the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Ice nucleation from drop-freezing experiments: Impact of droplet volume dispersion and cooling rates
Abstract. Because homogeneous ice nucleation is important for atmospheric science, special assays have been developed to monitor ultra-pure nanoscale water droplets for nucleation as the temperature is gradually lowered to deeply supercooled conditions. To analyze the experimental data and predict droplet freezing, we develop model that accounts for the cooling rate and the distribution of droplet sizes. We use the model to analyze two sets of experimental homogeneous nucleation data with carefully controlled cooling rates and droplet sizes. Rate expressions based on classical nucleation theory describes both experiments well and with rate parameters in approximate agreement with theoretical predictions based on the thermodynamics of water. We further demonstrate that a failure to account for dispersion in droplet volumes reduces the apparent barriers for ice nucleation. We provide an open source code to estimate nucleation parameters from drop-freezing assays, and another code to account for dispersion of droplet volumes and predict the outcome of drop-freezing experiments. We also present a sensitivity analysis to find the effect of temperature uncertainty on the measured nucleation spectrum. Our framework may be directly useful in accounting for droplet polydispersity and cooling rates for ice nucleation in clouds. Although our analysis pertains to homogeneous nucleation, we note that similar strategies may be applied to heterogeneous ice nucleation on minerals and organic particles with variable surface areas and nucleation sites.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2024-822', Anonymous Referee #1, 08 May 2024
The manuscript submitted by Addula et al., describes the development of a model to study homogeneous ice nucleation (nucleation rates, survival probabilities) for micrometer sized droplets. The model is based on the classical nucleation theory (CNT) and survival probability and optimizes two parameters (preexponential factor and energy barrier) for the representation of homogeneous ice nucleation based on experimental data obtained from Atkinson et al., 2016 and Shardt et al., 2022. The presented python codes (AINTBAD and IPA) seem to work properly on optimizing model parameters and predicting freezing behaviors. Furthermore, the authors expand their model to illustrate how droplet size variations and different cooling rates influence homogeneous ice nucleation. The presented work has implication in atmospheric chemistry and physics by explaining homogeneous ice nucleation in droplet experiments and clouds. The authors shared their code on github, which is very valuable for the community. The writing is quite concise and precise; however, some important information is missing, especially to guide unfamiliar readers through the manuscript. I believe that the model has a high potential to help fundamentally understand homogeneous ice nucleation and could even be developed further to analyze heterogeneous ice nucleation data. Therefore, I recommend this paper for publication in ACP after the authors address the following comments and questions.
Major comments:
- Link your results to literature and physical meaning: As the authors state in line 97 the parameters A and B are temperature-independent parameters and (B/[(1-deltaT)/deltaT^2] represents the free energy barrier of nucleation. What physical meaning does the pre-exponential factor represent? What values would you find for the properties of ice and water (e.g. interfacial free energy) when back calculating from the fitted values for B (e.g. from the global fit from Table 1)? Would these values line up with literature, e.g. Ickes et al., 2015?
- Define the variation in droplet sizes and size distributions better: I got confused with the droplet size variations since the authors use the terms variation, dispersity, and delta V interchangeable through the manuscript (e.g. section 5 and 6). What does delta V, dispersity and variation mean? How does size distribution (diameter?) link to volume distributions (function of r^3)? A clear definition of the variation parameter would be very helpful, maybe even plotting a size distribution from one of the analyzed datasets and showing how the fitted distribution matches the data (if possible).
- Expand the implications: I wish the authors could expand their implications in the abstract, section 9.1, and conclusions. To me it was not clear what we can learn from the model about homogeneous ice nucleation in the atmosphere. For example, the authors show in Figure 10c that the cooling rate influences the T50 values, however how does that relate to cooling rates observed in the atmosphere at different uplifts? I think the paper would be more suitable for ACP when the authors link their results better to implications in atmospheric science.
