Journal of Geophysical Research: Atmospheres

The role of time in heterogeneous freezing nucleation

Authors

  • Timothy P. Wright,

    1. Department of Marine, Earth, and Atmospheric Sciences, North Carolina State University, Raleigh, North Carolina, USA
    Search for more papers by this author
  • Markus D. Petters

    Corresponding author
    1. Department of Marine, Earth, and Atmospheric Sciences, North Carolina State University, Raleigh, North Carolina, USA
    • Corresponding author: M. D. Petters, Department of Marine, Earth, and Atmospheric Sciences, North Carolina State University, Campus Box 8208, Raleigh, NC 27695–8208, USA. (markus_petters@ncsu.edu)

    Search for more papers by this author

Abstract

[1] A small fraction of particles in the atmosphere can catalyze ice formation in cloud water drops through heterogeneous freezing nucleation at temperatures warmer than the homogeneous freezing temperature of approximately −38°C. The rate for heterogeneous freezing nucleation is dependent on several factors, including the type and surface area of dust that is immersed inside the drop. Although nucleation is an inherently stochastic process resulting from size fluctuations of the incipient ice germ, there is a growing body of literature that suggests that quasi-deterministic models of ice nucleation can describe laboratory experiments. Here we present new experiments and simulations that aim to better constrain theoretical models fitted to laboratory data. We collected ice nucleation data for Arizona Test Dust aerosol immersed in water using a droplet freezing assay setup that allows for the cooling rates to be changed between 10 and 0.01 K min−1. Discrete event simulations based on a variant of the multiple-component stochastic model of heterogeneous freezing nucleation were used to simulate different experimental procedures. The nucleation properties of the dust are specified by four material-dependent parameters that accurately describe the time dependence of the freezing process. We anticipate that the combination of discrete event simulations and a spectrum of experimental procedures described here can be used to design more meaningful laboratory experiments probing ice nucleation and will aid the development of better parameterizations for use in models.

1 Introduction

[2] Ice initiation in mixed-phase tropospheric clouds proceeds via heterogeneous freezing nucleation of water on aerosol surfaces. Particles that provide such surfaces are a subset of the atmospheric aerosol and are termed ice nuclei (IN). Primary ice in clouds can form on ice nuclei via several distinct pathways: (1) deposition of supersaturated water vapor with respect to ice on dry aerosol surfaces (deposition mode), (2) activation of an incipient ice germs on aerosol surfaces that are immersed in supercooled cloud droplets (immersion and condensation mode), (3) movement of an ice nucleus across the water/air interface (contact mode), and (4) any of the previous mechanisms aided by the presence of cooling (evaporation freezing) or electric charge (electrofreezing) [Baumgardner et al., 2012].

[3] An important open question in the field of cloud physics is the evolution of ice crystal concentrations with time. For example, the properties of persistent mixed-phase Arctic stratus clouds can only be explained if a continuous slow ice nucleation process is assumed, e.g., entrainment of IN from the free troposphere followed by contact nucleation [Fridlind et al., 2012]. In general, the time evolution of ice crystal concentrations in clouds can be attributed either to secondary processes such as rime splintering [Hallett and Mossop, 1974] or to the time dependence of the primary ice formation via freezing nucleation. Classical nucleation theory (CNT) of immersion freezing [Pruppacher and Klett, 1996] assumes that the ice phase forms on a germ that is located on a surface and that the germ continually fluctuates in size due to the fusion and melting of individual water molecules. Once the germ reaches a critical size, it becomes activated and grows rapidly at the expense of the surrounding liquid. This stochastic process gives rise to a nucleation rate which is expressed as the number of nucleation events per unit surface area and time. Classical nucleation theory therefore predicts that a sufficiently dispersed homogenous substrate, such as an aerosol of a single type, will result in an increasing number of ice crystals if the sample is given time to reside at a certain thermodynamic condition.

[4] One hypothesis is that ice nuclei surfaces are heterogeneous and that nucleation proceeds on preferred active sites [Fletcher, 1969]. These sites aid the formation of the lower entropy crystal structure of the newly formed ice phase and may correspond to crystal defects in a substrate [Vonnegut, 1947]; pores, cracks, or ledges etched on the surface [Knight, 1979; Sear, 2011]; or chemically primed insoluble carbonaceous surfaces [Gorbunov et al., 2001]. Most studies parameterize the free energy of germ formation on these sites in terms of either contact angle [Fletcher, 1969; Marcolli et al., 2007] or characteristic temperature [Vali, 2008; Broadley et al., 2012]. The distribution of the catalytic strength of these sites is generally assumed to be Gaussian [Marcolli et al., 2007] or exponential [Niemand et al., 2012; Hoose and Möhler, 2012].

[5] The fundamental challenge for ice nucleation studies is to separate the time dependence of the nucleation process from the properties of the active site distribution [Vali, 1994, 2008]. Although it is possible to derive nucleation rates from various experimental data [e.g., Archuleta et al., 2005; Eastwood et al., 2008], it remains unclear whether such derived rates can accurately predict the time dependence of the freezing process. This ambiguity arises because the residence time in most freezing apparatuses is short and difficult to systematically vary during experiments. For example, current continuous-flow diffusion chambers (CFDCs) [Rogers et al., 2001; Stetzer et al., 2008; Kanji and Abbatt, 2009] and similar instruments such as the Leipzig Aerosol Cloud Interaction Simulator (LACIS) [Hartmann et al., 2011] are limited to residence times on the order of a few seconds, and these times are not easily extended beyond a factor of 10 [Welti et al., 2012]. Cold-stage freezing assays [Durant and Shaw, 2005; Cantrell and Robinson, 2006; Vali, 2008; Fornea et al., 2009; Murray et al., 2011; Alpert et al., 2011; Wheeler and Bertram, 2012; Broadley et al., 2012; Pummer et al., 2012] and differential scanning calorimetry apparatuses [Marcolli et al., 2007] are better suited for exploring the time dependence since the cooling rate can be easily manipulated. These setups, however, may be more prone to artifacts because the aerosol/droplet is situated on a substrate and/or is immersed in oil, although clean work minimizes substrate effects [Murray et al., 2010; Broadley et al., 2012]. Further, control over the aerosol surface area inside the droplet is more difficult, especially when operating with submicron particles that are too small to be detected with standard optical microscopy. Thus, interpretation of the time dependence from cold-stage freezing assays can be hindered by the ability to uniquely describe the particle size and active site distribution probed during the experiment.

