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the Creative Commons Attribution 4.0 License.
| 15 Jan 2025
Homogeneous ice nucleation in adsorbed water films: A theoretical approach
Abstract. Ice nucleation plays a critical role in cloud formation and atmospheric processes, influencing precipitation and climate. In this study, we present a theoretical approach for describing homogeneous ice nucleation within adsorbed water films on insoluble substrates, and suggest that it may be a mechanism for deposition ice nucleation with non-porous ice nuclei that induce ice premelting. Our theory is based on the Frenkel-Halsey-Hill (FHH) adsorption model, which characterizes the substrate-adsorbate interaction, and the classical nucleation theory of homogeneous freezing, which describes the probability of ice formation. We use the theory to model the melting point, critical ice nucleus size, and nucleation rates as functions of adsorbed water film thickness and substrate properties. Our results indicate that the melting point depression can be as much as 5 K on hydrophilic substrates when the thickness of the water film is 1 nm. The onset temperature for homogeneous ice nucleation (235 K for cloud droplets) can shift 1–2 K lower in adsorbed films. At temperatures below 235 K, the humidity at which ice nucleation occurs is determined by the condition that the adsorbed water film must be thick enough to accommodate the critical ice nucleus. Comparisons of calculated relative humidity conditions with experimental ice nucleation data for silica particles show promising agreement, validating the FHH model as a framework for describing deposition ice nucleation in the atmosphere.
Competing interests: Ari Laaksonen is a member of the editorial board of Atmospheric Chemistry and Physics
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this preprint. The responsibility to include appropriate place names lies with the authors.- Preprint
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RC1: 'Comment on egusphere-2024-4095', Anonymous Referee #1, 05 Feb 2025
This is the review of the manuscript entitled “Homogeneous ice nucleation in adsorbed water films: A theoretical approach” by Laaksonen et al.
This study presents a theoretical approach based on the Frenkel-Halsey-Hill (FHH) adsorption model to describe deposition ice nucleation in thin films of adsorbed water in absence of pores. This is used to formulate equations to derive the homogeneous ice nucleation rate coefficients within adsorbed water films on insoluble substrates. This approach is then applied to derive the ice melting point, critical ice nucleus size, and nucleation rates as functions of adsorbed water film thickness and substrate properties. The theoretically derived thermodynamic conditions that result in ice nucleation are compared to experimental results.
The topic of this study fits well in Atmospheric Chemistry and Physics considering previously published theoretical and experimental ice nucleation articles in this journal.
Deposition ice nucleation is an understudied topic and still eludes complete understanding of the underlying physical processes that result in ice formation. How ice nucleates from the supersaturated vapor phase on non-porous substrates is understood little. Approaching this by invoking homogeneous ice nucleation in thin water films, typically not detected in ice nucleation experiments, provides a new conceptual model that can be further tested. The application of the FHH adsorption model to derive homogeneous ice nucleation rate coefficients is a neat way to think about this.
The manuscript is well written, and I enjoyed reading it. My comments mostly address clarifications to make reading this manuscript a bit easier for the reader. Although I feel that proposed model has validity and advances our understanding, I have a minor comment regarding the apparent agreement of the experimental data with the model.
Comments:
Line 28: Here and throughout the manuscript: You use monolayers, films and multilayer films. This may need more careful definition. One would assume that a water film consists of several monolayers of water. But what is meant by a multilayer water film?
Line 44 and following: I feel the expression of “film-wise” is unfortunate. I would try to find a better wording what is meant here.
Lines 65 -70: Here, we have A(T) and A’. The latter is then switched to A. This juggling of parameter definitions is also done on line 202 (there you use A for A_w…). Frankly speaking, it took extra effort to keep track of which parameters are temperature dependent and which not. I suggest not substitute but stick with a fix set of parameters, like A(T) and A^298 K, etc. This would facilitate reading of the manuscript. On line 68, I assume you meant the A’ parameter?
Lines 94-97: Maybe provide references for studies that show single and/or multiple monolayer water adsorption on substrates below 100% relative humidity.
