Theoretical calculation of collection efficiencies for collector-generator microelectrode systems

Steady-state mass transport is considered in a 'collector-generator' system of microelectrodes, in which an electroactive species is generated at the surface of one microelectrode (the 'generator') and is transported by diffusion to the surface of another (the 'collector'), where it reacts. Following its generation, the species may also be consumed by a homogeneous first-order reaction. Using a form of Green's theorem, the total current at the collector is obtained as an integral over the surface of the generator, in which the integrand contains the surface current density, multiplied by a weighting function that can be determined by solving a reaction-diffusion problem for the collector microelectrode alone. Therefore, the collection efficiency may be calculated without solving the detailed transport problem in the presence of both microelectrodes. As an illustration of these ideas, the collection efficiency is calculated numerically as a function of the rate constant of the homogeneous reaction, in the mathematically simplest case, where the current density on the surface of the generator is uniform. Three geometrical configurations are considered: (i) both collector and generator are circular disc microelectrodes of equal radii; (ii) the collector is a disc and the generator a concentric thin ring and (iii) the collector is a thin ring and the generator a concentric disc. Comparison with asymptotic approximations suggests that the collection efficiency is fairly insensitive to the distribution of current at the generator. Similar results are therefore to be expected in the more general situation of a non-uniform current density at the generator. This remains true if generation is enhanced by species produced at the collector or by the homogeneous reaction.