[6] On theoretical grounds, the nucleation rate and its time dependence in heterogeneous freezing nucleation are reasonably well understood. If the nucleation rate increases strongly with decreasing temperature, the time dependence (or cooling rate dependence) of the freezing process diminishes, and nucleation appears as a quasi-deterministic process on experimental time scales [Vali, 2008; Niedermeier et al., 2011; Broadley et al., 2012, Ervens and Feingold, 2012]. Several recent experimental studies have explicitly addressed the time dependence of the freezing process. These experiments varied time from ~20 s [Welti et al., 2012] to ~14 min [Broadley et al., 2012] to as long as ~70 min [Murray et al., 2012]. Combined, these studies suggest that the time dependence is weak and subordinate to temperature fluctuations for dust aerosol proxies. However, time in atmospheric clouds can extend from hours to days depending on the cloud type. Here we build on the studies exploring the time dependence by introducing a new cold-stage freezing assay setup that extends the residence time up to several days. Further, we develop a new discrete event simulation modeling framework. Coupled with a broad range of experiments, we show for the first time how cold-stage freezing assay experiments can be used to uniquely deconvolve the active site distribution and the temperature dependence of the nucleation rate for a given substrate. The resulting model parameters predict freezing behavior that is fully consistent with already published data from continuous-flow diffusion chamber (CFDC) measurements, thereby lending confidence that a unified set of parameters can be found to describe the size, time, and temperature dependence for a specific substrate.

2 Methods

2.1 Experimental Approach

[7] Measurements of immersion-mode freezing spectra were conducted with a newly constructed droplet freezing assay that is similar in design to that of Bigg [1953]. A schematic of the design is shown in Figure 1. In brief, an emulsion of water in a low-density immiscible fluid is placed on top of a glass cover slide that rests inside a custom-built aluminum dish. The liquid matrix protects the drops from coming into contact with airborne containments and prevents drops from interfering with each other. The cover slip is treated with AquaSil Siliconizing Fluid (TS-42799; Thermo-Scientific) to create a hydrophobic monolayer that minimizes freezing events induced by the substrate [Cantrell and Robinson, 2006; Fornea et al., 2009]. A thermistor (MP-3176; TE Technology, Inc.) is seated inside the aluminum dish to record the temperature of the metal. The block is set on a thermoelectric element (TE-127-1.4-2.5; TE Technology, Inc.) for cooling. The temperature of the aluminum block is controlled using a proportional integral differential controller (TC-36-25-RS232; TE Technology, Inc.) integrated into a LabView data acquisition and instrument control system. The heat sink for the thermoelectric element consists of a thermoelectric cooler (HP-1200CPV; TECA) that is outfitted with a two-stage cascade (CCP-31) and held at −25°C. With this setup, the aluminum dish can be cooled to −45°C at rates ranging from 0.01 to 10 K min−1.

Figure 1.

Schematic of the experimental setup.

[8] The observation area is enclosed in a clear acrylic box and flushed with dry nitrogen to prevent frosting. The cover slip was imaged with a stereomicroscope (EMZ8TR; Meiji) that was outfitted with a 2592 × 1944 pixel resolution camera (Infinity 1-5C; Lumenera). Images were recorded in ~0.17°C intervals and stored for postprocessing. The optical resolution of the system ranges from 0.8 to 10 µm/pixel with 4X at 0.7X zoom, respectively. A 0.5X objective (MA794; Meiji) is optionally used to extend the field of view by a factor of 2. The corresponding minimum and maximum fields of view are 2.1 × 1.6 mm and 25 × 19 mm.

[9] Prior to introducing the samples into the aluminum tray, a 10 µl aliquot of water sample is placed in a 10 ml vial containing 2 ml squalene (≥99% purity; Acros Organics). Before usage, the vial and syringe is cleaned in a base bath (saturated solution of KOH in isopropyl alcohol) followed by an acid bath (concentrated H2SO4) and then rinsed with ultrapure water [Stan et al., 2009]. The squalene aliquot is measured using an Eppendorf pipette whose tips are discarded after use. The vial is then shaken for ~5 s to disperse the water into droplets ~50–250 µm in diameter. The resulting suspension is poured onto the cover slip. This procedure results in 300–1500 droplets that are visible in the field of view with the 0.5X objective combined with 1.5X zoom. The block is then cooled at a preprogramed linear rate, e.g., 1 K min−1. When a water drop freezes, the drop darkens from a nearly transparent, white circle to a fully black circle. An in-house-developed algorithm processes the images to automatically detect potential freeze events. Suspected freeze events are inspected manually and determined to be either a true freeze event, a false positive, or a freeze event induced by drops coming in contact with each other. False positives can arise from jitters in the camera, from drops coalescing, or from noise in the image that the processing filters could not remove. False positives and droplet/droplet contact freeze events are discarded. For each true freeze event, the x/y coordinates, the pixel area of the drop, and the temperature are stored. The x/y coordinate data are used to identify and track drops across multiple refreeze experiments. The pixel area data are used to reconstruct the drop volume, by assuming that the drop is spherical, for modeling purposes.

[10] The temperature on the surface of the glass slide was calibrated by attaching a second thermistor to a glass slide. The slide was placed in the dish and covered with squalene in the same manner as during freezing experiments. The dish was then cooled at various cooling rates. The offset between dish and squalene temperature was ~0.9°C for cooling rates of 1 K min−1 and slower. For the fastest cooling rate, the offset temperature approached ~1.5°C. Temperatures reported in this study account for these offsets. Thermal gradients can exist within drops [Gurganus et al., 2011]. Our drops are too small for meaningful temperature gradients to persist, but thermal gradients in the squalene matrix are likely. These, however, were not further quantified.

[11] To test the experimental setup, freezing spectra of pure water from two sources, namely, in-house-generated ultrapure water (18.2 MΩ resistivity, <5 ppm organic material) and high-performance liquid chromatography (HPLC)–grade water (1 ppm maximum residue after evaporation; Fisher Chemical), were measured. For these, the temperature where 50% of the drops froze varied between −34°C and −36°C for 100–200 µm diameter droplets, which is in close agreement with the value expected for this size range [Langham and Mason, 1958].

