Abstract
Supercooled water droplets are widely used to study supercooled water1,2, ice nucleation3,4,5 and droplet freezing6,7,8,9,10,11. Their freezing in the atmosphere affects the dynamics and climate feedback of clouds12,13 and can accelerate cloud freezing through secondary ice production14,15,16,17. Droplet freezing occurs at several timescales and length scales14,18 and is sufficiently stochastic to make it unlikely that two frozen drops are identical. Here we use optical microscopy and X-ray laser diffraction to investigate the freezing of tens of thousands of water microdrops in vacuum after homogeneous ice nucleation around 234–235 K. On the basis of drop images, we developed a seven-stage model of freezing and used it to time the diffraction data. Diffraction from ice crystals showed that long-range crystalline order formed in less than 1 ms after freezing, whereas diffraction from the remaining liquid became similar to that from quasi-liquid layers on premelted ice19,20. The ice had a strained hexagonal crystal structure just after freezing, which is an early metastable state that probably precedes the formation of ice with stacking defects8,9,18. The techniques reported here could help determine the dynamics of freezing in other conditions, such as drop freezing in clouds, or help understand rapid solidification in other materials.
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Data availability
All optical image and X-ray scattering data used in this study have been deposited37 at the CXIDB repository. Source data are provided with this paper.
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Acknowledgements
Use of the Linac Coherent Light Source (LCLS), SLAC National Accelerator Laboratory, was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under contract no. DE-AC02-76SF00515. The experiments were supported by the US Department of Energy, Office of Science, Chemical Sciences, Geosciences, and Biosciences Division. A.K. and M.M.-M. acknowledge support from the National Science Foundation under grant no. 2123634 for developing the optimization algorithm used to determine the freezing parameters. Supplementary funding for this project was provided by the Rutgers University–Newark Chancellor’s Research Office. We thank E. H. Dao and S. Kim for experimental assistance, M. D. de Almeida and A. Abdalla for evaluation of the optical data, G. Blaj for information on the X-ray detector and H. A. Stone for critical reading of the manuscript.
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C.A.S. conceived and designed the study. C.A.S., S.B., P.R.W., J.E.K., M.J.H., R.H.P. and S.A.H.G. prepared the experiment. C.A.S., P.R.W., N.D.L., R.G.S., H.L., A.L.A., M.L. and S.B. conducted the experiment. T.F.K., M.M.-M., A.K., S.M. and C.A.S. analysed the optical image data. C.A.S., S.M., M.M.-M., A.K. and N.D.L. analysed the X-ray scattering data. A.K., M.M.-M. and C.A.S. developed the models and determined the freezing parameters. C.A.S. wrote the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Optical imaging and characterization of freezing processes.
a, Drops at different stages of freezing. The images show two exposures of the same drop, with the second (lower) exposure capturing the blowup resulting from the XFEL pulse. The droplets travelled from top to bottom in the images. b, Drop-splitting events captured using 12 exposures for each drop. The images illustrate the range of fragment velocities in binary fractures and a seven-fragment fracture. The velocities measured from images may be smaller than the true velocity owing to translations perpendicular to the image plane. c, The ellipticity of droplet images indicates that the freezing-induced deformation is mostly complete before stage 3 of freezing. d, Evolution of the distribution of spicule heights with the time of flight. The distributions evolved discontinuously, justifying the distinction between small and large spicules. e, Distribution of visible spicule numbers. f, Distribution of fragment velocities after binary fractures. The histogram shows the geometric mean of the fragment velocities.
Extended Data Fig. 2 Evaporative cooling and coarse solidification models.
a,b, Model geometries. c, Implicit and Crank–Nicolson schemes. d–f, The numerical convergence of the models. g, Comparison of the cooling model with the measurements of Goy et al.46. h, Comparison of the solidification model versus the analytical model of Wildeman et al.6. i–k, Modelled solidification times of an isolated 40.2-µm-diameter drop, after the completion of dendritic ice growth, in three scenarios: evaporative cooling in vacuum, atmospheric cooling of stationary and free-falling drops in a standard atmosphere at roughly 8 km altitude (236 K, 34 kPa) and cooling in an oil matrix.
Extended Data Fig. 3 Statistical and observational biases in the image data.
a, Vertically integrated image intensities for experiments with two optical exposures, recorded at different times of flight. The image intensity decreases as the number of drops observed at a given horizontal position increases. Freezing leads to an increase in the lateral spread of the drops36, which can affect the stage statistics. To mitigate this statistical bias, only data up to 6.89 ms were used to determine the freezing parameters. b, Illustration of the magnification of the image of the interior of the drop. c, Stage 2 of freezing can be observed only if the ice region overlaps with the imaged volume and its size in the image exceeds a minimum size. The liquid region must also exceed this minimum size to identify stage 2 in a freezing drop. d, Spicules are visible only if they extend outside the drop image. This is equivalent to the spicules being visible only if they grew on a specific region of the drop surface.
