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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 $i=1,\ldots,N$. 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 $C_{\rm T}$ of the structured target, which is valid for any $κ_i>0$. This target capacitance $C_{\rm T}$ 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 $N\gg 1$ 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.