Numerical Simulation of Capture of Diffusing Particles in Porous Media

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript entitled "Numerical simulation of capture of diffusing particles in porous media" studies the process of diffusing particles being captured in porous media through numerical simulation. The research covers the diffusion and capture dynamics of particles in porous media with randomly distributed traps, as well as the dynamic changes of trap filling. The manuscript is well-structured, the research methods are clear, and the results are analyzed in detail, which has certain reference value for research in related fields. However, there is still room for improvement in some aspects of the manuscript to enhance its academic contribution and readability.

1. It is hoped that the authors will add a description of the main findings in the abstract, such as "a step-like behavior of the particle distribution function after a long time is found" to highlight the uniqueness and novelty of the research. At the same time, the potential impact of these findings on related fields can be briefly mentioned.
2. The introduction can be further strengthened to add a systematic review of existing research and a clear positioning of the work of this paper. It is recommended that the authors discuss the shortcomings of existing research in more detail in the introduction and how this paper fills these gaps. For example, some descriptions of the practical application background of particle diffusion and capture in porous media can be added to highlight the practical significance of the research.
3. The article lacks theoretical explanations for the observed phenomena, especially the step-like behavior of the particle distribution function at long time scales and the logarithmic time dependence of the boundaries of the trap filling region. It is recommended that the authors try to explain these phenomena theoretically or cite relevant theoretical work to support the experimental results. For example, these phenomena can be explained by analyzing the statistical properties of the trap distribution or the stochastic process of particle diffusion.
4. The experimental design section can be further optimized and a detailed discussion of the selection of experimental parameters can be added. It is recommended that the authors add more parameter combinations in the experimental design, such as different pore shapes and trap distribution patterns, to more comprehensively study the kinetic behavior of particle capture. At the same time, it is possible to consider adding sensitivity analysis of the experimental results to determine which parameters have the most significant impact on the capture time.
5. It is recommended that the authors describe the specific implementation process of the numerical simulation in more detail, including the programming language used, the specific application of numerical methods (such as finite difference method, Monte Carlo method, etc.), and how to ensure the accuracy and stability of the simulation. In addition, a detailed discussion of the boundary condition treatment during the simulation process can be added, such as how to deal with the reflection and absorption of particles on the pore boundaries.
6. The results analysis section can be further deepened to provide more quantitative data and statistical analysis. It is recommended that the authors provide more quantitative data, such as determining the degree of influence of different pore sizes and trap concentrations on the capture time through statistical analysis. At the same time, the error analysis of the experimental results can be added to evaluate the reliability and repeatability of the experimental results. In addition, the statistical properties of the particle distribution function (such as mean, variance, etc.) can be calculated to gain a deeper understanding of the dynamic behavior of particle capture. At the same time, more advanced data analysis methods, such as machine learning or data mining techniques, can be considered to mine the potential laws in the simulation data.
7. The discussion of the selection of simulation parameters can be added, such as why the range of 1000 to 10000 traps is selected, and the impact of these parameters on the results.
8. The conclusion section can be further strengthened to add further discussion and prospects of the research results. It is recommended that the authors discuss the potential applications of the research results in the conclusion, such as practical applications in low-k materials, and how to use these findings to optimize the performance of the materials. At the same time, the direction of future research is proposed, such as further studying the effects of different pore structures and trap distributions on particle capture dynamics.
9. The number of references in the article is small, and the references cited are relatively old. It is recommended that the author add more references, especially the latest research results in the field of particle diffusion and capture in porous media in recent years. In addition, some relevant review literature can be cited to help readers better understand the research status and development trends in this field.

Author Response

Dear Reviewer !

Thank you very much for yours comments, which improve the quality of paper and understanding of the results.

Comments 1. It is hoped that the authors will add a description of the main findings in the abstract, such as "a step-like behavior of the particle distribution function after a long time is found" to highlight the uniqueness and novelty of the research. At the same time, the potential impact of these findings on related fields can be briefly mentioned.

Responce 1.  Yes, we agree and rewrite Abstract . After some discussions we decide to focus the main results of the paper 1) only the motion of the interface, which divide the region with fulfilled traps and the region of free traps. This problem directly is connected with the damage and mechanical properties of porous low k materials, used in nanoelectronics  

2)  the influence of the different radius of pores on the diffusion and capture on traps. namely the dependence of the extinction time on the number of traps at different radius.

