Sensitivities of immersion freezing: Reconciling classical nucleation theory and deterministic expressions


  • Barbara Ervens,

    Corresponding author
    1. Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, Boulder, Colorado, USA
    2. Chemical Sciences Division, NOAA Earth System Research Laboratory, Boulder, Colorado, USA
    • Corresponding author: B. Ervens, Chemical Sciences Division, NOAA Earth System Research Laboratory, Boulder, CO, USA. (

    Search for more papers by this author
  • Graham Feingold

    1. Chemical Sciences Division, NOAA Earth System Research Laboratory, Boulder, Colorado, USA
    Search for more papers by this author


[1] Ice particle number concentrations are often described deterministically, i.e., ice nucleation is singular and occurs on active sites unambiguously at a given temperature. Other approaches are based on classical nucleation theory (CNT) that describes ice nucleation stochastically as a function of time and nucleation rate. Sensitivity studies of CNT for immersion freezing performed here show that ice nucleation has by far the lowest sensitivity to time as compared to temperature, ice nucleus (IN) diameter, and contact angle. Sensitivities generally decrease with decreasing temperature. Our study helps to reconcile the apparent differences in stochastic and singular freezing behavior, and suggests that over a wide range of temperatures and IN parameters, time-independent CNT-based expressions for immersion freezing may be derived for use in large-scale models.

1 Introduction

[2] The explicit representation of microphysical processes in cloud models is a great challenge since it requires the simultaneous description of processes on spatial and temporal scales covering many orders of magnitude. Simplified expressions that properly reflect the underlying physical processes are needed. One of the largest uncertainties in assessing changes in radiative forcing in the atmosphere lies in the description of aerosol-cloud interactions. The interplay between the gas, liquid, and solid water phases in mixed-phase clouds includes the competition for water vapor by droplets and ice particles during formation and growth. Droplet formation and growth by water vapor condensation is a strong function of particle size and to a lesser extent of hygroscopicity [e.g., Ervens et al., 2005]. Heterogeneous ice nucleation is more complex since freezing occurs via different modes (immersion, deposition, or contact freezing) that all require the knowledge of ice nucleus (IN) properties that are much more poorly understood. For heterogeneous freezing, the relevant temperature (T) range is 234 K < T < 273 K; at lower T, homogeneous freezing becomes dominant [Jensen and Toon, 1997; Kanji and Abbatt, 2009].

[3] Many models include empirical expressions to deterministically predict the number of ice particle concentration (NIN) based on observations as a function of T and/or supersaturation [e.g., Cotton et al., 1986; Fletcher, 1969; Meyers et al., 1992]. More refined parameterizations include IN composition and particle size [DeMott et al., 2010; Phillips et al., 2008, 2013]. These expressions are based on continuous flow diffusion chambers (CFDC) in which ambient particles are exposed to conditions of controlled T and/or supersaturation. Particles that freeze within the exposure time (typically tens of seconds) are identified as IN at the prescribed conditions. This sampling strategy implies that the freezing behavior in the atmosphere is sufficiently well represented by these conditions. Similarly, laboratory-based deterministic parameterizations of NIN are derived by recording the fraction of frozen particles in an ice chamber as a function of T [Vali, 1971, 1994].

[4] In contrast to singular, time-independent freezing, classical nucleation theory (CNT) describes a stochastic freezing process and a time-dependent probability of freezing P, which is the fraction of particles that freeze within a given time t and is determined by a nucleation rate J. J is a function of particle diameter DIN, the activation energy at the solution-ice interface ΔFact, and the critical energy of ice germ formation ΔFcr (with k = Boltzmann constant) [e.g., Khvorostyanov and Curry, 2004]:

display math(1)
display math(2)

[5] The contact angle θ is a macroscopic description of the interaction of an IN with surrounding water (in the case of immersion freezing). It is a measure of ΔFcr at the IN surface; the smaller θ and ΔFcr are, the more readily the particle that is immersed in the droplet will initiate freezing. Several studies have suggested that particle surfaces exhibit multiple nucleation sites with different ΔFcr. With increasing DIN and/or number of nucleation sites, the probability of small θ on the IN surface increases [Fletcher, 1958; Niedermeier et al., 2011; Vali, 2008; Vali and Stansbury, 1966]. For such particles, a single good nucleation site will suffice to incur immediate freezing (P → 1); i.e., the process will appear to be time-independent.

