Diffusion of C-O-H Fluids in a Sub-Nanometer Pore Network: Role of Pore Surface Area and Its Ratio with Pore Volume
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsMinor Comments:
Abstract: Briefly define "S-major/Z-major" adsorbents for broader readability.
Fig. 3: Specify if distributions are for straight/zigzag channels in subcaptions. The logarithmic scale obscures subtle features; consider adding linear-scale insets for high-probability regions.
Figs. 4–5: Label data points by adsorbent type (e.g., S-major, Half) to clarify trends.
Eq. 1: Define the constant *a* explicitly (proportional to surface relaxivity?).
Why is the linear fit for "Half" ethane adsorbents poor (Table 1: *a* ≈ -1.91, large error)? Discuss briefly.
Page 5: "Volume normalized self-diffusion coefficient (VD_self)" – clarify if this is D_self × V or D_self / V. The latter seems likelier based on context.
Introduction: Streamline background on NMR/S/V (lines 45–58) to focus more on the knowledge gap in micropores.
Results: Avoid passive phrasing (e.g., "A notable observation is..." → "Notably,...").
Discussion: Emphasize why CO₂ shows stronger S/V dependence earlier (kinetic diameter + surface affinity).
Recommendation:
Accept after Minor Revisions.
The manuscript provides valuable insights into fluid dynamics under sub-nanometer confinement. With clarifications and minor edits, it will be a significant contribution to the field.
Author Response
Minor Comments:
Comment: Abstract: Briefly define "S-major/Z-major" adsorbents for broader readability.
Response: We have included a sentence explaining these terms in the abstract. (Lines 20-22)
Comment: Fig. 3: Specify if distributions are for straight/zigzag channels in sub captions. The logarithmic scale obscures subtle features; consider adding linear-scale insets for high-probability regions.
Response: As already marked in the original manuscript, the distributions in part (a) (left, vertical panels) are for straight channels while those in part (b) (right, horizontal panels) are for zigzag channels. Further, we have now specified in the revised manuscript that these distributions are for the system S4Z4, i.e. unmodified silicalite. (Line 196) Indeed we started by using linear scale of the intensity for Figure 3. However, we found that logarithmic scale brings out the features in a better way and we have therefore decided to keep the logarithmic scale maps. Further, as the linear scale maps do not show any additional features, we are not including them to avoid unnecessary crowding.
Comment: Figs. 4–5: Label data points by adsorbent type (e.g., S-major, Half) to clarify trends.
Response: The data points for different adsorbent types are already marked differently with legends explaining the difference.
Comment: Eq. 1: Define the constant *a* explicitly (proportional to surface relaxivity?).
Response: The constant a is a numerical factor (4/(3d√π); d being the spatial dimension). We have included the exact definition of a in the revised manuscript (Line 59). Further, we note that the relation between S/V with D as used in the current work differs from that with the transverse relaxation time T2, which is; 1/T2=ρ2(S/V); and involves the transverse surface reflexivity (ρ2) which is constant for a given setting of NMR experiments.
Comment: Why is the linear fit for "Half" ethane adsorbents poor (Table 1: *a* ≈ -1.91, large error)? Discuss briefly.
Response: We have added the following new text explaining this (Lines 277 – 286).
Among the three types of adsorbents, the adsorbent type with only half volume available for adsorption exhibits the weakest dependence. For ethane, this dependence is too weak and is overshadowed by the scatter in the data. The combined effect of a weak dependence and large scatter results in a small value of ‘a’ (Table 1) with large error margin for ethane in ‘Half’ adsorbents. It is noteworthy that the ‘Half’ adsorbents have the smallest number of open pore connections (0-12, compared to 48 for S4Z4; and >12 for S-major and Z-major), i.e. intersections which have open access to molecules from all sides [22]. The contribution of intersections to the diffusivity of adsorbed molecules is therefore suppressed even further in these adsorbents because of partial inaccessibility from either straight or zigzag channels.
