Thermodynamic Derivation of the Energy of Activation for Ice Nucleation

The activation key role in the radiative and of the upper troposphere. correct representation in atmospheric models requires understanding ::: of :::: the ::::::::::::: microscopic ::::::::::: processes :::::::: leading :: to :::: ice :::::::::::: nucleation. :: A :::: key ::::::::::: parameter in the ::::::::::: theoretical :::::::::::: description ::: of ::: ice ::::::::::: nucleation :: is :::: the ::::::::::: activation :::::::: energy, :::::: which : controls the ﬂux of water molecules from the bulk of the liquid to the solid during the early stages of ice formation. In most studies it is estimated by

Response: The derivation of the activation energy is based on the fluctuation theorem (FT) which is a result non-reversible statistical thermodynamics instead of "classical macroscopic thermodynamics" as the reviewer suggests. FT relates the macropcospic response of a thermodynamic system to its microscopic dynamics under the assumption that the latter is stochastic and Markovian. These are typically valid assumptions at conditions away from the glass transition temperature. A further assumption is that of microscopic reversibility, which requires that thermodynamic potentials can be locally defined within the liquid. This assumption was also used to write the work dissipated during interface transfer. This is guaranteed near equilibrium, however in non-equilibrium conditions only holds for systems starting at equilibrium. Thus the application of FT for interface transfer is valid only if each molecule can be considered in equilibrium with its local environment within the liquid, which is again valid away from the glass transition temperature. Finally, a heuristic approach was used to write an expression for the dissipated work, which basically involves counting the minimum number of different ways in which four-coordinated water can be built. This is a simple geometric argument (akin for example to bond-counting), and is based on the result of Adam and Gibbs (1965) who showed that the transition probability within liquids is determined by the size of the smallest cooperative region. To address the reviewer's concern section 2.1 has been rewritten putting additional emphasis on the model's approximations.
Comment by the Reviewer: Secondly, the statements about the violation of the Second Law of the thermodynamics do not seem to be correct. The Second Law of Thermodynamics (as all thermodynamics) is strictly speaking applicable only to physical system in the thermodynamic limit (with the number of molecules N → ∞ and volume V → ∞ so that N/V remains finite). It does not apply to microscopic systems (a few molecules) and it does not forbid the decrease in the entropy in a non-isolated microscopic sub-system of a macroscopic system.
Response: The picture of an "apparent" violation of the second law was used as resource to describe the spontaneous organization of molecules into icelike structures, which in a macroscopic system would be impossible (what the reviewer refers to as "thermodynamically unfavorable"), but that are possible in small systems. As mentioned above it is possible to write thermodynamic potentials for microscopic systems near the equilibrium where microscopic reversibility holds. The statements have been modified to clarify these points.
Comment by the Reviewer: In section 2, outlining the theoretical basis of the proposed model, it is assumed (as often done in CNT), that the ice crystal is formed away from the air-liquid interface so that it is not affected by surface tension effects. However, most of the experimental work on crystal nucleation in water is performed by observing the freezing of droplets... Therefore, the conventional "semi-empirical" (page 18154) application of CNT to ice nucleation is based on empirical values of theoretical parameters (such as σ iw and ∆G act extracted by fitting the experimental results for the crystal nucleation rate in droplets with a CNT expression... Typical sizes of experimental (as well as atmospherically relevant) droplets allow one to assume that the formation of a single crystal nucleus in a droplet immediately leads to the crystallization of the latter, i.e., the time of growth of a crystal nucleus to the size of the whole droplet is negligible ...
Response: A significant surface-to-volume ratio in small droplets does not guarantee a predominance of surface based nucleation. The reviewer has made several arguments in favor of a significant role of surface stimulated nucleation (SSN) in ice formation, however there are experimental results both supporting and challenging this view (Sigurbjörnsson and Signorell, 2008;Kay et al. 2003). SSN requires a germ growing in a particular orientation so that at least one its "facets" is aligned with the droplet-vapor interface. The reviewer has calculated the probability of such rare process and showed that SSN would still be thermodynamically favored over volume-based nucleation. However this result requires several assumptions that have not been shown unequivocally to hold. For example it is assumed that the exposing interface aligned with the dropletvapor interface has a well-defined interfacial tension with a value similar to that of the bulk. Also the flux of water molecules to the nascent germ in volume and surface based processes is assumed to be the same. However this is not guaranteed as one can imagine that the water molecules between the growing ice and the droplet-vapor interface would be subject to a confinement effect reducing their mobility. Finally, as it is shown in this work, at very low temperature the nucleation rate is increasingly less controlled by thermodynamics and more dependent on the preexponential factor (e.g., the activation energy), which would limit the effect of SSN on ice formation.
It is certainly out of the scope of this work to settle the debate on the role of SSN in ice formation. To address the reviewer's concern it has been emphasized in the revisited work that all expressions are applicable to cases where ice nucleation is predominantly volume-based, and that only experimental results where nucleation rates were interpreted as volume-based are used. It is however acknowledged in the revisited work that more research is needed on this topic.
Comment by the Reviewer: The goal of the author is to derive a thermodynamic expression for the activation energy ∆Gact in order to avoid considering it as an adjustable parameter in the CNT. However, the final equation (14) for ∆G act contains parameters E and T 0 which are themselves adjustable parameters in the Vogel-Fulcher-Tammann equation, eq.(12). The question arises if the goal has been achieved to the full extent or not...
Response: It has, to its full extent. E and T 0 are not part of CNT, but define the bulk diffusion coefficient. They can be measured and determined independently of CNT. They are akin to constants used other correlations for physical properties like viscosity, heat of fusion, density and the like. E and T o are not adjustable parameters; their values cannot be adjusted to match measured nucleation rates without losing their theoretical meaning. Moreover, E and T o are related to the configuration entropy of water, and in principle also admit a thermodynamic derivation.
