# Improved boundary homogenization for a sphere with an absorbing cap of arbitrary size

Denis S Grebenkov

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Denis S Grebenkov. Improved boundary homogenization for a sphere with an absorbing cap of arbitrary size. The Journal of Chemical Physics, 2025, 163 (8), pp.086101. 10.1063/5.0290891. hal-05344164

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Improved boundary homogenization for a sphere with an absorbing cap of arbitrary size

Denis S. Grebenkov¹, *

¹Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS - Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France

(Dated: Received: July 16, 2025/ Revised version:)

Finding accurate approximations for the effective reactivity of a structured spherical target with a circular absorbing patch of arbitrary size is a long-standing problem in chemical physics. In this Communication, we reveal limitations of the empirical approximation proposed in [J. Chem. Phys. 145, 214101 (2016)]. We show that the original approximation fails at large patch surface fractions $\sigma$ and propose a simple amendment. The improved approximation is validated against a semi-analytical solution and is shown to be accurate over the entire range of $\sigma$ from 0 to 1. This approximation also determines the probability of reaction on the patch and the capacitance of such a structured target.

PACS numbers: 02.50.-r, 05.60.-k, 05.10.-a, 02.70.Rr

Keywords: partial reactivity, diffusion-controlled reactions, reactive patch, capacitance, mixed boundary condition

We revisit an emblematic problem of steady-state diffusion outside a three-dimensional inert ball of radius $R$ with a perfectly absorbing circular patch $\Gamma$ of arbitrary radius $a$ on its boundary (Fig. 1(a)). Fixing a concentration $C_0$ at infinity, the diffusive flux $J$ onto the absorbing patch can be expressed as $J = \int_{\Gamma} (D\partial_r c)dS$, where $D$ is the diffusion coefficient. Here the steady-state concentration $c(r, \theta, \phi)$, written in spherical coordinates, satisfies the Laplace equation, $\Delta c = 0$, with $c \to C_0$ as $r \to \infty$, and mixed Dirichlet-Neumann boundary conditions on the sphere (at $r = R$): $c = 0$ on the absorbing patch ($0 \le \theta < \epsilon$) and $\partial_r c = 0$ on the remaining reflecting boundary ($\epsilon \le \theta \le \pi$), with $\epsilon = 2 \arcsin(a/(2R))$. The effective reactivity $\kappa$ of such a structured target is then defined by requiring that the diffusive flux $J$ is equal to the flux of reacted particles on the homogenized spherical boundary of reactivity $\kappa$ [1]

$$J = \frac{J_S}{1 + D/(\kappa R)} \quad \Rightarrow \quad \frac{R}{D}\kappa = \frac{1}{J_S/J - 1}, \quad (1)$$

where $J_S = 4\pi DRC_0$ is the Smoluchowski flux onto a perfectly absorbing sphere of radius $R$.

As the above boundary value problem does not admit an explicit analytical solution, various numerical schemes and approximations were developed. In particular, an empirical approximate formula for the effective reactivity $\kappa$ was proposed in [2]:

$$\frac{R}{D}\kappa_{\text{emp}} = \frac{2}{\pi}\sqrt{\sigma}\frac{1 + 2.6\sqrt{\sigma} - 3.2\sigma^2}{(1 - \sigma)^{3/2}}, \quad (2)$$

where $\sigma = \pi a^2/(4\pi R^2)$ is the fraction of the sphere covered by the absorbing patch. This formula, which was constructed from the numerical values of the reactivity

estimated from Brownian dynamics simulations, satisfies the well-known asymptotic behavior in the small-$\sigma$ limit. We aim at revealing and amending limitations of this formula in the limit $\sigma \to 1$ when only a small fraction of the sphere is not absorbing.

To inspect this limit, we recall that the effective reactivity $\kappa$ is inversely proportional to the blockage coefficient of a potential flow in a tube with a blocking disk [3–5]. In the limit of a small blocking disk, the approximation (32) from [3] yields the following asymptotic behavior (see also the explicit relation (11.39b) from [6]):

$$\frac{R}{D}\kappa \approx \frac{R}{\sqrt{S}} \frac{3\pi^{3/2}}{4(1-\sigma)^{3/2}} = \frac{3\pi}{8}(1-\sigma)^{-3/2} \quad (\sigma \to 1), \quad (3)$$

where $S = 4\pi R^2$ is the surface area. This behavior also follows from the asymptotic analysis of the steric factor in [7]. In turn, the empirical formula (2) yields $\frac{R}{D}\kappa_{\text{emp}}(\sigma) \approx \frac{4}{5\pi}(1 - \sigma)^{-3/2}$, with a different numerical prefactor. This difference has a dramatic effect onto the effective reactivity. In particular, the relative error of the empirical approximation, $1 - \kappa_{\text{emp}}(\sigma)/\kappa(\sigma)$, approaches $1 - 32/(15\pi^2) \approx 0.78$ as $\sigma \to 1$. In other words, the empirical formula (2) significantly underestimates the effective reactivity for large $\sigma$ that limits the range of its applicability.