The step-like behavior of survival probability on long time asymptotics requred the more place and detailed discussion, so in our opinion it will be included in future paper.  And we include the only two points in Abstract and Introduction. 

Comments 2. The introduction can be further strengthened to add a systematic review of existing research and a clear positioning of the work of this paper. It is recommended that the authors discuss the shortcomings of existing research in more detail in the introduction and how this paper fills these gaps. For example, some descriptions of the practical application background of particle diffusion and capture in porous media can be added to highlight the practical significance of the research.

Responce 2   Yes,  we agree. According Comment we extend the review of existing researchers, namely, include the problem of mass transfer (diffusion) in biotechnology field, closely connected with our paper. In all papers the study focus on the diffusion in the porous system, but in our paper as far as we know the study of influence of the radius of the pores on the diffusion characteristics was studied. It is important for the study the diffusion and capture on traps in real  porous systems with branched structure of pores with different sizes. We add these points in Introduction too.

Comments 3.  The article lacks theoretical explanations for the observed phenomena, especially the step-like behavior of the particle distribution function at long time scales and the logarithmic time dependence of the boundaries of the trap filling region. It is recommended that the authors try to explain these phenomena theoretically or cite relevant theoretical work to support the experimental results. For example, these phenomena can be explained by analyzing the statistical properties of the trap distribution or the stochastic process of particle diffusion.

Responce 3.  Yes. we agree. Of course it is better when there are some theoretical explanations , but it is requires the more place and introduction some theoretical models. We include in Conclusion Ref 26 for our theoretical work ,  which explain the logarithmic time dependence of the boundaries of the trap filling region by cylindrical symmetry of the problem. 

Comments 4. The experimental design section can be further optimized and a detailed discussion of the selection of experimental parameters can be added. It is recommended that the authors add more parameter combinations in the experimental design, such as different pore shapes and trap distribution patterns, to more comprehensively study the kinetic behavior of particle capture. At the same time, it is possible to consider adding sensitivity analysis of the experimental results to determine which parameters have the most significant impact on the capture time.

Responce 4.  Yew, we agree.  The answer to question 4 was added to Section 3.2. To describe the effect of the pore radius, two trap filling options were considered.

Option 1: The seeding of traps for different pores was carried out independently. In this case, the general trends described below persist. However, it is quite difficult to compare the results for a specific number of traps. Upon detailed assessment of the configurations, it was found that the concentration of traps along the Z-axis significantly affects trapping. The more traps are placed near the buffer zone, the faster trapping occurs. This fact can be logically explained: the probability of a change in the Z-coordinate is two times lower than that of a change in the X- or Y-coordinate.

Option 2: After selecting the trap coordinates for a pore with a radius of 3a, traps with similar coordinates are generated for pores with radii 9a, 27a, and 81a. The traps share the same Z-coordinates, while the X and Y-coordinates are calculated using the polar coordinate formula for the corresponding radius. These configurations were used for the simulations described below.

Comments 5.  It is recommended that the authors describe the specific implementation process of the numerical simulation in more detail, including the programming language used, the specific application of numerical methods (such as finite difference method, Monte Carlo method, etc.), and how to ensure the accuracy and stability of the simulation. In addition, a detailed discussion of the boundary condition treatment during the simulation process can be added, such as how to deal with the reflection and absorption of particles on the pore boundaries.

Responce 5.  Yes, we rewrite the text according comments. In Section 2.1, we have revised the text to provide a more detailed description of particle movement and reflection. 

The C++ programming language (ISO C++17) and the Visual Studio 2022 development environment were used for numerical calculations. To ensure the stability and correctness of the results, a series of experiments—20 for each trap configuration—were conducted. The obtained results were averaged, and statistical estimates (provided in Appendix A) demonstrate that the results are generally stable.

Comments 6.  The results analysis section can be further deepened to provide more quantitative data and statistical analysis. It is recommended that the authors provide more quantitative data, such as determining the degree of influence of different pore sizes and trap concentrations on the capture time through statistical analysis. At the same time, the error analysis of the experimental results can be added to evaluate the reliability and repeatability of the experimental results. In addition, the statistical properties of the particle distribution function (such as mean, variance, etc.) can be calculated to gain a deeper understanding of the dynamic behavior of particle capture. At the same time, more advanced data analysis methods, such as machine learning or data mining techniques, can be considered to mine the potential laws in the simulation data.

