Exploring an approximation for the homogeneous freezing temperature of water droplets

In this work, based on the well-known formulae of classical nucleation theory (CNT), the temperature TNc=1 at which the mean number of critical embryos inside a droplet is unity is derived from the Boltzmann distribution function and explored as an approximation for homogeneous freezing temperature of water droplets. Without including the information of the applied cooling rate γcooling and the number of observed dropletsNtotal_droplets in the calculation, the approximation TNc=1 is able to reproduce the dependence of homogeneous freezing temperature on drop size V and water activity aw of aqueous drops observed in a wide range of experimental studies for droplet diameter > 10 μm and aw > 0.85, suggesting the effect of γcooling and Ntotal_droplets may be secondary compared to the effect of V and aw on homogeneous freezing temperatures in these size and water activity ranges under realistic atmospheric conditions. We use the TNc=1 approximation to argue that the distribution of homogeneous freezing temperatures observed in the experiments may be partly explained by the spread in the size distribution of droplets used in the particular experiment. It thus appears that the simplicity of this approximation makes it potentially useful for predicting homogeneous freezing temperatures of water droplets in the atmosphere.


Introduction
Since the summary article of McDonald (1953), it has been widely observed that ice nucleation of water droplets does not occur at the ice melting temperature (e.g., 273.15K at 1 atm), and liquid water is frequently observed in clouds as cold as to 238 K (Rosenfeld and Woodley, 2000;Hu et al., 2010).Laboratory observations of homogeneous ice nucle-ation in pure water generally show that all droplets do not freeze at exactly the same temperature, and that the fraction of droplets that freeze in a given time is a function of temperature and time (hereafter we refer to this type of experiment as a fraction experiment) (e.g., Bigg, 1953;Carte, 1956;Broto and Clausse, 1976;Earle et al., 2010;Riechers et al., 2013).Here, experimental data of the freezing temperatures of pure water droplets from 15 independent studies over the past 60 years are collected (Fig. 1 and Table 1), showing a clear dependence of freezing temperature upon drop volume across different experiments.Over the investigated size interval (1-1000 µm diameter), observed freezing temperatures range from 232 to 240 K.The range of freezing temperatures and the volume dependence in Fig. 1 are consistent with the experimental data reviewed in Pruppacher (1995).
On the other hand, solutes, at sufficiently high concentrations, can suppress the homogeneous freezing temperature of water droplets.Koop et al. (2000) showed that the depression of freezing temperature strongly depends on the water activity a w of the solution droplet, which has been confirmed in several independent experimental studies (e.g., Knopf and Lopez, 2009;Knopf and Rigg, 2011).In this paper, two aforementioned features of homogeneous ice nucleation observed in the experimental data are examined -(1) the volume and water activity dependence of homogeneous freezing temperatures of water droplets T f (V , a w ) and (2) the distribution of homogeneous freezing temperatures observed in fraction experiments f (T f ).In this paper, we describe only volume-based nucleation and do not include the droplet surface effects on homogeneous ice nucleation as there remains considerable uncertainty about the importance of surface nucleation (Kay et al., 2003;Duft and Leisner, 2004).Unified explanations of the observed dependencies of the homoge-Published by Copernicus Publications on behalf of the European Geosciences Union.neous freezing temperature on droplet size and water activity have been proposed by several studies based on different theoretical frameworks such as ice nucleation rate J and density fluctuation (e.g., Pruppacher, 1995;Baker and Baker, 2004;Khvorostyanov and Curry, 2009;Barahona, 2014).In our study, based on a cornerstone of classical nucleation theory (CNT), namely that a critical embryo existing in a droplet triggers ice crystal formation, we explore a simple approxi-mation for the homogeneous freezing temperature and seek a simpler parameterization to describe homogeneous ice nucleation process in the atmosphere.Section 2 describes the approximation, Sect. 3 gives the comparisons between the theoretical estimates and the experimental data, Sect. 4 is the discussion, and Sect. 5 is the summary.
2.1 The approximation T N c =1 (V , a w ) According to CNT, the formation of a critical embryo inside a droplet can trigger the freezing process in the droplet.The critical embryo, defined as the i-mers having the highest formation energy, is formed by the critical fluctuation in orientation of hydrogen bonds (e.g., density fluctuation) (Baker and Baker, 2004), which is large enough to provide the formation energy of the critical embryo F c (T , a w ) and remove metastability of supercooled water.The probability of occurrence of the critical fluctuation is exp − F c (T ,a w ) k B T (Landau and Lifshitz, 1969, p. 472-473;Pruppacher and Klett, 1997), and thus the mean number of the critical embryos inside a water droplet in thermal equilibrium can be predicted by a Boltzmann distribution (Landau and Liftshitz, 1969, p. 107;Vali, 1999), where V is the volume of the droplet, ρ is the number density of water molecules, k B is Boltzmann's constant, T is the temperature of the droplet, and F c (T , a w ) is the formation energy of the critical embryo in the droplet with water activity a w at T , which will be discussed in detail in Sect.2.2.The Boltzmann distribution form of the critical embryo is