The Effective Reactivity for Capturing Brownian Motion by Partially Reactive Patches on a Spherical Surface

Abstract

We analyze the trapping of diffusing ligands, modeled as Brownian particles, by a sphere that has [Formula: see text] 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 [Formula: see text] for [Formula: see text]. 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 [Formula: see text] of the structured target, which is valid for any [Formula: see text]. This target capacitance [Formula: see text] is defined in terms of a Green’s matrix, which depends on the spatial configuration of patches, the local reactive capacitance [Formula: see text] of each patch and another coefficient that depends on the local geometry near a patch. The analytical dependence of [Formula: see text] on [Formula: see text] 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 [Formula: see text]. In the homogenization limit of [Formula: see text] identical uniformly-spaced patches with [Formula: see text], 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 [Formula: see text], even when there is only a moderately large number of reactive patches.

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BibTeX RIS

BibTeX

@article{grebenkov2026effective,
  title = {{The Effective Reactivity for Capturing Brownian Motion by Partially Reactive Patches on a Spherical Surface}},
  author = {Denis S Grebenkov and Michael Ward},
  journal = {Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal},
  year = {2026},
  month = may,
  doi = {10.1137/25m180562x},
  eprint = {2509.26381},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05765236v1},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - The Effective Reactivity for Capturing Brownian Motion by Partially Reactive Patches on a Spherical Surface
AU  - Denis S Grebenkov
AU  - Michael Ward
PY  - 2026
DA  - 2026/05/21
JO  - Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal
DO  - 10.1137/25m180562x
UR  - https://hal.science/hal-05765236v1
AB  - We analyze the trapping of diffusing ligands, modeled as Brownian particles, by a sphere that has [Formula: see text] 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 [Formula: see text] for [Formula: see text]. 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 [Formula: see text] of the structured target, which is valid for any [Formula: see text]. This target capacitance [Formula: see text] is defined in terms of a Green’s matrix, which depends on the spatial configuration of patches, the local reactive capacitance [Formula: see text] of each patch and another coefficient that depends on the local geometry near a patch. The analytical dependence of [Formula: see text] on [Formula: see text] 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 [Formula: see text]. In the homogenization limit of [Formula: see text] identical uniformly-spaced patches with [Formula: see text], 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 [Formula: see text], even when there is only a moderately large number of reactive patches.
SN  - 1540-3459
SN  - 1540-3467
ER  -

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