# Diffusion-Limited Reaction Kinetics of a Reactant with Square Reactive Patches on a Plane

## Abstract

**:**

## 1. Introduction

## 2. Reaction Models and Computational Methods

#### 2.1. Reaction Models

#### 2.2. Computational Methods

^{®}v.5.4, COMSOL AB, Stockholm, Sweden], an FEM software package. In our models, to achieve high accuracy in the FEM calculations, we use very fine meshes in the regions of and near the reactive patches; in the mesh construction, the mesh for the region of the reactive patches is “extremely fine” in the predefined setting of COMSOL or is finer than the “extremely fine” setting, while the mesh for the remaining regions is “fine” or has a more refined mesh setting. With this mesh setting, the calculations with COMSOL yield high accuracy. For example, in reaction models where the exact solutions are known, the relative error between the exact and numerical solutions is less than or approximately 1% [25]. The FEM calculation is typically finished within a few minutes on a standard desktop computer, and even for demanding cases, it is finished at most within a few hours. COMSOL is also used to visualize the results shown in the figures with xmgrace (http://plasma-gate.weizmann.ac.il/Grace/), which is used for plotting and fitting curves to the data.

## 3. Results and Discussion

#### 3.1. First Model Type: N× N SQUARE Reactant Model

#### 3.2. Second Model Type: N × N Square Reactant Model

#### 3.3. Dependence of Kinetics on the Fraction of the Total Reactive Patch Area

#### 3.4. Effect of the Reactive Patch Distribution on the Kinetics

_{81}C

_{9}$\approx 2.61\text{}\times \text{}{10}^{11}$; in fact, when we consider the symmetry of arrangements and PBCs, the actual number of symmetrically unique patch configurations is less than this number. Therefore, instead of considering all the cases, we consider 10 representative cases from the one with mostly aggregated patches to the one with largely separated patches, as shown in Figure 9a. We also use the average value of the pairwise distance as a way to measure the degree of scattering of patches over the reactant surface. For each case, we calculate the normalized rate constant $\overline{k}$ for the first and second model types. We display the results as a plot of the normalized rate constant $\overline{k}$ versus the dimensionless average pairwise distance ${\overline{d}}_{2}/a$ or ${\overline{d}}_{2}\left(\mathrm{PBC}\right)/a$, as shown in Figure 9b.

#### 3.5. Symmetric Reactive Patch Distributions

^{2}, 2

^{2}, 3

^{2}, 4

^{2}, 5

^{2}, 6

^{2}, 7

^{2}, and 8

^{2}and arrange the patches symmetrically (see Figure 10a). Note that the case with ${n}_{p}\text{}$= 3

^{2}is exactly the same as Case 6 of Figure 9a. For each case, we calculate the normalized rate constants $\overline{k}$ for the first and second model types, the results of which we present in Figure 10b. The general trend is that the normalized rate constant $\overline{k}$ increases with ${n}_{p}$. As a result, for the first type of model, the case with ${n}_{p}\text{}=\text{}$8

^{2}and $\sigma \text{}$= 1/9 has a high rate constant (0.807), comparable to the rate constant (0.827) of the case with ${n}_{p}\text{}=\text{}$1 and $\sigma \text{}$= 0.7 (see Figure 8). However, for the second type of model, even for ${n}_{p}\text{}=\text{}$1, the normalized rate constant $\overline{k}$ is very high (0.891); thus, the enhancement of $\overline{k}$ due to the division and separation of patches is not as dramatic as that in the first type of model. Another feature in the plots of Figure 10b is that as ${n}_{p}$ increased, the effect of the enhancement of $\overline{k}$ is reduced; the enhancement is largest when ${n}_{p}$ changed from 1 to 4.

#### 3.6. Multiple Divisions of Patches and the Power–Law Relationship

#### 3.6.1. Analysis for the Case of $\sigma =$ 1/9

#### 3.6.2. Analysis for the Cases of $\sigma =$ 0.01 and 0.001

## 4. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of our first chemical reaction model. (Left) One reactant molecule A with an N × N cellular surface (black lines) is located at the center of the coordinate system and is on the x-y plane at z = 0. The outer boundary condition is represented by the blue lines. (Right) The concentration profiles of reactant B from the Smoluchowski solutions for the N × N square models of N = 2 and 3 with certain reactive cell distributions are represented by color maps. The length of each side of reactant A is 1.8 $l$ ($l$: Unit length), and the distance from the center to the outer boundary (R

_{outer}) is 10 $l$. The red cells in the N × N square reactants indicate the reactive patches.

**Figure 2.**Schematic diagram of our second chemical reaction model. (Center) The arrangement of reactive patches on the bottom surface at z = 0 is repeated through periodic boundary conditions (PBCs) for the x-y dimensions. The blue top surface at z =$\text{}{R}_{outer}$ corresponds to the outer boundary in the first model type in Figure 1, and the concentration on the blue surface is normalized to 1. (Left) The concentration profile of reactant B obtained from the Smoluchowski solution for the 3 × 3 square model with a certain reactive cell distribution is represented by a color map. (Right) The concentration profile of reactant B for the 2 × 2 square model with a certain reactive cell distribution is represented. In this unit cell, the length of each side of the bottom surface at z = 0 is 1.8 $l$ ($l$: Unit length), and the distance from the bottom surface to the top surface (${R}_{outer}$) is 10 $l$. The red cells in the N × N square reactants indicate the reactive patches.

