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Critical Radius of Supercooled Water Droplets: On the Transition Towards Dendritic Freezing Tillmann Buttersack, and Sigurd Bauerecker J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b09913 • Publication Date (Web): 04 Jan 2016 Downloaded from http://pubs.acs.org on January 11, 2016

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The Journal of Physical Chemistry

January 4, 2016

Critical Radius of Supercooled Water Droplets: on the Transition Towards Dendritic Freezing Tillmann Buttersack and Sigurd Bauerecker* Institut für Physikalische und Theoretische Chemie, Technische Universität Braunschweig, Hans-Sommer-Strasse 10, 38106 Braunschweig, Germany, [email protected], +49 531 391-5336

Abstract The freezing of freely suspended supercooled water droplets with a diameter of bigger than a few micrometers splits into two rather different freezing stages. Within the first very fast dendritic freezing stage a spongy network ice with an ice portion of less than one third forms and more than two thirds of liquid water remain. In the present work the distribution of the ice portion in the droplet directly after the dendritic freezing phase as well as the evolution of the ice and temperature distribution has been investigated in dependence of the most relevant parameters as droplet diameter, dendritic freezing velocity (which correlates with the supercooling) and heat transfer coefficient to the surroundings (which correlates with the relative droplet velocity compared to the ambient air). For this purpose on the experimental side acoustically levitated droplets in climate chambers have been investigated in combination with high-speed cameras. The obtained results have been used for finite element method (FEM) simulations of the dendritic freezing phase under consideration of the beginning second, much slower heat-transfer dominated freezing phase. A theoretical model covering 30 layers and 5 shells of the droplet has been developed which allows one to describe the evolution of both freezing phases at the same time. The simulated results are in good agreement with experimental as well as with calculated results exploiting the heat balance equation. The most striking result of this work is the critical radius of the droplet which describes the transition of one-stage freezing of the supercooled water droplet towards the thermodynamically forced dendritical two-stage freezing where the droplet cannot sufficiently get rid of the formation heat anymore. Depending on the parameters named above this critical radius was found to be in the range of 0.1 to 1 micrometer by FEM simulation.

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Keywords: spongy network ice, stage-one freezing, recalescence, atmospheric processes, finite element method (FEM) modeling.


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I. Introduction In the atmospheres of the Earth, other planets and their satellites water droplets predominantly freeze at supercoolings being higher than -5°C. Their diameters are roughly between 0.1 and 10,000 μm with a main size of a few microns.1 Also in other research fields as food sciences,

2

cryobiology,3,4 and

pharmacy the freezing of supercooled pure water and aqueous solutions is of high importance.5,6 Although there exists a comprehensive literature in this field, the detailed freezing process of supercooled liquids (here especially of aqueous droplets) covering heat production, heat conduction and convection, dendritic growth, brine rejection, diffusion, charge separation, and others, is highly complicated and not sufficiently understood. 7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 The freezing of bigger supercooled water droplets (ΔT > 5 K) in the micrometer to millimeter size range splits into two stages: firstly, a fast adiabatic stage (kinetically controlled, during milliseconds) where a dendritic network is formed within an aqueous vicinity while the droplet is heated towards the melting point; this occurs asymmetrically from one side to the other. Secondly, a slow quasi-isothermal stage (limited by heat transport, during seconds) where the droplet completely freezes from outside to the interior; this proceeds symmetrically from the outer surface into the interior of the droplet.25,21,26,27,28,29 The portion of ice x which is formed during the first dendritic freezing stage, depends on the supercooling temperature ΔT, because the latent heat ΔH must be stored in the droplet (water/ice) since the transformation is so fast that nearly no heat ΔQ is released to the environment. This can be described in a simple heat balance equation: 1,21 𝑥 ∙ ∆𝐻 = 𝑥 ∙ 𝑐𝑝,𝑖𝑐𝑒 ∙ ∆𝑇 + (1 − 𝑥) ∙ 𝑐𝑝,𝑤𝑎𝑡𝑒𝑟 ∙ ∆𝑇 + ∆𝑄.