Line 98-99: What is the difference between bulk ice and hexagonal ice? Can hexagonal ice form in a thin water film? As you outlined below, it needs some extra layers. If the film has the same number of water molecules, the ice film is larger in surface area? I assume this does not matter when treating it as truly 2D? So, is it a postulate or an assumption to make things work?
Line 128: Could you provide references for typical LJ values?
Lines 150, Eq. 23: Please explain further why you can just replace the exponent “3” with “B”. Above you argue that B=3 for liquid and solid phase. Why can you substitute and allow for its variation in this case? Is it for the sake of having a free parameter?
Line 158, 160: Explain how you derived those two equations. Just looking at the above equations, it is difficult to follow.
Line 164 and following: We would assume that Jhom and ice properties are not decoupled from the water saturation vapor pressures. Can you use different sets of parameters for ice nucleation and properties of water and ice? Espinosa et al. is somehow only chosen to yield better Jhom? In this regard, there might be an even better description of Jhom (Knopf and Alpert, 2023).
Line 202: See comment above on parameter naming.
Line 260: What do you mean by “where V_A denotes the volume of adsorbed water on a single ice nucleus”? How is this volume defined? Does it include additional layers of water?
Figure 8: The discussion of Fig. 8 does not mention much the difference in results in response of using Murphy and Koop vs. Wagner and Pruss saturation vapor pressures. In the former case, the critical humidity drops to 110% for some cases but not when applying the latter. Is this just because the water saturation line by Wagner and Pruss is a bit steeper at lower temperatures? This is surprising. So, it is very sensitive to the saturation vapor pressure? Would you expect this?
Line 287-289: Above you mentioned to ignore the Kelvin effect for this study. Did you account for the Kelvin effect when modeling 100 nm particles (Fig. 8)?
Figure 9 discussion: Your model sensitivity to R is stated as quite weak: an order of magnitude change in R for 3% RHice variation. However, the measurement points have more than 10% RH uncertainty. In other words, uncertainties in R cannot explain the trend in the data? It “has to” be due to variation in A and B parameters and amount of water present? Maybe this could be clearer stated.
Line 327: I am not sure, if “promising agreement” is the right wording. I appreciate that the authors clearly state the caveat that this apparent agreement can only be achieved by using the vapor pressure equation derived by Wagner and Pruss (2002), though the vapor pressure equation of Murphy and Koop has held up well for ice nucleation studies across many disciplines. Would it be worthwhile to consider uncertainties in those vapor pressure formulations and perform some sensitivity tests? What does it mean for the theory, if somehow greater vapor pressures are needed below the homogeneous freezing limit to achieve apparent agreement? Which parameters are affected by this? This discussion could point to further research needs.
References
Knopf, D. A. and Alpert, P. A.: Atmospheric ice nucleation, Nat. Rev. Phys., 10.1038/s42254-023-00570-7, 2023.
Citation: https://doi.org/10.5194/egusphere-2024-4095-RC1 -
RC2: 'Comment on egusphere-2024-4095', Anonymous Referee #2, 19 Jun 2025
In the paper "Homogeneous ice nucleation in adsorbed water films: A theoretical approach" by Laaksonen et al. 2025 the authors present a theoretical framework to describe homogeneous freezing within adsorbed water films on insoluble substrates based on the Frenkel-Halsey-Hill (FHH) adsorption model and Classical nucleation theory (CNT). The developed model is tested against laboratory measurements of homogeneous freezing of adsorbed water on silica. The theory and the assumptions are well-described, and the sensitivity results of the theoretical framework are interesting. The study is well-suited for ACP, and I suggest publishing it after minor revisions.
Major comments:
- It is a lot of formulas to process when reading that paper, and a lot of jumping around between formulas. It would have made reading a lot easier if there were a variable list in the appendix, with the variables, names, and units. Especially, the units would make it clearer in some equations to fully grasp them.
- The naming of the variables are sometimes a bit confusing, for example switching A to A' for temperature independence, the difference between mu_ia and mu_A, the x in equation 19 for the two variables that are the same for water and ice (while for v one has to replace x with w or i).
- Some concepts used and mentioned are not explained. I would suggest adding some explanations to some basic concepts/terms since the readers for this paper might come from a variety of fields and might not have the background on all terms, e.g., CCN, CCN activation, LJ molecules, adsorption isotherms, Laplace pressure, Nitrogen adsorption isotherms.
