Numerical Simulation of Diffusion in Cylindrical Pores: The Influence of Pore Radius on Particle Capture Kinetics

Abstract

The diffusion and trapping of particles in complex porous media are fundamental processes in materials science and bioengineering. This study systematically investigates the influence of pore radius on particle capture kinetics within a three-dimensional cylindrical pore containing randomly distributed absorbing traps. Numerical simulations were performed for a wide range of pore radii (from 3a to 81a, a is a minimal length of the problem, arbitrary unit) and trap concentrations M (from 100 to 5090, these numbers are determined by the pore geometry) using a random walk algorithm. The particle lifetime (τ), characterizing the capture rate, was calculated and analyzed. Results reveal three distinct capture regimes dependent on trap concentration: a diffusion-limited regime at low concentration M (<1000), a transition regime at medium M (1000 < M < 2000), and a trap-density-dominated saturation regime at high M (>2000). For each regime, optimal approximating functions for τ(M) were identified. Furthermore, empirical relationships between the approximating coefficients and the pore radius were derived, which enable the prediction of particle lifetimes. The findings demonstrate that while the pore radius significantly impacts capture kinetics at low trap densities, its influence diminishes as trap concentration increases, converging towards a universal behavior dominated by trap density.

1. Introduction

The foundational theory of random walks on lattices with traps was established in [1,2,3,4], while the diffusion-reaction kinetics among traps was further developed in [5,6]. Interest is also driven by the expected new features in particle transport, and its applications [7,8,9,10,11]. A particularly relevant application involves investigating diffusion and trapping processes of active particles within porous dielectric low-k materials [12,13,14,15]. This problem is relevant for stability, mechanical strength and process compatibility of low k materials [16].

Additionally, the problem of diffusion in complex heterogeneous media and trapping also appeared in emerging bioengineering fields, e.g., in bone nanostructure [17,18], cartilage [19], or brain extracellular space [20]. This problem is studied by different methods, including numerical simulation [21,22,23]. Furthermore, the process of diffusion in pores has a directed form and this driven transport is called active and it is described in some experimentally important situations. This process was recently considered in detail in [24,25,26].

In parallel, significant research attention has recently focused on the dynamic aspects of capture processes. For instance, studies of active (self-propelled) particles in confined geometries have demonstrated a marked deviation of capture kinetics from passive diffusion, largely due to effects like boundary accumulation and motility-induced clustering [27,28]. Furthermore, research into the influence of external fields (e.g., electrical, flow) has revealed complex, nonlinear dependencies in capture probability [29,30]. Modern hybrid and accelerated numerical methods (e.g., [31]) now enable the detailed simulation of such intricate dynamic scenarios within complex geometries. The present study addresses a fundamental yet insufficiently explored question within this broader context: the role of a purely static geometric parameter—the pore radius—in the absence of external fields. Establishing this baseline understanding serves as a necessary foundation for the future analysis of more complex, dynamic cases involving active transport or field-driven processes.

Recently the theory of the capture on traps in the electric field was developed [32] and the numerical simulation of influence of the electric field in two-dimensional homogeneous media was performed [33]. However, these results are not included in this paper, because the driven transport in porous media requires a separate study.

In our previous study [34], we reported numerical simulations of particle diffusion in a three-dimensional cylindrical pore with traps for some values of radius of pore R = 3a, 9a, 27a, and 81a, where a denotes the characteristic system length scale. Notably, the behavior observed for R = 3a, 9a, and 27a differed significantly from that for R = 81a.

In this article, we study particle capture in cylindrical pores to account for the influence of pore radius on particle capture kinetics in a wide range of radius such as R = 3a, 6a, 9a, 15a, 18a, 21a, 27a, 30a, 40a, 50a, 60a, 70a, 81a in more details. Through comprehensive numerical experiments spanning a broad range of pore radii, we systematically investigate the influence of pore radius on particle capture kinetics by analyzing particle lifetime statistics, developing mathematical approximations for these lifetimes, and characterizing the behavior of the corresponding approximating functions as a function of pore radius.

