dependent variable changes this transport equation to an equivalent linear form (without approximation); the transformation allows one to use superposition methods to formulate the problem as a nonlinear integral equation on the surface of the electrode. The proposed method is general in the sense that it can be applied to any electrode and container geometry once the kernel function associated with this geometry is known. The transformation allows one to express the ratio of the limiting current to the diffusion limiting current as a simple geometry-independent relationship. Specific calculations are given for disk and hemisphere electrodes. Simulation results for the disk electrode illustrate the nonlinear effects of migration on the reaction distribution over the electrode surface below the limiting current, and emphasize the advantage of a microelectrode for determining kinetic parameters for the electrode reaction.