2 Referee 2 I thank the reviewer for the comments on the manuscript. They are addressed in detail below.
Comment by the Reviewer: My second major remark refer to the desig-nation of the new formulation for the activation energy as a "phenomenological model". In my understanding, "phenomenological" means being based on observations. However, the author stresses that there is no empirism entering this expression (which I'm not too sure about, see below). Wikipedia gives the following definition: "A phenomenological model (sometimes referred to as a statistical model) is a mathematical expression that relates several different empirical observations of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory. In other words, a phenomenological model is not derived from first principles." -I don't think this is what describes the approach of the author, and the wording should be changed (or justified).
Response: The proposed model is phenomenological in the sense that it is derived from ascribing certain characteristics to the process of interface transfer (e.g., collective behavior, work dissipation, and a defined interface) as heuristics to reach a thermodynamic view of the ice germ growth. A true mechanistic description of the interface transfer process requires an molecular dynamics approach. The definition given by the reviewer is somehow too strict, and may stem from an older view where molecular dynamics simulations were not available and approximations to the behavior of microscopic systems were considered full mechanistic theories. To address the reviewer's concern the model is referred simply as "theoretical" in the revisited work.
Comment by the Reviewer: Thirdly, the derived expression oddly is very similar to the Zobrist et al (2007) formulation (compare equations 14 and 18). When eq. 14 is evaluated at aw = aw eq , the two expressions differ only by the factor T=(T .. 118K). This similarity is certainly no coincidence and should be discussed further.
Response: The two expressions are fundamentally different. Equation 14 is never evaluated at a w = a w,eq , since it implies equilibrium conditions for which nucleation is not possible. Secondly, the apparent similarity originates because the bulk diffusion coefficient is expressed by the same relation, that is the VFT equation. However in the case of expressions like the one derived by Zobrist et al. (2007) the relation between D ∞ and ∆G act is hypothesized a priori while in this work it results from the explicit consideration of the thermodynamics of interface transfer. This has been addressed in the revisited work.
Comment by the Reviewer: Furthermore, this means that the new expression contains the same empirical fit parameters (E, T0) which are criticized in the Zobrist formulation. Response: The criticism raised on expressions like the one formulated by Zobrist et al (2007) refers to the a priori assumtion that the activation energy for interface transfer must have the same form as that of the bulk, neglecting the dynamics of the interface, not on the usage of the diffusion coefficient. The parameters E and T o describe the bulk diffusion coefficient, a physical property of water. They are not degrees of freedom of CNT and are not found by matching nucleation rate measurements. Other physical properties like the equilibrium water activity, the water density and the enthalpy of fusion also have fitted parameters. It is not a claim of this work that besides the activation energy all other physical properties of water can also be obtained from the proposed model.

Comments by the Reviewer:
• page 18158, line 15: 'the probability of such collective arrangement is given by f (T, a w ).' This is a fundamental point for the further derivations, but it is not well explained why this probability should be exactly the same as the factor in the diffusion coefficient (eq. 5).
• page 18160, line 6: Again, why is f (T, a w ) = P (W ) Response: The function f (T, a w ) is introduced to distinguish between the bulk diffusion coefficient D ∞ and the diffusion coefficient across the interface, D. The latter must be taken as an "effective diffusivity", since it parameterizes procesess that are not necessarily diffusive in nature (e.g., molecular rearragement). In the view proposed in this work the breaking of hydrogen bonds is a necessary but not sufficient condition for the incorporation of water molecules into the ice germ. Molecular rearragement is required for interface transfer and requires surpassing an energy barrier. The molecules in the liquid fluctuate in different ways, some of which lead to spontaneous organization. The probability of a spontaneous process occurring in a given direction is determined by the work required for such fluctuation, W . Since interface transfer requires the spontaneous organization of water molecules into ice-like structures with probability described by f (T, a w ), it follows that if W describes the work required for collective rearragement, the probability of fluctuation P (W ) must be equal to f (T, a w ). This explanation has been added to Section 2.1.
Comments by the Reviewer: page 18163, line 24ff: As discussed here, it was shown by Ickes et al (2015) that the combination of the Z07 activation energy and the Reinhardt and Doye (2013) surface tension gives the best agreement to observations of the freezing rate, including observations at T ¡ 200K. So if this combination was used in-stead of Z07 together with the B14 surface tension, this would agree much better to observations than what is shown in Fig. 4. This figures displays an unfair comparison.
Response: The surface tension from B14 was used in all expressions to highlight differences due solely to the activation energy. The comparison is not unfair. Using the expression of Reinhardt and Doye (2013) leads to lower nucleation rates than when using the expression from B14. At 220 K and a w = 1, Fig  . The latter is closer to the experimental results. To address the reviewer's concern J hom calculated using the expression of Reinhardt and Doye (2013) has been added to Comments by the Reviewer: Please add more details to the caption of Fig. 1 (e.g. what are the bright and dark blue spheres? what are states 1 und 2? Why is Gice,eq higher than Gice,1 and Gice,2 ?).
Response: Figure 1 represents an idealization of the process described in this work. However I agree that the cartoon adds little to the discussion and may instead lead to confusion. It has been removed from the plot and Fig. 1 replaced with a simpler Figure. Comments by the Reviewer: Why is the temperature dependence of the data shown in Fig. 4b very different from the predicted temperature dependence?
Response: This is already discussed in Section 3. The theoretical reasons are unclear. However another possibility may be a slight drift in a w during the experimental measurements. The data shown in Fig. 4b was obtained with similar techniques in which the initial a w is set but is not controlled during the experiment (in fact in both cases the initial a w is reported, instead of a w at the point of freezing). It is shown in Fig. 4b and discussed in Section 3 that a decrease in a w of 0.02 during the experiments would introduce a spurious temperature dependency and be enough to explain the discrepancy between the theory and the measurements.