Responce 6.  Yes, we agree. Information about the number of active particles has been added to Section 3.2.   

Let us consider how the median value of the number of active particles changes after 1,000,000 iterations. For configurations with more than 780 traps, complete extinction occurs for all pores, meaning traps captured all particles. The figure shows the dependence of the median number of surviving particles on the number of traps.

Comments 7.  The discussion of the selection of simulation parameters can be added, such as why the range of 1000 to 10000 traps is selected, and the impact of these parameters on the results

Responce 7.  Yes, thank you for this comment. It is necessary to clear the number of traps, which was used at the modeling. It is obvious that in the case of the small number of traps there are long times will be required to obtain the temporal dependencies of survival probability of particles, so the low bound of number of traps begins from 1000 traps. The upper bound of traps is 10 000. This number corresponds to total number of places for traps on the cylindrical surface: At R=3a and H=1000 the number of possible places for traps is equal to 20 000.  This paragraph is included at page 5 after table 1  

Comments 8. The conclusion section can be further strengthened to add further discussion and prospects of the research results. It is recommended that the authors discuss the potential applications of the research results in the conclusion, such as practical applications in low-k materials, and how to use these findings to optimize the performance of the materials. At the same time, the direction of future research is proposed, such as further studying the effects of different pore structures and trap distributions on particle capture dynamics.

Responce 8.   Thank you. According yours Recomendation we discuss the possible applications of obtained results in Conclusion.

As a result, there is a possibility of escaping the "narrow neck," and the percolation approach must be used to determine the effective extinction time for pores with a complex branched structure. Thus, the obtained results are considered as one of the steps toward studying complex systems with branched pore structures.

One of the important issues in using low-k dielectric materials is the effect of the penetration of active diffusing particles due to porosity on the reliability of microcircuits and the mechanical strength of these materials. From this point of view, our results on diffusion inside pores and capture are also useful for reliability problems.

The investigation of diffusion processes and capture by traps is also important for cell metabolism and protein activity [26]. 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 [27, 28].

Comments 9.  The number of references in the article is small, and the references cited are relatively old. It is recommended that the author add more references, especially the latest research results in the field of particle diffusion and capture in porous media in recent years. In addition, some relevant review literature can be cited to help readers better understand the research status and development trends in this field.

Responce 9.   Yes, we agree. According the recomendations of Reviewers we add more references  Ref 6,7  and Ref. 17-24 , and Ref 26-31 about  numerical simulation and diffusion in living systems in the biotechnologies.  Our paper will be useful for understanding the diffusion processes in the complex systems with branched structure of pores.

 

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The study is relevant to computational modelling of diffusion processes in porous media which is important for various applications, including for instance material science, microelectronics.  The study explores the impact of pore size and trap density on diffusion dynamics, which is crucial for understanding transport phenomena. The use of a 3D cylindrical pore geometry makes the study more realistic compared to previous 1D analyses. The figures, including graphs and tables, effectively illustrate the relationship between the number of traps, the survived particles, and the extinction time (i.e. capture time on traps). The numerical approach used in the study aligns with the journal’s focus on computational engineering making it highly suitable for publication.

However, I propose some suggestions that could improve this work.

I recommend expanding the Introduction to provide a more detailed discussion of diffusion and particle capture in porous media. A more comprehensive review of the state of the art would significantly enhance the paper's accessibility for readers.

Additionally, the authors should consider referencing the work of Professor Torquato's research group, which extensively explores diffusion and trapping in porous media: for example, the paper: S. Torquato, Diffusion and Reaction Among Traps: Some Theoretical and Simulation Results, Journal of Statistical Physics, 65, 1173 (1991)

and related publications.

Moreover, including examples of diffusion modelling in various research fields would strengthen the paper's broader relevance. Studies on diffusion in complex heterogeneous media are also conducted in bioengineering, e.g. in bone nanostructure [1], cartilage [2] or brain extracellular space [3]. Consider citing the work of Bini et al. [1], that investigates the influence of structural constraints on diffusion properties, using a Monte Carlo random walk approach, a methodology closely related to the present study.

References:

[1] Bini, F. et al. 2021. 3D random walk model of diffusion in human Hypo- and Hyper- mineralized collagen fibrils. Journal of Biomechanics, Volume 125, 110586. doi: 10.1016/j.jbiomech.2021.110586

[2] Momot, K.I.. 2011. Diffusion tensor of water in articular cartilage. Eur. Biophys. J. 40, 81–91. doi: 10.1007/s00249-010-0629-4.