derived from the partitioning function of the grand canonical ensemble, and it should be noted that the derived particle number of the Boltzmann distribution function is not a "constant" but a "mean" number (detailed derivation and explanations can be found in Landau andLiftshitz, 1969, p. 107, andSadovskii, 2012, Chapter 3.1).The total freezing time τ freezing of a water droplet can be split conceptually into three stages -(1) τ meta_remove (∼ 1 J ), the time needed for the occurrence of the critical fluctuation; (2) τ formation , the time needed to form a critical embryo; and (3) τ growing , the growing time for the critical embryo expanding to the whole droplet body.These depend on V , a w and T of the droplet (Pruppacher and Klett, 1997;Bauerecker et al., 2008).To observe freezing of droplets with volume V and water activity a w occurring at temperature T , the residence time of freezing experiments τ residence at T has to be longer than τ freezing (V , a w , T ), resulting in a dependence of the homogeneous freezing temperature on the cooling rate γ cooling of droplets in principle.According to the theoretical estimates (see Pruppacher and Klett, 1997, p. 678), the timescale of τ formation + τ growing for the size of the droplets investigated here is short compared with the typical residence times in the laboratory studies.Thus, the dominant factor determining the homogeneous freezing temperatures is τ meta_remove .Because τ meta_remove is the time needed for the occurrence of the critical fluctuation among water molecules, τ meta_remove is shorter in a larger droplet with more molecules Vρ or at lower temperature when the fluctuation probability exp( − F c (T ,a w ) k B T ) is higher; τ −1 meta_remove ∝ N c_mean (V , a w , T ).Embryo creation and growth is a stochastic process and N c_mean (V , a w , T ) simply expresses the mean state, so there is always a spread of τ meta_remove among droplets even in an idealized case that all the droplets used in the experiment have exactly the same V and a w and are at exactly the same temperature T .The spread of τ meta_remove can be wider when there are more observed droplets N total_droplets , which in principle can explain the fraction experiments that some droplets with shorter τ meta_remove can always be frozen at higher temperature, or in shorter time for droplets at the same temperature even when the droplets have a monodisperse size distribution and exactly the same a w .Hereafter we refer to the distribution of homogeneous freezing temperatures owing to N total_droplets when all the droplets have exactly same V and a w as a stochastic feature.Based on the abovementioned principles, the homogenous freezing temperature of water droplets and τ meta_remove can each be written as a function of V , a w , γ cooling and N total_droplets , namely T f (V , a w , γ cooling , N total_droplets ) and τ meta_remove (V , a w , γ cooling , N total_droplets ).Koop et al. (1998) reported that observed homogeneous freezing temperatures do not significantly depend on γ cooling of the droplets for γ cooling smaller than 20 K min −1 (corresponding to vertical velocities 33.3 m s −1 in clear air).The results of Koop et al. (1998) actually indicate that the slope of is very steep at the temperature when the scale of τ meta_remove is close to τ residence in most practical experiments and realistic atmospheric conditions, resulting in the small dependence of T f on γ cooling as suggested by Brewer and Palmer (1951).Based on that, in most of the practical freezing experiments and realistic atmospheric conditions (γ cooling < 20 K min −1 ), the observed homogeneous freezing temperatures can be considered as a threshold temperature when ∂τ meta_remove ∂T → ∞.In this study, we intend to find this threshold temperature directly from the information given by N c_mean (V , a w , T ).The number of critical embryos derived from the Boltzmann distribution is a mean value and does not provide any information regarding freezing time, so it cannot be used to study the dependence of the homogeneous freezing temperature on cooling rate (i.e., time dependence) and number of droplets used in the experiments (i.e., stochastic feature).Nevertheless, since the formation of one critical embryo is required to trigger the ice nucleation process in a droplet, T N c =1 may be a good approximation for the threshold temperature, the temperature at which the mean number of the critical embryos inside a droplet is unity, which can be given by According to the formula of F c (T , a w ), T N c =1 is determined by V and a w of the droplet, namely T N c =1 (V , a w ). Figure 2 shows the mean number of critical embryos inside a pure water droplet (a w = 1) at different temperatures using Eq.(1) (see next section for details of F c (T , a w ) used in the calculation).It indicates that smaller droplets require lower temperatures to reach the state that N c_mean = 1, showing the volume dependence of T N c =1 (V , a w ). Figure 3 shows the mean number of critical embryos inside a solution droplet with different values of water activity.The result indicates that more concentrated solution droplets (lower a w ) need lower temperature to reach the state that N c_mean = 1.This represents the solution effect on T N c =1 (V , a w ).The sensitivity of T N c =1 (V , a w ) to the variation of diameter δd and water activity δa w of droplets can be written as where d is the diameter of droplet (µm).As shown in Fig. 1, the dependence of T N c =1 on log 10 d is nearly linear, so the decadal log is used here to simply derive the linear dependence.The values of ∂T Nc=1 ∂a w and ∂T Nc=1 ∂log 10 d are about 216 and 2.5 K, respectively, over the investigated interval of water activity and drop size, which are derived numerically from Eq. (2).