[12] The droplet freezing assay was used to probe IN activity of Arizona Test Dust (ATD) (Powder Technology, Inc.; ISO 12103-1, A4 Coarse) particles. We selected ATD because it has been extensively studied in previous work [Niedermeier et al., 2010; Marcolli et al., 2007; Sullivan et al., 2010a] and its chemical composition and hygroscopic properties are relatively well known [Vlasenko et al., 2005]. Four distinct types of experiments were performed.

2.1.1 Refreeze Experiments

[13] A 0.1 wt % suspension of ATD in ultrapure water was dispersed onto a glass slide at room temperature. The temperature of the dish was cooled to −5°C at the fastest possible rate and held steady for 90 s. Then, the temperature was reduced at a rate of 1 K min−1 to −45°C while images were captured at 0.5 Hz, corresponding to 0.17°C intervals. After reaching −45°C, the sample was warmed to 3°C at the fastest possible rate and held constant for 180 s. Subsequently, the temperature was reduced to −5°C in approximately 30 s and held for an additional 180 s to allow the temperature to stabilize in preparation for the next cycle. The freeze/thaw cycle was repeated 40 times.

[14] An additional set of refreeze experiments was performed with the same methods as before, except the squalene/water emulsion was sonicated for 60 s prior to introduction to the sample dish. The effect of this treatment was to produce smaller droplets ranging in size from 15 to 75 µm in diameter.

[15] Tracking each drop across the 40 experiments occurred in postprocessing. The experiment that contains the most identified drops is used as a reference for calculating the average motion of the drops between each experiment. The relative changes in the x and y position of the drops in each experiment were found by manually comparing an image tile from each experiment to the reference experiment. For each drop in the reference image, a nearest neighbor search is performed to identify the drop in the other 39 experiments that most likely aligns with the reference drop. These 40 images were then inspected manually to cull mismatched drops.

2.1.2 Mass Concentration Experiments

[16] For these experiments, a suspension of ATD in ultrapure water was dispersed into several hundred droplets on the glass slide. The weight fraction of ATD in water was varied between 0.01 and 1 wt %, and freezing spectra were recorded. For each experiment, the temperature was cooled rapidly from room temperature to −5°C and then held constant at this temperature for 90 s until the temperature stabilized. The temperature was then cooled from −5°C to −45°C at a cooling rate of 1 K min−1.

2.1.3 Cooling Rate Experiments

[17] These experiments were performed using a 0.1 wt % suspension of ATD in ultrapure water and varying the cooling rate between 0.01 and 5 K min−1. The temperature was cooled from −5°C to −45°C at the designated cooling rate. Between each measurement, the sample was warmed to 3°C and allowed to thaw for 5 min. The total run time for the combined experiments was approximately 4 days.

2.1.4 Hold Experiments

[18] Two separate hold experiments were performed. In the first experiment, a suspension of 0.1 wt % ATD was cooled from 3°C to a target temperature of −26.1°C, where ~20% of the droplets had frozen, at a cooling rate of 1 K min−1. The temperature was then held constant at the target temperature, with less than 0.05° fluctuations over a period of ~13.3 h, after which the linear cooling rate resumed to chill the drops to −45°C. During the portion of the experiment where the temperature was held constant, images were recorded once every 30 s. A second experiment was performed with 1 wt % ATD. The procedure was the same, except that the hold temperature was −22°C, where ~8% of the drops froze before the hold started, and the hold time was ~15.9 h.

2.2 Modeling Approach

[19] Our modeling approach is a variant of the multiple-component stochastic model of heterogeneous freezing nucleation [e.g., Marcolli et al., 2007; Niedermeier et al., 2011; Wheeler and Bertram, 2012]. The model consists of a parameterization of the temperature dependence of the nucleation rate for a single active site, a Gaussian distribution of the catalytic strength of active sites, and the assumption that active sites have a uniform spatial distribution over the aerosol surface area.

[20] According to classical nucleation theory [Pruppacher and Klett, 1996, equation (9-37)], the temperature (T) dependence of the heterogeneous nucleation rate (J) can be formulated as

display math(1)

where c1 depends on the radius of the particle and the concentration of single water molecules adsorbed on the surface in metastable equilibrium with the embryo of the new phase, and c2 depends on the energy of activation for the diffusion of a water molecule across the ice/water interface, the surface tension of the ice/water interface, the equilibrium freezing temperature of a spherical ice germ in the supercooled droplet that is in equilibrium with the humid environment, the average density of ice over the temperature interval between 0°C and the equilibrium freezing temperature, the compatibility parameter of ice on the substrate, and the latent heat required to melt ice. In general, nucleation rates vary by orders of magnitude over a temperature interval of a few degrees [Vali, 2008]. The term exp(−c2T) is more dependent on T than the c1T term; thus, c1T can be taken as approximately constant. Many of the quantities entering c2 have known temperature dependence (e.g., the energy of activation for diffusion or the latent heat of melting) [Pruppacher and Klett, 1996], but to a first approximation, c2 can be also assumed constant for a specific compatibility parameter (i.e., a single active site). Based on this and consistent with similar assumptions made in previous studies [Murray et al., 2011; Broadley et al., 2012], we assume that the temperature dependence of the nucleation rate for a single active site can be expressed as

display math(2)

where Tc is a characteristic temperature that parameterizes the catalytic strength of the active site (conceptually identical to contact angle or compatibility parameter descriptions), T is the temperature, u expresses the slope of the temperature dependence, and J expresses the expected number of freeze events per unit time if an ensemble of identical active sites was exposed to a given temperature. The parameters u and Tc are algebraically related to c1 and c2 and are used mainly for clarity and convenience.

[21] The value of u determines the relative sensitivities of time versus temperature in heterogeneous freezing nucleation. For large values of u, the nucleation rate varies so dramatically with temperature that it can be approximated as a step function, effectively selecting Tc as the freezing temperature of the active site. In the limit, this corresponds to the deterministic (or singular) model of ice nucleation. From a physical perspective, u subsumes properties of both the fluid (e.g., latent heat, surface tension, or activation energy for self-diffusion) and the substrate (e.g., the compatibility parameter of the active site, adsorbed water in equilibrium with the germ, or size of the active site). For this reason, we point out that u may vary for different active sites having the same characteristic temperature and may also systematically vary with absolute temperature or droplet diameter.