[6] In the current study, we perform a systematic sensitivity analysis of CNT and identify parameter ranges of T, t, DIN, and θ over which freezing behavior is particularly sensitive. We then explore the extent to which small variability in T translates to changes in P similar to those inferred due to variability in IN properties (θ and DIN). The results from the sensitivity analysis are discussed in the context of t-independent NIN parameterizations, and conditions are identified for which t dependence can be neglected without considerable effect on P.

2 Sensitivity Studies of Probability of Freezing P to Individual Parameters in CNT

2.1 Sensitivity of Freezing Probability

2.1.1 Definition of the Sensitivity Parameter S(X)

[7] IN properties (DIN and θ) depend on the source and nature of the IN material. DIN can vary from several nanometers to tens of micrometers, and contact angles vary over a range of ~0° < θ < ~180°, with the smallest θ (“best IN”) typically observed for biological material (e.g., bacteria, spores) and larger ones for more abundant IN such as dust or soot [Hoose et al., 2008; Phillips et al., 2009]. While in ice chamber experiments, the exposure time of aerosol particles at a specified T can be properly controlled, in the atmosphere, this parameter is hard to determine. Even if clouds exist for extended periods of time with stable average T, fluctuations on small scales exist, and particles are exposed to different T during their residence time in a cloud.

[8] In the following, we explore the sensitivity to P of the four parameters in equations ((1)) and ((2)) (T, DIN, θ, t). We quantify sensitivity S to P to X (∈ {T, θ, DIN, t} as

display math(3)

with the aid of numerical sensitivity studies, where we vary X over wide ranges while keeping other parameters constant. In this way, we can compare S(X) values in a relative sense.

2.1.2 Sensitivities of P to Properties of Individual Particles

[9] We assume particles with DIN = 500 nm (unless otherwise noted in Figure 1c) that are exposed to prescribed conditions for t = 100 s (unless otherwise noted in Figure 1d). In Figures 1a–1c, we restrict our analysis to results for 0.1 < P < 0.9 since the maximum in S(X) occurs in this range. At lower or higher P, a change in X leads to only small changes in P; S(X) → 0 when P → 1 or P → 0. S(X) values outside this P range would occupy the white space between the data points shown and S(X) = 0; e.g., at a given T, particles with lower θ freeze faster than t = 100 s (P > 0.9) whereas those with higher θ require more than 100 s to reach the minimum P (0.1).

Figure 1.

Sensitivities of freezing probability P, S(X) = ∂ P / ∂ lnX (equation ((3))) based on classical nucleation theory for immersion freezing to (a) temperature (DIN = 500 nm, t = 100 s), (b) contact angle (DIN = 500 nm, t = 100 s), (c) IN diameter (t = 100 s), and (d) time (T = 250 K, DIN = 500 nm). Patterns in the model data are due to the ranges of selected parameters and incomplete sampling of the parameter space. (e) S(T)p for four IN populations that exhibit IN size (DIN,g, σDIN) and contact angle distributions (θg, σθ) (t = 100 s); (f) S(t)p for the same distributions (T = 254 K). Note that for Figures 1e and 1f, all S(X)p are shown whereas for Figures 1a–1d, only absolute maximum S(X) values (0.1 < P < 0.9) are displayed.

[10] Figure 1a shows that the absolute values of S(T) decrease from ~ –2000 at T ~270 K to S(T) ~ –100 at T ~240 K. Similarly, Figure 1b shows T,θ pairs that fulfill the P constraints for constant DIN and t. The values are within the range of ~ –60 < S(θ) < ~–10, and thus, S(θ) is roughly four times smaller than S(T) for T < ~260 K but about 40 times smaller than |S(T)| at T > ~260 K. Higher S(X) at higher T can also be seen for S(DIN) for a diameter range of 20 nm < DIN < 2000 nm (Figure 1c); however, S(DIN) does not exceed ~8 for any conditions, and thus, differences in DIN do not translate into significantly different P. S(DIN) decreases with DIN toward a value of ~2, i.e., at least two and one orders of magnitude smaller than S(T) and S(θ), respectively. The most common known IN types, e.g., dust, soot, and bacteria, usually have DIN > 100 nm. Consequently, while IN types differ in their composition and surface properties (θ), differently sized IN of a given type and with a single nucleation site will not show great variability in freezing behavior. For the results in Figure 1d, we extend the P range to 10–5 < P < 0.9999. One can show analytically that S(t) has a maximum value of e–1 (0.36) at P = (1 – e–1). S(t) is smaller when P → 1 or P → 0, as an increase in t does not lead to any further change in P. Figure 1d verifies that S(t) < e–1. Comparison to all other S(X) shows that among the four parameters P is least sensitive to t.