Extended Data Fig. 4 Simulations of freezing drop ensembles and determination of the freezing parameters.
a,b, Verification of the statistics of the ensemble simulations. The numbers of droplets observed in each stage have binomial distributions and the standard deviations of stage numbers are equal to the binomial standard deviations. c, The radial distribution of ice-nucleation events. d,e, Two-stage fitting of the parameters of the freezing model. Error-function values were sampled near the minimum using a pattern-search algorithm, then the error-function dependencies on the parameters were fitted with parabolic functions to find the minimum of the error function. f, Comparison of the numbers of drop fragments from experiments and simulations. The fragment numbers provide an independent test of the model because they were not used for the determination of the freezing parameters.
Extended Data Fig. 5 Freezing-model results.
a, The modelled ice-nucleation rate is a parametrization valid between 233.7 and 235.6 K. It is consistent with several other measurements at both lower and higher temperatures and with a parametrization based on a Vogel–Fulcher–Tammann temperature dependence of self-diffusion50. It is not consistent with another similar study conducted at an X-ray laser21 or with parametrizations with constant51 or power-law52 dependencies of the self-diffusion. b, The modelled dendritic ice growth velocity lies close to the extrapolation of Pruppacher’s data53 for the bulk growth of dense dendrites below 264 K, which scales linearly with the temperature. The error bars and bands represent the uncertainty of the freezing parameters (Extended Data Table 1) and of the temperature. For the literature data, the error bars are reproduced from the original work.
Extended Data Fig. 6 Separation of X-ray scattering into crystal and liquid components.
a, Accumulated X-ray detector image from stage 3 drops (top half of the detector). b, Corresponding separated image containing only the crystal diffraction. The intensity scale is shifted by −0.5 compared with a. c, Corresponding separated image containing only the liquid-scattering component. The intensity scale is the same as in a. d, Total and separated scattering profiles.
Extended Data Fig. 7 X-ray crystal diffraction and liquid scattering during freezing.
a, X-ray diffraction from drops in the last four stages of freezing, recorded with 0.01 mJ XFEL pulses. These data were not affected by detector saturation. The panel also shows simulated diffraction pattern from a nanocylinder of hexagonal ice; the relative heights of the first three peaks are different from those simulated for bulk hexagonal ice9. b, Evolution of X-ray diffraction from ice crystals, 0.13 mJ XFEL pulses. The height of some peaks was affected by saturation. c, Evolution of diffuse X-ray scattering from the liquid, 0.42 mJ XFEL pulses. The liquid-scattering data were not affected by saturation. In all panels, the standard deviations of experimental data, evaluated by means of bootstrapping over bands with ΔQ = 0.0025 Å−1, are shown as lower confidence bands at one standard deviation (Methods).
Extended Data Fig. 8 Freezing after heterogeneous nucleation.
a, Diffraction from stage 2 drops of pure water (0.13 mJ XFEL pulses), doped with ATD (0.14 mJ) and doped with AgI (0.12 mJ). b, Diffraction from stage 5 drops of pure water (0.13 mJ), doped with ATD (0.03 mJ) and doped with AgI (0.04 mJ). The left-side graphs in a and b show the entire Q range and the right-side graphs are zoom-ins on the lower-height peaks at large Q. In panels a and b, the standard deviations, evaluated by means of bootstrapping over bands with ΔQ = 0.0025 Å−1, are shown as lower confidence bands at one standard deviation (Methods). c, Stages of freezing for drops doped with ATD and AgI. See the Supplementary Information for a description of how they differ from the freezing stages of pure water drops.
Extended Data Fig. 9 Distribution and shapes of diffraction spots at different freezing temperatures and time delays.
From left to right, the figure shows accumulated diffraction images of the first three diffraction rings during stages 2 and 5 and of the rings at medium and high diffraction angles during stage 5. a, Pure water drops. During stage 2, the rings are not well defined because of spots from strongly strained crystals that appear between the rings. The spots at large diffraction angles show a large radial elongation owing to inhomogeneous strain within single crystals. b, Drops with ATD. The spots are similar to those from pure water. c, Drops with AgI. There are substantially fewer diffraction spots per drop and the first three rings are already well defined during stage 2, indicating a higher degree of long-range order. The brightest spots at large angles are less elongated radially, indicating less inhomogeneous strain over the approximately 1-µm-diameter regions investigated by the XFEL beam.
Supplementary information
Supplementary Information
The Supplementary Information text file contains Supplementary Methods (modelling and data processing) and Supplementary Discussion (heterogeneous freezing). It includes associated Supplementary Equations and Supplementary References.
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Kalita, A., Mrozek-McCourt, M., Kaldawi, T.F. et al. Microstructure and crystal order during freezing of supercooled water drops. Nature 620, 557–561 (2023). https://doi.org/10.1038/s41586-023-06283-2
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DOI: https://doi.org/10.1038/s41586-023-06283-2
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