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

TL;DR

The paper develops a rigorous three-term asymptotic expansion for the capacitance ${\mathcal{C}}_{\rm T}$ of a sphere with many small partially reactive patches, capturing inter-patch diffusion interactions and boundary curvature through a Green’s function framework and Steklov expansions.Key contributions include expressing ${\mathcal{C}}_{\rm T}$ in terms of local reactive capacitances $C_i(\kappa_i)$ and monopole corrections $E_i(\kappa_i)$, and providing explicit formulas for circular patches alongside a generalization to arbitrary patch shapes.A homogenization analysis yields a scaling law for the effective capacitance ${C}_{\rm eff}$ and effective reactivity ${k}_{\rm eff}$ in the small-patch, low-coverage limit, with corrections dependent on patch geometry and reactivity, and consistent limits with classical results.The authors validate the theory with a novel Monte Carlo algorithm tailored to spherical geometry, and demonstrate accurate agreement with exact, semi-analytical, and spectral solutions across diverse configurations, highlighting robustness beyond the formal asymptotic regime.Overall, the work provides a comprehensive mathematical treatment and practical computational tools for predicting diffusion-limited trapping by structured, partially reactive targets on curved boundaries.

Abstract

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, 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.

Figures (10)

• Figure 1: Geodesic normal coordinates $(\xi_1,\xi_2,\xi_3)^T$ at ${\bf{x}}_i\in \partial\Omega$, with the geodesics (orange and blue curves) indicated.
• Figure 1: Schematic two-dimensional illustration of a thin layer of width $a$ near the boundary ${\partial {\mathcal{B}}}$ with a reactive patch ${\partial {\mathcal{B}}}_i$ (thick red interval). A random path $\hat{{\bf X}}_1 = {\bf X}_{\tau_1} \rightsquigarrow {\bf X}_{\tau_2} = \hat{{\bf X}}_2$ from the first arrival point $\hat{{\bf X}}_1$ to the escape point $\hat{{\bf X}}_2$ on the surface ${\partial {\mathcal{B}}}^a$ is shown. Two arrows indicate the boundary ${\partial_{e} {\mathcal{B}}}_{i}$ of the patch ${\partial {\mathcal{B}}}_i$ (in three dimensions, ${\partial_{e} {\mathcal{B}}}_{i}$ is a curve but here it is reduced to two endpoints of the shown interval).
• Figure 1: (a): Dimensionless effective reactivity $k_{\rm eff}$ versus $\kappa = {\varepsilon} {\mathcal{K}} R/D$ for $N = 12$ identical circular patches of radius ${\varepsilon}$ centered at the vertices of an icosahedron. Symbols are the Monte Carlo results (with $M = 10^5$ realizations and $a = 10^{-2}$), thick lines show (\ref{['eq:kappa_J']}) with $C_{\rm T}$ from the asymptotic formula (\ref{['homo:c0_orig_1']}), and thin lines are the homogenized asymptotic formula (\ref{['eq:keff']}). (b): Same plot but with a logarithmic scale on the vertical axis to better show the small $\kappa$ comparison.
• Figure 2: The reactive capacitance $C_i(\kappa_i)$ for the circular patch ${\Gamma}_i$ of unit radius ($a_i = 1$). (a): A comparison of $C_i(\kappa_i)$ numerically computed from (\ref{['eq:Cmu_def0']}), with the one-, two-, and three-term approximations obtained from (\ref{['eq:Cmu_Taylor']}) and (\ref{['eq:cn_exact']}), valid for $\kappa_i \ll 1$. (b): The sigmoidal approximation (\ref{['mfpt:sigmoidal_2']}) provides a decent approximation of the numerical result for $C_i(\kappa_i)$ on the full range $\kappa_i >0$.
• Figure 2: Rescaled effective reactivity ${\mathcal{K}}_{\rm eff} R/D$ as a function of the patch surface fraction $f = {\varepsilon}^2/4$ for a single absorbing patch of radius ${\varepsilon}$ on the unit sphere (with ${\mathcal{K}} = \infty$). Comparison between Monte Carlo simulations (with $M = 10^5$ realizations and $a = 10^{-2}$), the empirical approximation (\ref{['eq:keff_Dagdug']}), and semi-analytical solution from Traytak95.

Theorems & Definitions (4)

• Lemma 2.1
• Lemma 2.2
• Proposition 1
• Lemma C.1