Diffusion-controlled reactions: Mathematical formulation, variational principles, and rigorous bounds

This paper is concerned with the problem of predicting the effective rate constant k associated with diffusion-controlled reactions in media composed of static and reactive traps (sinks) which are generally distributed randomly throughout a region containing reactive particles. The effective equation for diffusion-controlled reactions is derived using the method of homogenization. This leads to a rigorous definition of k. General variational principles are then formulated to obtain rigorous upper and lower bounds on k. These variational principles are applied by evaluating them for three different types of admissible fields. The upper and lower bounds which result are computed for both random and periodic arrays of equisized spherical sinks.