Imperfect diffusion-controlled reactions on a torus and on a pair of balls

TL;DR

The paper addresses steady-state diffusion toward partially reactive targets in 3D and shows that the shape-reaction interplay is governed by the exterior Steklov spectrum. It introduces a general spectral representation for the concentration and diffusive flux, J = D C_A |∂Ω| Σ_k F_k /(μ_k^{-1} + D/κ), where μ_k are Steklov eigenvalues (geometric lengthscales) and F_k are mode weights, with special cases recovering the Smoluchowski and partial reactivity limits. For two geometries— a torus and a pair of equal spheres—the authors compute the Steklov spectrum efficiently using toroidal and bispherical coordinates, reveal that the first two eigenmodes dominate, and propose a robust two-term flux approximation J_app that matches the exact flux across reactivity regimes. The results illuminate the nontrivial coupling between target geometry and reactivity, provide practical tools for predicting reaction rates in complex shapes, and offer a pathway to encounter-based and heterogeneous reactivity modeling in diffusion-controlled processes.

Abstract

We employ a general spectral approach based on the Steklov eigenbasis to describe imperfect diffusion-controlled reactions on bounded reactive targets in three dimensions. The steady-state concentration and the total diffusive flux onto the target are expressed in terms of the eigenvalues and eigenfunctions of the exterior Steklov problem. In particular, the eigenvalues are shown to provide the geometric lengthscales of the target that are relevant for diffusion-controlled reactions. Using toroidal and bispherical coordinates, we propose an efficient procedure for analyzing and solving numerically this spectral problem for an arbitrary torus and a pair of balls, respectively. A simple two-term approximation for the diffusive flux is established and validated. Implications of these results in the context of chemical physics and beyond are discussed.

Figures (5)

• Figure 1: (Left) Illustration of a torus with major radius $H = 1$ and minor radius $R = 0.25$ (top) and its projection onto $xz$ plane (bottom). (Right) Illustration of a pair of identical balls of radius radius $R = 0.25$, whose centers are separated by distance $2H$ (top) and its projection onto $xz$ plane (bottom).
• Figure 2: The diffusive flux (rescaled by $DC_A H$) onto the torus with the fixed major radius $H$ and three minor radii $R$ (see the legend), as a function of $\kappa H/D$. Symbols present the spectral expansion (\ref{['eq:J_spectral']}), referred to as the exact solution. Solid lines indicate its two-term approximation (\ref{['eq:J_approx']}), dashed line shows the asymptotic relation (\ref{['eq:J_asympt']}) for the case $R/H = 0.1$, whereas dash-dotted lines illustrate the limiting values (\ref{['eq:JSmol_general']}), with the capacitance given by Eq. (\ref{['eq:Rc_torus']}).
• Figure 3: The diffusive flux (rescaled by $DC_A H$) onto a pair of equal spheres of radii $R$ (see the legend), whose centers are separated by a fixed distance $2H$, as a function of $\kappa H/D$. Symbols present the spectral expansion (\ref{['eq:J_spectral']}), referred to as the exact solution. Solid lines indicate its two-term approximation (\ref{['eq:J_approx']}), dashed line shows the asymptotic relation (\ref{['eq:J_asympt2']}) for the case $R/H = 0.1$, whereas dash-dotted lines illustrate the limiting values (\ref{['eq:JSmol_general']}), with the capacitance given by Eq. (\ref{['eq:Samson']}).
• Figure 4: Several axially symmetric Steklov eigenfunctions $V_{0,n}^{\pm}$ for the torus with $R = 0.5$ and $H = 1$. The cross-section in the $xz$ plane with $x > 0$, that corresponds to $\phi = 0$, is shown. Top row: eigenfunctions $V_{0,n}^{+}$ that are symmetric with respect to the horizontal plane; bottom row: eigenfunctions $V_{0,n}^{-}$ that are antisymmetric.
• Figure 5: (a) Rescaled eigenvalues $R \mu_{m,0}^+$ of the exterior Steklov problem for the torus with major and minor radii $H$ and $R$. Symbols present the numerical results obtained by diagonalizing the truncated matrices $\mathbf{M}^{+}_m$ of size $10\times 10$, whereas lines show the asymptotic relation (\ref{['eq:mu_ext_torus']}). (b) The difference $|R\mu_{m,n}^{\pm} - n|$ between the eigenvalues $\mu_{m,n}^{\pm}$ (rescaled by $R$) and their limits $n$. Symbols present the results for $\mathbf{M}^{+}_m$, lines show the results for $\mathbf{M}_m^{-}$. In both cases, the truncated matrices $\mathbf{M}^{\pm}_m$ of size $10\times 10$ were numerically diagonalized.