[22] We hypothesize that a specific aerosol type, e.g., “Arizona Test Dust,” can be conceptualized as a surface that has active sites of differing catalytic strength. These sites are assumed to be dispersed uniformly in space and have a Gaussian distribution of their characteristic temperature. The model does not account for movement of the dust particles within the droplets over time. The corresponding parameters of this distribution are {nsite, μTc, σTc}, where nsite is the number density of active sites per unit surface area, μTc is the average characteristic temperature, and σTc is the standard deviation. A specific particle having some surface area then becomes a random sample from the larger idealized surface.

[23] Not being able to observe the nucleation process directly is a fundamental problem faced by most ice nucleation experiments. Typical observations present the fraction of particles that cause freezing events at the end of a well-characterized thermodynamic trajectory. For example, consider the mass concentration experiment described in the previous section. An unknown number of dust particles are placed randomly inside a droplet. An ensemble of several hundreds of these drops is then cooled at a linear rate, and one of the observations is the temperature where 50% of the droplets froze. Although it is easy to fit these data with the four-parameter model outlined above, it is difficult to do so unambiguously because the parameter space is underdetermined by the experiment. To address this inversion problem, we propose to apply discrete event simulation [Banks et al., 1996] to describe the ice nucleation experiments. Specifically, our simulation generates an instance of the experiment based on random sampling from the model domain and then decomposes the experiment into process segments. For the experiment described above, the approach is to generate a discrete number of ATD particles immersed in a droplet. These particles have surface area and mass that are consistent with the underlying ATD size distribution and the volume of the droplet. The number and strength of the active sites are consistent with the ATD active site distribution, and drop-to-drop variability is modeled as a series of Poisson processes. A droplet is passed along the thermodynamic cooling trajectory in discrete temperature or time bins, where temperature and time are connected via the cooling rate. Determining whether a freeze event has taken place is decided based on the probability derived from the nucleation rate in the time interval [t,  t + δt] and the outcome of a die roll. Iterating this procedure over all droplets and integrating over the thermodynamic trajectory results in the freezing spectrum and provides a direct comparison to the original experimental observation. The model is inherently stochastic and relies on pseudorandom number generation. Expected variability in experimental outcomes can be estimated by reinitializing the random number generator with a different seed and computing ensemble model statistics. A formal description of the model algorithm is provided in Appendix A.

[24] By recreating the experiment in the model, analytical inversion algorithms needed to interpret freezing assay experiments [Vali, 1994] are not required. The fundamental model material parameters {u, Nsite, μTc, σTc}, together with the particle and droplet size distributions, can be applied directly to simulate experiments. Simple modifications to the computational procedure allow for simulation of other experimental setups without changing the fundamentals of the model mechanics.

3 Results

[25] Since a priori information about appropriate values for the parameter set {u, nsite, μTc, σTc} was not available, we performed Monte Carlo simulations over reasonable expected ranges to isolate if specific experiments can uniquely constrain part of the parameter space. The bounds were determined through trial and error. When values are picked outside the bounds described below, the model returns physically unrealistic results (i.e., all drops freezing at near the melting point, no drops freezing before the homogeneous limit, or drops freezing over the entire span of freezing temperatures). The vertices of the hypercube parameter space were discretized over the corresponding intervals as follows: u = [0.75 °C− 1, 10 °C− 1] in 10 geometrically spaced steps, μTc = [−40°C, –30°C] in four steps, σTc = [1°C, 2.5°C] in four steps, and nsite = [1011 m− 2, 5 × 1012 m− 2] in four steps. In addition to these four material parameters, the following experimental parameters were varied: cooling rate (cr) = [0.01 K min−1, 5 K min−1] in five steps and wt % = [0.1%, 1%] in three steps. Refreeze experiments were modeled by generating a unique set of drops using Table A1 and repeating the simulation over the thermodynamic trajectory defined by cooling the drops from −5°C to −45°C. In each model run, the default parameters were set to {u, nsite, μTc, σTc, cr, wt %} = {2 °C− 1, 5 × 1011 m− 2, − 40 °C, 2 °C, 1 K min− 1, 0.1%}, except for the parameter being investigated which was varied over the search space described above. For each parameter set, a set of 100 drops were produced, and the above freezing scenario was repeated 100 times. The drop parameters were maintained for the 100 refreezes, but the seed for the random number generator was varied. Summary statistics were computed to find the mean (μrefreeze) and standard deviation (σrefreeze) of the freezing temperature for an individual drop. The ensemble average over the 100 drops for these quantities is denoted by inline image and inline image.

Table 1. Active Site Generation
inline imageFor all (∀) droplets (i): Generate the droplet diameter from a lognormal distribution OR (∨) use experimental data to generate drop diameter distribution.
inline imageCompute the expected number of dust particles in the ith droplet from the sample weight percent (wt %) of particles per volume of solution, the volume of the water droplet (inline image, and the particle number to mass ratio (NMR).
inline imageGenerate the number of dust particles inside the ith droplet from a Poisson distribution.
inline imageFor all droplets (i) AND all dust particles (npart,i): Generate the dust particle sphere equivalent diameter from the empirical size distribution.
inline imageFor all droplets (i): Compute the surface area of dust inside the ith droplet.
inline imageFor all droplets (i): Compute the expected number of active sites in the ith droplet from the active site density and the dust surface area.
inline imageFor all droplets (i): Generate the number of active sites from a Poisson distribution.
inline imageFor all droplets (i) AND for all active sites (Nsite,i): Generate the set of characteristic temperatures from a normal distribution.

[26] The results for the above set of simulations are summarized in Figure 2. The average variability in the refreeze temperature, inline image, is strongly dependent on u and can be summarized by the power law relationship inline image. The main result, perhaps not surprisingly, is that the random variability in the observed freezing temperatures decreases as u increases, because larger u values imply that the slope of the nucleation rate with respect to temperature becomes steeper and nucleation approaches deterministic behavior. Less obviously, our simulations show that the values of the parameters {nsite,μTc,σTc}, cooling rate, and wt % have only a marginal effect on the relationship between u and inline image. As a consequence, we believe that measurements of inline image can be used to uniquely constrain u and thus deconvolve the roles of time and active site distribution using observations.