- One aspect of the paper, which is a bit unclear to me as a reader, is the definition that B=3 and when/where it is used and where not. In Eq. 18 for example, it is directly replaced while other equations keep the B. Then there is a sensitivity for different B values in a lot of plots, which is kind of contradicting that it was assumed to be 3 in some derivations? This could be better explained.- There are many assumptions made in the course of the paper. This is necessary and often clearly explained, but some assumptions I found difficult to follow:
1.) Why is there a 1/T dependence assumed in eq. 4?
2.) Another assumption that I could not connect with the rest of the paper is the statement on page 6, line 152 that the LJ potential is spherically symmetric -> where or in which formulas does that play a role, and how does that affect the framework?
3.) I would also like to have had a bit more background on the LJ values used here.
4.) I could not derive or grasp how equations 24 and 25 were derived.
5.) In section 4.4, a threshold of AF of 1% is chosen. Is that identical to the detection limit of SPIN?
6.) In section 4.,5 a size of 400 nm is chosen even if the size distribution looks very different from that, and a value of 70 nm (or something around 100 nm) would have made more sense. It is explained that the model is not very sensitive to that, but still, I found the value of 400 nm very arbitrary and difficult to understand. If it hardly makes a difference and there is no strong argument for the 400 nm, why not use the value from the size distribution measurements?- Sometimes, there could be more reflection on the assumptions used and the consequences. I was surprised by the big difference between the left and the right plot in Fig. 8 - especially the curve for A=3, B=1.5 that bends down when using Murphy and Koop. I would like to see a bit more discussion on this and the related uncertainties (in section 4.4), since Wagner and Pruss was used for Fig. 9 (?, see below).
- Section 4.5: It is not mentioned nor explained (why) that the e_w equation from Wagner and Pruss is used (?).
- Conclusions: can be a bit more elaborate, especially the statement on the importance of the adsorption of a multilayer film in deposition ice nucleation in the atmosphere (either remove the atmospheric relevance or explain how you come to this conclusion).
- Code and data availability: I suggest making this public and not upon request.Minor comments:
- Some figures have a slightly different style in the axis labels, legends, units/variables (italic or not), e.g., Fig. 2 vs. 3, but also the legend has a different size for Fig. 7 a and b. Make it uniform.
- If possible, replace "references therein" by the most important references needed for your context.
- Can you give some context on the film thickness? What are "typical values" and why did you choose what you did?
- Can you add some more explanation on A and B and what it means, also in the discussion, for example, on page 10 (line 233-235).
- page 3, line 75: This sentence is a bit difficult to read since the formula (= for equals) is breaking up the sentence. Also, I was wondering if an adsorbed layer means one adsorbed layer?
- Eq. 12 and 13: add that P_w/e_w=S_w and P_i/e_i=S_i.
- Eq. 16: I had a bit of trouble understanding this equation (maybe related to the naming of the chemical potentials and some confusion on my side). If possible, add more explanation/steps here.
- Eq. 19: v_x is the molecular volume?
- page 6, line 151: split up this very long and complex sentence.
- page 7, line 171: be more specific equations for ... .
- page 9, line 200: P_w=P_i is also true at the phase boundary? It is the more trivial argument in my opinion.
- Fig. 4: A symbol for the intersection point would help to see that quicker.
- Increase legends in Fig. 4, 8, and potentially 9 (it is the smallest/worst in Fig. 4).
- Fig. 5: M. P. temperature is nowhere explained - use melting point temperature instead (fewer abbreviations, less confusion).
- Fig. 5 caption: the last sentence = interpretation is not needed here.
- Fig. 9: Is pink vs. green a good choice when it comes to colorblind-friendliness?
- Fig. 9: The sensitivity bar is hard to see. Is that independent of T?
- page 15, line 309: I don't understand why this is explicitly stated here/in the context.
- page 15, line 311: But the measurements look steeper?
- page 16, line 321-322: Conclusions? What do these specific values mean?
- page 16, line 324: Can it not be measured or confirmed how hydrophilic the silica is here?
