Diffusion-controlled reactions. II. Further bounds on the rate constant

We apply variational principles derived earlier by us to obtain rigorous upper and lower bounds on the rate constant k associated with diffusion-controlled reactions among static, reactive traps. We derive a phase-interchanged interfacial-surface lower bound on k and a new type of lower bound which we term a "void" bound. Among other results, we compute the phase-interchanged interfacial-surface bound on k for a distribution of fully penetrable (i.e., randomly centered) spheres in which the trap region is the space exterior to the spheres (the "holes" between the spheres) and diffusion occurs in the sphere region. We also derive an upper bound on k for identical spherical traps (randomly or periodically arranged), and evaluate it for a simple cubic lattice.