2.1.3 Sensitivities of P to Properties of Particle Populations

[11] In order to generalize our results, we calculate sensitivities for particle populations S(X)p that exhibit wide DIN and θ ranges, characterized by geometric mean values DIN,g and θg, and geometric standard deviations σDIN and σθ. For populations, P is calculated as the sum of individual P(DIN, θ), weighted by the relative contributions of the individual DIN and θ to the two distributions. Overall S(T)p (Figure 1e) and S(t)p (Figure 1f) are smaller than those in Figures 1a and 1d because we include all P(DIN) values and by definition S(X) = 0 if P ~0 or P ~1. S(T)p values are very sensitive to σθ, all else equal (red and blue lines; Figure 1e) with the highest S(T)p for the narrowest θ distribution. For such distributions, S(T)p and S(T) are of the same order of magnitude. Maximum |S(T)p| values occur at the same T where (θ, T) data pairs for S(T) are predicted for the most frequent θ values in the distribution (θ ~50° ± 10° at ~254 K). Different DIN distributions but identical θ distributions (red and black lines in Figure 1e) lead to very similar S(T)p, which confirms our finding that DIN does not greatly affect P (Figure 1c). All four particle populations show constant S(t)p over a wide t range. As in the case of S(T)p, the highest S(t)p is found for narrow θ distributions, i.e., when a large fraction of particles has P ~e-1 and can significantly change P at a given T (blue line). Overall, the trends in Figures 1e and 1f confirm results in Figures 1a and 1d: While absolute values for S(X)p are smaller, the ratio of the sensitivities is robust, i.e., S(T)p/S(t)p ~ S(T)/S(t) ~104.

2.2 How Much Does X ∈ {T, θ, DIN, t} Have to be Changed for a Prescribed Change in P?

[12] While the sensitivities in Figure 1 already suggest that small variability in T and θ causes much more significant changes in P than variability in DIN and t, in the following, we explore in a more heuristic way X ranges that result in a change in P from P1 = 0.999 to P2 = 0.001. Since Figures 1a–1c suggest a strong T dependence of S(X),we chose three T regimes (cases I–III; T ~269 K, ~258 K, and ~246 K with θ(Ι)  = 24°, θ(ΙΙ) = 51°, and θ(ΙΙΙ) = 79°). Similar to the analysis in Figure 1, we use DIN = 500 nm and t = 100 s and vary T1 to T2 to achieve the prescribed P1 and P2. Results from these sensitivity studies are summarized in the second column of Table 1. It is more practical to express the changes in T and θ as absolute differences ΔX, whereas the ratio X1/X2 is more meaningful to illustrate changes in DIN and t (fourth and fifth columns in Table 1).

Table 1. Change in Parameter X (= T, θ, DIN, t) That Causes a Reduction in Freezing Probability From P1 = 0.999 to P2 = 10-3
  1. One parameter X is varied at a time; all other X are set to values as specified (Cases I-III)
 T ≠ constΤ = 269 ΚΤ = 269 ΚΤ = 269 Κ
Case Iθ = 24°θ ≠ constθ = 24°θ = 24°
 DIN = 500 nmDIN = 500 nmDIN ≠ constDIN = 500 nm
 t = 100 st = 100 st = 100 st ≠ const
 T / Kθ / °DIN / nmt / s
 ΔX = 0.4ΔX = 1.2X1 / X2 = 6X1 / X2 = 6839
 T ≠ constT = 258 KT = 258 KT = 258 K
Case IIθ = 51°θ ≠ constθ = 51°θ = 51°
 DIN = 500 nmDIN = 500 nmDIN ≠ constDIN = 500 nm
 t = 100 st = 100 st = 100 st ≠ const
ΔXΔX = 1.4ΔX = 2.9X1 / X2 = 19X1 / X2 = 6854
 T ≠ constT = 246 KT = 246 KT = 246 K
Case IIIθ = 79°θ ≠ constθ = 79°θ = 79°
 DIN = 500 nmDIN = 500 nmDIN ≠ constDIN = 500 nm
 t = 100 st = 100 st = 100 st ≠ const
 ΔX = 2.1ΔX = 5.7X1 / X2 = 21X1 / X2 = 7000