(1a)

∆𝑄 = (𝛼 ∙ ∆𝑇 ∙ 𝐴 ∙ ∆𝑡)/𝑚 = (6 ∙ 𝜆 ∙ ∆𝑇)/(𝑢 ∙ 𝑟 ∙ 𝜌),

(1b)

The released heat can be approximated by the flowing term:

where λ is the heat transfer coefficient of the air, u the freezing speed, r the radius of the droplet, α the heat transfer coefficient from water to air, A the surface area of the droplet, m the mass of the droplet, Δt the time for the SOF and ρ the density of water. To experimentally determine the portion of dendritic ice which is generated during the first freezing stage, calorimeters can be employed. 30,31,32,33 3 ACS Paragon Plus Environment


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If the droplet size is more and more decreased the dendritic freezing process can no longer be considered as adiabatic because the heat release via the drop surface towards the surrounding medium, e.g. air, gets increasingly important for the freezing process. This is due to the fact that the heat production during the dendritic freezing time span is proportional to d3 while the heat release through the drop surface only scales with d2 which means that the bigger droplets cannot get rid of the formation heat as the smaller droplets do. So there should be a critical droplet radius rcrit at which the heat production rate is equal the heat release rate with the result that the droplet completely freezes in one stage only. The split into two freezing phases is thermodynamically no longer forced at rcrit. Within the transition zone towards bigger droplets (d > 0.1 mm) a twofold freezing behavior can be expected where both freezing stages superpose. Although Macklin mentioned the dendritic freezing as "wet" freezing in opposite to "dry" freezing and growing of hailstones already in the early sixties, 34, 35, 36 and although one can clearly expect that such a fundamental transition should have a strong impact on the forming ice structure and therefore on the physical properties of the ice and on the ice chemistry, to our knowledge there is no information in the literature about such a critical droplet size. This was a strong motivation to establish a semi empirical model based on finite element method (FEM) simulation tools and experimental data from the literature and our own laboratory. Apart from the supercooling temperature, the freezing velocity of the dendritic front and the heat transition coefficient at the drop surface obviously are two further crucial parameters which are not simply available (as, e.g. the heat conduction coefficient) and therefore have experimentally to be determined and/or varied in the model. Due to the constantly increasing computation power, in the last years even personal computers have become powerful enough to perform the needed ambitious FEM simulations within hours. So FEM software packages as Comsol Multiphysics, Ansys and OpenFoam which have been technically perfected to a high degree now, are increasingly employed not only in the engineering sciences but also in physico-chemical fields.37,38,39,40,41,42,43 Precondition for the application of Comsol multiphysics, which has the advantage to freely combine different partial differential equations and therefore is able to model complex physico-chemical problems as the present two-stage freezing process, is that a water droplet with a micron in diameter includes about 2⋅1010 molecules and therefore can be regarded as homogeneous matter. 44 4 ACS Paragon Plus Environment

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II. Methods A. Experimental For monitoring the freezing process of supercooled water droplets and especially the velocity of the stage one (dendritic) freezing front, vacuum-isolated cooling and climate chambers have been used in combination with highspeed monitoring.

21,25

Droplets with a diameter of 1 - 3 mm have been

positioned within an ultrasonic levitator (20 kHz and 58 kHz) or between two thin wire loops in the cooling chamber. 45,46,47 The freezing process was recorded with a high-speed-camera at frame rates in the range of 500 - 5000 frames per second. Two short-arc lamps with Y-shaped optical fibers and focal lenses have been used to illuminate the droplet from four directions. In the most cases the experiments have been carried out in a smaller (19 L) cooling chamber so that camera and lighting are positioned outside at room temperature. Here, the droplet temperature was calculated from the cooling curve of the droplet under consideration of the droplet volume and cooling time which was checked by FEM simulation of the heat transport processes. In other cases a bigger cooling chamber with a 500 L volume has been used, which included the complete recording and illumination technique together with an additional infrared camera for temperature monitoring. Further details of the experimental setup is described in Refs. 21,25,48,49

B. Computational For the simulation of the two-stage freezing process of (water) droplets a three-dimensional model was developed which can be reduced to a two-dimensional rotational model employing the symmetry around the axis of the dendritic freezing direction (z axis), see Fig. 1. The partitioning of the spherical volume is twofold. Thirty parallel layers provide for the dominating dendritic freezing process in z direction starting from one point at the drop surface according to the most probable case of external heterogeneous nucleation while five equidistant spherical shells consider the much slower second stage freezing process which symmetrically proceeds from outside to inside, see Figure 1. In sum the total number of 90 ring zones results. This model was implemented with the finite elements software program-package Comsol Multiphysics on personal computers (i7, 32 GB Ram).



