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The complete series solution for two diffusion-controlled chemically reactive sites, each of radius a, located on an inert plane an arbitrary center-to-center distance d apart, is presented. Simple analytical forms are developed to calculate the site reaction rates in terms of da. The impact of these results on a number of important problems in chemical physics is discussed.

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Competitive interactions within diverse mixed populations of chemically active sites are prevalent throughout nature, science, and engineering. Their effects are readily seen in the distribution of dead and surviving aerobic cells within a thick biofilm and particle shape changes during the growth and coarsening of crystals. Even in the most dilute case, competition for a reactant requires at least two spheres/cells, and the solution of the two-spherical sink problem is of interest for several reasons. The solution accurately describes lower cell concentration behavior (10(8) cells/l), and like the Smoluchowski diffusion-reaction treatment for a single sphere, the analysis is extremely helpful in understanding the fundamental phenomena of the effect on the first spherical sink of the presence of a second different spherical sink. In addition these exact solutions are required for the systematic extension to higher density behavior by rigorous expansions in the spherical sink densities. The method of the twin spherical expansion is used with a formal matrix elimination scheme to generate an exact solution for two distinct spherical sinks of differing sizes and kinetics. The two sinks exist in a medium, which supplies a reactant to the sinks via Fickian diffusion. The two sinks compete for the same reactant with different first-order reactions occurring at the surface of each sink. Earlier work focused on two spherical sinks of the same size with identical surface reaction kinetics. This work has been advanced to allow for diversity in the theory of cellular or reactive sink competition. A number of interesting higher order interactive phenomena are observed in this paper when the different reactive sinks are in close proximity. (c) 2004 American Institute of Physics.