2. Problem Statement and Simulation Algorithm

We numerically investigate the diffusion of active particles within a cylindrical pore of radius R and height H as a simplified model of porous media. The inner pore surface contains randomly distributed absorbing traps, with the trap concentration varied systematically. The initial condition corresponds to a uniform density of N₀ particles released at the pore entrance (the Z = 0 plane), simulating a constant particle flux. This setup is motivated by applications such as the diffusion of active species (e.g., oxygen, termed O-particles) through porous materials containing trapping sites (C-particles) [35,36]. While prior studies have employed one-dimensional approximations with multiple particle types [37,38], this work focuses explicitly on the influence of the pore radius R on the particle capture kinetics in a three-dimensional geometry.

A comprehensive set of pore radii R was selected to systematically investigate the geometric influence across a broad spectrum. The initial series R = 3a, 9a, 27a, 81a [34], chosen as a geometric progression, uncovered a qualitative shift in capture kinetics at R = 81a. To perform a detailed analysis of this transitional regime and to establish continuous parameter dependencies on R, the present study employs an extended series of radii. This expanded set incorporates both values that are multiples of the fundamental scale 3a (e.g., 15a, 18a, 21a, 30a) and values that are multiples of 10a (i.e., 40a, 50a, 60a, 70a). This approach ensures uniform coverage of the range from highly confined (3a) to relatively large (81a) pores and facilitates the detection of potential nonlinearities in the scaling relationships.

A critical feature of our comparative analysis is that for each fixed trap number M, an identical set of angular and longitudinal coordinates (θ,z) was used to position traps on the surface for every pore radii R. This methodological choice guarantees that any differences in the calculated mean lifetime τ arise solely from changes in the pore geometry (i.e., the radius R) and are not confounded by random variations in the trap layout. As a direct consequence, the fluctuations in τ as a function of M observed in Figure 1 (and related figures) are not statistical noise but instead reflect the deterministic influence of specific, shared spatial patterns in the trap distributions—such as local clusters or voids along the pore axis—on the capture kinetics. These patterns are identical across all radii, leading to the synchronized fluctuations visible in the plots.

In this paper, we use the same algorithms as in our previous work [34], which include two main components: the Algorithm for Simulations of Diffusion Processes in Media with Traps and the Algorithm for Placing Traps. We described these algorithms in Appendix A.

3. Modeling Process and Results of Numerical Simulation

3.1. Particle Lifetime and the Three Capture Regimes

To characterize the rate of particle capture by traps, we define the lifetime parameter, denoted as τ, based on the exponential decay model:

$$N_t=N_0\exp \left(-{\displaystyle \frac{t}{\tau }}\right)$$

(1)

Here, N₀ is the initial number of particles, and N_t is the number of active particles at iteration t. The lifetime τ corresponds to the characteristic time (in iteration steps) at which the number of particles decreases by a factor of e. Equation (1) describes the simple process of capture on traps with “death” of particle after capture, and the trap can capture particle only one time. However, it differs from CTRW transport [1]. In the CTRW the particles only delays on the traps and there is a possibility to leave the traps, usually described by delay function with the power law kernel.

Numerical experiments revealed that this exponential dependence stabilizes for pores with radii 3a, 9a, and 27a when the number of traps satisfies M ≥ 180 [34]. In contrast, for the larger pore radius 81a, stabilization requires a higher number of traps (M ≥ 220). Qualitatively, this behavior arises due to the increased diffusion volume and the consequently reduced probability of particles reaching the pore surface where traps are located.

In this study, we investigated cylindrical pores with radii R ∈ {3a, 6a, 9a, 15a, 18a, 21a, 27a, 30a, 40a, 50a, 60a, 70a, 81a}, where a denotes the characteristic length scale of the system. The computed lifetime values τ for each pore radius, derived from numerical simulations under consistent trap concentration and distribution conditions, are comprehensively tabulated in Appendix B.