Formation energy of the critical embryo
F c (T , a w ) The formation energy of the critical embryo F c (T , a w ) can be written as where σ i/w (T , a w ) is the interfacial energy between liquid water and solid ice, s is the shape factor of the embryo (∼ 21 by assuming the shape is hexagonal prism), r c is the radius of the critical embryo, v water 1 is the volume of single water molecule, e sw and e si are the saturation vapor pressures over water and ice, respectively (Murphy and Koop, 2005), and a w is the water activity of the solution droplet (see detailed derivations of Eq. 4 in Defour andDefay, 1963, andPruppacher andKlett, 1997).It should be noted that the term k B T ln(a w ) in r c (Eq. 5) is the entropy of unmixing, which originates from the change in the Gibbs free energy of the bulk solution during freezing and has generally been neglected in the previous theoretical studies (Bourne and Davey, 1976;Black, 2007).Barahona (2014) pointed out that although this term is small for dilute solution, it should not be neglected when applying to high-concentration solution droplets (see Eq. 8 in Barahona, 2014).
The value of interfacial energy between liquid water and solid ice σ i/w (T , a w ) is needed for our calculation of Eqs. ( 4) and ( 5).As most studies suggest that the temperature dependence of σ i/w (T , a w ) should be linear (Ickes et al., 2015), and that increasing the concentration of the solution droplet increases the value of σ i/w (T , a w ) (Jones and Chadwick, 1971;Alpert et al., 2011), σ i/w (T , a w ) can be written as where σ i/w,e is the interfacial energy at the equilibrium temperature of pure ice water and T 0 is the equilibrium temperature.The direct measurement of σ i/w (T , a w ) is extremely difficult, so most of the estimations are based on combinations of CNT and laboratory measurements of T f and observed freezing rate to retrieve the values of σ i/w (T , a w ) (e.g., Zobrist et al., 2007;Murray et al., 2010).These studies have shown considerable diversity in the reported estima- tions of σ i/w (T , a w ) (Ickes et al., 2015).Instead, we use values of σ i/w,e and ∂T derived from a state-of-the-art molecular dynamics model that explicitly simulates the molecular configurations under supercooling conditions.Benet et al. (2014) give values of σ i/w,e from the TIP4P (σ i/w,e = 26.5 × 10 −3 J m −2 ), TIP4P/2005 (σ i/w,e = 27 × 10 −3 J m −2 ), and TIP4P-Ew water models (σ i/w,e = 27.5 × 10 −3 J m −2 ), and these three values will all be used in our calculations.According to Ickes et al. (2015), the values of σ i/w,e used here are close to the median of all the values derived from the previous studies.Regarding Espinosa et al. (2014) provided an average value of 0.25 × 10 −3 (J m −2 K −1 ) from three different water molecular models (TIP4P/ICE, TIP4P and TIP4P/2005) down to a supercooling of about 30 K. Regarding ∂σ i/w ∂a w , Barahona (2014) proposed a new thermodynamic framework approximating the the interfacial energy between ice and aqueous solution by assuming the interface between solid ice and liquid water is made of liquid molecules trapped by the solid matrix, which gives the relationship between σ i/w and a w .Based on this approximation, the solution effect on the interfacial energy can be written as where w is the surface excess of water (∼ 1.46) (Spaepen, 1975) and s area is the surface area parameter (∼ 1.105 mol 2 3 ) (see Barahona, 2014, for details).The values of σ i/w (T , a w ) estimated from above studies are used to derive the numerical result T N c =1 (V , a w ) presented here.To test our approximation, we aim to compare the observed T f (V , a w ) and f (T f ) with T N c =1 (V , a w ) derived using the constraint in Eq. ( 2).First, T N c =1 (V , a w = 1) of pure water droplet is derived.Figure 1 shows the comparison between the experimentally determined homogeneous freezing temperatures T f (V , a w = 1) (details of the experiments are provided in Table 1) and the approximations T N c =1 (V , a w = 1).For droplet diameters > 10 µm, the theoretical values of T N c =1 (V , a w = 1) derived by the value of σ i/w,e from TIP4P water model agree very well with most of the experimental data T f (V , a w = 1).Using the values of σ i/w,e from TIP4P/2005 and TIP4P-Ew leads to a shift downward of about 1-2 K of T N c =1 (V , a w = 1).There is one study regarding the time dependence that should be mentioned.The laboratory observation of Murray et al. (2010) (black triangle in Fig. 1) showed that varying of cooling rate from 2.5 to 10 K min −1 corresponds to a shift of 0.5 to 1 K in observed freezing temperatures of pure water droplets, and our best agreement estimates that T N c =1 (V , a w = 1) can only explain the experimental data with the slowest cooling rate (2.5 K min −1 ).The finding of Murray et al. (2010) will be discussed in Sect. 4. For droplets smaller than 10 µm (diameter), there are obvious deviations of observed freezing temperatures among the experimental studies.These studies do not provide enough information regarding γ cooling , N total_droplets and the spread in drop size, so we cannot evaluate what causes the disparity.We suggest that freezing experiments of pure droplets smaller than 10 µm (diameter) need more refinement and should report the potentially important dependencies such as applied cooling rate, size distribution  (Koop et al., 2000;Knopf and Lopez, 2009;Knopf and Rigg, 2011) and the approximation T N c =1 (V , a w ) (using σ i/w,e from TIP4P model).
of droplets and number of observed droplets used in experiments.
Second, the solution effect on homogeneous freezing temperature T f (V , a w ) is explored by changing the water activity in Eqs. ( 5) and ( 6) to derive the approximation T N c =1 (V , a w ), which will be compared with the experimental data collected in Koop et al. (2000), Knopf and Lopez (2009) and Knopf and Rigg (2011).Droplet sizes used in the collected experimental data range from 1 to 10 µm in Koop et al. (2000), from 10 to 80 µm in Knopf and Lopez (2009) and from 20 to 80 µm in Knopf and Rigg (2011), and these sizes are included to calculate the approximation T N c =1 (V , a w ). Figure 4 shows the comparison between the experimental data and the approximation T N c =1 (V , a w ).Without considering the time dependence (γ cooling varying from 1 to 10 K min −1 among all the experiments) and the stochastic feature (i.e., N total_droplets ), the result shows that the approximation T N c =1 (V , a w ) is in good agreement with the experimental data for a w > 0.85.The scattering of the experimental data between the theoretical estimates for a w > 0.85 (i.e., T N c =1 for d = 1 to 80 µm) suggests that the spread of droplet size applied in the experiments may play an important role in the spread of homogeneous freezing temperatures.For the solution droplets with high concentration (a w < 0.85), the observed freezing temperatures show considerable spread.Abbatt et al. (2006) suggest that the disparity of the experimental data for low a w can be partly attributed to a variety of heterogeneous process, which can result in the higher observed freezing temperatures.In addition, as suggested by Knopf and Lopez (2009), the deviations at low water activity may be most likely due to our incomplete understanding of a w for certain aqueous solutions and the corresponding uncertainties.Future experimental study, we suggest, should focus on the freezing process of solution droplets with high solute concentration (a w < 0.85) to clarify the causes of the disparity.
Regarding the experimental uncertainty, Knopf and Lopez (2009) reported that the value of a w for supercooled aqueous solutions has the experimental uncertainty δa w of about ±0.01, which can result in the variation in T N c =1 of about ±2 K based on Eq. ( 3).Riechers et al. ( 2013) reported that the size of droplets produced by the microfluidic device used in their experiment has 3 standard deviations (99.7%) of about 18 to 33 µm in diameter, which can cause a variation in T N c =1 of about ±0.2 to ±0.5 K based on Eq. (3).Therefore, the variation in T N c =1 caused by the experimental uncertainties δa w and δd can be substantial and should not be neglected.We suggest that future experimental studies should provide detailed information regarding experimental uncertainties δa w and δd for the purpose of better constraining the observed freezing temperatures.

