Electrodes of microscopic dimensions play an increasingly important role in many electrochemical systems of industrial significance, e.g., thin-film batteries (with electrodes and electrolyte layers of micron dimensions), and microelectrodes (with micron or submicron dimensions) used as sensors or tools for electroanalytical studies. As the size of an electrode decreases, the thin charge layer adjacent to its surface, often on the order of angstroms, exerts an increasing influence on the current characteristics of the electrode. A mathematical model is used to study the impact of the charge layer on a microelectrode immersed in a dilute concentration of binary electrolyte. The transport-limited current density on the electrode depends on a dimensionless parameter , corresponding to the quotient of the Debye length by the electrode radius. (The Debye length characterizes the charge-layer thickness.) As becomes small, the pde's describing charge transport become singularly perturbed, and numerical solution of the equations becomes increasingly difficult. Matched asymptotics were used to calculate the current in the limit < < 1, and the results were compared with numerical solutions for larger -values. This talk will focus mostly on the matched asymptotic procedure, with some discussion of how a knowledge of the boundary-layer structure can be used to aid in the numerical solution of the equations.
This is joint work with Mark W. Verbrugge.