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The basic task is to calculate the maximum portion of ice forming in small droplets (0.1 - 1000 µm) of supercooled water (-25 °C) while within the ring zones the dendritic ice / water mixture heats up to the melting point (0°C) under consideration of the internal heat conductivity λ and heat capacity cp as well as the heat transition via the surface into the surrounding medium as air. The heat transfer coefficient (water-air) α itself depends on the diameter of the droplet d, the heat conductivity of air λair 𝛼=

𝑁𝑢∙𝜆𝑎𝑖𝑟

(2)

𝑑

and the Nusselt number Nu, which is defined as:50,51 𝑑∙𝑢𝑎𝑖𝑟

𝑁𝑢 = 2 + 0.6 ∙ 𝑃𝑟 1/3 ∙ 𝑅𝑒 1/2 = 2 + 0.53 ∙ �

𝜈

.

(3)

In contrast, the thermal conductivity of water λ, the viscosity ν, the Prandtl number Pr and the Reynolds number Re are dependent on the temperature but not dependent on the drop diameter.52,53 Equation 3 simplifies to Nu = 2 if convection is neglected. In the experiment where acoustic levitators have been used the convection cannot be neglected. 54 Here the heat transition coefficient is estimated to αexp ≈ 200 W/(m²·K) for a d = 3 mm droplet by exploitation of the cooling behavior of levitated droplets described in Ref. 21 The phase change is simulated by a stepwise, local production of heat and by modifying the physical properties of the volume elements from liquid to solid including temperature dependencies. The maximum portion of ice in the actual volume element depends on the mean temperature of that element (heat storage, eq. 1) and the temperature of the surrounding elements (heat transfer). The freezing speed u is set constant for the whole dendritic freezing process which is justified by the experiments described above, see Fig. 2. We use a main value of 0.1 m/s in the simulations, and also 0.02 m/s and 0.5 m/s to cover the range which is spanned by the available experimental and theoretical results which are summarized in Fig. 3. Further details of the model can be found in the additional materials part.


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Figure 1. Scheme of the semi-empiric 2D-symmetric model. The droplet is divided into 90 ring zones ordered in 30 equal thin layers and 5 spherical shells. The layer wise freezing starts always from the bottom surface element. The black arrows mark the freezing direction (big arrow: dendritic freezing, small arrows: second step freezing) and the red arrows symbolize the heat transfer (big arrow: heat conduction during dendritic freezing, small arrows: heat conduction and heat transition predominantly during second step freezing).



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III. Experimental Results The dendritic freezing of the supercooled droplets starts at a random position but with a high probability (> 90 %) at the drop surface. Two typical stage-one freezing (SOF) processes at high supercoolings are shown in Figure 2. The dendritic freezing front often is planar (see lower trace in Fig. 2), but can also be slightly concave or convex (see upper trace in Fig. 2). In these examples the dendritic freezing of the 1.6 mm droplet starts with a supercooling of -21 °C (top) and needs around 18 ms at a constant speed of 0.09 m/s. The droplet with the diameter of 2.6 mm and a supercooling of -23 °C (bottom) freezes with 0.16 m/s considerably faster which demonstrates a relatively high variability of the dendritic freezing velocity in general, compare Figs. 3 and 4.

Figure 2. Dendritic freezing of levitated pure water droplets monitored by highspeed imaging. Top trace: drop diameter 1.6 mm, recording speed 1000 fps, supercooling -21 °C, freezing velocity 0.09 m/s, and freezing direction in the droplet from left to right; note, that the clear white zones are reflections of the illumination. Bottom trace: drop diameter 2.6 mm, recording speed 2000 fps, supercooling -23 °C, freezing velocity 0.16 m/s and freezing direction is from right to left.

The propagation speed of the dendritic freezing front u during SOF strongly depends on the supercooling. Figure 3 shows an overview of this dependence resulting from three experimental 1,55,56 and three theoretical studies. 57,58,59 The general tendency seems to mark a maximum of the dendritic freezing speed between 10 and 20 cm/s apart from the experimental results of Shibkov et al. which show an accelerating increase of the freezing speed even up to 30 K supercoolings (the reason for this divergent behavior may lie in the fact that thin water layers on metallic surfaces and not droplets have been investigated in that studies Kusalik

58

60,56

). The theoretical studies of Weiss et al.