Fraction of frozen pure water droplets as a function of temperature f (T f )
To further examine the application of T N c =1 (V , a w ) in homogeneous ice nucleation, T N c =1 (V , a w ) is compared to the experimental data of the fraction experiment of Riechers et al. (2013).According to CNT, the stochastic feature of the ice nucleation process can basically explain the distribution of freezing temperatures observed in the fraction experiments (Pruppacher and Klett, 1997, Eqs. 7-71;Koop et al., 1998;Niedermeier et al., 2011).However, current technology to produce water droplets for such experiments introduces a spread of sizes, and the freezing temperatures show a clear dependence on droplet volume (Fig. 1), so the spread in sizes of water droplets used in the experiments may be important for explaining the distribution f (T f ).In other words, the size distribution of droplets used in a given experiment  may be an important factor governing the observed spread of freezing temperatures (i.e., dotted line shown in Fig. 1).
To test this, we incorporate the reported droplet size distribution width into the numerical calculation.Unique among such studies, Riechers et al. ( 2013) report both the spread of homogeneous freezing temperatures and the mean µ and standard deviation σ of droplet size.According to Eq. ( 3), the spread in the size distribution of water droplets will result in a spread in the fraction of frozen droplets because larger droplets have higher T N c =1 (V , a w ) (i.e., require less supercooling to freeze).Given the droplet size width, the distribution of the approximations T N c =1 (V , a w ) of droplets can be derived from Eq. ( 2).Given a Gaussian distribution of drop sizes, we estimate the fraction of drops that will freeze at a given temperature solely by assuming that the spread in freezing temperatures arises from the spread in droplet sizes based on Eq. (3).For example, we estimate T N c =1 (V , a w ) of the droplets with size of µ + 3σ (∼ the largest 0.15 % of the drops) as the theoretical onset freezing temperature T onset f , T N c =1 (V , a w ) of the droplets with size of µ + 1.64σ (≈ the largest 10 % of the drops) as the theoretical estimates T 10 % f , T N c =1 (V , a w ) of the droplets with mean size as the theoretical estimates T 50 % f , T N c =1 (V , a w ) of the droplets with size of µ − 1.64σ (≈ the smallest 10 % of the drops) as the theoretical estimates T 90 % f , and T N c =1 (V , a w ) of the droplets with size of µ − 3σ (≈ the smallest 0.15 % of the drops) as the theoretical estimates T end f .The results presented in this section only use the value of σ i/w,e from the TIP4P water model, which has the best agreement with the experimental data shown in Sect.3.1 (Fig. 1).
There are five experimental results from Riechers et al. ( 2013), each with different µ and σ .The comparisons (Fig. 5 and Table 2) show that our estimates match the experimental data fairly well.The slope of the freezing fraction vs. temperature in the theoretical results is driven entirely by the reported spread in the size distribution of drops and matches fairly well with the observed slope, although across the experiments the theoretical slope is somewhat greater (observed values are shifted to the right of the blue curve at the higher temperatures but mostly to the left at the lower temperature), which might be attributable to the stochastic feature of the ice nucleation process.That said, the observational uncertainties in the experimental values of T onset , T 10 % , T 50 % and T 90 % more or less span the theoretical values derived from Eq. (2).Riechers et al. ( 2013) also reported that, during cooling, the majority of the droplets are frozen over a temperature interval of 0.84-0.98K, which is consistent with the range between the theoretical estimates T onset f and T end f derived here, namely 0.42-1.06K from five different droplet size distributions, suggesting the spread in droplet size (i.e., a disperse distribution) may be an important factor governing the spread of the homogeneous freezing temperatures observed in a given fraction experiment.
The comparisons made in Sect.3.1 to 3.2 shows that the distribution of the freezing temperatures among the data can mostly be explained by the dependence of T N c =1 (V , a w ) on V and a w for droplet diameter > 10 µm and a w > 0.85 without considering the dependence of homogeneous freezing temperature on N total_droplets and γ cooling in the calculations.It suggests that, in most of the practical experi- ments and for most atmospheric conditions, the timescale of τ residence is shorter than τ meta_remove at the temperatures higher than T N c =1 (V , a w ) (i.e., τ residence < τ meta_remove , when T > T N c =1 (V , a w )), and when the temperature of the droplets is close to T N c =1 (V , a w ), the timescale of τ meta_remove decreases strongly with temperature decreases and becomes shorter than τ residence of the experiments (i.e., τ residence > τ meta_remove when T < T N c =1 (V , a w )).This leads to the result that most of the homogeneous ice nucleation process can only be observed at temperatures close to T N c =1 (V , a w ) even though, in principle, droplets can be frozen at any temperature.