However I agree that the discrepancy is troubling. Additional experimental results from Larson and Swanson, (2006) for the homogeneous freezing of ammonium sulfate were added to the Figure. They show a stronger dependency on T than the data of Alpert et al. (2011), although still lower than the predicted by the model. However the uncertainty in the temperature in the Larson and Sanwson, (2006) data is too large to establish a statistically significant difference. It is acknowledged that further research is required to elucidate this point.

Comments by the Reviewer:
• Please give units for the variables in Table 1.
• It should be mentioned that the B14 formulation of surface tension is also a fit to observations.
• page 18164, line 10: insert 'of' before 'Jeffery and Austin' • page 18179, Fig. 4: Please use a distinct line style and line color instead of the minuscule crosses for 'CNT, this work'.
Response: All technical comments have been addressed in the revisited work.
3 Referee 3 I thank the reviewer for his/her assessment.
Comment by the Reviewer: The manuscript is well written throughout, except in the Activation energy section 2.1. The reviewer found this section confusing to follow and thus recommends a clearer discussion and mathematical development in the text with corresponding clarifications reflected in Figure  1. In particular, identifying and labeling the connection between the specific activation energy for interface transfer (Dmu), DG, and W.The free energies Figure Response: Section 2.1 has been rewritten to clarify several points raised during this discussion. Figure 1 has been simplified to make it clearer, explicitly showing the relation between W and ∆G.
Comment by the Reviewer: ... discuss how difficult it is, from a fundamental point of view, to get the nucleation free energetics of these processes correct.
Response: This was dicussed in Section 4. It is acknowledged that the specification of water properties at very low temperature is very difficult and in general all studies use some form of thermodynamic continuation to define a w,eq and ∆h f for T < 235 K. This point has been emphasized in the revisited paper.
Comment by the Reviewer: It would be nice to see the differences in the critical germ sizes predicted between the models as well as those deduced from experiment.
Response: The activation energy does not affect the critical germ size. The requested comparison is shown in Barahona (2014, Figure 5). The activation energy does affect the measured freezing temperature, which may translate into a different critical size estimated at the point of freezing. However this is highly dependent on the nucleation threshold chosen to calculate the freezing temperature and therefore may be misleading.
Comment by the Reviewer: Furthermore, the author can make a correspondence between the CNT interfacial free energy and the NNF formalism and plot the effective surface tension of both for comparison.
Response: This is shown in Figure 2 of Barahona (2014). The interfacial tension in NNF is not modified by the activation energy since it is obtained without fitting nucleation rates. Thus the requested figure would remain unchanged from B14.
Comment by the Reviewer: The author expresses the significance in the NNF compared to CNT, in that the former if free from the bias induced by uncertainties in the parameterization of the interfacial free energy between water and ice. However, the NNF model has expanded the number of variables (i.e., degrees of freedom) compared to CNT, and hence it isn't too surprizing that better agreement over a broad temperature range is found between prediction and experiment.
Response: The number of degrees of freedom is not increased. None of the parameters of the model presented is obtained by fitting nucleation rates and therefore are not degrees of freedom in the same sense as in CNT. Certainly the physical properties of water are not degrees of freedom either since they are determined independently. It was shown in B14 that the parameters Γ w and s used in the definition of the interfacial energy must be restricted to narrow ranges to be physically valid. They are not found by fitting nucleation rates but instead from physical arguments. Varying them over a wider range than discussed in Section 3.5 of B14 would invalidate the theory.

Comment by the Reviewer:
The reviewer suggests the author consider a sensitivity analysis, similar to previous work by the author, of his new NNF model on the relevant variables. This will help to better constrain the parameters as well as determine which variables have the most profound influence on the homogeneous nucleation rate.
Response: The sensitivity analysis regarding the effect of uncertainty in the interfacial energy on J hom was carried out in Barahona (2014). As shown in Figure 8 of B14 the uncertainty in J hom from variation in Γ w and s is about two orders off magniture and decreases with decreasing T since the nucleation rate becomes more dependent on ∆G act and less dependent on the interfacial energy (this is emphasized in the revisited paper). Regarding the uncertainty in the activation energy, it is moslty a function of a w,eq and n t . It is acknowledged that a w,eq may be uncertain at low T . The approximation used in this work is supported by experimental results (e.g., Koop et al . (2000)). Regarding n t it is estimated that the preexponential factor would increase by about two orders of magnitude by a change in n t from 16 to 15. However a plausible range of variability for n t is hard to estimate since the characteristics of the transient state are not known. Essentially a value of n t less than 16 would indicate that some of the rearrangement routes to form four-coordinated water are prohibited. More research is needed to elucidate this point. Another assumption that may impact the model is that of microscopic reversibility which becomes weaker at low T since water dynamics becomes slower and it cannot be always assumed that the water molecules are in equilibrium within the bulk liquid. Unfortunately giving a plausible range of variability is challenging since deviations from equilibrium are difficult to quantify, even with molecular dynamics methods. This analysis has been included in Section 3 of the revisited paper.
Comment by the Reviewer: As a minor issue, the data points represented in Figure 4 are difficult to discern. Perhaps some arrows might help?
Response: This has been corrected.

Comment by the Reviewer:
Finally, the author should provide some comments on the connection between the phenomenological thermodynamics in the new NNF and a more rigorous statistical mechanics formulation in terms of configurational partition functions of nucleating clusters from the liquid. This can aid in the identification of relevant reaction coordinates, interaction energies, fields, etc. so as to bridge the continuum and molecular scales.