57

and Rozmanov and

indicate a qualitatively similar characteristics where the Weiss study is in better agreement

with the experimental results of Pruppacher et al. 1 and our results presented here and in Refs.

21,25

Compared to the Pruppacher data which reach to supercoolings up to about -16°C with no evident 8 ACS Paragon Plus Environment

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relative maximum, our measurements cover data up to -24°C supercooling and seem to demonstrate a relative maximum of the freezing rate at around -17 to -19°C supercooling, see Fig. 3. One physical reason, for the decrease of the freezing rate at higher supercoolings is the viscosity which increases with decreasing temperature and increasingly hinders the orientation movement of the water molecules needed to take the according position in the dendritic network which is known as WilsonFrenkel kinetics. 61 A further remarkable result is that the freezing speed seems to be not a function of the droplet size, see Fig. 4. Here the freezing velocities of droplets with supercoolings in the 15 ± 1 K interval at different diameters in the range of 2.5 - 4 mm are depicted. A considerable spread is obvious but no dependence on the drop size. The Pearson correlation coefficient for linear regression is with 0.018 almost zero of a linear plot (red line). In view of lacking data for smaller drop diameters we see this as a resilient hint that the present size independence is valid for macroscopic droplet sizes (relative small surface volume ratio) in general.



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Figure 3. Freezing velocity u of dendritically freezing droplets depending on the supercooling: white squares: Shibkov (experimental, 56 25 thin layers), red dots: Bauerecker, Buttersack including this work (experimental, droplets), blue triangles: Pruppacher et al. and Hallet 1,55 57,62 (experimental, droplets), green squares: Weiss et al. (MD simulation, TIP4P/Ice, prismatic surface), green tilted triangels: 58 Rozmanov et al. (MD simulation, TIP4P-2005, basal plane), pink stars: Carignano (MD simulations, different speed for basal and 59 prismatic growth at about 35 and 50 cm/s, six-site model). The three black triangles at -25°C supercooling mark the values used in the presented FEM model. Experimental data within the range of the grey oval region would be eligible but are not available in the moment.


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Figure 4. Size dependence of the dendritic freezing speed u of supercooled droplets: droplets with a supercooling of 15±1 K show no size dependence. The Pearson correlation coefficient is 0.018.



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IV. Computational Results Based on the experimental results, in the simulation the drop size, the dendritic freezing velocity and the heat transfer coefficient (drop surface) have been varied as the main parameters. In Fig. 5a the portion of the dendritic ice directly after the first freezing stage is depicted for the three dendritic freezing speeds of 0.02 m/s (slow), 0.1 m/s (main value) and 0.5 m/s (fast) which cover the range spanned in Fig. 3 (compare the three black triangles at -25°C supercooling). In Fig. 5b again this resulting dendritic ice portion is shown at the main dendritic freezing velocity of 0.1 m/s for three heat transfer coefficients expressed by the relative speed between the droplet and the surrounding medium (air in this case) which is intuitively well imaginable: 0 m/s (main value, is most important for real situations, e.g. in the atmosphere), 10 m/s (droplet suspended in an acoustic levitator according to "rough" atmospheric situations), and 100 m/s (strong thunderstorms). In Fig. 5 it can clearly be seen that droplets with a radius between 100 - 1000 µm are frozen evenly after the dendritic freezing stage and contain a portion of 27 to 28 percent (value in right bottom edge of each half-slice) of dendritic ice as predicted with equation 1, no matter how fast the water freezes (0.02 - 0.5 m/s). However for smaller droplets than about 100 µm this portion of dendritic ice increases significantly with decreasing droplet size and the distribution of dendritic ice becomes uneven, namely in the bottom half of the droplet the ice portion is up to 100 percent while in the upper half it can stay between 0 and mostly below 30 percent. Further, it can be noted that the portion of ice in the outer shell is mostly increased. This is most impressive for the droplet size of 1 µm at 0.02 m/s dendritic freezing velocity where an outer shell of almost massive ice has formed while in the inner upper half most of the water is still liquid. For the smaller droplets the averaged portion of dendritic ice (right bottom slice corner) considerably decreases with increasing freezing velocity (first two columns) which impressively demonstrates the relatively increased heat transport through the drop surface. In Fig. 5b can directly be realized by comparison of the three rows that the influence of the heat transfer coefficient (or the relative speed of the droplet) on the freezing water/ice system is rather small especially for drop radii bigger than 1 µm. However, there is a remarkable influence for the smaller droplets, e.g. for the 1 µm droplet where the averaged portion of dendritic ice after the first freezing stage increases with the heat transfer coefficient from 0.37 to 0.45 as intuitively expected. 12 ACS Paragon Plus Environment