Discussion
As mentioned in Sect.2, the observed freezing temperatures with γ cooling ∼ 2.5 K min −1 reported in Murray et al. (2010) can be well described by T N c =1 (V , a w ), but there is a shift of 0.5 to 1 K in observed freezing temperatures when varying the cooling rate from 2.5 to 10 K min −1 .One possibility is that the total freezing time τ freezing needed to freeze a droplet at T N c =1 (V , a w ) is longer than the timescale of τ residence when γ cooling is higher than 2.5 K min −1 , which may be attributed to τ meta_remove , τ formation or τ growing .Without considering the experimental uncertainty associated with the thermal equilibrium time τ thermal , these 0.5 to 1 K shifts correspond to 3 to 6 s shifts (for γ cooling = −10 K min −1 ), which may be partly caused by τ formation + τ growing .Bauerecker et al. (2008) (hereafter Ba08) explored an advanced method providing time series of water droplet temperature during the entire cooling and freezing process (from supercooled wa-ter to completely freezing) using an infrared camera.The results of Ba08 showed for the droplet sized 3 mm (diameter) that τ growing is around 20 s and τ thermal is around 60 s.The droplet used in Ba08 is much larger than the size normally used in the freezing experiments because of the limitation of IR camera sensitivity.If τ growing linearly depends on drop radius, we may expect it to be several tenths of a second for the drops sized 10-100 µm in diameter.We suggest that the infrared camera technique should be used more widely in the future experimental studies of ice nucleation with smaller droplets, which can add significant insights into the time dependence study of ice nucleation, and clarify the importance of τ meta_remove , τ formation and τ growing observed in the experiments.On the other hand, Koop et al. (1998) suggested that mass transport of water can take place between the frozen ice particles and supercooled droplets when the cooling rate is smaller than about 2 K min −1 , but if the cooling rate is too large, it can cause an offset between the measured temperature and the actual temperature of the drops, which can cause a bias in the observed freezing temperatures.Therefore, we suggest that in future experimental studies, in order to precisely measure ∂T f ∂γ cooling , potential biases at high cooling rate and the shift caused by τ formation + τ growing should be better constrained.Since Koop et al. (1998) and Murray et al. (2010) showed different dependencies of homogeneous freezing temperatures on γ cooling , future experiments should reexamine and perform the same experiments for γ cooling > 2.5 K min −1 .The results shown in Figs. 1 and  4 suggest that the time consideration may be more important when droplet volume and water activity are low where the experimental data show considerable inconsistency (i.e., www.atmos-chem-phys.net/16/7239/2016/Atmos.Chem.Phys., 16, 7239-7249, 2016 a w < 0.85 and d < 10 µm), and future experiments are suggested to emphasize these droplet size and water activity ranges.

