THE EFFECTIVE REACTIVITY FOR CAPTURING BROWNIAN MOTION BY PARTIALLY REACTIVE PATCHES ON A SPHERICAL SURFACE^*

We analyze the trapping of diffusing ligands, modeled as Brownian particles, by a sphere that has N partially reactive boundary patches, each of small area and arbitrary shape, on an otherwise reflecting boundary. For such a structured target, the partial reactivity of each boundary patch is characterized by a Robin boundary condition, with a local boundary reactivity κ_i for i = 1, …, N. For any spatial arrangement of well-separated patches on the surface of the sphere, the method of matched asymptotic expansions is used to derive explicit results for the capacitance C_T of the structured target, which is valid for any κ_i > 0. This target capacitance C_T is defined in terms of a Green's matrix, which depends on the spatial configuration of patches, the local reactive capacitance C_i (κ_i) of each patch and another coefficient that depends on the local geometry near a patch. The analytical dependence of C_i (κ_i) on κ_i is uncovered via a spectral expansion over Steklov eigenfunctions. For circular patches, the latter are readily computed numerically and provide an accurate fully explicit sigmoidal approximation for C_i (κ_i). In the homogenization limit of N ⨠ 1 identical uniformly-spaced patches with κ_i = κ, we derive an explicit scaling law for the effective capacitance and the effective reactivity of the structured target that is valid in the limit of small patch area fraction. From a comparison with numerical simulations, we show that this scaling law provides a highly accurate approximation over the full range κ > 0, even when there is only a moderately large number of reactive patches.