Figure 2.

Modeled dependence on the variation in the refreeze temperature (inline image) versus the slope of the nucleation rate with respect to temperature (u parameter). Each point is the average inline image of 100 modeled refreeze experiments. Vertical bars correspond to ±1 standard deviation. The solid line is a power law fit to the average values.

[27] Figure 3 shows the results for the observed variability of four individual drop freezing temperatures across 40 refreeze experiments. The top two panels show results for two exemplary selected drops for which the variability appears completely random. This is in contrast to the two droplets shown in the lower panels, which exhibit systematic shifts in the freezing temperature. These step changes have been observed in previous studies and attributed to morphological changes to the surface [Vali, 2008; Sear, 2011; Wang et al., 2012] or diffusion of the ice nucleus to the edge of the droplet [Shaw et al., 2005; Durant and Shaw, 2005; Fornea et al., 2009]. We quantified the number of drops that exhibited a systematic shift in the freezing temperature by testing the hypothesis of randomness using autocorrelation analysis. This analysis decomposes the n-point (here n = 40) sample into two blocks that are offset by k steps, where k = 1, …, n − 2 is the lag. For each lag, the Pearson correlation coefficient, R(k), between the two blocks is computed. Completely random samples, defined as a successive series of freeze events that are statistically independent, have an expected R(k) = 0 for all k considered. For a large sample size, the distribution of R(k) can be approximated as Gaussian with variance 1/N, where N is the number of points used to compute R(k) [Anderson, 1942]. Using the effective sample size N = n − k − 1, the confidence limits for R(k) can be approximated as inline image, where z is the z score at significance level α. If the data are random and α = 0.05 (z = 1.96), we would expect that, on average, 5% of |R(k)| > c. For an n = 40 point refreeze series, we therefore expect that not more than two values of R(k) exceed c. Droplets that did not meet this criterion did not exhibit completely random freezing pattern and may have undergone active site modification. The top two droplets in Figure 3 pass the autocorrelation test with 1 and 0 values of R(k) exceeding c, while the bottom two do not with 9 and 11 values of R(k) exceeding c. Visual inspection of the refreeze temperature series that failed the autocorrelation test suggests that the autocorrelation test accurately captures the nonrandom patterns in the data. It is unclear, however, if drops that barely failed the test, i.e., those with three or four points having |R(k)| > c, have truly undergone active site modification.

Figure 3.

Observed freezing temperatures of four individual drops containing 0.1 wt % ATD over 40 refreeze experiments. Horizontal lines correspond to the mean (blue) and ±1 standard deviation around the mean (red) of the average freezing temperature. The top two panels are drops that passed the autocorrelation test, and the bottom two panels are drops that failed the autocorrelation test for randomness.

[28] Figure 4 shows the results for two sets of refreeze experiments. For the two experiment sets, these data span active sites ranging from −36.66°C < μrefreeze < −22.52°C and −33.76°C < μrefreeze < −20.01°C. The droplet diameters range in size from 15 to 75 µm and from 50 to 120 µm. σrefreeze varied between 0.21°C and 3.52°C and 0.17°C and 2.91°C, with a majority of droplets in both experiments having σrefreeze < 1.0°C. There are no apparent trends with temperature and particle size, suggesting that σrefreeze and, hence, u are solely controlled by the substrate and do not depend on the characteristic temperature of the active site. The results of the autocorrelation test show that, in the first sample, 82 of 161 drops failed the test, whereas in the second sample, 137 of 264 drops failed the test. As shown in Figure 4, the drops that failed the autocorrelation test are interspersed with those that passed the test. The statistics for the refreeze experiments can be summarized as inline image = 0.77°C ± 0.51°C, corresponding to u = 1.56°C−1 and inline image = −29.05°C ± 3.58°C for the refreeze experiment with 50–150 µm droplets. For the experiment with the larger droplets, the statistics are inline image = 0.62°C ± 0.34°C, corresponding to u = 1.91°C−1 and inline image = −24.38°C ± 1.60°C. If the droplets that fail the autocorrelation test are removed from the analysis, inline image changes to 0.56°C and 0.55°C, corresponding to a decrease of ~28% and 11%, respectively.

Figure 4.

Mean drop freezing temperature versus the standard deviation of the freezing temperature. Open circles represent drops that failed the autocorrelation test.

[29] The autocorrelation test only flags droplets in which a particle changes from one freezing mode to another but does not identify samples in which some subset of particles is permanently located at the water/oil interface. These particles may initiate freezing at a warmer temperature and thus have larger nucleation rates. We can neither affirm nor rule out this possibility in our experiments. It remains unclear, however, if IN serving in the contact mode would fundamentally have a different inline image value compared to IN serving in the immersion mode. If the sites leading to freezing in the contact and immersion are identical, but only the nucleation is shifted toward warmer temperature, the retrieved inline image should, in principle, be identical. Additional studies are required to test this hypothesis.

[30] Figure 5 shows the results of the mass concentration experiments. As the concentration of ATD increases, the droplets freeze at warmer temperatures. This is to be expected due to the increase in the dust surface area immersed within the droplet, thus raising the likelihood that an active site with a warmer Tc is present. Evidence of homogeneous freezing occurs in experiments where the concentration of ATD is 0.01 wt %. In this case, ~30% of the drops did not contain IN with active sites whose characteristic temperature is warmer than the homogeneous freezing temperature. We point out that the pure water control run shows a few droplets that freeze well before the homogeneous freezing limit. This tail was evident in experiments with both HPLC and in-house-generated ultrapure water and is consistent in magnitude across a number of pure water experiments. It is not entirely clear whether this tail is caused by random impurities in the silanized glass slide, impurities in the squalene, or impurities in the water. The earliest freeze event of pure water typically occurs at T < −25°C and fractions frozen before the onset of homogenous freezing range from 5% to 10%, indicating the noise floor of the experimental design.

Figure 5.

Drop freezing temperature versus percent of drops frozen. Overlaid are model results using the model presented in section 2.2 and Appendix A.