Summary
The limitation of our method proposed here is that the time dependence and the stochastic feature of homogeneous freezing temperature cannot be considered because the Boltzmann distribution applied here is a average distribution and does not provide any information regarding time.Combining the well-known Boltzmann distribution function for the mean number of critical embryos N c_mean (V , a w , T ) and their formation energy F c (T , a w ) from CNT formulae, T N c =1 (V , a w ) is derived as a function of volume and water activity of water droplets.With the comparisons made in Sect.3.1 to 3.2, it can be summarized that, under most atmospheric conditions, homogeneous freezing temperatures can be well described by the new approximation T N c =1 (V , a w ) proposed here without considering information of the applied cooling rate (i.e., time dependence) and the number of droplets used in the experiment (i.e., stochastic feature) for d > 10 µm and a w > 0.85.Future experimental study is suggested to focus on the homogeneous freezing process of droplets with high solute concentration (a w < 0.85) and small volume (d < 10 µm).The experimental spread in homogeneous freezing temperatures of water droplets may be partly explained by the size distribution of droplets used in the experiments.The advantage of our approximation in the cloud modeling is that the "temperature history" of droplets is not required to calculate the homogeneous freezing temperature as it is when using the ice nucleation rate (i.e., Eqs.7-71 in Pruppacher and Klett, 1997).When using the ice nucleation rate J (T (t)), the complete temperature history of droplets is needed to calculate the integration of J (T (t)) with respect to time in order to consider the time dependence and the stochastic feature, which can introduce considerable complexity in cloud modeling.However, based on the experimental studies of homogeneous freezing temperature collected and discussed in our study, we suggest that, in most of the practical experiments and realistic atmospheric conditions (i.e., γ cooling < 20 K min −1 ), the time dependence and the stochastic feature of homogeneous freezing temperature may be a secondary factor compared to the effect of volume and water activity for droplet diameter > 10 µm and a w > 0.85.The approximation proposed here is relatively simple to implement into cloud models and may improve the representation of homogeneous ice nucleation in the atmosphere.