References
[ Sigurbjörnsson and Signorell(2008)  of water molecules from the bulk of the liquid to the solid during the early stages of ice formation. In most studies it is estimated by direct association with the bulk properties of water, typically viscosity and self-diffusivity. As the environment in the ice-liquid interface may differ from that of the bulk this approach may introduce bias in calculated nucleation rates. In this work a phenomenological ::::::::::: theoretical model is proposed to describe the trans- 10 fer of water molecules across the ice-liquid interface. Within this framework the activation energy naturally emerges from the combination of the energy required to break hydrogen bonds in the liquid, i.e., the bulk diffusion process, and the work dissipated from the molecular rearrangement of water molecules within the ice-liquid interface. The new expression is introduced into a generalized form of classical nucleation theory. Even though no nucle- 15 ation rate measurements are used to fit any of the parameters of the theory the predicted nucleation rate is in good agreement with experimental results, even at temperature as low as 190 K where it tends to be underestimated by most models. It is shown that the activation energy has a strong dependency on temperature and a weak dependency on water activity. Such dependencies are masked by thermodynamic effects at temperatures typical 20 of homogeneous freezing of cloud droplets, however may affect the formation of ice in haze aerosol particles. The phenomenological model introduced in this work :::: new ::::::: model provides an independent estimation of the activation energy and the homogenous ice nucleation rate, and it may help to improve the interpretation of experimental results and the development of parameterizations for cloud formation. 25 2 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Introduction
Ice nucleation in cloud droplets and aerosol particles leads to cloud formation at low temperature ::::::::::::: temperatures : and promotes cloud glaciation and precipitation (Pruppacher and Klett, 1997). In absence of ice nuclei, it proceeds by homogeneous freezing. Modeling and experimental studies suggest a significant contribution of homogeneous freezing to the formation 5 of clouds in the upper troposphere (Barahona and Nenes, 2011;Barahona et al., 2014;Gettelman et al., 2012;Jensen et al., 2013). The parameterization of ice nucleation is critical to the proper representation of clouds in atmospheric models. In most cloud models it is done using empirical correlations (e.g., Lohmann and Kärcher, 2002;Kärcher and Burkhardt, 2008;Barahona et al., 2010Barahona et al., , 2014. The most common approach uses the so-called water 10 activity criterion (Koop et al., 2000) where the homogeneous nucleation rate, J hom , is parameterized in terms of the difference between the water activity, a w , and its equilibrium value, a w,eq . The greatest advantage of the water activity criterion is that it is independent of the nature of the solute and therefore facilitates the formulation of general parameterizations of ice nucleation (Barahona and Nenes, 2008;Kärcher and Lohmann, 2002;15 Liu and Penner, 2005).
Empirical correlations provide a simple way to parameterize ice nucleation however provide limited information on the nature of ice formation. Theoretical models help to elucidate the mechanism of ice nucleation and to explain and extent experimental results. Over the last decade molecular dynamics (MD) and other detailed methods have provided 20 an unprecedented look at the microscopic mechanism of ice formation (Espinosa et al., 2014). It is known now that the formation of stable ice germs requires the cooperative rearrangement of several molecules (Matsumoto et al., 2002;Moore and Molinero, 2011) and is preceded by structural transformations within the liquid phase (Moore and Molinero, 2011;Bullock and Molinero, 2013). Detailed experiments and MD simulations have shown that in- 25 stead of forming a single stable structure, several metastable ice structures likely exist during the first stages of ice nucleation (Moroni et al., 2005;Malkin et al., 2012;Russo et al., 2014). There is also a profound relation between anomalies in the properties of water at low temperature and the formation of ice (Buhariwalla et al., 2015), and the relation between low and high density regions within supercooled water and the onset of ice nucleation is starting to be elucidated (Kawasaki and Tanaka, 2010;Singh and Bagchi, 2014;Bullock and Molinero, 2013). Phenomenological :::::: Some ::::::::::: theoretical : models use mechanistic assumptions to describe 5 the formation of ice. Although less detailed in nature than MD, they are more amenable to the development of parameterizations and to the interpretation of experimental results. The quintessential example of such models is the classical nucleation theory, CNT. According to CNT ice formation proceeds by spontaneous density fluctuations within the liquid phase forming an initial stable ::::::::::: ice germ, which then grows by incorpo-10 ration of water molecules from an equilibrium cluster population (Kashchiev, 2000). CNT provides a framework to understand ice nucleation and has been instrumental in the development of parameterizations from experimental data (e.g., Pruppacher and Klett, 1997;Khvorostyanov and Curry, 2009;Murray et al., 2010). On the other hand, J hom estimated with CNT and using independent estimates of thermodynamic parameters typically results 15 in stark disagreement with measurements (Pruppacher and Klett, 1997;Kawasaki and Tanaka, 2010;Barahona, 2014). Thus CNT is commonly used semi-empirically, fitting several parameters of the theory, most commonly the liquid-ice interfacial tension, σ iw , and the activation energy, ∆G act , to measured nucleation rates (e.g., Jeffery and Austin, 1997;Khvorostyanov an 2004;Murray et al., 2010;Ickes et al., 2015). 20 Using CNT semi-empirically has the disadvantage that the theory cannot be decoupled from experimental measurements of J hom . It has been shown that σ iw obtained by fitting CNT to measured nucleation rates tends to be biased high to account for mixing effects neglected in common formulations of CNT (Barahona, 2014). Moreover, the dependency of σ iw on temperature tends to depend on the value of other fitted parameters of the theory 25 (Ickes et al., 2015). Recently Barahona (2014) (hereinafter B14) introduced a mechanistic model of the ice-liquid interface in terms of thermodynamic variables, without fitting CNT to measured nucleation rates. This was done by hypothesizing the existence of a transition layer around the germ with chemical potential defined by the entropy of the ice and 4 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | the enthalpy of the liquid, and using the model of Spaepen (1975) to define the interface thickness. This approach was termed the negentropic nucleation framework (NNF). Recent MD simulations showing the existence of a low density region around the ice germ support the NNF model (Singh and Bagchi, 2014). Introducing NNF into CNT and correcting the nucleation work for mixing effects resulted in good agreement of predicted J hom with ex- 5 perimental results (Barahona, 2014). NNF was also shown to be consistent with the water activity criterion. On the other hand, even with the inclusion of NNF into :: in : CNT, the theory predicts a maximum in J hom for pure water at around 210 K. Such behavior is at odds with experimental results (Manka et al., 2012), and is ascribed to a strong increase in the activation energy as temperature decreases. 