[31] To model the expected number of active sites present in a droplet, the number-to-mass ratio (NMR) for the dust aerosol must be specified (Appendix A). An initial value of 1.8 × 1013 particles/kg dust was estimated from the manufacturer-provided volume distribution. However, the smallest particle diameter included in these calculations is 0.87 µm. Particles with D < 0.87 µm were undoubtedly present in the sample, thereby rendering NMR uncertain. We therefore revised the initial estimate of NMR upward to 2 × 1014 particles/kg to obtain better agreement between the model and the data at low dust wt %, where the calculations are most sensitive to the value.

[32] Modeled freezing spectra generated with the discrete event simulation approach described in section 2.2 and Appendix A are superimposed on the graph. Here u = 2.2°C−1, determined from the refreeze experiment, was assumed. The material parameters used were {nsite,μTc,σTc} = {8 × 1012 m− 2, − 42 °C, 4.2 °C}.These parameters were determined by an iterative procedure to minimize the residual between the model and the data. Homogeneous freezing in the model occurs when the model temperature reaches −36°C and the simulation did not produce a freezing event. This happens when there are no active sites in the drop or the assigned characteristic temperature of the active sites within the drop were so cold as to make the probability of freezing before homogenous freezing nearly zero. In the case of the 0.01 wt % model run, 83 of the 406 drops were not assigned dust particles. An additional 12 drops in the model were assigned dust particles, but the particle surface area within the droplet did not have active sites having sufficiently warm values of Tc. We point out that raggedness in the model simulations is caused by the stochasticity introduced by random sampling. Repeating the model with different random seeds and computing the ensemble average would result in a smoother prediction.

[33] Results from cooling rate experiments are shown in Figure 6. For each cooling rate, the drops were cooled from 0°C to −40°C at the prescribed cooling rate. The temperature where 50% of the droplets froze (T50) decreased from −24.02°C to −25.35°C as the cooling rate increased from 0.01 to 5 K min−1. Experiments performed at 0.01 K min−1 correspond to a total time of ~2.8 days, thereby bracketing the maximum time-related effect that can be expected for slow-rising and long-lasting clouds. To ensure that multiday experiments did not alter the IN properties, we restricted the multicooling rate experiment to 138 droplets that could be positively identified across all 11 experiments. Additionally, three measurements were made at cooling rates of 1 K min−1 at the beginning, middle, and end of the experiment runs. The T50 temperatures of these experiments were as follows: –24.75°C, –25.67°C, and –25.12°C.

Figure 6.

Temperature at which 50% of drops containing 0.1 wt % ATD froze as a function of cooling rate (red filled circles). Open circles correspond to the average prediction from 20 simulations. Vertical bars correspond to ±1 standard deviation. Numbers indicate the order in which the cooling rates were performed. Solid lines are included to guide the eye.

[34] The cooling rate dependence was modeled using the same parameter set as for the mass concentration experiment. To estimate the expected variability in T50 due to random chance, we repeated the model freezing simulation 20 times with different random seeds. Results from the model are superimposed on the data in Figure 6. The model predicts that the T50 temperature changes from −23.8°C to −26.6°C over the cooling rates of 0.01–5 K min−1.Due to the discrepancy between predicted and observed data, model runs were performed to ascertain what value of u would be needed to account for the observed slope in the T50 versus cooling rate values. The value of u that matches the observed slope was found to be 4.4°C−1. This corresponds to a change in the average of the standard deviation in the refreeze experiment from 0.63°C to 0.28°C. This suggests either unaccounted thermal variability up to 1°C in our setup or that the slope of the nucleation rate is smaller than what is predicted by the refreeze experiments.

[35] The hold experiment demonstrates that even when the temperature of the droplets is held constant, freezing still occurs. A single-component stochastic model predicts a linear increase in the fraction of droplets frozen when graphed in the manner shown in Figures 7a and 7b. The slope is a direct measure of the nucleation rate at the hold temperature. Contrary to the expectation of CNT, the rate of observed freezing events tapers off as the hold time progresses. The deviation from linearity is an indication of heterogeneity of active sites [Niedermeier et al., 2011] and the finite number of droplets during the experiment. Particles that only have active sites with critical temperatures that are much lower than the hold temperature will nucleate ice less readily. Thus, an increase in the heterogeneity, which is characterized and controlled by σTc, will result in stronger deviations from linearity. This behavior is illustrated by the model calculations shown in Figures 7c and 7d. In these illustrative calculations, u and μTc are identical to the values obtained from the mass concentration and refreeze experiment, σTC was specified, and the hold temperature was selected such that ~30% of particles remain unfrozen after ~15 h. Since the rate at which droplets freeze depends much more strongly on temperature than time, it is difficult to directly apply the model parameters found by the previous experiments together with the actual hold temperature to predict the fraction of unfrozen drops. Slight discrepancies between the model-predicted rate and the actual nucleation rate at the hold temperature would result in seemingly large deviations between the observation and the model. The key point revealed by the experiment and the model shown in Figure 7 is that ATD is a heterogeneous surface and σTc ~ 4°C is broadly consistent with the nonlinearity during the hold experiment.

Figure 7.

Fraction of unfrozen drops over time while the temperature is held constant. (a) Two separate hold experiments. (b) Same as Figure 7a but with data restricted to the drops that froze in the first 200 min. (c) Modeled drop freezing when σTc is varied and Thold is adjusted so that ~30% of the drops remain unfrozen at the end of the hold. (d) Same as Figure 7c but with data restricted to the drops that froze in the first 200 min. In all four panels, straight lines are provided to help gauge the linearity of the curves.

4 Discussion and Conclusions

[36] Our results show that a set of four parameters comprising the slope of the nucleation rate with temperature (u parameter) and a Gaussian distribution characterizing the characteristic temperature of active sites can model the experimental data presented here. One might critique the experimental approach used here based on the fact that multiple dust particles, some with sizes much larger than are typically found in the atmosphere, are present in each drop. Further, the number and particle surface area in each individual drop are unknown, and all results rely on interpretation through statistical simulation. Finally, the drops rest on a solid substrate and are embedded in an oil matrix. This begs the question: Is the inferred parameter set consistent with prior experiments that more closely mimic the ice nucleation process as it occurs in the atmosphere?