Figure 1 .
Figure1.Freezing temperatures of pure water droplets: comparison between the approximations T N c =1 (V , a w = 1) and the collected experimental data.Experimental data: the uncertainties and ranges of the drop size and the freezing temperatures are presented by the dotted lines if information is provided by the studies (details in Table1).The approximations T N c =1 (V , a w = 1): blue line -σ i/w,e from the TIP4P model; green line -σ i/w,e from the TIP4P/2005 model; and red line -σ i/w,e from the TIP4P-Ew model.See Sect. 2 for the definition and calculation of T N c =1 .

Figure 2 .
Figure 2. Mean number of critical embryos N c_mean (by Eq. 1) in a pure water droplet (a w = 1) with different size (diameter) as a function of temperature.Solid circle: the approximations T N c =1 (V , a w ) derived by Eq. (2) (using σ i/w,e from the TIP4P model).

Figure 3 .
Figure 3. Mean number of critical embryos N c_mean (by Eq. 1) in a solution droplet (diameter = 10 µm) with different water activity as a function of temperature.Solid circle: the approximations T N c =1 (V , a w ) derived by Eq. (2) (using σ i/w,e from TIP4P model).

3
Results -comparison between the approximation and the experimental data 3.1 Volume and water activity dependence of T f (V , a w )

Figure 4 .
Figure 4. Comparison between the experimental data of freezing temperatures of solution droplets(Koop et al., 2000;Knopf and Lopez, 2009;Knopf and Rigg, 2011) and the approximation T N c =1 (V , a w ) (using σ i/w,e from TIP4P model).

Figure 5 .
Figure 5.Comparison between the experimental results of the fraction experiment from Riechers et al. (2013) and the theoretical estimates derived here.Red: experimental results with uncertainties from Riechers et al. (2013).Blue: theoretical estimates (σ i/w,e from the TIP4P model).

Table 1 .
Information regarding the details of the homogeneous ice nucleation experiments used in the comparison, including the size, the freezing temperature, and the cooling rate and uncertainty of the experiments.

Table 1 .
Continued.Mode ; and d not defined or provided by the experiments.Diameter of water droplets used in the experiments: + median size; m mean size; and n not provided by the experiments.
Homogeneous freezing temperature T f : a Freezing temperature when half of the water droplets are frozen T 50 % ; b freezing temperature when 95 % of the water droplets are frozen T 95 % ; c freezing temperature when most of the droplets are frozen (peak signal) T

Table 2 .
Comparison between the experimental results of the fraction experiment from Riechers et al. (2013) and the theoretical estimates T N c =1 derived here.