Competitive diffusion into two reactive spheres of different reactivity and size

Exact series solutions for the reaction rates of two reactive spheres of different reactivity and size are presented. The reaction rates are explicitly expressed in terms of the interparticle distance from reaction- to diffusion-limited conditions. The competitive diffusion favors the larger particle and the reaction rate of the smaller one is significantly reduced as the size ratio increases. For interactions between a diffusion-limited sink and a reaction-limited one, the diffusive interaction favors the reaction rate of the former and hinders that of the latter. Several approximate expressions have been proposed and compared with the exact results. The monopole approximation always underestimates the rate of the smaller sink for diffusion-limited reaction. The asymptotic limit of large size ratio is also analyzed. The asymptotic expression gives a quite accurate result if the smaller particle is regarded as a point sink plus a point dipole.