[37] Specifically, these experiments use the CFDC or similar techniques [Welti et al., 2009; Niedermeier et al., 2010; Sullivan et al., 2010b], immerse a single dust particle of a known size inside a droplet, and process it through air that is supersaturated with respect to ice. At the outlet of the instrument, an optical particle counter characterizes the fraction of droplets that form ice at a given temperature [Rogers et al., 2001]. To represent this setup, we reconfigured our statistical simulation model to place a single ATD particle within a drop at the specified particle size and seeded the model with active sites using the material parameters presented earlier. A total of 106 drops were generated and kept at constant temperature for a specified residence time. The fraction of droplets frozen was computed as a function of particle diameter and temperature for an assumed residence time of 5 s. Comparisons between these model results and literature data are presented in Figure 8. The model demonstrates that irrespective of particle size, the fraction of frozen drops increases by 3 orders of magnitude for an approximate 10° decrease in the instrument temperature. This is also comparable to Saharan dust whose activated fraction increases by 2 orders of magnitude over 10° [DeMott et al., 2011, Figure 6].

Figure 8.

Comparison of model extrapolation to data from literature. Data are taken from Niedermeier et al. [2010] (blue squares), Sullivan et al. [2010a] (red circle), Sullivan et al. [2010b] (blue circle), and Welti et al. [2009] (diamonds). Colors indicate the particle diameter used in the respective study. Lines show model predictions based on the same parameter set used to model Figure 5.

[38] The CFDC model results in Figure 8 demonstrate the impact of particle size on the fraction of droplets that activate at any given temperature. Because the number of active sites is directly proportional to particle surface area, larger-particle active sites have warmer characteristic temperatures, thus leading to larger activated fractions. The agreement between the data and the model is not perfect, but the trends are present. ATD data presented by Welti et al. [2009, Figure 5d] (100 and 200 nm points between the “water saturation” line and the “breakthrough” line) show that for a given activated fraction, there is an increase in the freezing temperature with increasing particle size. Additionally, data from Sullivan et al. [2010a, 2010b] show an increase in activated fraction at a given temperature for an increase in particle size. There are clear discrepancies in the data that suggest that current experimental uncertainties surrounding the CFDC measurements are too large to either prove or disprove the validity of the moderate size dependence predicted by the model extrapolation of our experimental results. This further confounds efforts to separate the surface area effect from that of particle composition (e.g., mineralogy) that may also change with particle size and may be different between the ATD samples used in the respective studies. For example, Broadley et al. [2012] demonstrated how the particles from illite-rich powder consist of agglomerates of smaller particles that are tens to hundreds of nanometers across.

[39] The CFDC-type measurements do offer additional constraints on the active site density. Since the particle size is known and there is only one particle present in the drop, the average number of active sites, nsite, on the particles can be constrained. In actuality, the value nsite = 1.6 × 1012 m−2 applied in section 3 to model the mass concentration data presented in Figure 5 was selected with prior knowledge of the CFDC model comparison. This tuning was required because the conversion from aerosol mass to number and surface area is approximate and varying nsite and number-to-mass ratios have competing effects in the fitting of the data. However, we believe that precise control of the particle size distribution in our type of experiments would eliminate this ambiguity.

[40] The observed σrefreeze values for ATD (0.17°C to 3.58°C) are within the range of values reported in previous studies. Specifically, various IN types can be characterized as having inline image ~ 1°C (soil) [Vali, 2008], inline image ~ 1°C to 3.4°C (volcanic ash, peat, and black carbon) [Fornea et al., 2009], and inline image ~ 0.2°C (nonadecanol) [Zobrist et al., 2007]. Using the analysis presented here, it is possible to predict inline image from u for other materials. For example, kaolinite has a nucleation rate slope of 2.92°C−1 [Murray et al., 2010], which corresponds to a predicted inline image°C. Nonrandom fluctuations in the freezing temperature reported here and elsewhere [Vali, 1994, 2008; Fornea et al., 2009; Durant and Shaw, 2005] are not accounted for in the stochastic model. Excluding obvious nonrandom samples from our data set did not lead to a significant change in σrefreeze, and thus, at least for the dust sample presented here, nonrandom variability does not significantly bias the inferred u value. This effect, however, could be large for organic compounds [Wang et al., 2012].

[41] A separate but important question is whether the inferred u values from σrefreeze data are sufficient to predict the time dependence of the freezing process. Our results are encouraging inasmuch as the u value inferred from the refreeze experiments (Figure 4), together with the active site distribution inferred from the mass concentration experiments (Figure 5), was approximately predictive of the cooling rate (Figure 6) without the need to modify the parameter set. This finding is consistent with the experiments reported by Broadley et al. [2012], who performed experiments on illite-rich powder, and could also reconcile mass concentration and time dependence based on a similar nucleation rate parameterization. A crucial test for this emerging understanding on time dependence will be to identify and test substances other than mineral dust that have larger σrefreeze (e.g., the black carbon sample reported by Fornea et al. [2009]) and thus should feature much stronger cooling rate dependence compared to ATD or illite-rich powders. Further, studies on ambient IN are needed to better understand the role of time for primary ice formation processes in the atmosphere.