Minor comments:
- Titel: the title does not really highlight that this manuscript describes a model study, nor that homogeneous ice nucleation was investigated. Suggestion: “Modeling homogeneous ice nucleation from drop-freezing experiments: Impact of volume dispersion and cooling rates”
- Abstract: Could you describe why homogeneous ice nucleation is important for atmospheric science?
- Line 3-5: Can you state what the model predicts? Nucleation rates? Survival probability? Predicts Kinetics?
- Check for typos and missing words within the manuscript: For example, abstract line 3: “… we develop a model that accounts for …”
- Equation (3), Tm only gets explained later. Please define Tm when talking about equation 3.
- Line 86: It might be helpful to explain the physical origin of the classical nucleation theory in two sentences to understand where equation 5 comes from. Especially, since the factors in this equation are referred to later on.
- Line 112: Do you refer to the droplets diameter when talking about the size? A clear definition would be helpful including the calculated volume (in micro m^3 or pL).
- Line 137: How many droplets did Atkinson et al., 2016 measure?
- Figure 3a and line 141: Why did you decide to only show the global fit and not the individual fits with different droplet sizes? How would the individual fit compare with the global fit?
- Line 165: Very impressive to see that experiment-to-experiment variations results in more scattering in the predictions for A and B. Would a “perfect” experiment result in a dataset which gives the same values for A and B? Does the difference in A and B for different droplet sizes (Table 1) only originate from experimental uncertainty and errors?
- Figure 4: How would the pooled dataset overlap with the superposition model? Can the authors pool the data from Atkinson et al., 2016 and compare it with the model predictions?
- Line 215: What does sufficiently narrow mean? Is there a quantitative measure of how narrow the distribution is?
- Figure 5: One datapoint is off the line (x=y), why?
- In line 225 you state that in Figure 7a we see data from Riechers at al., 2013. However, in the legend of Figure 7a you wrote Shardt et al., 2022. Please clarify.
- Figure 7: It is hard to see the difference between this work and the literature. Could you consider having a second panel showing a “zoom-in” to the dataset, e.g. x-axis spanning from 235K to 240K and y-axis spanning 10^3 to 10^9.
- Is there a reason why the cooling rate dependency is explained in section 7? If not, you might want to consider presenting the fitting of the cooling rate dependency (Shardt et al. 2022) prior the literature comparison of nucleation rates.
- Line 256: Clarify why the results contradict the nucleation theory. Experimental uncertainty and errors?
- Line 296: You state a cooling rate of 1K/min, but the legend in Figure 10b says 1K/ns. Please correct.
- Line 307: Why would you anticipate smaller delta G for smaller droplets and why is the opposite the case?
- Line 336: “Through the HUB-backward code from de Almeida Ribeiro et al., (2023) …”
- Line 322: -0.4K to +0.4K?
- Conclusion: Could the authors explain in 1-2 sentences how the model could be developed further to predict heterogeneous freezing? What could we learn form applying the model to heterogeneous ice nucleation data, assuming very uniform ice nucleating particles (characteristic nucleation temperature) are measured in a defined concentration (defined surface area and thus nucleation site density)?
References:
Atkinson, J.D., Murray, B.J. and O’Sullivan, D., 2016. Rate of homogenous nucleation of ice in supercooled water. The Journal of Physical Chemistry A, 120(33), pp.6513-6520.
Shardt, N., Isenrich, F.N., Waser, B., Marcolli, C., Kanji, Z.A., deMello, A.J. and Lohmann, U., 2022. Homogeneous freezing of water droplets for different volumes and cooling rates. Physical Chemistry Chemical Physics, 24(46), pp.28213-28221.
Atkinson, J.D., Murray, B.J. and O’Sullivan, D., 2016. Rate of homogenous nucleation of ice in supercooled water. The Journal of Physical Chemistry A, 120(33), pp.6513-6520.
