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Reactive capacitance of flat patches of arbitrary shape

Denis S. Grebenkov, Raphael Maurette

TL;DR

This work develops a spectral framework for diffusion-controlled reactions on flat patches of arbitrary shape by introducing the reactive capacitance $C(\mu)$, which quantifies the total flux of reacting particles onto the patch. The authors formulate and analyze an exterior Steklov problem, derive two complementary spectral expansions, and establish monotonicity relations, probabilistic interpretations, and bounds that connect geometry, reactivity, and diffusion. A key practical contribution is an efficient numerical method to compute Steklov eigenpairs on arbitrary patches and a simple sigmoidal approximation $C^{\rm app}(\mu)$ that depends only on the patch area and electrostatic capacitance, with small relative error across shapes. The results enable accessible predictions of diffusion-controlled reaction characteristics in domains with multiple patches and offer insights for design and optimization in chemical physics and related fields. The approach provides a bridge between geometric shape, reactive kinetics, and diffusion, with potential applications to multi-patch targets and shape-optimization problems in heterogeneous media.

Abstract

We investigate the capacity of a flat partially reactive patch of arbitrary shape to trap independent particles that undergo steady-state diffusion in the three-dimensional space. We focus on the total flux of particles onto the patch that determines its reactive capacitance. To disentangle the respective roles of the reactivity and the shape of the patch, we employ a spectral expansion of the reactive capacitance over a suitable Steklov eigenvalue problem. We derive several bounds on the reactive capacitance to reveal its monotonicity with respect to the reactivity and the shape. Two probabilistic interpretations are presented as well. An efficient numerical tool is developed for solving the associated Steklov spectral problem for patches of arbitrary shape. We propose and validate, both theoretically and numerically, a simple, fully explicit approximation for the reactive capacitance that depends only on the surface area and the electrostatic capacitance of the patch. This approximation opens promising ways to access various characteristics of diffusion-controlled reactions in general domains with multiple small well-separated patches. Direct applications of these results in statistical physics and physical chemistry are discussed.

Reactive capacitance of flat patches of arbitrary shape

TL;DR

This work develops a spectral framework for diffusion-controlled reactions on flat patches of arbitrary shape by introducing the reactive capacitance , which quantifies the total flux of reacting particles onto the patch. The authors formulate and analyze an exterior Steklov problem, derive two complementary spectral expansions, and establish monotonicity relations, probabilistic interpretations, and bounds that connect geometry, reactivity, and diffusion. A key practical contribution is an efficient numerical method to compute Steklov eigenpairs on arbitrary patches and a simple sigmoidal approximation that depends only on the patch area and electrostatic capacitance, with small relative error across shapes. The results enable accessible predictions of diffusion-controlled reaction characteristics in domains with multiple patches and offer insights for design and optimization in chemical physics and related fields. The approach provides a bridge between geometric shape, reactive kinetics, and diffusion, with potential applications to multi-patch targets and shape-optimization problems in heterogeneous media.

Abstract

We investigate the capacity of a flat partially reactive patch of arbitrary shape to trap independent particles that undergo steady-state diffusion in the three-dimensional space. We focus on the total flux of particles onto the patch that determines its reactive capacitance. To disentangle the respective roles of the reactivity and the shape of the patch, we employ a spectral expansion of the reactive capacitance over a suitable Steklov eigenvalue problem. We derive several bounds on the reactive capacitance to reveal its monotonicity with respect to the reactivity and the shape. Two probabilistic interpretations are presented as well. An efficient numerical tool is developed for solving the associated Steklov spectral problem for patches of arbitrary shape. We propose and validate, both theoretically and numerically, a simple, fully explicit approximation for the reactive capacitance that depends only on the surface area and the electrostatic capacitance of the patch. This approximation opens promising ways to access various characteristics of diffusion-controlled reactions in general domains with multiple small well-separated patches. Direct applications of these results in statistical physics and physical chemistry are discussed.

Paper Structure

This paper contains 26 sections, 160 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Reactive capacitance
  3. Exterior Steklov problem
  4. Spectral expansion and basic properties
  5. Alternative spectral expansion
  6. Sigmoidal approximation and several bounds
  7. Probabilistic interpretations
  8. Domain monotonicity
  9. Large-reactivity limit
  10. Bounds for the principal eigenvalue
  11. Relation to the patch shape
  12. Numerical results
  13. Steklov eigenpairs
  14. Effect of patch anisotropy onto the eigenvalues
  15. Spectral weights
  16. ...and 11 more sections

Figures (11)

  • Figure 1: (a) A domain $\Omega$ with reflecting boundary $\partial\Omega_0$ (in gray) A flat patch $\Gamma$ on the horizontal plane of the upper half-space.
  • Figure 2: First 9 Steklov eigenfunctions $\Psi_k$ restricted to the circular patch of unit radius (the associated eigenvalues $\mu_k$ and weights $F_k$ are shown in the titles). These eigenfunctions were obtained by using a mesh with 688 triangles and 375 nodes.
  • Figure 3: First 9 Steklov eigenfunctions $\Psi_k$ restricted to the elliptic patch of semiaxes $0.5$ and $1$ (the associated eigenvalues $\mu_k$ and weights $F_k$ are shown in the titles). The eigenfunctions were obtained by using a mesh with 756 triangles and 413 nodes.
  • Figure 4: First 9 Steklov eigenfunctions $\Psi_k^N$ restricted to the elliptic patch of semiaxes $0.5$ and $1$ for the Neumann version (the associated eigenvalues are shown in the titles). The eigenfunctions were obtained by using a mesh with 756 triangles and 413 nodes.
  • Figure 5: First 9 Steklov eigenfunctions $\Psi_k$ restricted to the square patch $(-1,1)\times (-1,1)$ (the associated eigenvalues $\mu_k$ and weights $F_k$ are shown in the titles). The eigenfunctions were obtained by using a mesh with 674 triangles and 372 nodes.
  • ...and 6 more figures