Diffusion-controlled reaction rate to an active site in a spherical cavity: Extension of Berg's theory

TL;DR

This work addresses diffusion-controlled reactions inside a finite spherical cavity with an immobile reactive particle bearing an axially symmetric active patch. It generalizes Berg's classic theory to bounded domains by incorporating geometric confinement and surface anisotropy, using a one-patch Solc-Stockmayer model. A semi-analytical solution is developed via the dual series relations method and linked to a generalized separation of variables approach, reducing the problem to an infinite system of linear equations that is solved by truncation. The authors define a rate-correction factor J and an effective radius R_eff = R J, and provide numerical results showing how J depends on patch size and shell thickness, with validation against limiting cases and simple approximations, offering a framework for testing diffusion-controlled reactions in finite domains relevant to cellular contexts.

Abstract

This study is due to various applications in physics, chemistry and especially in biology, where both bounded configuration domain and chemical anisotropy could play a great part. In fact we generalize the well-known Berg theory, which describes diffusion-controlled reactions occurring within a spherically symmetric absorber-cavity system. The trapping probability and the reaction rate at which a small diffusing particle is captured by an axially symmetric one reactive patch absorber inside a spherical cavity were found semi-analytically and numerically by means of the dual series relations method. This approach leads to such incredibly fast convergence, that it may rightly be referred to as exact one. The results obtained can be used to test numerical programmes that describe diffusion-controlled reactions in real physical systems for reactants with arbitrary anisotropic reactivity, which are located inside of various cavities as well as in the unbounded domains. Moreover, we managed to find a close connection between the dual series relations method and the generalized method of separation of variables.

Figures (8)

• Figure 1: Geometric sketch of the axially symmetric one-active patch Solc-Stockmayer model. Panel (a): A test $B$-reactant and their source located at the outer boundary $\partial \Omega_0$ are given in black. Panel (b): The RP comprises an inert sphere (blue) and an active site, which is modeled by a spherical cap with opening angle $\theta_0$ (red). The inner boundary $\partial \Omega$ (blue) 2-partition: $\left\{ \partial \Omega_{D}, \partial \Omega_{N}\right\}$. In the both panels Cartesian coordinates are depicted in green.
• Figure 2: Geometry of the spherically symmetric Berg's model Berg93 for the limiting case when the RP (red) is chemically isotropic and ideally absorbing. The spherical shell adsorber is also shown in red. Hereinafter $\epsilon:=R/R_0$ (\ref{['thickness2']}).
• Figure 3: Schematic description of the decomposition procedure (\ref{['GenS']}): (a) solution $u^+_0$ inside the cavity sphere $\Omega_{\xi}^{0}$ (green); (b) solution $u^-$ outside the RP $\Omega_{\xi}^{-}$ (blue); (c) general solution $u=u^+_0+u^-$ in the intersection $\Omega_{\xi}^{+}=\Omega_{\xi}^{0} \cap \Omega_{\xi}^{-}$ (green).
• Figure 4: The rate correction factor $J\left( \theta_0, \epsilon\right)$ as a function on the angular size $\theta_0$ at four different magnitudes of the thickness ratio: $\epsilon = 10^{-4};\, 0.25;\, 0.5;\, 0.8$. Straight dashed lines indicate appropriate horizontal asymptotes as $\theta_0 \to \pi-$ (\ref{['Lim1']}).
• Figure 5: The correction factor $J\left( \theta_0; \epsilon \right)$ (red solid curves) with its zeroth $J^{(0)}\left( \theta_0 \right)$ (green dash-dot curves) and first $J^{(1)}\left( \theta_0; \epsilon \right)$ (blue dotted curves) approximations as function of $\theta_0$ at: (a) $\epsilon = 0.25$, (b) $\epsilon = 0.5$ and (c) $\epsilon = 0.8$.

Theorems & Definitions (23)

• Definition 3.1
• Remark 3.1
• Remark 4.1
• Remark 4.2
• Remark 5.1
• Remark 5.2
• Remark 6.1
• Remark 9.1
• Definition 10.1
• Definition 10.2