[13] At T ~269 K, ΔT + 0.4 K achieves the same change in P from P1 to P2 as Δθ + 1.2° or DIN1/DIN2 ~6, or t1/t2 ~7000. At the lowest T (case III, T ~246 K), the corresponding ranges required to change P1 to P2 are ΔT ~2.1 K, Δθ ~5.7°, or a decrease in DIN by a factor 20. This trend is in agreement with the lower sensitivity at lower T (Figure 1); i.e., a larger variation in X is needed in order to cause the change from P1 to P2. In all cases, a very large range in t (t1/t2 ~7000, with little (2%) sensitivity to T) is required to change P1 to P2. This reinforces the weak sensitivity of P to t. The weaker sensitivity of T, DIN, and θ at lower T suggests that the sensitivity to t might become relatively more important at low T. However, one should keep in mind that S(t) is always smaller by several orders of magnitude than S(T) and S(θ). The values in Table 1 allow us to derive some empirical relationships to compare sensitivities in terms of absolute numbers. In all three T regimes, the change in θ is within 2 · ΔΤ [Κ] < Δθ [°] < 3 · ΔΤ [Κ]; i.e., variability of ΔT = 1 K will allow particles within 2° < Δθ < 3° to nucleate. Given that θ distributions of common IN (e.g., dust) cover several tens of degrees [e.g., Marcolli et al., 2007; Wheeler and Bertram, 2012], it can be concluded that only a small fraction of a particle population of a given IN type (characterized by a θ distribution) might be affected by such T variability.

3 Discussion

3.1 Comparison of CNT and the Concept of Ice Nucleating Active Surface Sites (INAS)

[14] The concept of ice nucleating active surface sites (INAS) allows one to evaluate laboratory data from different sources [Connolly et al., 2009; Hoose and Möhler, 2012]. P is expressed as a function of the density of INAS ns [cm–2] and the particle surface area A [cm2]:

display math(4)

[15] The similarity to equation ((1)) is obvious

display math(5)

if J [cm2 s–1] is replaced by J ′ [s–1] = J / DIN2. The time-independent approach in equation ((4)) and the CNT-based equation ((5)) lead to the same P(T) if the terms on the right-hand side in equations ((4)) and ((5)) are equivalent:

display math(6)

[16] The logarithm of ns(T) values shows an approximate linear increase with decreasing T with the slope dependent on the IN bulk composition. Keeping to a conceptual analysis, we choose a slope of d(log ns) / dT ~0.4 / K, which is approximately the slope found for kaolinite [Hoose and Möhler, 2012, Figure 11c]. We do not attempt to reproduce data accurately but present the values of θ that fulfill equation ((6)) for combinations of values of DIN, t, and T, and for three different T values along a line with this slope (Table 2).

Table 2. Contact Angles θ [°] That Fulfill the Equivalency J ′(T,DIN) · t ≡ ns (equation (6))
T236 K241 K246 Κ
ns107 cm–2105 cm–2103 cm–2
  1. Values for ns(T) are approximated by linear fits of log (ns) vs. T from data for kaolinite in Hoose and Möhler [2012, Figure 11c].
tDIN = 100 nm
104 s117.1101.186.2
102 s109.496.683.4
1 s102.692.380.6
DIN = 1000 nm
104 s117.7101.787.0
102 s109.997.384.2
1 s103.293.081.4

[17] Unlike our findings in section 2.1, here, a change in DIN only leads to marginal differences in the θ required to fulfill equation ((6)), even smaller than a change in t by a factor of 100. This smaller sensitivity is due to the smaller sensitivity of J ′ to DIN as compared to J, which is directly dependent on DIN2 (equation ((1))). DIN is included in J ′ in the geometric factor that describes ΔFcr as a function of the IN curvature and the surface tension between the substrate and ice [Fletcher, 1958, 1969]. It should be noted that this analysis refers only to the equivalency in equation ((6)); P in both approaches should show a similar sensitivity to A as in Figure 1c. The θ required to match ns increases with decreasing T. This trend confirms that freezing behavior cannot be represented with a single θ per particle [e.g., Welti et al., 2012]. Based on such experimental studies, observed freezing curves (P vs. T) were fitted to obtain θ distributions over the particle surface area. While the P integrated over the surface of a particle with n nucleation sites (Pn) can be reproduced by a single θ for a given set of T and t, it has been shown that the use of a single θ as compared to multiple ones leads to different NIN = f (T,t) [Ervens and Feingold, 2012].

