Imperfect diffusion-controlled reactions on a torus and on a pair of balls

We employ a general spectral approach based on the Steklov eigenbasis to describe imperfect diffusion-controlled reactions on bounded reactive targets in three dimensions. The steady-state concentration and the total diffusive flux onto the target are expressed in terms of the eigenvalues and eigenfunctions of the exterior Steklov problem. In particular, the eigenvalues are shown to provide the geometric lengthscales of the target that are relevant for diffusion-controlled reactions. Using toroidal and bispherical coordinates, we propose an efficient procedure for analyzing and solving numerically this spectral problem for an arbitrary torus and a pair of balls, respectively. A simple two-term approximation for the diffusive flux is established and validated. Implications of these results in the context of chemical physics and beyond are discussed.