10 The activation energy controls the flux of water molecules from the bulk of the liquid to the ice germ (Kashchiev, 2000). Most studies estimate ∆G act either by direct fit of CNT to measured nucleation rates, or from bulk estimates of viscosity, self-diffusivity and dielectric relaxation time (Ickes et al., 2015). The association of bulk properties with ∆G act relies on the assumption that the diffusion across the liquid-ice interface is similar to the 15 molecular diffusion in the bulk of the liquid (Kashchiev, 2000). MD results however suggest that the properties of water in the vicinity of the ice germ differ from the bulk, casting doubt into such approach (e.g., Kawasaki and Tanaka, 2010;Singh and Bagchi, 2014). Unlike for the interfacial energy where several theoretical models have been proposed (e.g., Spaepen, 1975;Digilov, 2004;Barahona, 2014), the phenomenological ::::::::::: theoretical treat-20 ment of ∆G act has been limited. One possible reason is that interface transfer is associated with random fluctuations near the ice-liquid interface, and therefore difficult to treat in terms of macroscopic variables. However several relations allow to describe the evolution of fluctuating systems in terms of measurable variables and their relaxation rates. Among them the fluctuation-dissipation theorem that describes the relation between global and local pertur- 25 bations (Jou et al., 2010), and the fluctuation theorem describing the work distribution in a fluctuating system (Crooks, 1999) have found widespread application in describing the evolution of small systems (Bustamante et al., 2005). With few exceptions (e.g., Røsjorde et al.,5 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 2000), such relations however have not made their way into descriptions of the ice nucleation process.
In this work a phenomenological ::::::::::: theoretical description of the diffusional process leading to the growth of ice germs during ice nucleation is advanced. The proposed model relies on a non-equilibrium view of the interface transfer and leads to the first phenomenological 5 :::::::::::::::: thermodynamic description of the activation energy for ice nucleation.

Theory
This section presents the theoretical basis of the proposed model. The ice germ is assumed to form away from the air-liquid interface so that it is not affected by surface tension effects. The water molecules in the liquid phase are assumed to be liquid water can be neglected. This is justified as it is energetically more favorable to incorporate molecules 15 close to ice germ than those far away from it. Direct interface transfer is thus the dominant growth mechanism of the ice germ (Kashchiev, 2000). Following these considerations the homogeneous nucleation rate can be written in general form as (Kashchiev, 2000), where v w is the molecular volume of water in the bulk, f * is the impingement factor of the 20 water molecules to the ice germ, and Z is the Zeldovich factor given by (Kashchiev, 2000), 6 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | where n * ins the number of water molecules in the ice germ. Other symbols are defined in Table 1. The nucleation work is given by (Barahona, 2014), (3) where Γ w = 1.46 is the coverage of the ice-water interface, and s = 1.105 defines the lattice geometry of the ice germ. The value of Γ w results from the explicit construction of the 5 interface following the rules: (i) maximize the density, (ii) disallow octahedral holes and (iii) preference for tetrahedral holes (Spaepen, 1975). The value of s is obtained assuming that the germ has a staggered structure lying somewhere between cubic and hexagonal ice (Malkin et al., 2012). Compared to common expressions for ∆G hom derived from CNT, Eq.
(3) has the advantage that it does not depend on an explicit parameterization of σ iw , for 10 which there is large uncertainty. Even though it is formulated on a purely theoretical basis, application of Eqs.
(1) to (3) has been shown to reproduce observed freezing temperatures (Barahona, 2014). The impingement factor is the frequency of attachment of water molecules to the ice germ. For steady state nucleation it is given by (Kashchiev, 2000), where γ ≈ 1 is the sticking coefficient, D the diffusion coefficient for interface transfer, Ω the surface area of the germ, d 0 the molecular diameter and Z 1 ≈ v −1 w , : the monomer concentration. germ. Uncertainty in the determination of f * results mostly from the calculation of D, which may differ from the bulk self-diffusivity of water. The most commonly used approximation to 7 D was derived from transition state theory by Turnbull and Fisher (1949) (see Section 2.3), who assumed that the activation energy for interface transfer is similar to that of the bulk ::::: liquid, however the vibration frequency is that of an elemental reaction in the gas phase. This approximation tends to underpredict the preexponential factor in Eq.
(1) at low temperature (Ickes et al., 2015). Here an alternative expression is proposed assuming that D can be 5 expressed in the form, where D ∞ is self-diffusivity of water in the bulk. :::: The ::::::::: function :::::::::: :: is :::::::::::: D. : Since D ∞ has been measured to T ∼ 180 K (Smith and Kay, 1999), can be found by fitting nucleation rate measurements. It is however desirable to obtain an expression for f (T, a w ) independent of J hom . To this end a heuristic approach is developed as follows. 15 Similarly to Turnbull and Fisher (1949) it is assumed that interface transfer requires the formation of a transient state. However instead of each molecule moving independently across the interface, the formation of the transient state requires the collective rearrangement of several water molecules. The probability of such collective arrangement is given by

Activation Energy
. This view does not imply that water is incorporated in clusters to the ice, but rather 20 that the rearrangement of the molecules facilitates the incorporation of each molecule into the preexisting ice lattice (Fig. 1). : . : Such lattice is assumed to be the exposing surface of a metastable ice germ. barrier. 25 8 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | This view is supported by MD simulations showing the increase in the fraction of fourcoordinated water prior to nucleation (e.g., Moore and Molinero, 2011;Matsumoto et al., 2002) and theoretical models where the self-diffusion of supercooled liquids is controlled by their configurational entropy (Adam and Gibbs, 1965).