[42] Assuming that the experimentally determined properties for ATD are representative of atmospherically relevant dust, what does this imply for the expected time dependence in ice crystal evolution in updrafts? To estimate the effect, we compute the fraction of dust particles that are expected to nucleate ice crystals assuming (1) that the atmospheric lapse rate is constant at 6.8 K km−1, (2) that updrafts move coherently with constant velocity, (3) that the properties of ATD are representative of atmospheric IN, and (4) that particle size is distributed lognormally with Dg =1.8 µm and σg = 2.4 µm, consistent with observations of the coarse mode in remote continental air [Seinfeld and Pandis, 2006, Table 8.3]. The fraction of droplets frozen was computed by randomly seeding a population of 105 particles and processing the ensemble along the thermodynamic trajectory dictated by the combination of the lapse rate and the updraft velocity. Figure 9 shows the expected difference in frozen fraction by comparing two assumed updraft velocities of 0.1 and 10 m/s to the expected fraction frozen if freezing is assumed to proceed deterministically. The deterministic model is evaluated in the same manner as the time-dependent model but assuming u = 100°C−1, which corresponds to a negligible inline image and, thus, negligible random variability. The calculations presented in the figure show that the expected fraction frozen increases by approximately 3 orders of magnitude over a temperature range of ~10°C. For the deterministic case, freezing occurs at the characteristic temperature Tc of the warmest active site immersed in the droplet. The slope of fraction frozen versus height is thus governed by the underlying active site distribution. If time dependence is factored in, there is a finite probability that particles freeze at temperatures warmer than Tc, and the shift to warmer temperatures is proportional to the residence time in cloud and thus inversely proportional to updraft velocity. The ATD value of u = 2.2°C−1 corresponds to an increase in nucleation rate of approximately 8 orders of magnitude for a decrease in temperature of 10°C. This relatively steep temperature dependence of the nucleation rate results in a low freezing probability at temperatures that are much warmer than Tc. The low freezing probability limits the maximum shift of the fraction frozen versus height curve toward warmer temperatures at 0.1 m/s updraft to ~4°C. These calculations therefore express the error introduced by neglecting the time dependence as equivalence in temperature and represent an upper limit due to the slight overestimate of cooling rate dependence by the selected u value relative to the experiment. Whether a 4°C error is acceptable or not may depend on the specific case. For most atmospheric modeling studies, however, total dust (or IN) loadings as well as specifying the exact thermodynamic conditions of the cloud will likely supersede the error that is introduced by neglecting time dependence. A corollary to this finding is that IN concentrations measured with the CFDC or a similar apparatus, which counts IN at residence times ranging between 1 and 10 s, can likely be used without modification to estimate primary ice formation in updrafts. Further studies, however, are needed to ensure that the time dependence found for ATD can be safely extrapolated to atmospherically relevant ice nuclei.

Figure 9.

Modeled fraction of droplets freezing versus height above cloud base, temperature, and time. Model calculations are based on the parameters used in Figure 5 and include time-dependent treatment for updrafts of 0.1 m/s (blue line) and 10 m/s (red line), as well as the deterministic time-independent treatment (black line).

5 Summary

[43] We performed measurements on ATD using a drop freezing assay that allowed for experimental times to last up to several days. The nucleation rate equation for freezing was derived using a set of material parameters, the distribution of active site characteristic temperatures, and u. We demonstrated through modeling that if all things remain the same, the variability in the freezing temperature of like drops can be described solely with the slope of the nucleation rate equation. A statistical model using four material parameters ({u,nsite,μTc,σTc} = {2.2 °C− 1, 1.6 × 1012 m− 2, − 42 °C, 4.2 °C}) could accurately model the observed random variability of freezing temperatures. Additionally, we could model the change in freezing spectra with variable dust concentrations and approximate the dependence on variable cooling rates. The model and the dust parameters were extended to model fraction of frozen drops versus temperature from prior CFDC experiments. When the simulation and parameters are used to model a hypothetical dust aerosol dispersed in a cloud, our results imply a modest time dependence that is equivalent to a few degree error in the freezing temperature of water droplets.

Appendix A

[44] Experiments are modeled by simulating i number of droplets, each having j active sites. The number of active sites per droplet depends on the volume of water, the mass concentration, and the size distribution of the dust. The size of the water droplet, the number of particles and the sizes (or surface areas) of the dust particles, and the number and catalytic strength distribution of the active sites are the outcomes of a random sampling experiment (random variates) from the underlying distributions. We refer to random variates as Xd, where subscript d denotes a particular probability density function. Specifically, inline image denotes a uniform distribution over the interval [0,1], inline image denotes a Gaussian distribution, inline image denotes a Poisson distribution, inline image denotes a lognormal distribution, inline image denotes an empirical distribution, and μ and σ denote the mean and the standard deviation, respectively. Uniform pseudorandom variates were generated using the Mersenne Twister algorithm [Matsumoto and Nishimura, 1998] within the MATLAB software package (MathWorks, Inc.; v7.9.0.529, 2009). MATLAB's internal algorithms were used to generate inline image, inline image, and inline image from inline image. The inverse transform method [Banks et al., 1996] was used to generate inline image for the dust size distribution. In brief, the manufacturer-supplied ATD volume distribution (D > 0.86 µm) was converted to a probability density function (PDF) of the particle number distribution assuming that particles are spherical. We assumed that the smallest particle diameter in the sample is D = 0.05 µm. The resulting PDF is integrated to produce a cumulative distribution function (CDF). A uniformly distributed number between 0 and 1 is generated, and the corresponding diameter is interpolated by inverting the CDF.

[45] The algorithm for the discrete event simulations is formalized in Tables A1 and A2. Table 1 describes the generation of i drops containing j active sites. The set of characteristic temperatures are dependent on the drop diameter (Ddrop), weight percent of ATD particle to water (wt %), the particle number to mass ratio of ATD (NMR), the mean active site characteristic temperature (μTc), and the standard deviation of the active site characteristic temperature (σTc). Table 2 performs the discrete event freezing simulation along the thermodynamic trajectory.

Table 2. Discrete Freezing Simulator
inline imageCompute the number of discrete simulation steps (k) based on a selected temperature resolution (δT) and initial and final temperatures.
inline imageCompute the simulation time step from the temperature resolution and the cooling rate (cr).
inline imageFor all droplets (i) and active sites (j) and temperature/time steps (k): Compute the expected number of freeze events based on the characteristic temperatures within the drop, u, current temperature, and the time step.
inline imageFor all droplets (i) and temperature/time steps (k): Generate the number of freeze events that take place in the droplet from the Poisson distribution.
inline imageFor all droplets (i) and simulation time steps (k): Outcome of discrete event simulation that determines if the ith droplet froze in the interval [t, t + δt].

[46] Key parameters are the model temperature resolution (δT) and the cooling rate (cr = δTt). At each temperature/time step, the number of freeze events for each active site is determined from a Poisson distribution whose argument is the expected number of freeze events determined from the nucleation rate equation (J), u parameter, and the size of the time bin (δt). The choice of δt determines the resolution of the model output (i.e., the smoothness of the curve) but does not affect the model outcome.

Acknowledgments

[47] This research was funded by the National Science Foundation (NSF) award NSF-AGS 1010851. We thank Chris Osburn for providing us ultrapure water. We also thank Paul DeMott and Ryan Sullivan for useful discussion and John Hader and Travis Morton for assistance with collection and processing of a portion of the data.