Multidimensional integral equations Part 1. A new approach to solving microelectrode diffusion problems

The diffusion problems for a wide class of microelectrodes are shown to be reducible to multidimensional integral equations (equations containing multiple integrals). The formulation and solution of these problems by means of integral transformations are given for several types of microelectrode systems: a microdisk embedded in an insulating plane or in an insulator of a finite thickness, a microband, a scanning electrochemical microscope and an array of inlaid planar arbitrarily-shaped electrodes. Every solution obtained is suitable for a simple electrode reaction with any values of heterogeneous electron transfer kinetic parameters and for any shape of signal applied to the electrode. Analogous equations may be derived for some other mechanisms (e.g. electrochemical). The equations are similar to each other, and quite similar algorithms are required to solve them. No solutions of integral equations of this type have previously been published. An algorithm and FORTRAN programs for computing the polarization curves for the inlaid microdisk were devised, and chronoamperograms and cyclic voltammograms were computed. The derived equations and results are compared with known analytical solutions and simulations. INTRODUCTION The complexity of the mathematics used by electrochemists is always determined by the character of the objects they want to study. Up to the early 195Os, when the objects of electrochemical research were relatively simple (e.g. steadystate currents; uncomplicated electrode reactions at spherical liquid (mercury) electrodes or at planar electrodes), analytical solutions were available for most electrochemical systems considered [l]. Later, the incorporation of more complicated mechanisms and more complicated electrochemical methods led to the development of various types of semi-analytical techniques, such as integral equations 12-41 and infinite series 151. These have largely been replaced by different variations of digital simulation [6,7], which appeared to be quite straightforward in the case of a complicated mechanism or special electrode geometry. Even these 0022-0728/92/$05.00