These authors contributed equally to this work.

Homogeneous nucleation is the prominent mechanism of glaciation in cirrus and other high-altitude clouds. Ice nucleation rates can be studied in laboratory assays that gradually lower the temperature of pure water droplets. These experiments can be performed with different cooling rates, with different droplet sizes, and often with a distribution of droplet sizes. We combine nucleation theory, survival probability analysis, and published data on the fraction of frozen droplets as a function of temperature to understand how the cooling rate, droplet size, and size dispersity influence the nucleation rates. The framework, implemented in the Python code AINTBAD (Analysis of Ice nucleation Temperature for

The thermodynamics and kinetics of ice formation from water are important for atmospheric science (

Special assays have been developed to study ice nucleation kinetics by monitoring hundreds of small supercooled water droplets (

For droplets subjected to constant supercooling, the induction time is exponentially distributed. Several analyses have modeled the exponential decay to understand how nucleation rates depend on supercooling (

To justify new elements of our approach to introduce the cooling rate and droplet polydispersity into the interpretation and prediction of experimental data, we briefly discuss the capabilities and gaps in existing models for analyzing the experiments with steadily cooled droplets. Analyses of drop-freezing experiments can be grouped according to two distinguishing criteria. The first distinction pertains to the models used for interpreting the nucleation rate.

A second distinction pertains to the analysis and interpretation of the droplet nucleation data themselves. Some studies focus on the fraction of droplets that nucleate in a specific supercooling range, i.e., the nucleation spectrum (

In this work, we combine survival probability analysis with classical nucleation theory to quantitatively predict the effects of different droplet volumes (

The probability that a single droplet of volume

Now we need theoretical models or experimental data for the nucleation rate to predict the survival probability. Classical nucleation theory gives the rate for homogeneous nucleation as (

The nucleation rate

The prefactor is related to both the frequency at which water molecules at the ice–water interface attach to the critical nucleus and the number of ice molecules that must attach to surmount the barrier. The prefactor is proportional to the self-diffusivity of water, and therefore it depends on temperature. However, over the small range of nucleation temperatures in this study (ca. 2 K), we assume that the prefactor is temperature independent.

Note that both parameters,

Using Eq. (

Across the range of nucleation temperatures observed in experiments (

The noisy estimates of

We implemented the numerical integration in Eq. (

Flowchart of the AINTBAD (Analysis of Ice nucleation Temperature for

We use the minimize function from the “scipy.optimize” module in Python to optimize the difference between the target survival probability and the predicted one by adjusting the parameters

Computed nucleation rate parameters

The measurements of

At a temperature of 235.5 K, the global fit yields a prediction of

Although the rate predictions show remarkable internal consistency, the inferred barriers are scattered and larger than barriers which have been inferred from other data sets (

Section

Experiments that report on droplet diameter dispersity (

The steep sigmoidal survival probabilities for droplets of a specific size, when superimposed, result in a more gradual sigmoid. The gradual sigmoid looks deceptively like the theoretical prediction in Eq. (

The joint survival probability distribution with volume and temperature variables is given by

Here we provide an example calculation to illustrate the effects of a broad droplet volume distribution. Let the normalized (gamma-type) distribution of droplet sizes be

We set

Illustrating the effects of the distribution of droplet diameter on survival probabilities. Open symbols are the experimental survival probabilities for different groups of size-selected droplets from

If we had been unaware of the droplet polydispersity or had not accounted for it, we might have interpreted the black curve in Fig.

Once we know the variation in droplet diameters, resultant survival probabilities can easily be computed with the help of the Python code presented in Sect.

First, we ask whether the range of droplet diameters in each experiment by

Parity plots showing how droplet diameter dispersion influences the inferred

When droplet diameter dispersity is ignored, the inferred

Figure

The combined survival probability and nucleation theory expression, as shown in Eq. (

Survival probability fits for homogeneous-nucleation data with different droplet diameters and cooling rates. Open symbols represent experimental data from

The predictions of

The computed free-energy barriers for various droplet diameters across the cooling rates.