Citation: https://doi.org/10.5194/egusphere-2024-822-RC1 -
RC2: 'Comment on egusphere-2024-822', Anonymous Referee #2, 10 May 2024
I support publication. The authors present a framework for analysis of homogeneous freezing data that addresses many of the experimental issues encountered in the sorts of experiments that are typically used in homogeneous freezing.
I was surprised to see just how much change the difference of a few microns in the diameter of a droplet in a freezing experiment can make in deriving the kinetic prefactor and the activation barrier.
The most surprising point to me was just how big the energy barrier is at nucleation. If the free energy barrier is 77kT, as referenced for one of the runs in a dataset presented in the paper, the exponential in Eqn 12 is 3.6e-34. But the prefactor is so large that the nucleation rate is easily observable. I was always taught that the exponential factor (the energy barrier) is really the only thing to worry about because it is an exponential. This analysis clearly shows that the prefactor has to be considered.
The authors specify the variables that make up the free energy (equn 13) or B (equn 5). A is never specified that I can see, and this makes getting physical insight from the analysis more difficult. I was really intrigued to see Fig. 3b, showing that A and B are clearly correlated. The discussion of that point could be improved in the manuscript, and being able to see the parameters that go into each one would help. A and B should be correlated because they both depend on temperature; seeing it so clearly in Fig. 3b puts a fine point on it.
I would also appreciate a more comprehensive discussion of the variation of the parameters with volume. I _think_ that B is independent of volume, but that it _seems_ to vary with volume because A does change with volume. Because of that, the nucleation rate becomes higher or lower, shifting the observed temperature range over which nucleation is observed, which means that the observed free energy barrier is different.
Minor points
Line 73: “p” in ln(p(t|V) should be capitalized, I think
Line 89: “... latent heat of ice,...” ice doesn’t have a latent heat. I would rephrase to latent heat of freezing or latent heat of fusion.
Line 306: “...one might anticipate smaller DELTA G for smaller droplets…” why might one expect this? What parameter in the expression for the free energy difference depends on the size of the droplet?
Lines 317,318: I think there’s a pair of parentheses missing in the reference here
Citation: https://doi.org/10.5194/egusphere-2024-822-RC2 - AC1: 'Response document for reviewer comments egusphere-2024-822', Ravi Kumar Reddy Addula, 31 Jul 2024
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2024-822', Anonymous Referee #1, 08 May 2024
The manuscript submitted by Addula et al., describes the development of a model to study homogeneous ice nucleation (nucleation rates, survival probabilities) for micrometer sized droplets. The model is based on the classical nucleation theory (CNT) and survival probability and optimizes two parameters (preexponential factor and energy barrier) for the representation of homogeneous ice nucleation based on experimental data obtained from Atkinson et al., 2016 and Shardt et al., 2022. The presented python codes (AINTBAD and IPA) seem to work properly on optimizing model parameters and predicting freezing behaviors. Furthermore, the authors expand their model to illustrate how droplet size variations and different cooling rates influence homogeneous ice nucleation. The presented work has implication in atmospheric chemistry and physics by explaining homogeneous ice nucleation in droplet experiments and clouds. The authors shared their code on github, which is very valuable for the community. The writing is quite concise and precise; however, some important information is missing, especially to guide unfamiliar readers through the manuscript. I believe that the model has a high potential to help fundamentally understand homogeneous ice nucleation and could even be developed further to analyze heterogeneous ice nucleation data. Therefore, I recommend this paper for publication in ACP after the authors address the following comments and questions.
Major comments:
- Link your results to literature and physical meaning: As the authors state in line 97 the parameters A and B are temperature-independent parameters and (B/[(1-deltaT)/deltaT^2] represents the free energy barrier of nucleation. What physical meaning does the pre-exponential factor represent? What values would you find for the properties of ice and water (e.g. interfacial free energy) when back calculating from the fitted values for B (e.g. from the global fit from Table 1)? Would these values line up with literature, e.g. Ickes et al., 2015?