In diffusion processes within media containing traps, a characteristic length scale emerges—the mean distance between traps—which introduces an associated with diffusion the timescale. For one-dimensional diffusion, this timescale is proportional to the square of the inter-trap distance. In higher dimensions, geometric constraints and pore curvature further influence the capture kinetics, necessitating the systematic analysis presented in this work. For one-dimensional diffusion the characteristic time is equal to

$$t_c={\displaystyle \frac{1}{D{c}^2}}$$

(2)

In this context, D represents the diffusion coefficient, and c denotes the concentration of traps.

To investigate the influence of pore radius on particle capture kinetics, we analyzed the asymptotic behavior of the lifetime (2) for pores with radii 3a, 15a, 18a, and 21a. We determine the dependence of the average lifetime ⟨τ⟩ on the number of traps M.

Analysis of the curves in Figure 1 reveals that for a fixed pore radius R, a relative reduction in the lifetime (e.g., by 50%) corresponds to a comparable relative increase in the trap number M. This behavior suggests that the dependence τ(M) for a given R can be approximated by a power-law function within the identified concentration intervals. Specifically, in the regime M < 1000, the relationship closely follows τ∝1/M², characteristic of a diffusion-limited capture process. In contrast, in the saturation regime M > 2000, the effective power-law exponent approaches zero, indicating a weak dependence on M. This observed scaling confirms the universality of the identified regimes and provides a means to predict kinetic changes due to variations in M within a single regime.

The relationship between ⟨τ⟩ and M, illustrated in Figure 1, exhibits two different regimes:

1.

Low trap concentration regime (M < 1000):

The lifetime ⟨τ⟩ decreases rapidly with increasing M, consistent with the scaling according to (2). This behavior reflects a diffusion capture process where the mean distance between traps governs the timescale of particle trapping.

2.

Trap-density-dominated saturation regime (M > 1000):

The lifetime reaches a plateau, indicating that the capture process becomes less sensitive to further increases in trap concentration. In this regime, the system transitions to a state where capture kinetics are dominated by other factors, such as spatial coverage of traps or geometric constraints, rather than by diffusion alone.

These results highlight the critical role of trap concentration and pore radius in determining capture efficiency and provide insight into the underlying mechanisms governing particle trapping in confined geometries.

We analyze the dependence of particle lifetime τ on trap concentration M over the extended range M ∈ [1000; 5090], as shown in Figure 2.

A detailed examination of the plots (Figure 1 and Figure 2) and numerical data allows us to specify the following key observations:

• The observed increase in the mean lifetime τ with pore radius R for a fixed trap number M directly results from the longer average time required for a particle to diffuse from the pore entrance to the cylindrical wall where the traps reside.
• The functional form of the τ(M) dependence shows a high degree of similarity across different pore radii, pointing to a universal character of the capture kinetics, particularly in the regime of high trap concentrations.
• The oscillatory behavior of the τ(M) curves for neighboring M values stems from the stochastic nature of the trap placement algorithm. This leads to inherent fluctuations in the local trap density—specifically, variations in their distribution along the longitudinal coordinate z—for each individual spatial configuration.

To elucidate the influence of pore radius on particle capture kinetics, we examine in detail the lifetime dynamics within small-radius pores. Figure 3 presents the dependence of the lifetime τ on the number of traps M for pore radii 3a, 6a, and 9a. The behavior of τ for these small radii is nearly identical, suggesting that geometric effects are minimal in this regime.

The nearly identical behavior of the curves for different radii in both regimes demonstrates the unified capture kinetics in small pores, where geometric differences have minimal effect on the diffusion-limited trapping process.

For quantitative analysis, we compute the lifetime difference Δτ = τ(3a) − τ(9a) to compare capture efficiency between pores of radii 3a and 9a. Figure 4 illustrates the dependence of Δτ on M. Positive values of Δτ (above the abscissa) indicate a longer lifetime in the 3a pore, implying slower capture, while negative values reflect a longer lifetime in the 9a pore. The observed variation in Δτ with M highlights subtle differences in capture kinetics due to pore size, even when radii are relatively small.