Equation
: :: Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | ::::::::::::::::::::::::::: ∆G = G liq − G ice = W = 0, being G liq and G ice,eq :::: G ice the Gibbs free energy of bulk liquid and ice, respectively(red line, Fig. 1). As the system . and an energy barrier for interface transfer is created, i.e., W > 0 and ∆G < 0(blue and black lines, Fig. 1). .  then :::::::::::::::::::::::::: That is, the energy dissipated when water molecules are incorporated into the ice germ is equal to their activation energy, i.e., interface transfer is a dissipative process. Considering only those subsystems that move across the interface we assume P (W ) + P (−W ) 20 Using this ::::: Using ::::: this :::: into Eq. (6) can be rearranged into, 11 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Using f (T, a w ) = P (W ) and W − ∆G = −n t ∆µ act :: 7) : we obtain, result mostly from collective rearrangement , the subsystem can be approximated as internally reversible. This means that there is no activation energy for movement confined within the boundaries of the subsystem.Within this framework a molecule moving from the bulk of the ice to the bulk of the liquid will experience a change in chemical potential equal to the excess 10 free energy of fusionof water, i. e. , ∆µ act ≈ −∆µ f . Thus we write,  a w , a w a w,eq ::::: where a w,eq is the equilibrium water activity.
To complete the derivation :: of f (T, a w ) it is necessary to specify the size of the subsystem, n t . Unlike ∆G, W is not a thermodynamic potential and therefore depends on the trajectory 10 of the system. Thus if there are n molecules involved in interface transfer, we need to account for all possible subsets of n molecules crossing the interface. MD simulations show that the onset of nucleation is accompanied by an increase in the number of fourcoordinated molecules (Moore and Molinero, 2011;Matsumoto et al., 2002). In the view proposed ::::: here this means that for each molecule that is incorporated into the ice germ at 15 least four neighboring molecules would rearrange . Thus it is natural to assume the base subsystem as having four molecules, and the number of possible subsets equal to : into Eq. (1) we obtain, where J 0 is referred as the preeexponential factor. Since water is a glass-forming substance, the temperature dependency of D ∞ can be described by the Vogel-Fulcher-Tammann (VFT) equation, 10 where D 0 , E and T 0 are fitting parameters (Table 1, Smith and Kay (1999)). At temperatures relevant for homogeneous ice nucleation the exponential term in Eq. (8) is expected to be much greater than one (although such is not the case when a w ∼ a w,eq ). Using this and substituting Eq.( ::::::::: replacing :::::: Eqs. :::: ::::: and : (12) into Eq.(11) we obtain, 15 Equation(13) has the form proposed by Turnbull and Fisher (1949). Thus the activation energy can be derived as, 14 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Equation (14) shows two contributions to the energy barrier for water transfer to the ice germ. The first term on the right hand side of Eq. (14) results from the breaking of hydrogen bonds in the liquid phase, i.e., the bulk diffusion process. The second term represents an additional energy barrier resulting from the entropy cost of molecular rearrangement within the ice-liquid interface. Substituting Eq. (14) into Eq. (13) we finally obtain,

Common form of CNT
In most studies CNT is used in a more simplified form than presented in Eq.
(1) (e.g., Khvorostyanov and Curry, 2004;Zobrist et al., 2007;Murray et al., 2010;Ickes et al., 2015). Typically, the expression of Einstein (1905) is used to relate diffusivity and viscosity and the 10 energy of activation of water is assumed to have the same value as in the bulk (Kashchiev, 2000). Other assumptions include a semi-spherical ice germ, and negligible mixing effects during the germ formation (Barahona, 2014). These considerations lead to the commonly used CNT expression for J hom (Turnbull and Fisher, 1949), 15 where N c is the number of atoms in contact with the ice germ, and ρ w and ρ i are the bulk liquid water and ice density, respectively. ∆G CNT is the energy of formation of the ice germ, which is commonly written in the form (Pruppacher and Klett, 1997), 15 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | where σ iw is th ice-water interfacial energy, and S i the saturation ratio with respect to ice. Other symbols are defined in Table 1. When using Eqs. (16) and (17), ∆G act and σ iw are typically considered free parameters.

Discussion
As temperature decreases the configurational entropy of water decreases increasing the 5 energy required to break hydrogen bonds, thus the self-diffusivity of water decreases (Adam and G 1965). Similarly, as T decreases the energy associated with the molecular rearrangement within the interface increases, which results from a more negative excess energy of fusion. The latter can also be understood as an increase in the irreversibility of the liquid-ice transformation as the system moves away form thermodynamic equilibrium, therefore increasing 10 the dissipated work, W diss . As a result, ∆G act increases monotonically as T decreases (Fig.  2). By definition, the rearrangement component of ∆G act , W diss , for a w = 1 is equal to zero at T = 273.15 K, i.e., the equilibrium temperature the bulk ice-water system. For T < 250 K it corresponds to about half of ∆G act .