Comment: Page 5: "Volume normalized self-diffusion coefficient (VD_self)" – clarify if this is D_self × V or D_self / V. The latter seems likelier based on context.
Response: The reviewer is right. We have now added the definition of both VDself (Line 204) as well as SDself (Line 214) in the revised manuscript.
Comment: Introduction: Streamline background on NMR/S/V (lines 45–58) to focus more on the knowledge gap in micropores.
Response: We have now added the following new text along with new relevant references related to NMR measurements in micropores and the knowledge gap (Lines 66 – 71).
In an experiment using powdered samples, it is difficult to distinguish between the signals coming from fluids confined in micropores and that in the relatively larger inter-crystalline pores that might exist between particles. Indeed, while several studies have reported fluid diffusivity in microporous materials like silicalite using NMR, the dependence of diffusivity on S/V is not addressed [17 - 19].
Comment: Results: Avoid passive phrasing (e.g., "A n
otable observation is..." → "Notably,...").
Response: We have revised this sentence. (Line 179)
Comment: Discussion: Emphasize why CO₂ shows stronger S/V dependence earlier (kinetic diameter + surface affinity).
Response: We agree with the reviewer on the need to emphasize the reason for stronger S/V dependence of diffusivity of CO2. As in the original manuscript, we explain this difference in the first few lines of the discussion section (Lines 245 – 248 in the revised manuscript). Further, even in the results section before discussion, the difference between CO2 and ethane in terms of kinetic size and interaction with the host had been explained in lines 215 – 224.
Recommendation:
Accept after Minor Revisions.
The manuscript provides valuable insights into fluid dynamics under sub-nanometer confinement. With clarifications and minor edits, it will be a significant contribution to the field.
Response: We thank the reviewer for the time spent in reading our manuscript and for the positive assessment along with the comments and suggestions that helped us improve the quality of the manuscript.
Reviewer 2 Report
Comments and Suggestions for Authors1. Abstract
The abstract mentions that molecular simulation methods were employed to investigate the influence of pore surface area and pore volume on the diffusion behavior of ethane and carbon dioxide. However, the manuscript lacks sufficient descriptions related to the molecular models and does not include molecular structure diagrams of ethane, carbon dioxide, or the siliceous rock.
2. Introduction
The introduction does not provide an adequate review of previous studies that used molecular simulation methods to investigate the diffusion behavior of ethane and carbon dioxide in silica nanopores. It also lacks a discussion on the existing gaps in current research.
3. Method
(1) The method section mentions that the model consists of a 2×2×3 siliceous rock supercell. It is recommended to add a diagram of the supercell model to Figure 2.
(2) Lines 115–116 mention the use of a spherical probe with a radius of 1.2 Å to test the pore surface area and volume. Please elaborate on why a 1.2 Å radius was chosen—does it relate to the molecular dynamic diameters of ethane and carbon dioxide?
(3) The manuscript indicates that immobile methane molecules were inserted to systematically alter the pore structure characteristics, but it does not explain how the methane blocking sites were selected. Were the blocking sites chosen randomly?
4. Results
(1) Lines 167–168 claim that "greater S-major surface area leads to higher mobility," but Figure 4 does not support this conclusion. Please verify and revise accordingly.
(2) The manuscript only provides a qualitative description of the pore structure and does not calculate the volumetric proportions. It is recommended to supplement quantitative parameters such as tortuosity, the volume and surface area of the main channels and intersections, and the S/V ratio.
(3) A sensitivity comparison is lacking. The manuscript does not quantify the differences in sensitivity of CO₂ and ethane to the S/V ratio.
5. Discussion
(1) The discussion merely repeats the results and does not delve into the physical mechanisms underlying diffusion under sub-nanometer confinement. The explanation for why diffusion coefficients significantly decrease despite the intersections resembling "bulk-like regions" is insufficient.
(2) The manuscript attributes the data scattering in Figure 6 solely to “intersection distortion,” but it does not quantify the relationship between the degree of distortion and the diffusion coefficient.
Author Response
We thank the reviewer for spending time reading our manuscript and for the useful and constructive comments.