- Define the variation in droplet sizes and size distributions better: I got confused with the droplet size variations since the authors use the terms variation, dispersity, and delta V interchangeable through the manuscript (e.g. section 5 and 6). What does delta V, dispersity and variation mean? How does size distribution (diameter?) link to volume distributions (function of r^3)? A clear definition of the variation parameter would be very helpful, maybe even plotting a size distribution from one of the analyzed datasets and showing how the fitted distribution matches the data (if possible).
- Expand the implications: I wish the authors could expand their implications in the abstract, section 9.1, and conclusions. To me it was not clear what we can learn from the model about homogeneous ice nucleation in the atmosphere. For example, the authors show in Figure 10c that the cooling rate influences the T50 values, however how does that relate to cooling rates observed in the atmosphere at different uplifts? I think the paper would be more suitable for ACP when the authors link their results better to implications in atmospheric science.
Minor comments:
- Titel: the title does not really highlight that this manuscript describes a model study, nor that homogeneous ice nucleation was investigated. Suggestion: “Modeling homogeneous ice nucleation from drop-freezing experiments: Impact of volume dispersion and cooling rates”
- Abstract: Could you describe why homogeneous ice nucleation is important for atmospheric science?
- Line 3-5: Can you state what the model predicts? Nucleation rates? Survival probability? Predicts Kinetics?
- Check for typos and missing words within the manuscript: For example, abstract line 3: “… we develop a model that accounts for …”
- Equation (3), Tm only gets explained later. Please define Tm when talking about equation 3.
- Line 86: It might be helpful to explain the physical origin of the classical nucleation theory in two sentences to understand where equation 5 comes from. Especially, since the factors in this equation are referred to later on.
- Line 112: Do you refer to the droplets diameter when talking about the size? A clear definition would be helpful including the calculated volume (in micro m^3 or pL).
- Line 137: How many droplets did Atkinson et al., 2016 measure?
- Figure 3a and line 141: Why did you decide to only show the global fit and not the individual fits with different droplet sizes? How would the individual fit compare with the global fit?
- Line 165: Very impressive to see that experiment-to-experiment variations results in more scattering in the predictions for A and B. Would a “perfect” experiment result in a dataset which gives the same values for A and B? Does the difference in A and B for different droplet sizes (Table 1) only originate from experimental uncertainty and errors?
- Figure 4: How would the pooled dataset overlap with the superposition model? Can the authors pool the data from Atkinson et al., 2016 and compare it with the model predictions?
- Line 215: What does sufficiently narrow mean? Is there a quantitative measure of how narrow the distribution is?
- Figure 5: One datapoint is off the line (x=y), why?
- In line 225 you state that in Figure 7a we see data from Riechers at al., 2013. However, in the legend of Figure 7a you wrote Shardt et al., 2022. Please clarify.
- Figure 7: It is hard to see the difference between this work and the literature. Could you consider having a second panel showing a “zoom-in” to the dataset, e.g. x-axis spanning from 235K to 240K and y-axis spanning 10^3 to 10^9.
- Is there a reason why the cooling rate dependency is explained in section 7? If not, you might want to consider presenting the fitting of the cooling rate dependency (Shardt et al. 2022) prior the literature comparison of nucleation rates.
- Line 256: Clarify why the results contradict the nucleation theory. Experimental uncertainty and errors?
- Line 296: You state a cooling rate of 1K/min, but the legend in Figure 10b says 1K/ns. Please correct.
- Line 307: Why would you anticipate smaller delta G for smaller droplets and why is the opposite the case?
- Line 336: “Through the HUB-backward code from de Almeida Ribeiro et al., (2023) …”
- Line 322: -0.4K to +0.4K?
