Diffusion-controlled reactions on an active site in a spherical cavity: Extension of Berg's theory

This study is due to various applications in physics, chemistry, and especially biology, where both the bounded configuration domain and chemical anisotropy could play a great part. In fact, we generalize the well-known Berg's theory, which describes diffusion-controlled reactions occurring within a spherically symmetric absorber-cavity system. The local concentration and the reaction trapping rate at which a small diffusing particle is captured by an axially symmetric one-reactive-patch absorber inside a spherical cavity were found semi-analytically and numerically by means of the dual series relations method. This approach leads to such incredibly fast convergence that it may rightly be referred to as an exact one. The results obtained can be used to test numerical programs that describe diffusion-controlled reactions in real physical systems for reactants with arbitrary anisotropic reactivity, which are located inside various cavities as well as in the unbounded domains. Moreover, we managed to find a close connection between the dual series relations method and the generalized method of separation of variables.