Technical corrections:- Add . after equation 3, 20, 21, 22 and a comma after equation 4, 26 and double check the writing of equations (: before etc.) - this is inconsistent in the current version of the manuscript.
- page 5, line 116: I believe it must be Eq. 11 and 13.
- page 6, line 149: There is a ) too much.
- Fig. 6 caption: remove the () around the variables, inconsistent.
- Fig. 8 legend: there is a space missing between the value and unit for R_p.
- A1, use {} for the exp function in the Murphy and Koop e_w equation to enhance readability.
- page 19, line 374: a space is missing after water.
Citation: https://doi.org/10.5194/egusphere-2024-4095-RC2 -
AC1: 'Response to comments by reviewers 1 and 2', Ari Laaksonen, 03 Jul 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2024-4095/egusphere-2024-4095-AC1-supplement.pdf
Status: closed
-
RC1: 'Comment on egusphere-2024-4095', Anonymous Referee #1, 05 Feb 2025
This is the review of the manuscript entitled “Homogeneous ice nucleation in adsorbed water films: A theoretical approach” by Laaksonen et al.
This study presents a theoretical approach based on the Frenkel-Halsey-Hill (FHH) adsorption model to describe deposition ice nucleation in thin films of adsorbed water in absence of pores. This is used to formulate equations to derive the homogeneous ice nucleation rate coefficients within adsorbed water films on insoluble substrates. This approach is then applied to derive the ice melting point, critical ice nucleus size, and nucleation rates as functions of adsorbed water film thickness and substrate properties. The theoretically derived thermodynamic conditions that result in ice nucleation are compared to experimental results.
The topic of this study fits well in Atmospheric Chemistry and Physics considering previously published theoretical and experimental ice nucleation articles in this journal.
Deposition ice nucleation is an understudied topic and still eludes complete understanding of the underlying physical processes that result in ice formation. How ice nucleates from the supersaturated vapor phase on non-porous substrates is understood little. Approaching this by invoking homogeneous ice nucleation in thin water films, typically not detected in ice nucleation experiments, provides a new conceptual model that can be further tested. The application of the FHH adsorption model to derive homogeneous ice nucleation rate coefficients is a neat way to think about this.
The manuscript is well written, and I enjoyed reading it. My comments mostly address clarifications to make reading this manuscript a bit easier for the reader. Although I feel that proposed model has validity and advances our understanding, I have a minor comment regarding the apparent agreement of the experimental data with the model.
Comments:
Line 28: Here and throughout the manuscript: You use monolayers, films and multilayer films. This may need more careful definition. One would assume that a water film consists of several monolayers of water. But what is meant by a multilayer water film?
Line 44 and following: I feel the expression of “film-wise” is unfortunate. I would try to find a better wording what is meant here.
Lines 65 -70: Here, we have A(T) and A’. The latter is then switched to A. This juggling of parameter definitions is also done on line 202 (there you use A for A_w…). Frankly speaking, it took extra effort to keep track of which parameters are temperature dependent and which not. I suggest not substitute but stick with a fix set of parameters, like A(T) and A^298 K, etc. This would facilitate reading of the manuscript. On line 68, I assume you meant the A’ parameter?
Lines 94-97: Maybe provide references for studies that show single and/or multiple monolayer water adsorption on substrates below 100% relative humidity.
Line 98-99: What is the difference between bulk ice and hexagonal ice? Can hexagonal ice form in a thin water film? As you outlined below, it needs some extra layers. If the film has the same number of water molecules, the ice film is larger in surface area? I assume this does not matter when treating it as truly 2D? So, is it a postulate or an assumption to make things work?
Line 128: Could you provide references for typical LJ values?
Lines 150, Eq. 23: Please explain further why you can just replace the exponent “3” with “B”. Above you argue that B=3 for liquid and solid phase. Why can you substitute and allow for its variation in this case? Is it for the sake of having a free parameter?
Line 158, 160: Explain how you derived those two equations. Just looking at the above equations, it is difficult to follow.
Line 164 and following: We would assume that Jhom and ice properties are not decoupled from the water saturation vapor pressures. Can you use different sets of parameters for ice nucleation and properties of water and ice? Espinosa et al. is somehow only chosen to yield better Jhom? In this regard, there might be an even better description of Jhom (Knopf and Alpert, 2023).