**Figure 3.**Eight configurations are generated from the configuration of reactive patches are shown on the left by symmetric operations of rotations around the z-axis out of the page and reflections through the planes represented by the dotted lines.

**Figure 4.**(

**a**) Average pairwise distance between reactive patches (cells) (${\overline{d}}_{2}$), the number of all configurations kinetically equivalent to a representative configuration (${n}_{c}$), and the normalized rate constant ($\overline{k}$) of a representative configuration in the 2 × 2 square model of the first type. All representative configurations from 0 cells (no reactive) to 4 cells (full reactive) are considered. (

**b**) ${\overline{d}}_{2}$, ${n}_{c}$, and $\overline{k}$ of a representative configuration in the 3 × 3 square model of the first type. All representative configurations from 0 cells (no reactive) to 9 cells (full reactive) are considered. For a given number of reactive cells, we arrange the configurations in order of increasing ${\overline{d}}_{2}$. Note that the unit of ${\overline{d}}_{2}$ is the length of unit cell $a$ of the N × N square model. $\langle \overline{k}\rangle $ represents the ensemble-averaged value of $\overline{k}$ for a given number of reactive cells.

**Figure 5.**(

**a**) Ensemble-averaged normalized rate constant $\langle \overline{k}\rangle $ as a function of the fraction of reactive cells over the entire reactant surface area ($\sigma $) for the 2 × 2 and 3 × 3 square reactant models of the first type. $\sigma $ is easily calculated by the number of reactive cells over the total number of cells. (

**b**) For the numbers of reactive cells from two to nine in the 3 × 3 square model, the plot of the normalized rate constant ($\overline{k}$) versus the dimensionless average pairwise distance (${\overline{d}}_{2}/a)$, which is the average pairwise distance (${\overline{d}}_{2}$) over the length of the unit cell ($a$). The dotted lines represent linear regression lines obtained from the xmgrace program. In the linear regressions, the correlation coefficients are 0.972 (2 cells), 0.978 (3 cells), 0.977 (4 cells), 0.974 (5 cells), 0.978 (6 cells), 0.971 (7 cells), and 0.968 (8 cells).

**Figure 6.**(

**a**) Average pairwise distance between reactive patches (cells) (${\overline{d}}_{2}$), the number of all configurations kinetically equivalent to a representative configuration (${n}_{c}$), and the normalized rate constant ($\overline{k}$) of a representative configuration in the 2 × 2 square model of the second type with PBCs. All representative configurations from 0 cells (no reactive) to 4 cells (full reactive) are considered. (

**b**) ${\overline{d}}_{2}$, ${n}_{c}$, and $\overline{k}$ of a representative configuration in the 3 × 3 square model of the second type with PBCs. All representative configurations from 0 cells (no reactive) to 9 cells (full reactive) are considered. For a given number of reactive cells, we arrange the configurations in order of increasing ${\overline{d}}_{2}$. Note that the unit of ${\overline{d}}_{2}$ is the length of unit cell $a$ of the N × N square model. $\langle \overline{k}\rangle $ represents the ensemble-averaged value of $\overline{k}$ for a given number of reactive cells.

**Figure 7.**(

**a**) Ensemble-averaged normalized rate constant $\langle \overline{k}\rangle $ as a function of the fraction of reactive cells over the entire reactant surface area ($\sigma $) for the 2 × 2 and 3 × 3 square models of the second type with PBCs. $\sigma $ is easily calculated by the number of reactive cells over the total number of cells. (

**b**) For numbers of reactive cells from two to nine in the 3 × 3 square model with PBCs, the plot of the normalized rate constant ($\overline{k}$) versus the dimensionless average pairwise distance (${\overline{d}}_{2}\left(\mathrm{PBC}\right)/a)$, which is the average pairwise distance (${\overline{d}}_{2}$) over the length of the unit cell ($a$) in the presence of PBCs. The dotted lines represent linear regression lines obtained from the xmgrace program. In the linear regressions, the correlation coefficients are 0.978 (3 cells), 0.918 (4 cells), 0.966 (5 cells), and 0.997 (6 cells).

**Figure 8.**Normalized rate constant $\overline{k}$ as a function of the surface coverage ratio of reactive patch $\sigma $ for a single reactive patch at the center of a reactant surface. $\sigma $ is calculated as the reactive square area over the total reactant square area. The insets show the color maps of the concentration of reactant B on the reactant A surface, where a reactive patch lies without PBCs (first model type, upper panel) and with PBCs (second model type, lower panel). The filled squares and circles indicate the values from Figure 4 and Figure 6, respectively.