Comment: The abstract mentions that molecular simulation methods were employed to investigate the influence of pore surface area and pore volume on the diffusion behavior of ethane and carbon dioxide. However, the manuscript lacks sufficient descriptions related to the molecular models and does not include molecular structure diagrams of ethane, carbon dioxide, or the siliceous rock.
Response: As a ‘communication’ article, we wanted to keep the length of the manuscript short, referring previous publications for details wherever possible. However, we agree with the reviewer on including the details in the current manuscript for the ready reference of the reader. We have therefore now included more details in the manuscript, including a Figure of the systems simulated as the reviewer suggested (Updated Figure 2).
Comment: The introduction does not provide an adequate review of previous studies that used molecular simulation methods to investigate the diffusion behavior of ethane and carbon dioxide in silica nanopores. It also lacks a discussion on the existing gaps in current research.
Response: We have now included more studies in the introduction dealing with the behavior of carbon dioxide and ethane in silica nanopores. We note that several of earlier publications from our group also deal with these systems. However, we decided not to include them to avoid self-referencing.
Comment: The method section mentions that the model consists of a 2×2×3 siliceous rock supercell. It is recommended to add a diagram of the supercell model to Figure 2.
Response: As the reviewer suggested, we have now added a simulation snapshot showing the supercell as also a schematic showing the pore topology and molecular models in Figure 2.
Comment: Lines 115–116 mention the use of a spherical probe with a radius of 1.2 Å to test the pore surface area and volume. Please elaborate on why a 1.2 Å radius was chosen—does it relate to the molecular dynamic diameters of ethane and carbon dioxide?
Response: To efficiently probe the surface area and pore volume available with reasonable resolution, a small spherical probe is used. With a spherical probe of radius 1.2 Å, we make sure that the calculation provides better resolution than the use of either fluid molecule which have the kinetic radius larger than this and also that two probes can fit the pores with diameter of 5.5 Å. We have now added the following new text explaining this (Lines 140 – 144).
A small spherical test particle is used to efficiently probe the surface area and pore volume available with reasonable resolution. With spherical probe of radius 1.2 Å, we make sure that the calculation provides better resolution than the use of either fluid molecule which have the kinetic radius larger than this and also that two probes can fit the pores with diameter of 5.5 Å.
Comment: The manuscript indicates that immobile methane molecules were inserted to systematically alter the pore structure characteristics, but it does not explain how the methane blocking sites were selected. Were the blocking sites chosen randomly?
Response: To block the pores, we started with loading methane molecules to saturation in the unmodified silicalite (S4Z4) by using GCMC simulations. While this resulted in random populating of adsorption sites, at high pressures for saturation, all adsorption sites were occupied by methane molecules. Methane molecules were then removed selectively from individual pores to make them available for adsorbing CO2 or ethane. The following new text explaining this has been added to the revised manuscript (Lines 105 – 108).
For this, methane was initially adsorbed at 200 K and 200 atm pressure using GCMC simulations, filling the entire pore space available to saturation. Subsequently, methane molecules were removed from some channels selectively to make them open and available for adsorbing other fluids.
Comment: Lines 167–168 claim that "greater S-major surface area leads to higher mobility," but Figure 4 does not support this conclusion. Please verify and revise accordingly.
Response: Figure 4 does indeed show that a greater S-major surface area leads to higher mobility, although this increase is not monotonic. To clarify this, we have now modified the sentence (Line 208).
Comment: The manuscript only provides a qualitative description of the pore structure and does not calculate the volumetric proportions. It is recommended to supplement quantitative parameters such as tortuosity, the volume and surface area of the main channels and intersections, and the S/V ratio.
Response: The volumetric proportions of different adsorbents were indeed estimated and reported in Figure 2 in the original manuscript. Further, in the revised manuscript, we estimated the S/V ratios exclusively for the three types of unblocked pores in S4Z4 – straight and zigzag channels and their intersections. The newly added text discussing these quantities appears at the end of section 2.4 (Lines 157 -165) and is reproduced below.