Figure 1. Dependence of particle lifetime (τ) on the number of traps (M) for different pore radii. The top figure (a) corresponds to the low trap concentration regime (M < 1000), while the bottom figure (b) shows the high concentration regime (M > 1000). Figures illustrate the transition from a steep, radius-dependent decrease in lifetime at low M to a convergence towards a plateau at high M, highlighting the influence of trap density.

Figure 1. Dependence of particle lifetime (τ) on the number of traps (M) for different pore radii. The top figure (a) corresponds to the low trap concentration regime (M < 1000), while the bottom figure (b) shows the high concentration regime (M > 1000). Figures illustrate the transition from a steep, radius-dependent decrease in lifetime at low M to a convergence towards a plateau at high M, highlighting the influence of trap density.

Figure 2. Dependence of particle lifetime (τ) on trap concentration (M) for different pore radii. The top figure (a) displays the medium concentration regime (1000 < M < 2000), while the bottom figure (b) shows the high concentration regime (M > 2000). The plots demonstrate the continuing convergence of lifetime values across different pore radii with increasing M, indicating the transition to a trap-density-dominated capture process where geometric parameters become less significant.

Figure 2. Dependence of particle lifetime (τ) on trap concentration (M) for different pore radii. The top figure (a) displays the medium concentration regime (1000 < M < 2000), while the bottom figure (b) shows the high concentration regime (M > 2000). The plots demonstrate the continuing convergence of lifetime values across different pore radii with increasing M, indicating the transition to a trap-density-dominated capture process where geometric parameters become less significant.

Figure 3. Dependence of particle lifetime (τ) on trap concentration (M) for small-radius pores (3a, 6a, 9a). The top figure (a) shows the low concentration regime (M < 1000), while the bottom figure (b) presents the high concentration regime (M > 1000).

Figure 3. Dependence of particle lifetime (τ) on trap concentration (M) for small-radius pores (3a, 6a, 9a). The top figure (a) shows the low concentration regime (M < 1000), while the bottom figure (b) presents the high concentration regime (M > 1000).

Figure 4 shows that the absolute magnitude of the lifetime difference between the 3a and 9a pores, ∣Δτ∣, decreases monotonically as the number of traps M increases. The maximum absolute values of Δτ occur in the low trap concentration regime (M < 1500), while for M→3000, the difference diminishes sharply. The observed decrease in the absolute difference ∣Δτ∣ with increasing M (Figure 4) can be largely attributed to the reduction in the overall magnitude of the mean lifetime τ and the corresponding decrease in the variance of its estimate. At low M, τ values are large and exhibit considerable dispersion, resulting in significant fluctuations in Δτ. A more direct indication of a transition to a regime where pore geometry exerts a diminished influence is the universalization of the functional form of the τ(M) dependence across different radii. This is evident in Figure 1, where the curves for various R values converge, showing a similar functional trend.

Figure 4. Difference in particle lifetime (Δτ = τ_3a − τ_9a) between pores of radii 3a and 9a as a function of trap concentration (M). The top (a) shows the low concentration regime (M < 1000), where significant differences in lifetime are observed due to geometric effects. The bottom (b) displays the high concentration regime (M > 1000), demonstrating the convergence of lifetime values as trap density becomes the dominant factor in capture kinetics. Positive Δτ values indicate longer lifetime in the 3a pore.

Figure 4. Difference in particle lifetime (Δτ = τ_3a − τ_9a) between pores of radii 3a and 9a as a function of trap concentration (M). The top (a) shows the low concentration regime (M < 1000), where significant differences in lifetime are observed due to geometric effects. The bottom (b) displays the high concentration regime (M > 1000), demonstrating the convergence of lifetime values as trap density becomes the dominant factor in capture kinetics. Positive Δτ values indicate longer lifetime in the 3a pore.