**Figure 9.**(

**a**) Ten representative arrangements of patches in the 9 × 9 square reactant model with the average pairwise distance between reactive patches (cells) (${\overline{d}}_{2}$) for the first model type (without PBCs) and with ${\overline{d}}_{2}\left(\mathrm{PBC}\right)$ for the second model type (with PBCs). Here, $a$ is the length of the unit cell of the 9 × 9 square model. (

**b**) Plot of the normalized rate constant $\overline{k}$ versus the dimensionless average pairwise distance ${\overline{d}}_{2}/a$ for the first model type or ${\overline{d}}_{2}\left(\mathrm{PBC}\right)/a$ for the second model type, where $a$ is the length of the unit cell. Note that the cases giving the lowest and highest rate constants for the first type of model are Case 1 and Case 10. For the second type of model, Case 1 and Case 6 give the lowest and highest rate constants, respectively. The dotted lines represent linear regression lines obtained from the xmgrace program. In the linear regressions, the linear equations with correlation coefficients are $y\text{}\left(\overline{k}\right)\text{}=\text{}0.0726\text{}x\left({\overline{d}}_{2}/a\right)+\text{}0.217$ with 0.990 for the first model type model and $y\text{}\left(\overline{k}\right)\text{}=\text{}0.0303\text{}x\left({\overline{d}}_{2}\left(\mathrm{PBC}\right)/a\right)+\text{}0.844$ with 0.940 for the second model type.

**Figure 10.**(

**a**) A square reactant model with symmetric reactive patch distributions based on the uniform patch distributions. In this model, once the number of patches ${n}_{p}$ is determined, the spatial arrangement of patches is completely determined. The arrangements of the reactive patches (red) for ${n}_{p}\text{}$= 1

^{2}, 2

^{2}, 3

^{2}, 4

^{2}, 5

^{2}, 6

^{2}, 7

^{2}, and 8

^{2}are displayed. (

**b**) The normalized rate constant $\overline{k}$ as a function of ${n}_{p}$ for the first (red) and second (green) model types. The upper and lower insets present color maps of the concentration of reactant B for the first and second model types, respectively.

**Figure 11.**Sequential division-separation procedures of reactive patches on a reactant for a fixed value of $\sigma $ (

**a**) In the first division, a single reactive patch is broken into four smaller identical patches, and the smaller patches are separated and symmetrically arranged on the x-y plane. This division-separation procedure can be carried out any number of times $M$. (

**b**) Log-log plot of the normalized rate constant $\overline{k}$ versus the number of patches ${n}_{p}$ for the case of $\sigma \text{}$= 1/9 shown in Figure 10b. From the linear regressions, we obtain the following linear equations for ${\mathrm{log}}_{10}\overline{k}$ ($y$) versus ${\mathrm{log}}_{10}{n}_{p}$ ($x$): $y\text{}=\text{}0.2158x-0.4569$ for the first model type (correlation coefficient: 0.985) and $y\text{}=\text{}0.02348x-0.04335$ for the second model type (correlation coefficient: 0.960). These linear equations are represented by the dotted lines. The inset shows a comparison between $\overline{k}$ from Figure 10b and $\overline{k}$ from the linear regression.

**Figure 12.**Sequential division-separation procedures of reactive patches for the first model type with $\sigma \text{}$= 0.01 and the second model type (PBC) with $\sigma \text{}$= 0.01 and 0.001 (

**a**) The normalized rate constant $\overline{k}$ as a function of ${n}_{p}$ for the first (red) and second (green) model types. (

**b**) Log-log plot of the normalized rate constant $\overline{k}$ versus the number of patches ${n}_{p}$. From the linear regressions, we obtain the following linear equations for ${\mathrm{log}}_{10}\overline{k}$ ($y$) versus ${\mathrm{log}}_{10}{n}_{p}$ ($x$): $y\text{}=\text{}0.3923x-0.9914$ for the first model type with $\sigma \text{}$= 0.01 (correlation coefficient: 0.997), $y\text{}=\text{}0.1004x-0.1869$ for the second model type with $\sigma \text{}$= 0.01 (correlation coefficient: 0.970), and $y\text{}=\text{}0.2672x-0.4510$ for the second model type with $\sigma \text{}$= 0.001 (correlation coefficient: 0.988). These linear equations are represented by the dotted lines. The inset shows a comparison between $\overline{k}$ from (

**a**) and $\overline{k}$ from the linear regressions.

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**MDPI and ACS Style**

Eun, C.
Diffusion-Limited Reaction Kinetics of a Reactant with Square Reactive Patches on a Plane. *Symmetry* **2020**, *12*, 1744.
https://doi.org/10.3390/sym12101744

**AMA Style**

Eun C.
Diffusion-Limited Reaction Kinetics of a Reactant with Square Reactive Patches on a Plane. *Symmetry*. 2020; 12(10):1744.
https://doi.org/10.3390/sym12101744

**Chicago/Turabian Style**

Eun, Changsun.
2020. "Diffusion-Limited Reaction Kinetics of a Reactant with Square Reactive Patches on a Plane" *Symmetry* 12, no. 10: 1744.
https://doi.org/10.3390/sym12101744