An important aspect of Eq. (14) is that it predicts an effect of water activity on the activa- 15 tion energy. The dependency of ∆G act on a w is however much weaker than on T . Decreasing a w from 1.0 to 0.9 leads only to about 10% decrease in ∆G act (Fig. 2). This is caused by a lowering in the dissipated work, W diss = −n t ∆µ f , with decreasing a w . Lowering a w reduces the chemical potential of water but not that of ice as it is likely that no solute is incorporated into the ice germ during the early stages of ice formation (Barahona, 2014), 20 therefore reducing ∆µ f . ∆G hom (Eq. 3) is much more sensitive to a w and dominates the dependency of J hom on a w . Empirical estimates of ∆G act have been developed in several studies, and were recently reviewed by Ickes et al. (2015). The authors found that the usage of the correlation ::::::::::: expression derived by Zobrist et al. (2007) from self-diffusivity measurements (Smith and Kay,25 1999), along with the fit of Reinhardt and Doye (2013) for σ iw , into Eq. (16)  used : as it is the only available correlation that includes an explicit dependency of σ iw on a w . Usage of the B14 correlation also ensures that ∆G CNT ≈ ∆G hom since it empirically accounts for mixing effects. The Zobrist et al. (2007) correlation results from taking the derivative of the exponential argument of Eq. (12) in the form, Equation (18) gives ∆G act around the mean of common models used in the literature (see Fig. 1 of Ickes et al. (2015)). Thus the model of Zobrist et al. (2007) will be used as benchmark for comparison. However ∆G act calculated using the correlation :: of : Jeffery and Austin (1997) is also presented in Fig. 2 for reference. Although the latter is also derived from the 10 bulk properties of water, it typically results in values of ∆G act lower than ∆G act,Z07 . (Section ::::: 2.1); at T = 180 K ∆G act,Z07 is greater than ∆G act by almost a factor of two. Figure 3 compares the preexponential factor calculated from Eq. (11) against the common CNT formulation, Eq. (16). Equation (18) was used to calculate ∆G act in the latter. For differ by less than a factor of two. Thus the 5 difference between J 0 and J 0,CNT is almost entirely due to ∆G act . For T > 230 K usage of either ∆G act,07 or Eq. (14) introduces less than two orders of magnitude difference in J 0 . However for T < 230 K using ∆G act,07 leads to a much faster decrease in J 0 than with Eq. (14), which is explained by the quadratic increase in ∆G act,07 as T decreases. At 180 K, they differ by almost 10 orders of magnitude. As expected, lowering the water activity 10 slightly increases J 0 since ∆G act is slightly reduced. Despite the noticeable dependency of ∆G act on T , J hom is only ::::::: mostly sensitive to variation in ∆G act at low T . This is illustrated in Fig. 2. For a w = 1 and T > 230 K, ∆G hom >> ∆G act , i.e., the nucleation rate is completely controlled by the nucleation work. As T decreases ∆G hom and ∆G act become comparable and for T < 200 K, J hom is mainly con-15 trolled by ∆G act . Since most experimental measurements of J hom are carried out around 235 K (Fig. ?? : 4), the lack of sensitivity of J hom to ∆G act at these conditions may lead to the incorrect notion that ∆G act is constant. Such misconception may not be critical for the homogeneous freezing of pure water at atmospheric conditions since it rarely occurs at T < 230 K. However it may introduce error in J hom for a w < 1 (Fig. 2, black lines) since 20 ∆G hom and ∆G act become comparable at temperatures relevant to the formation of cirrus from haze aerosol particles (Barahona and Nenes, 2008).
As direct measurements of ∆G act are not available, the skill of ∆G act in reproducing experimental measurements is assessed through evaluation of J hom . For common formulations of CNT (Section 2.3) this has the caveat that such comparison is influenced by 25 specification of other parameters of the theory. This is not the case when using the NNF formulation (Eq. 3) since it does not explicitly depend on σ iw . It was shown in B14 that using ∆G act,07 and Eq. (3) into Eq. (16) reproduced measured J hom for T > 230 K. The results Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | of B14 ::: for :::::::: a w = 1 are shown in Fig. ?? :: 4 along with several experimental measurements, empirical correlations, and results from the formulation of CNT presented in Section 2.3.  Compared to the formulation of B14, J hom from Eq. (11) only differs in the specification of 15 J 0 which mainly depends on ∆G act . As expected, for T > 230 and a w = 1 the formulation of B14 and Eq. (11) produce similar J hom , and within experimental variability and model uncertainty (typically about 3 orders of magnitude) of measured values. Notably J hom predicted by NNF is very close to the data of Riechers et al. (2013) who used a microfluidic device to obtain an accurate estimation of T . For T < 230 K, J hom from B14 is much lower 20 than measured values (by up to 9 orders of magnitude), which is also the case for the CNT formulation, Eq. (16), when using ∆G act,07 . In both formulations J hom decreases for T below 210 K, which results from an strong increase in ∆G act,07 and a decrease in J 0 . Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Most experimental measurements of J hom have been carried out for a w = 1. However homogeneous freezing for a w < 1 is likely important for the formation of cirrus at low T (e.g., Koop et al., 2000). Figure ?? (right panel) shows J hom for a w = 0.9 from Eqs. (16) and (11), and using ∆G act,07 and Eq. (14) to compute the activation energy. The correlation derived by Koop et al. (2000) is also reproduced along with available experimental data 5 (Alpert et al., 2011;Knopf and Rigg, 2011). For the latter only data reported for T < 221 K is shown to avoid heterogeneous freezing effects.  a w = 0.9, J hom from all formulations agree within three orders of magnitude, and within experimental uncertainty ::::::::::::::::::::::::::::::::::::::::::::::::: (Alpert et al., 2011;Larson and Swanson, 2006) o the measured rates. However for ::: T at a w = 0.9 than at a w = 1.0. Another possibility may be a slight decrease in a w during the experiments. freezing.
: More research and further experimentation is required to clarify this point.