Subtracting the S and V of S4Z0 and/or S0Z4 from those for S4Z4, an estimate for the S/V ratio for the straight, zigzag channels and their intersections in S4Z4 can be obtained. We estimate the S/V ratio of intersections in S4Z4 to be 2.88 × 1010 m-1 while that for the straight and zigzag channels are respectively 1.07 × 1010 m-1 and 1.06 × 1010 m-1. The void space in the intersections is wider than the dimater of straight or zigzag channels. Further, while the straight and zigzag channels are cylindrical, intersections are more sphere-like. Note that while the S/V for a cylindrical geometry with radius r is proportional to 2/r, that for the spherical geometry is 3/r. This makes the S/V ratio of the intersections significantly larger.
Comment: A sensitivity comparison is lacking. The manuscript does not quantify the differences in sensitivity of CO₂ and ethane to the S/V ratio.
Response: We have added the following new text to address this (Lines 267 – 270).
Further, the linear fits for CO2 in all adsorbents are significantly better than those for ethane (rightmost column, Table 1) suggesting a more systematic S/V dependence of diffusivity of CO2 compared to that for ethane.
Comment: The discussion merely repeats the results and does not delve into the physical mechanisms underlying diffusion under sub-nanometer confinement. The explanation for why diffusion coefficients significantly decrease despite the intersections resembling "bulk-like regions" is insufficient.
Response: We have now expanded the discussion section, including newly added text to explain the results. In fact, the diffusion coefficients are expected to decrease even in larger pores with presence of significantly large bulk-like regions (according to Eq.1). In the sub-nanometer pores investigated here, with an absence of the bulk region, the reduction in diffusivity is expected to be significantly stronger than that suggested by Eq. 1, with probably no S/V dependence. However, presence of the intersections that present a small fraction of the total pore space, gives rise to the S/V dependence seen in Figures 6 and 7. The following new text is added to the discussion section.
...For such porous media, the reduction in the diffusion coefficient of the confined fluid compared to the bulk state is expected to be stronger than that suggested by Eq. 1 and independent of S/V....(line 239 - 242)
...This is because of the presence of intersections that provide a fraction of pore space wide enough to accommodate non-overlapping adsorption layers and a very small bulk-like region. The presence of this small bulk-like region in a small fraction of the pore space gives rise to the S/V dependence of the diffusion coefficient seen in Figure 6.... (lines 250 - 254).
...Further, the linear fits for CO2 in all adsorbents are significantly better than those for ethane (rightmost column, Table 1) suggesting a more systematic S/V dependence of diffusivity of CO2 compared to that for ethane.... (lines 267 - 270).
...Among the three types of adsorbents, the adsorbent type with only half volume available for adsorption exhibits the weakest dependence. For ethane, this dependence is too weak and is overshadowed by the scatter in the data. The combined effect of a weak dependence and large scatter results in a small value of ‘a’ (Table 1) with large error margin for ethane in ‘Half’ adsorbents. It is noteworthy that the ‘Half’ adsorbents have the smallest number of open pore connections (0-12, compared to 48 for S4Z4; and >12 for S-major and Z-major), i.e. intersections which have open access to molecules from all sides [22]. The contribution of intersections to the diffusivity of adsorbed molecules is therefore suppressed even further in these adsorbents because of partial inaccessibility from either straight or zigzag channels.... (lines 277 - 286)
Comment: The manuscript attributes the data scattering in Figure 6 solely to “intersection distortion,” but it does not quantify the relationship between the degree of distortion and the diffusion coefficient.
Response: We agree with the reviewer that the relationship between the degree of distortion and the diffusion coefficient has been left unquantified. However, since as explained in the manuscript (original as well as the revised versions), because of the difference in the way different intersections are blocked from the side of straight or zig-zag channels, quantifying the non-uniform properties of these intersections is not straightforward. This is especially so because of the random nature of the distribution of blocking methane molecules. This could be an issue that can be addressed in a future study.
Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors presented a detailed simulation-based analysis on how surface area/volume affects diffusivity of ethane and CO2 in porous media. The manuscript is well-written with solid scientific relevance. In my opinion it possess readiness for acceptance. Below are my comments/questions and it would be great if authors can taken them into consideration either in current manuscript or future work
- What are the applicable situations for Eq. (1), especially if the data suggests that diffusion is not strongly correlated with S/V for C2H6, as suggested by Fig 6 and 7?
- If the relative size between channel width and adsorption layer thickness is what affects the residence/diffusion of molecules(Fig. 1), why is there a difference between intersections and non-intersections? I would imagine if adsorption layer thickness overlaps in non-intersection areas, it will also hold true for intersections. Am I missing something here?
- What is the reason behind that in Fig.4 and 5, only partial range of surface area/volume is tested for Half and S-major cases?
- Is there typo in line 174? Do authors mean that "higher volume" instead of "higher surface area" exhibit higher diffusivities?
- Is there a way to de-couple effects from kinetic diameter and interaction with surface atoms to see which has dominant effects on the delta between the behaviors observed for C2H6 and CO2?
Author Response
Comment: The authors presented a detailed simulation-based analysis on how surface area/volume affects diffusivity of ethane and CO2 in porous media. The manuscript is well-written with solid scientific relevance. In my opinion it possess readiness for acceptance. Below are my comments/questions and it would be great if authors can taken them into consideration either in current manuscript or future work.
Response: We thank the reviewer for spending time reading our manuscript and for the positive assessment as well as useful and constructive comments.
Comment: What are the applicable situations for Eq. (1), especially if the data suggests that diffusion is not strongly correlated with S/V for C2H6, as suggested by Fig 6 and 7?
Response: As explained in the introduction of the original (as well as the revised) manuscript, Eq. 1 is applicable for pores that are significantly wider than the size of the confined molecule (Figure 1a). The silicalite nanopores we studied in the current work resembles the case of pores being comparable in size to the confined molecule (Figure 1b). In this case, Eq. 1 is not expected to be applicable, and diffusivity is not dependent on S/V. However, our analysis shows that while Eq. 1 is not applicable to our results as expected, some correlation of the diffusion of CO2 and somewhat less so with ethane (Figure 6 and 7) still exists due to the presence of intersections.
Comment: If the relative size between channel width and adsorption layer thickness is what affects the residence/diffusion of molecules(Fig. 1), why is there a difference between intersections and non-intersections? I would imagine if adsorption layer thickness overlaps in non-intersection areas, it will also hold true for intersections. Am I missing something here?
Response: The intersections of straight and zigzag channel are ellipsoidal in shape and are wider than the diameter of either of these channel types. Also, unlike the cylindrical shapes of the channels, the intersections are more sphere-like making the adsorption layers on the opposite surfaces remain distinct and not overlap as in the channels.
Commment: What is the reason behind that in Fig.4 and 5, only partial range of surface area/volume is tested for Half and S-major cases?
Response: In this work, the variation in surface area and volume emerges because of blocking/unblocking of pores. This means that different types of adsorbents can have different ranges of surface area/volume. For Z-major adsorbents, the variation in surface area/volume resulting from blocking/unblocking pores happens to cover a wider range. This range was not chosen intentionally but was a result of the blocking/unblocking of pores.
Comment: Is there typo in line 174? Do authors mean that "higher volume" instead of "higher surface area" exhibit higher diffusivities?
Response: The reviewer is right. It was a typo, and we have corrected it in the revised manuscript.
Comment: Is there a way to de-couple effects from kinetic diameter and interaction with surface atoms to see which has dominant effects on the delta between the behaviors observed for C2H6 and CO2?
Response: We thank the reviewer for this interesting comment. The effects of kinetic diameter and interaction with surface atoms can be de-coupled by selectively modifying/weakening the guest-host interactions. However, this is out of scope of the current work and will be investigated in a future study.
Reviewer 4 Report
Comments and Suggestions for Authors- The title is informative but slightly long.