- Conclusion: Could the authors explain in 1-2 sentences how the model could be developed further to predict heterogeneous freezing? What could we learn form applying the model to heterogeneous ice nucleation data, assuming very uniform ice nucleating particles (characteristic nucleation temperature) are measured in a defined concentration (defined surface area and thus nucleation site density)?
References:
Atkinson, J.D., Murray, B.J. and O’Sullivan, D., 2016. Rate of homogenous nucleation of ice in supercooled water. The Journal of Physical Chemistry A, 120(33), pp.6513-6520.
Shardt, N., Isenrich, F.N., Waser, B., Marcolli, C., Kanji, Z.A., deMello, A.J. and Lohmann, U., 2022. Homogeneous freezing of water droplets for different volumes and cooling rates. Physical Chemistry Chemical Physics, 24(46), pp.28213-28221.
Atkinson, J.D., Murray, B.J. and O’Sullivan, D., 2016. Rate of homogenous nucleation of ice in supercooled water. The Journal of Physical Chemistry A, 120(33), pp.6513-6520.
Citation: https://doi.org/10.5194/egusphere-2024-822-RC1 -
RC2: 'Comment on egusphere-2024-822', Anonymous Referee #2, 10 May 2024
I support publication. The authors present a framework for analysis of homogeneous freezing data that addresses many of the experimental issues encountered in the sorts of experiments that are typically used in homogeneous freezing.
I was surprised to see just how much change the difference of a few microns in the diameter of a droplet in a freezing experiment can make in deriving the kinetic prefactor and the activation barrier.
The most surprising point to me was just how big the energy barrier is at nucleation. If the free energy barrier is 77kT, as referenced for one of the runs in a dataset presented in the paper, the exponential in Eqn 12 is 3.6e-34. But the prefactor is so large that the nucleation rate is easily observable. I was always taught that the exponential factor (the energy barrier) is really the only thing to worry about because it is an exponential. This analysis clearly shows that the prefactor has to be considered.
The authors specify the variables that make up the free energy (equn 13) or B (equn 5). A is never specified that I can see, and this makes getting physical insight from the analysis more difficult. I was really intrigued to see Fig. 3b, showing that A and B are clearly correlated. The discussion of that point could be improved in the manuscript, and being able to see the parameters that go into each one would help. A and B should be correlated because they both depend on temperature; seeing it so clearly in Fig. 3b puts a fine point on it.
I would also appreciate a more comprehensive discussion of the variation of the parameters with volume. I _think_ that B is independent of volume, but that it _seems_ to vary with volume because A does change with volume. Because of that, the nucleation rate becomes higher or lower, shifting the observed temperature range over which nucleation is observed, which means that the observed free energy barrier is different.
Minor points
Line 73: “p” in ln(p(t|V) should be capitalized, I think
Line 89: “... latent heat of ice,...” ice doesn’t have a latent heat. I would rephrase to latent heat of freezing or latent heat of fusion.
Line 306: “...one might anticipate smaller DELTA G for smaller droplets…” why might one expect this? What parameter in the expression for the free energy difference depends on the size of the droplet?
Lines 317,318: I think there’s a pair of parentheses missing in the reference here
Citation: https://doi.org/10.5194/egusphere-2024-822-RC2 - AC1: 'Response document for reviewer comments egusphere-2024-822', Ravi Kumar Reddy Addula, 31 Jul 2024
Peer review completion
Journal article(s) based on this preprint
Data sets
AINTBAD and IPA input data Ravi Kumar Reddy Addula, Ingrid de Almeida Ribeiro, Valeria Molinero, and Baron Peters https://github.com/Molinero-Group/volume-dispersion
Model code and software
AINTBAD and IPA codes Ravi Kumar Reddy Addula, Ingrid de Almeida Ribeiro, Valeria Molinero, and Baron Peters https://github.com/Molinero-Group/volume-dispersion
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Ravi Kumar Reddy Addula
Ingrid de Almeida Ribeiro
Valeria Molinero
Baron Peters
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.