[3] Jin, S.,et al. 2008. Random-Walk model of diffusion in three dimensions in brain extracellular space: comparison with microfiberoptic photobleaching measurements. Biophys. J. 95, 1785–1794. doi: 10.1529/biophysj.108.131466.

Citing these works would strengthen the study’s connection to existing literature and place it within the broader context of diffusion modelling.

In the results or conclusion sections, the authors should include a comparison with similar computational models from previous studies.

I also recommend that the authors better highlight the limitations of their study.

Additionally, in the conclusion section I suggest providing a brief discussion on future applications.

Please review the paper with careful attention to detail and make the necessary revisions to address these concerns. Once these issues are resolved, your paper will be even stronger.

Comments on the Quality of English Language

I have noticed a few typos and grammatical errors throughout the paper. I recommend thorough proofreading to correct these issues to ensure the highest clarity.

Author Response

Comment 1 I recommend expanding the Introduction to provide a more detailed discussion of diffusion and particle capture in porous media. A more comprehensive review of the state of the art would significantly enhance the paper's accessibility for readers. Additionally, the authors should consider referencing the work of Professor Torquato's research group, which extensively explores diffusion and trapping in porous media: for example, the paper: S. Torquato, Diffusion and Reaction Among Traps: Some Theoretical and Simulation Results, Journal of Statistical Physics, 65, 1173 (1991)and related publications. Moreover, including examples of diffusion modelling in various research fields would strengthen the paper's broader relevance. Studies on diffusion in complex heterogeneous media are also conducted in bioengineering, e.g. in bone nanostructure [1], cartilage [2] or brain extracellular space [3]. Consider citing the work of Bini et al. [1], that investigates the influence of structural constraints on diffusion properties, using a Monte Carlo random walk approach, a methodology closely related to the present study.

References: [1] Bini, F. et al. 2021. 3D random walk model of diffusion in human Hypo- and Hyper- mineralized collagen fibrils. Journal of Biomechanics, Volume 125, 110586. doi: 10.1016/j.jbiomech.2021.110586 

[2] Momot, K.I.. 2011. Diffusion tensor of water in articular cartilage. Eur. Biophys. J. 40, 81–91. doi: 10.1007/s00249-010-0629-4.

[3] Jin, S.,et al. 2008. Random-Walk model of diffusion in three dimensions in brain extracellular space: comparison with microfiberoptic photobleaching measurements. Biophys. J. 95, 1785–1794. doi: 10.1529/biophysj.108.131466.

Citing these works would strengthen the study’s connection to existing literature and place it within the broader context of diffusion modelling.

Responce 1 Dear Reviewer !  Thank you very much for yours cooments, they are very useful and made our paper is more interstinf for broader circle of readers, especially specilaists in bioenginering field. So we include all References Ref 6,7 and 18-24,  which you recommended and one more Ref 26-31. It is really well for readers of our paper. I well know the papers of professor Torquato concerning conductivity of inhomegeneous media, and less about diffsuion, with my pleasure I include the Ref for papers prof. Torquato Ref 6, 7 

Comment 2  In the results or conclusion sections, the authors should include a comparison with similar computational models from previous studies.

Responce 2 As far as we know the most of similar computaional models Ref 26 study as a rule one- dimensioanl case or more specific cases  Ref 27,28 , so our simulations are useful for reserachers from different fields. 

Comment 3 I also recommend that the authors better highlight the limitations of their study.

Responce 3 

Briefly we want to discuss the limitations of our paper. It is connected with simple geometry of studied object – the single pore in the form of cylinder. Of course the real porous low k materials have a branched structure with different pores of different radius with intersections. Another issue is connected with using the random number counter as Mersenne Twister (mt19937)" algorithm to generate random numbers.

    Comment 4 Additionally, in the conclusion section I suggest providing a brief discussion on future applications.

Responce 4 

One of the important issues in using low-k dielectric materials is the effect of the penetration of active diffusing particles due to porosity on the reliability of microcircuits and the mechanical strength of these materials. From this point of view, our results on diffusion inside pores and capture are also useful for reliability problems.

The investigation of diffusion processes and capture by traps is also important for cell metabolism and protein activity [17-20]. 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 [21-24], [29-31].

 

 

 

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The author has made the necessary changes in response to the reviewer's comments and therefore recommends acceptance of the manuscript.