Line 202: See comment above on parameter naming.
Line 260: What do you mean by “where V_A denotes the volume of adsorbed water on a single ice nucleus”? How is this volume defined? Does it include additional layers of water?
Figure 8: The discussion of Fig. 8 does not mention much the difference in results in response of using Murphy and Koop vs. Wagner and Pruss saturation vapor pressures. In the former case, the critical humidity drops to 110% for some cases but not when applying the latter. Is this just because the water saturation line by Wagner and Pruss is a bit steeper at lower temperatures? This is surprising. So, it is very sensitive to the saturation vapor pressure? Would you expect this?
Line 287-289: Above you mentioned to ignore the Kelvin effect for this study. Did you account for the Kelvin effect when modeling 100 nm particles (Fig. 8)?
Figure 9 discussion: Your model sensitivity to R is stated as quite weak: an order of magnitude change in R for 3% RHice variation. However, the measurement points have more than 10% RH uncertainty. In other words, uncertainties in R cannot explain the trend in the data? It “has to” be due to variation in A and B parameters and amount of water present? Maybe this could be clearer stated.
Line 327: I am not sure, if “promising agreement” is the right wording. I appreciate that the authors clearly state the caveat that this apparent agreement can only be achieved by using the vapor pressure equation derived by Wagner and Pruss (2002), though the vapor pressure equation of Murphy and Koop has held up well for ice nucleation studies across many disciplines. Would it be worthwhile to consider uncertainties in those vapor pressure formulations and perform some sensitivity tests? What does it mean for the theory, if somehow greater vapor pressures are needed below the homogeneous freezing limit to achieve apparent agreement? Which parameters are affected by this? This discussion could point to further research needs.
References
Knopf, D. A. and Alpert, P. A.: Atmospheric ice nucleation, Nat. Rev. Phys., 10.1038/s42254-023-00570-7, 2023.
Citation: https://doi.org/10.5194/egusphere-2024-4095-RC1 -
RC2: 'Comment on egusphere-2024-4095', Anonymous Referee #2, 19 Jun 2025
In the paper "Homogeneous ice nucleation in adsorbed water films: A theoretical approach" by Laaksonen et al. 2025 the authors present a theoretical framework to describe homogeneous freezing within adsorbed water films on insoluble substrates based on the Frenkel-Halsey-Hill (FHH) adsorption model and Classical nucleation theory (CNT). The developed model is tested against laboratory measurements of homogeneous freezing of adsorbed water on silica. The theory and the assumptions are well-described, and the sensitivity results of the theoretical framework are interesting. The study is well-suited for ACP, and I suggest publishing it after minor revisions.
Major comments:
- It is a lot of formulas to process when reading that paper, and a lot of jumping around between formulas. It would have made reading a lot easier if there were a variable list in the appendix, with the variables, names, and units. Especially, the units would make it clearer in some equations to fully grasp them.
- The naming of the variables are sometimes a bit confusing, for example switching A to A' for temperature independence, the difference between mu_ia and mu_A, the x in equation 19 for the two variables that are the same for water and ice (while for v one has to replace x with w or i).
- Some concepts used and mentioned are not explained. I would suggest adding some explanations to some basic concepts/terms since the readers for this paper might come from a variety of fields and might not have the background on all terms, e.g., CCN, CCN activation, LJ molecules, adsorption isotherms, Laplace pressure, Nitrogen adsorption isotherms.
- One aspect of the paper, which is a bit unclear to me as a reader, is the definition that B=3 and when/where it is used and where not. In Eq. 18 for example, it is directly replaced while other equations keep the B. Then there is a sensitivity for different B values in a lot of plots, which is kind of contradicting that it was assumed to be 3 in some derivations? This could be better explained.- There are many assumptions made in the course of the paper. This is necessary and often clearly explained, but some assumptions I found difficult to follow:
1.) Why is there a 1/T dependence assumed in eq. 4?
2.) Another assumption that I could not connect with the rest of the paper is the statement on page 6, line 152 that the LJ potential is spherically symmetric -> where or in which formulas does that play a role, and how does that affect the framework?