Figure

All parametrizations predict nucleation rates within an order of magnitude of each other and for experiments for temperatures between 235 and 240 K. However, there is a small gap in the data near 237 K. Figure

We developed a versatile code capable of taking various parametrizations for the homogeneous-nucleation rate

Flowchart of the IPA (Inhomogeneous Poisson Analysis) code to obtain the survival probability of water droplets as a function of temperature for any distribution of droplet diameters, cooling rate, and homogeneous-nucleation rate parametrization.

To extend

Utilizing the CNT parametrization from

Typical rates of cooling in clouds span

We use the CNT parametrization from

Each of the two data sets, those from

Alternatively, larger nucleation barriers at the lower temperatures (larger supercooling levels) may result from the combined effects of the nucleation barrier (estimated with the AINTBAD code) and diffusion barriers within the prefactor (not yet considered). We analyze this possibility by first predicting the survival probability of liquid droplets upon cooling for a proposed narrow distribution of droplet volumes using the IPA code with a cooling rate of 1 K min

Another important factor that has a significant effect on the measured nucleation spectrum is the measurement of droplet temperature. The estimations of the droplet temperatures in freezing experiments show large variability (

In this section, we conduct a sensitivity analysis using our model to quantify the impact of temperature measurement uncertainty on estimated free-energy barriers. To perform this analysis, we utilized data from frozen 75

Illustration of the effect of temperature uncertainty on the nucleation spectrum.

The resulting differential freezing spectra are illustrated by the blue squares and green triangles in Fig.

The computed free-energy barriers and nucleation rates for various uncertainties in the temperature measurements. The mean and standard deviation of

Homogeneous ice nucleation is the predominant mechanism of glaciation in cirrus and other high-altitude clouds, making the accurate representation of cloud microphysics highly dependent on the homogeneous-nucleation rates. It has been shown that the predictions of cloud models are sensitive to the rate of cooling and variations in the slope of the nucleation rate with temperature (

Homogeneous-nucleation rates can be obtained from experiments that record the freezing temperature while pure water droplets are gradually cooled to temperatures far below 0 °C. Prior studies have analyzed these experiments using Poisson statistics to infer rates at different temperatures, and then they have fitted the rate vs. temperature data to empirical or theoretical rate expressions. Here we directly analyze the fraction of frozen droplets vs. temperature data to estimate rate parameters within a stochastic survival probability framework. We implement our approach in a Python code, AINTBAD (Fig.

Diagram illustrating the usage of the AINTBAD and IPA codes.

We applied the AINTBAD code to analyze the homogeneous-nucleation data obtained from two different studies:

We derived a superposition formula to predict how the distribution of droplet sizes causes a broadening in the distribution of nucleation temperatures. The broadening causes the AINTBAD analysis to underestimate the

Our analysis suggests that the barrier obtained with AINTBAD is the sum of the nucleation and diffusion barriers. As the nucleation barriers decrease monotonously with temperature, we infer that the increase in the overall barrier for nucleation at 235.5 K originates in a steeper temperature dependence of the diffusion coefficient of water

While laboratory experiments strive to study nucleation in the narrowest possible distribution of droplet sizes to avoid spurious impacts on the parametrization of nucleation rates, clouds can have a relatively broad distribution in the size of water droplets. We developed the Python code IPA to predict the nucleation spectrum for any given distribution of droplet diameters at any cooling rate. As input, IPA uses the distribution of droplet diameters and a parametrization of the nucleation rate

We restrict our discussion in this article to homogeneous nucleation, but it might be possible to develop similar methods for analysis of heterogeneous-nucleation data. A key challenge is that pure water droplets vary only in volume, while heterogeneous-nucleation sites may vary in the surface chemistry, pore geometry, and size (area) of the active region. These differences lead to sites with different barriers and also different prefactors. Except for special cases of highly regular surfaces, the estimated

The codes and data that support the findings of this study are available in the GitHub repository at

The supplement related to this article is available online at:

RKRA, IdAR, VM, and BP designed the project and prepared the manuscript. RKRA, IdAR, and BP developed the models and performed the analysis.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We thank Geoffrey Poon, Max Flattery, and Conrad Morris for helpful discussions. This work was supported by the Air Force Office of Scientific Research through MURI award no. FA9550-20-1-0351.

This research has been supported by the Air Force Office of Scientific Research (MURI award no. FA9550-20-1-0351).

This paper was edited by Hinrich Grothe and reviewed by two anonymous referees.