To further quantify the variability in lifetime among small-radius pores (3a, 6a, 9a), we analyzed the range of τ values, defined as

$${\mathrm{R}}_{\mathsf{\tau }} \left(\mathrm{M}\right)=\max \left({\mathsf{\tau }}_{3a},{\mathsf{\tau }}_{6a},{\mathsf{\tau }}_{9a}\right)-\min \left({\mathsf{\tau }}_{3a},{\mathsf{\tau }}_{6a},{\mathsf{\tau }}_{9a}\right),$$

(3)

Figure 5 presents the dependence of R_τ on M. The range R_τ provides a measure of the dispersion in lifetimes due to pore size differences, complementing the pairwise comparison in Figure 4.

For small radii (3a, 6a, 9a), the absolute magnitude of this range is indeed substantial: in the low-concentration regime (M < 1000), Rτ reaches up to ~50,000 iterations (Figure 5a). Expressed relative to the mean lifetime τ for a given M, this corresponds to a spread of approximately 10–25%. This suggests that while the qualitative shape of the kinetic curves is similar for these small pores, the quantitative efficiency of capture differs notably. For M > 1000, the value of Rτ(M) decreases, reflecting both a faster overall capture process and a diminishing influence of the pore radius as trap density becomes the dominant factor.

To further elucidate these observations, we analyze the pore size dependence of the maximum lifetime τ_max.

Table 1. Pore radius corresponding to the maximum lifetime τmax for varying trap concentrations M.

Pore Radius	Number of Configurations with Maximum Lifetime τmax	%
3a	40	8.1
6a	56	11.3
9a	399	80.6

summarizes the pore radius at which the maximum lifetime τ_max is achieved for each trap concentration M, providing insight into how optimal capture conditions shift with system parameters.

For trap concentrations exceeding M > 2000, the maximum lifetime τ_max was predominantly observed in pores of radius 9a, indicating that larger pores can sustain longer particle lifetimes under high trap density conditions. This result highlights the critical interplay between geometric confinement and trap density in determining capture kinetics. Specifically, while higher trap concentrations generally enhance capture efficiency, the pore radius modulates this effect by influencing diffusion paths and particle-surface interaction probabilities.

These findings underscore the necessity of simultaneously optimizing both geometric parameters (e.g., pore radius) and trap density when designing porous materials for applications requiring controlled particle capture or retention. Future work could extend this analysis to more complex pore geometries and broader parameter ranges to further refine design principles.

3.2. Approximation of Lifetime Dependence on Trap Concentration

As demonstrated in previous work [29], the lifetime τ, which characterizes the capture rate by traps, exhibits a strong dependence on trap concentration M. To quantitatively analyze this relationship, the range of M was divided into three distinct intervals based on concentration levels:

• Low concentration: M < 1000
• Medium concentration: 1000 < M < 2000
• High concentration: M > 2000

For each interval, three candidate functions were evaluated to approximate τ(M):

• Hyperbola: τ(M) = a/M + b
• Quadratic hyperbola: τ(M) = a/M² + b
• Parabola: τ(M) = a + bM + cM²

Approximation was performed using nonlinear least-squares optimization via the curve_fit function from the scipy module in Python 3.8. The quality of each fit was assessed using the mean absolute error (MAE) between predicted and simulated values. Detailed numerical results and error metrics are provided in Appendix C.

For all pore radii studied, the optimal approximating functions are established:

• M < 1000: Quadratic hyperbola (captures initial steep decay)
• 1000 < M < 2000: Hyperbola (balances simplicity and accuracy)
• M > 2000: Parabola (describes saturation behavior effectively)

The average approximation error for the selected functions was below 15% for all pores, meeting the acceptable accuracy threshold.

These results indicate a transition in the functional form of τ(M) across concentration regimes, reflecting underlying changes in the dominance of physical mechanisms, such as diffusion-limited capture at low M and transition to density-driven saturation at high M.

The dependence of the approximating function coefficients on pore radius was analyzed for both low and medium trap concentration regimes. Figure 6 illustrates the relationship between the coefficients A and B of the quadratic hyperbola τ(M) = A/M² + B, used to model lifetime τ for M < 1000, and the pore radius R.