3.) I would also like to have had a bit more background on the LJ values used here.
4.) I could not derive or grasp how equations 24 and 25 were derived.
5.) In section 4.4, a threshold of AF of 1% is chosen. Is that identical to the detection limit of SPIN?
6.) In section 4.,5 a size of 400 nm is chosen even if the size distribution looks very different from that, and a value of 70 nm (or something around 100 nm) would have made more sense. It is explained that the model is not very sensitive to that, but still, I found the value of 400 nm very arbitrary and difficult to understand. If it hardly makes a difference and there is no strong argument for the 400 nm, why not use the value from the size distribution measurements?- Sometimes, there could be more reflection on the assumptions used and the consequences. I was surprised by the big difference between the left and the right plot in Fig. 8 - especially the curve for A=3, B=1.5 that bends down when using Murphy and Koop. I would like to see a bit more discussion on this and the related uncertainties (in section 4.4), since Wagner and Pruss was used for Fig. 9 (?, see below).
- Section 4.5: It is not mentioned nor explained (why) that the e_w equation from Wagner and Pruss is used (?).
- Conclusions: can be a bit more elaborate, especially the statement on the importance of the adsorption of a multilayer film in deposition ice nucleation in the atmosphere (either remove the atmospheric relevance or explain how you come to this conclusion).
- Code and data availability: I suggest making this public and not upon request.Minor comments:
- Some figures have a slightly different style in the axis labels, legends, units/variables (italic or not), e.g., Fig. 2 vs. 3, but also the legend has a different size for Fig. 7 a and b. Make it uniform.
- If possible, replace "references therein" by the most important references needed for your context.
- Can you give some context on the film thickness? What are "typical values" and why did you choose what you did?
- Can you add some more explanation on A and B and what it means, also in the discussion, for example, on page 10 (line 233-235).
- page 3, line 75: This sentence is a bit difficult to read since the formula (= for equals) is breaking up the sentence. Also, I was wondering if an adsorbed layer means one adsorbed layer?
- Eq. 12 and 13: add that P_w/e_w=S_w and P_i/e_i=S_i.
- Eq. 16: I had a bit of trouble understanding this equation (maybe related to the naming of the chemical potentials and some confusion on my side). If possible, add more explanation/steps here.
- Eq. 19: v_x is the molecular volume?
- page 6, line 151: split up this very long and complex sentence.
- page 7, line 171: be more specific equations for ... .
- page 9, line 200: P_w=P_i is also true at the phase boundary? It is the more trivial argument in my opinion.
- Fig. 4: A symbol for the intersection point would help to see that quicker.
- Increase legends in Fig. 4, 8, and potentially 9 (it is the smallest/worst in Fig. 4).
- Fig. 5: M. P. temperature is nowhere explained - use melting point temperature instead (fewer abbreviations, less confusion).
- Fig. 5 caption: the last sentence = interpretation is not needed here.
- Fig. 9: Is pink vs. green a good choice when it comes to colorblind-friendliness?
- Fig. 9: The sensitivity bar is hard to see. Is that independent of T?
- page 15, line 309: I don't understand why this is explicitly stated here/in the context.
- page 15, line 311: But the measurements look steeper?
- page 16, line 321-322: Conclusions? What do these specific values mean?
- page 16, line 324: Can it not be measured or confirmed how hydrophilic the silica is here?
Technical corrections:- Add . after equation 3, 20, 21, 22 and a comma after equation 4, 26 and double check the writing of equations (: before etc.) - this is inconsistent in the current version of the manuscript.
- page 5, line 116: I believe it must be Eq. 11 and 13.
- page 6, line 149: There is a ) too much.
- Fig. 6 caption: remove the () around the variables, inconsistent.
- Fig. 8 legend: there is a space missing between the value and unit for R_p.
- A1, use {} for the exp function in the Murphy and Koop e_w equation to enhance readability.
- page 19, line 374: a space is missing after water.
Citation: https://doi.org/10.5194/egusphere-2024-4095-RC2 -
AC1: 'Response to comments by reviewers 1 and 2', Ari Laaksonen, 03 Jul 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2024-4095/egusphere-2024-4095-AC1-supplement.pdf
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