To illustrate these limitations, one has to solve the mixed boundary value problem for the concentration and then to compute the flux $J$. Traytak reformulated it in terms of dual series and found its formal semi-analytical solution, from which the effective reactivity follows as

$$\frac{R}{D}\kappa = (1/P - 1)^{-1}, \quad P = \frac{J}{J_S} = \sum_{k=0}^{\infty} [(\mathbf{I} - \mathbf{M})^{-1}]_{0,k} \mathbf{M}_{k,0}, \quad (4)$$

where $\mathbf{I}$ is the identity matrix, and the elements of the

<footnote>* denis.grebenkov@polytechnique.edu</footnote>
<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) A perfectly absorbing circular patch of radius a located on the North pole of the otherwise reflecting sphere of radius R. (b) Rescaled effective reactivity $\kappa R/D$ as a function of the patch surface fraction $\sigma$. Solid line shows the semi-analytical solution (4) obtained by truncating the matrix $M$ to the size $100 \times 100$, dash-dotted line indicates the asymptotic formula (3), dashed line presents the empirical approximation (2), while circles give the improved empirical approximation (6). (c) Relative error of empirical approximations (2, 6) as compared to the semi-analytical solution (4).</figcaption></figure>

infinite-dimensional matrix $M$ are

$$M_{l,m} = \frac{1}{2\pi(m+1)} \left( \frac{\sin((l+m+1)\epsilon)}{l+m+1} + \frac{\sin((l-m)\epsilon)}{l-m} \right), \quad (5)$$

with $l, m = 0, 1, 2, \dots$ (note that the second term should be understood as $\epsilon$ for $l = m$) [8]. We emphasize that this is the exact solution (see further extensions in [9]). A truncation of the matrix $M$ to a finite size $N \times N$ yields

an accurate approximation for $P$ and thus for $\kappa$, whose error decreases very rapidly as $N \to \infty$. Note that the ratio $P$ can be interpreted as the probability of reaction on the patch (before escaping to infinity).

The knowledge of the correct asymptotic behavior (3) and the possibility of an accurate computation of $\kappa(\sigma)$ via Eq. (4) allow us to remediate the empirical formula (2). For this purpose, we keep the structure of this formula but replace numerical coefficients 2.6 and -3.2 by adjustable constants $A$ and $B$. By imposing the asymptotic behavior (3), we first relate $A$ and $B$ in the leading order of the limit $\sigma \to 1$: $(2/\pi)(1+A-B) = 3\pi/8$, from which $B = 1 + A - 3\pi^2/16$. Then fitting $(1-\sigma)^{3/2}\kappa(\sigma)$ against the semi-analytical solution (4) over $\sigma$ from 0 to 1, we determine the optimal value $A \approx 2.32$, from which $B \approx 1.47$. We arrive thus to another empirical approximation,

$$\frac{R}{D} \kappa_{\text{new}}(\sigma) = \frac{2}{\pi} \sqrt{\sigma} \frac{1 + 2.32\sqrt{\sigma} - 1.47\sigma^2}{(1 - \sigma)^{3/2}}, \quad (6)$$

which correctly reproduces both limits $\sigma \to 0$ and $\sigma \to 1$.

We use the truncated semi-analytical solution (4) as a benchmark for illustrating the accuracy of both empirical approximations. To control the effect of the truncation order, $\kappa$ was computed via Eq. (4) for $N = 20$ and $N = 100$, and the maximal relative error between these two computations was 0.2% over the considered range of $\sigma$ from 0.01 to 0.99 (note that larger truncation orders may be needed for smaller $1 - \sigma$). Figure 1(b) illustrates the behavior of the effective reactivity $\kappa$ (rescaled by $R/D$) versus $\sigma$. First of all, one can appreciate the correct form of the asymptotic behavior (3) as $\sigma \to 1$. For small $\sigma$, the original approximation (2) is close to the semi-analytical solution, as expected. However, deviations start to emerge for $\sigma \gtrsim 0.6$ and become dramatic as $\sigma \to 1$, as discussed above. Finally, the improved empirical approximation (6) is accurate over the entire range of $\sigma$, not exceeding the relative error of 6%. This is clearly illustrated on Fig. 1(c).

In summary, we inspected limitations of the empirical approximation for the effective reactivity proposed in [2]. In the limit $\sigma \to 1$, this approximation captured correctly the exponent of the diverging behavior, $\kappa \propto (1 - \sigma)^{-3/2}$, but failed to get the correct prefactor. As a consequence, the relative error of this approximation was exceedingly high as $\sigma \to 1$. Using recent asymptotic results [3, 6], we amended this problem and proposed an improved version (6) of this approximation, which reproduces correctly both limits $\sigma \to 0$ and $\sigma \to 1$. Its accuracy was validated by comparison with the semi-analytical solution (4). Beyond its own interest, the effective reactivity $\kappa$ gives access to several important quantities such as the probability of reaction, $P = 1/(1 + D/(\kappa R))$, the capacitance of the structured target with a reactive patch, $C = \kappa/(4\pi D)$, as well as its hydrodynamic Stokes drag [2, 10]. Our improved formula (6) allow one to access these quantities for the entire range of the covered fraction $\sigma$ that would be difficult to achieve by other means.
ACKNOWLEDGMENTS

The author thanks Dr. A. T. Skvortsov for fruitful dis-
cussions and acknowledges the Alexander von Humboldt
Foundation for support within a Bessel Prize award.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are
available from the corresponding author upon reasonable
request.

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