- The abstract doesn't clearly present the main research question or hypothesis upfront.
- In abstract: While the simulations show that CO₂ diffusion decreases at high S/V, the mechanism or why it differs from wider pores isn’t explained.
- correct below mistake:
"fluid" is singular, "their" is plural from
"both a fundamental as well as applications point of view" - The main goal of the paper (studying S and S/V effects via simulation) is buried near the end. Introduce it earlier to orient the reader better.
- GCMC and MD simulation details (e.g., temperature, pressure) are appropriately given.
However: simulation box dimensions, number of molecules, time steps, and cutoff distances are not mentioned. These are crucial for reproducibility. - Fix 9510 × 10⁻¹⁰ m²/s to scientific form
Split long sentences for clarity: "However, for highly mobile fluids, if the confinement is strong enough, the time scales involved in the motion of the fluid molecules in the pores and the inter-particle voids and surfaces can differ significantly, thus providing a means of excluding the contribution of the inter-particle and particle surface adsorbed fluid in a study of the S/V dependence of the diffusion coefficient of the confined fluid"
Author Response
Comment: The title is informative but slightly long.
Response: We agree with the reviewers’ comment on the length of the title. We have therefore modified the title. The new title is
Diffusion of C-O-H Fluids in a Sub-nanometer Pore Network: Role of Pore Surface Area and its Ratio with Pore Volume
Comment: The abstract doesn't clearly present the main research question or hypothesis upfront.
Response: We respectfully disagree with this comment. The main objective of the work (probing the S/V dependence of diffusion of fluids confined in sub-nanometer pores) is clearly stated in the abstract.
Comment: In abstract: While the simulations show that CO₂ diffusion decreases at high S/V, the mechanism or why it differs from wider pores isn’t explained.
Response: We thank the reviewer for this comment. A sentence explaining this has now been added to the end of the abstract (lines 25, 26).
Comment: correct below mistake:
"fluid" is singular, "their" is plural from
"both a fundamental as well as applications point of view"
Response: We have now corrected these mistakes.
Comment: The main goal of the paper (studying S and S/V effects via simulation) is buried near the end. Introduce it earlier to orient the reader better.
Response: We have now clearly expressed the goal of the paper earlier, at the end of the introduction section. The following new text has been added at lines 89- 91
In particular, we investigate whether the diffusion in sub-nanometer pores can have an S/V dependence as expected for wider pores and described by Eq. 1.
Comment: GCMC and MD simulation details (e.g., temperature, pressure) are appropriately given.
However: simulation box dimensions, number of molecules, time steps, and cutoff distances are not mentioned. These are crucial for reproducibility.
Response: We have now included these details in the methods section. The newly added text is reproduced below.
A simulation cell made up of 2 × 2 × 3 unit cells of silicalite (40.044 × 39.798 × 40.149 Å3) was prepared with the visualization software VESTA [24] using the atomic coordinates provided by Koningsveld et al [25]. (lines 97-99)
... This corresponded to 128 molecules of either fluid adsorbed in the cell. A time step of 1 fs was used and long-range interactions were cut-off at 14 Å. (lines 134 - 135)
Comment: Fix 9510 × 10⁻¹⁰ m²/s to scientific form
Response: We agree with the reviewer on the importance of using the proper scientific form for numerals. However, in this case, we presented the diffusion coefficient in terms of × 10⁻¹⁰ m²/s for ease of comparison with other values of diffusion coefficient that are reported in the same unit and exponent.
Comment: Split long sentences for clarity: "However, for highly mobile fluids, if the confinement is strong enough, the time scales involved in the motion of the fluid molecules in the pores and the inter-particle voids and surfaces can differ significantly, thus providing a means of excluding the contribution of the inter-particle and particle surface adsorbed fluid in a study of the S/V dependence of the diffusion coefficient of the confined fluid"
Response: We thank the reviewer for this suggestion. We have now split this sentence.
Round 2
Reviewer 4 Report
Comments and Suggestions for Authorsnew version is acceptable to be published.