The nonlinear relationships demonstrate how geometric scaling affects capture kinetics parameters, with notable deviations observed for larger pore radii (R > 60a) due to finite-size effects and transition in diffusion regimes. A notable shift in the behavior of these coefficients is observed as the pore radius increases, with significant deviations occurring for larger radii (60a, 70a, 81a). These deviations can be attributed to the following factors:

• Transition to a qualitatively distinct diffusion regime, where bulk diffusion characteristics become more dominant;
• Manifestation of finite-size effects, as the pore dimensions approach the scale of the system;
• Alterations in the nature of particle interactions due to reduced confinement.

The relationship between the coefficients and pore radius is nonlinear, particularly for coefficient B. A second-order polynomial approximation was applied to quantify this dependence:

For coefficient A: A(R) = (175.637 R² – 604.571 R + 2.53 × 10⁶) × 10⁴

with a coefficient of determination R² = 0.9973.

For coefficient B: B(R) = 10.441 R² – 28.387 R – 311.530

with R² = 0.9989.

The high R² values confirm the accuracy of the polynomial approximations, providing a reliable empirical basis for predicting coefficient behavior across varying pore radii.

Figure 7 illustrates the dependence of the coefficients A and B of the hyperbolic function τ(M) = A/M + B (used to approximate the lifetime τ for 1000 < M < 2000) on the pore radius R.

The observed trends highlight the influence of pore size on capture kinetics in the medium concentration regime. To quantify this dependence, a second-order polynomial approximation was applied to both coefficients:

For coefficient A: A(R) = (0.827 R² + 1.567 R + 3856.749) × 10⁴

For coefficient B: B(R) = 1.396 R² – 42.698 R – 13,788.25

The dependencies of the coefficients A and B on the pore radius R (Figure 6 and Figure 7) provide a quantitative characterization of geometric effects on the capture kinetics. The monotonic increase in coefficient B with R shows that the asymptotic lifetime in the saturation regime is governed by the time required for diffusion to the pore wall—a timescale that grows with the pore radius. Coefficient A captures how the sensitivity of the lifetime τ to the trap concentration M is itself modulated by the pore size.

The high-order polynomial model effectively captures the nonlinear relationship between the approximating function coefficients and the pore radius, underscoring the complex interplay between geometric confinement and trap distribution in the medium concentration regime. These empirical formulas serve as a predictive tool for estimating lifetime behavior across diverse pore sizes, highlighting the critical role of pore geometry in modulating capture efficiency.

The derived coefficients allow for the direct calculation of lifetime approximating functions, eliminating the need for further numerical simulations. To validate this approach, we assessed the performance of these functions against data for pore radii ranging from a to 81a. A comparison of the approximation errors is presented in Table 2. The first row of the table indicates the trap concentration range for which the lifetime was approximated, while the subsequent rows provide the following data:

• SO: The mean approximation error obtained from direct numerical fitting of the lifetime data.
• SO₂: The mean approximation error resulting from the application of the coefficient formulas derived above.

This comparison demonstrates the accuracy and utility of the empirical coefficient formulas in predicting lifetime behavior, reducing computational costs while maintaining high fidelity to the full numerical results.

Table 2. Comparison of approximation errors between direct fitting (SO) and formula-based (SO₂) methods.

	M < 1000		1000 < M < 2000
R	SO	SO2	SO	SO2
3a	13.854	13.822	13.845	14.131
6a	13.498	13.485	13.478	13.612
9a	13.462	13.551	13.474	13.432
15a	-	13.38	12.697	12.595
18a	12.275	12.283	11.757	11.589
21a	12.799	12.776	10.762	10.456
27a	12.026	12.12	10.322	10.302
30a	11.299	11.335	10	9.937
40a	11.164	11.1	8.216	8.373
50a	9.617	9.647	7.431	7.422
60a	10.427	10.173	6.113	6.117
70a	9.611	9.721	5.494	5.527
81a	8.768	9.023	4.653	4.63

Table 2. Comparison of approximation errors between direct fitting (SO) and formula-based (SO₂) methods.