3.2 Interpretation of Atmospheric Observations Using CNT Sensitivities

[18] In addition to the laboratory-based INAS approach, empirical relationships of NIN and T (and/or supersaturation) have been derived based on ambient observations (section 1) for use in atmospheric models. Many of these expressions predict absolute values of NIN rather than frozen fractions (P) that are related to an upper limit of the particle concentration. Implications of the lack of an upper limit have been discussed [Eidhammer et al., 2009]. The sampling procedure to determine NIN usually involves a CFDC that aspirates particles into a chamber where they are exposed to a preset T for ~10 s.

[19] Based on our analysis in Figure 1 and Table 1, we can attempt some quantification of the fraction of particles that are not identified by the CFDC as IN but do initiate freezing in the atmosphere at the same T over longer time scales: If t is 100 times longer in a cloud (at constant T), particles within 1° < Δθ < 5° might be missed in the CFDC, with the lower value valid at T ≥ 260 K. Since T in the CFDC measurements has a variability of ±0.1 K [Kanji et al., 2011], a 100-fold increase in t could cause a similar uncertainty, in particular at high T. Indeed, correlations of predicted and observed NIN show highest deviations at high T [DeMott et al., 2010]. This fact could either be explained by the higher sensitivity or simply by the much lower NIN that exist at high T (NIN(bacteria) ≪ NIN(dust)). A more sophisticated NIN parameterization for deposition/condensation freezing includes different IN types (i.e., some measure for θ distributions), in addition to DIN and T [Phillips et al., 2008, 2013]. Given the large uncertainty associated with ambient IN measurements, the small variability caused by different exposure times can likely be neglected.

[20] Our suggestion that CFDC measurements do not give significantly biased NIN is in contrast to recent findings that suggest that stochastic freezing occurs over several hours (~1 day) in clouds without any significant IN entrainment [Westbrook and Illingworth, 2013]. The authors suggest that CFDC measurements might significantly underestimate the true NIN. Based on our analysis, we estimate that differences of five orders of magnitude in t (10 s vs. 1 day) are equivalent to a change of ΔT ~ –2 K (at T ~258 K, i.e., the approximate T of the observed clouds). Even though the average T within the cloud might have been constant, small-scale fluctuations could have caused the observed increase in NIN. While our findings based on CNT analysis could explain the observed behavior, other effects might also be responsible for an apparent continuous freezing. For example, physicochemical modification of IN during exposure to ambient conditions could cause changes in θ (and possibly to a lesser extent in DIN) and enhance the IN ability of particles.

4 Summary and Conclusions

[21] Sensitivity studies of classical nucleation theory (CNT) reveal that variabilities in time (t) and ice nucleus diameter (DIN) have less impact on the frozen fraction (P) than variations in temperature (T) and surface properties (contact angle θ). In general, all sensitivities decrease with decreasing T. Case studies show that a change in T of ~1 K has a similar impact on P as θ changes of Δθ = 2° whereas a similar change is only caused by an increase in DIN by one order of magnitude, or in t by three orders of magnitude. This high sensitivity to T can explain the feasibility of empirical expressions that neglect t as a controlling parameter. The weak sensitivity to t suggests that common IN samplers might only miss a small fraction of particles by restricting the sample time to a few seconds. Small uncertainties in T in the instrument might significantly bias the sampling; e.g., an uncertainty of ΔT ±0.2 K translates to the same uncertainty as differences in t of about two orders of magnitude. We show that sensitivity trends for single particles (one DIN and θ) also hold for IN populations that cover ranges of DIN and θ distributions. While most expressions that predict ice number concentrations as currently implemented in models are empirically based, it seems feasible to develop more physically (CNT) based relationships that include T, DIN, and IN type (i.e., θ range). Assuming different values for time over atmospherically relevant ranges will only lead to minor differences in the parameterization coefficients.


[22] The authors acknowledge support from NOAA's Climate Goal.