At low temperature (T < 210 K) the usage of Eq. (14) leads to a higher J hom than when 5 ∆G act,07 is used, for both formulations of CNT. For a w < 1 Eq. (16) and Eq. (11) do not overlap as is the case for a w = 1, which results from the different sensitivity to a w of both formulations . ::::: :::

Conclusions
This work advances a phenomenological :::::::::: theoretical : description of the process of interface transfer of water molecules from the liquid phase to the ice during the early stages of nu-5 cleation. Unlike previous approaches, the model presented here does not assume that the water properties in the liquid-ice interface are the same as those of the bulk. Instead a theoretical approach is proposed where the interaction of several water molecules is required for interface transfer. Application of this model resulted in a thermodynamic definition of ∆G act . As D ∞ and σ iw can also be defined on a thermodynamic basis (Adam and Gibbs,10 1965; Barahona, 2014), this work gives support to the assertion of Koop et al. (2000) that the ice nucleation rate can be determined entirely by thermodynamics.
The approach proposed here elucidates two contributions to the activation energy. The first one is the self-diffusion process in the bulk water, that is, the breaking of hydrogen bonds in the liquid phase. The second is the work dissipated during interface transfer, as- 15 sociated with the rearrangement of the water molecules within the ice-liquid interface. The commonly used model of Turnbull and Fisher (1949) neglected the latter. However since homogeneous ice nucleation occurs away from equilibrium, interface transfer implies an energy cost to the system. At temperatures relevant for homogeneous ice nucleation it represents about half of ∆G act . 20 It was shown that at low temperature interface transfer has the largest effect on the nucleation rate. For such conditions ∆G act ∼ ∆G hom and variations in the preexponential factor may dominate the variation in J hom . On the other hand moderate variation in ∆G act will have a limited effect on J hom for pure water droplets since they typically freeze at T > 230 K where ∆G hom >> ∆G act . However ∆G act may have a marked influence for the homo- 25 geneous freezing of haze aerosol which occurs at very low temperature. Also ∆G act may 22 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | impact the nucleation rate when the same formulation is used for heterogeneous ice nucleation as the nucleation work is typically lower than in the homogeneous case.
For T > 230 K the formulation of ∆G act presented here predicts values close to those obtained using empirical correlations, particularly that of Zobrist et al. (2007). However for T < 230 K, Eq.(14) predicts a linear increase in ∆G act with decreasing T , and differs from 5 the nonlinear tendency typically found when ∆G act is assumed to be determined solely by self-diffusivity of bulk water (Ickes et al., 2015). As a result, at low T the preexponential factor, hence the nucleation rate, predicted using empirical formulations of ∆G act tends to be lower than found in this work.
Introducing the new formulation of ∆G act into a generalized form of CNT (Eq. 1) and 10 using the NNF framework to define ∆G hom , resulted in good agreement of J hom with observations, even at very low T where it is underestimated by most models. This is remarkable since no parameters of the theory were found by fitting nucleation rates. Introducing Eq. (14) into a common formulation of CNT and with σ iw constrained as in B14 also led to a good agreement of J hom with measured values. For a w = 0.9 and T > 218 K predicted J hom is 15 in agreeement ::::::::::: agreement : within experimental uncertanity with reported experimental values, however it tends to be higher than measurements at lower T . It is not clear whether systematic deviation in a w during the experiments, or unkown ::::::::: unknown : factors not considered in the theoretical models are the source of this discrepancy and more research is needed to elucidate this point. The NNF model, which can be independently constrained 20 and evaluated, may be more suitable to investigate such differences between theory and measurements than common formulations of CNT where ∆G act and σ iw must be fitted to measured J hom . :::: D ∞ :::: has :::::: been ::::::::::::::: independently ::::::::::: measured ::::::::::::::::::::::::::::: (e.g., Smith and Kay, 1999),

2.
: Guided by MD results, it was assumed that a molecule crossing the interface would interact with four other molecules, so that n t = 16. This is expected at low T 5 since the water structure becomes more ice-like, however n t may be a function of the temperature. For example, the size of cooperative regions in water is known to be a function of the configurational entropy and therefore of temperature (Adam and Gibbs, 1965). It is not clear whether that should also be the case for interface transfer. Another source of uncertainty has to do with the specification of Several studies (e.g, Johari et al., 1994;Koop and Zobrist, 2009) have used some form of thermodynamic continuation below T ∼ 235 K to define a w,eq and ∆h f , which is also used in this work. These functions are not unique since several combinations of parameters 15 can lead to thermodynamically consistent solutions. This work centers on the activation energy as a fundamental parameter. Equation (15) 20 however suggest that the flux of water molecules from the bulk to the ice may be better understood in terms of the bulk self-diffusivity of water and the probability of interface transfer, f (T, a w ). These two quantities have a more specific physical meaning than ∆G act . D ∞ has been independently measured (e.g., Smith and Kay, 1999), whereas f (T, a w ) is related to the work dissipated during ice nucleation and can in principle be obtained from 25 MD simulations.
From their analysis of different models Ickes et al. (2015) concluded that at low T either σ iw is thermodynamically undefined or the temperature dependency of ∆G act reverses. Such predictions are mistaken. This work shows that both ∆G act and σ iw can be defined 24 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | on a thermodynamic basis. The work of Ickes et al. (2015) however shows the difficulties in ascribing physical behavior to the parameters of CNT by fitting experimental results. The independent phenomenological ::::::::::: theoretical : formulation presented here may be more amenable to testing and expansion. In turn, a physically-based definition of the parameters of CNT may improve the development of parameterizations of ice formation in cloud models, Equilibrium a w between bulk liquid and ice (Koop and Zobrist, 2009) E, T 0 Parameters of the VFT equation, 892 K and 118 K, respectively (Smith and Kay, 1999 Koop et al., 2000Bartell & Chushak, 2003Manka et al., 2012Hagen et al., 1981Pruppacher, 1995Riechers et al., 2013Taborek et al., 1985Larson and Swanson, 2006 a w =1