	M < 1000		1000 < M < 2000
R	SO	SO2	SO	SO2
3a	13.854	13.822	13.845	14.131
6a	13.498	13.485	13.478	13.612
9a	13.462	13.551	13.474	13.432
15a	-	13.38	12.697	12.595
18a	12.275	12.283	11.757	11.589
21a	12.799	12.776	10.762	10.456
27a	12.026	12.12	10.322	10.302
30a	11.299	11.335	10	9.937
40a	11.164	11.1	8.216	8.373
50a	9.617	9.647	7.431	7.422
60a	10.427	10.173	6.113	6.117
70a	9.611	9.721	5.494	5.527
81a	8.768	9.023	4.653	4.63

During the approximation procedure using the curve_fit function for the pore radius R = 15a in the low concentration regime (M < 1000), a non-convergence error of the covariance matrix was encountered. However, application of the empirical coefficient formulas described in Section 3.2 yielded a satisfactory result, with a mean approximation error of 13.38.

In the low concentration range (M < 1000), the maximum discrepancy between the direct numerical approximation error (SO) and the formula-based error (SO₂) was 0.255 for R = 81a, while the minimum discrepancy was 0.004 for R = 50a. The average discrepancy in this regime was 0.084. For the medium concentration range (1000 <M <2000), the maximum discrepancy reached 0.306 (R = 21a), the minimum was 0.015 (R = 50a), and the average discrepancy was 0.103.

Analysis of the accuracy of the two approximation methods (direct numerical fitting, SO, and the formula-based approach, SO₂) across different trap concentration ranges revealed the following key patterns:

• High Consistency: Both methods exhibit exceptionally high consistency across all pore radii, with correlation coefficients r > 0.998 in both concentration ranges, confirming the reliability of the proposed empirical approach.
• Radius-Dependent Accuracy: The mean approximation errors decrease systematically with increasing pore radius, reflecting the influence of geometric parameters on modeling accuracy.
• Small Discrepancies: The maximum discrepancies between SO and SO₂ are 0.084 for M < 1000 and 0.306 for 1000 < M < 2000, representing less than 3% of the average error values in each regime.
• Statistical Validation: The Wilcoxon signed-rank test confirmed the absence of statistically significant differences between the methods (p > 0.38), supporting the use of the formula-based approach for calculating approximating function coefficients.

These results demonstrate that the empirical coefficient formulas provide a computationally efficient and accurate alternative to direct numerical fitting, enabling reliable estimation of particle lifetimes without the need for extensive numerical experiments.

4. Discussion

Let us discuss the results obtained from the numerical simulation of particle diffusion in porous media. Our investigation reveals several key patterns regarding the influence of pore radius and trap concentration on the capture rate of diffusing particles:

1.

Behavior with Geometric Modulation: The lifetime (τ) of particles exhibits a similar dependence on the number of traps (M) across various pore radii. However, for significantly larger radii (R > 50a), this dependence undergoes substantial modifications. These changes are attributed to a transition into a qualitatively different diffusion regime, where finite-size effects and diminished confinement alter particle dynamics.

2.

Concentration-Driven Convergence: The difference in lifetime values for different pore radii markedly decreases as trap concentration increases. This convergence indicates that at high trap densities, the capture process becomes dominated by the abundance of traps rather than by geometric constraints.

3.

Identification of Capture Regimes: Three distinct operational regimes for particle capture were identified and characterized based on trap concentration:

• Low concentration regime (M < 1000): Capture is strongly influenced by diffusion limitations and pore geometry.
• Medium concentration regime (1000 < M < 2000): A transition region where both trap density and geometry play significant roles.
• High concentration regime (M > 2000): Capture is primarily controlled by trap density, with geometric parameters having a diminished effect.

4.

Physical Origin of the Observed Trapping Regimes: Role of Discrete Geometry and Trap Coverage. The empirical boundaries between the trapping regimes (at M ≈ 1000 and M ≈ 2000) can be attributed to the discrete geometry of the system and the trap placement algorithm. Since the cylinder height is H = 1000 lattice nodes and the traps are placed at integer z-coordinates, a condition M << 1000 means many discrete z-levels contain no traps. This creates localized regions with low trap density, allowing particles entering from the pore opening to diffuse deeper before being captured. This effect accounts for the relatively long lifetimes τ observed in the low-concentration regime. When M ≈ 1000, there is on average one trap per discrete z-value, leading to a near-continuous coverage of the pore wall. At this point, the system transitions to a regime where the search for a trap on this now densely populated surface becomes the rate-limiting step in the capture kinetics. A further increase in M to ~2000 results in an average of two traps per z-coordinate. This significantly minimizes local fluctuations in coverage and effectively eliminates the possibility of extended, trap-free regions. Consequently, the system enters a saturation regime where the wall can be considered uniformly and densely coated. In this regime, the lifetime τ becomes largely independent of M and is instead governed primarily by the time required for a particle to diffuse from the pore entrance to the wall—a timescale dependent on the pore radius R. Thus, the qualitative division into three distinct regimes is a general consequence of the chosen cylindrical geometry and trap-placement algorithm. The specific numerical values of the thresholds are directly linked to the discrete spatial scale of the model, particularly the cylinder height H.

5.

Optimal Approximation Functions: For each regime, an optimal approximating function was determined:

• Low concentrations (M < 1000): τ(M) = b + a/M²
• Medium concentrations (1000 < M < 2000): τ(M) = b + a/M
• High concentrations (M > 2000): τ(M) = a + bM + cM²

6.

Empirical Coefficient Calculation: We proposed a method for calculating the coefficients of these approximating functions using empirical relationships derived from the pore radius. A comparative evaluation with directly fitted functions demonstrates the high accuracy of this method, providing a reliable and computationally efficient predictive tool.

7.

Prospects for Application and Further Research. The quantitative relationships and empirical approximations derived for particle lifetime in cylindrical pores provide a foundational basis for engineering estimates in designing porous materials with tailored sorption or catalytic properties, where precise control over active particle delivery and capture rates is crucial. Specifically, these results are applicable to optimizing the microstructure of low-permittivity dielectrics (low-k materials) and analyzing transport phenomena in biomineralized tissues. To extend the model’s applicability, several important directions for future development are identified: (1) incorporating external force fields (e.g., electric, chemical potential gradients) to simulate driven or active transport; (2) considering dynamic (fluctuating) or partially reflective traps; (3) including interactions among diffusing particles; and (4) generalizing the geometry to more complex, fractal, or anisotropic pore networks. Pursuing these extensions will facilitate the creation of a more universal model, applicable to a broader range of challenges in filtration, separation, targeted drug delivery, and the analysis of intracellular transport.

The findings of this study are significant for the design and engineering of porous materials with tailored sorption properties. The developed approximation dependencies enable the prediction of particle capture kinetics across a wide range of geometric parameters, facilitating the optimization of material performance for specific applications.

The investigation of diffusion processes and capture by traps is also important for cell metabolism and protein activity [39]. The processes of water transfer are responsible for the diffusion of nutrients, metabolic exchange, and ion transport within bone structures. They contribute to the mechanism of bone adaptation, the stabilization of the mineral structure, and the interaction between minerals and collagen [40,41,42].

Another promising application is the modeling of migration and entrapment for heavy organic components, such as asphaltene clusters, within porous hydrocarbon reservoirs. Diffusive transport under a gravitational field and the subsequent precipitation of asphaltenes onto active mineral surface sites—which can be represented as traps in our framework—are potential key mechanisms in the formation of tar mats (high-viscosity zones critical for reservoir development) [43]. Our core model, extended to account for an external driving field (e.g., a concentration or gravity gradient), could provide a foundation for the quantitative assessment of the conditions and kinetics governing such capture processes.