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In Fig. 6 the evolution of the temperature distribution during the dendritic freezing stage is illustrated for the two droplet radii of 0.5 and 1 µm which are close to the critical size. The abscissa linearly marks the freezing front through the droplet in 20 % steps. As in Fig. 5 the ordinate is determined by the factor-five variation of the dendritic freezing velocity around the mean value of 0.1 m/s. Due to the heat production the freezing front is the hottest zone of the freezing droplet and reaches nearly 0°C. Below that front is a warm zone and above a colder one. The faster the freezing speed the stronger the T gradients appear. It can be seen (better for the 1 µm droplet) that the freezing fronts tend to develop convexly during the about first 40 %, then planar at the middle and rather concavely during the end phase. The reason for this behavior is that the heat can flow into an opening volume at the beginning but is restricted to a confined space in the end phase (heat bottleneck). The critical droplet radius which allows the formation of 100 % ice within the first freezing step - which means in just one step only - is between 0.1 and 0.5 µm and depends mainly on the freezing rate and secondly on the heat transfer coefficient towards the surrounding (see Figure 7). Lower freezing rates and higher heat transfer coefficients cause a bigger critical radius. In addition to the dendritic freezing of a droplet in an aerial environment, we also investigated the ambience of oil which shows a much bigger thermal conductivity (λoil ≈ 6 · λair = 0.15 W/(m·K) ) and heat transfer coefficient (see white diamonds in Fig. 7). In this case the critical radius is with 0.5 µm as that under atmospheric conditions at the lower freezing velocity of 0.02 m/s. In all simulated configurations the averaged portion of dendritic ice after the first freezing stage changes from about 27 to 100 % in the size range of 0.1 to 10 μm droplet radius, compare Fig. 7. The model is verified by comparing the surface temperature of a cooling droplet in the described experimental climate chamber held at -25°C and which was recorded with a calibrated infrared highspeed camera. Furthermore, the accurate prediction of the 27 % portion of ice after the dendritic freezing stage which is stated by the basic heat balance equation (equation 1a) for the bigger droplets is a sound verification of the model (compare also equation (1) and (7) and Pruppacher's simple formula for the dendritic ice content being ∆T / 80 K ≈ 31 % 1).



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Figure 5. Distribution of the ice portion directly after the fast dendritic freezing phase in 90 individual ring-zones of droplets with a radius between 0.1 and 1000 µm. In each case the dendritic freezing starts at the bottom point of the symmetry axis so that all droplets freeze from bottom to top. The values near the right bottom edge of the half-circles mark the average portion of ice after the dendritic freezing front has crossed the whole droplet. Note that ice portions between 0 and 1 represent spongy network ice. Due to symmetry reasons only the right half slices of the droplets are shown. a) Freezing velocities are varied at a heat transfer coefficient of zero. Bottom row: 0.02 m/s (slow), middle row: 0.1 m/s (medium speed, main value) and top row: 0.5 m/s (fast). b) Heat transfer coefficients are varied at a freezing rate of 0.1 m/s expressed as relative velocities of the droplet against the ambient air. Bottom row: 0 m/s (no convection); middle row: 10 m/s (medium convection as within an ultrasonic levitator); top row: 100 m/s (strong convection). Note that the bottom row of b) contains the same information (half slices) as the middle row in a). Within the chosen parameter ranges the droplet size and the dendritic freezing speed have a stronger influence on the final dendritic ice distribution than the heat transfer coefficient through the droplet surface.


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Figure 6. Evolution of the temperature distribution of a dendritically freezing droplet with 0.5 μm radius (left) and 1 μm radius (right) without convection at varied dendritic freezing velocities (compare Fig 3.): 0.02 m/s (slow, bottom row), 0.1 m/s (moderate, middle row), 0.5 m/s (fast, top row). The heat transfer coefficient α is 26 kW/(m²·K) for the 1 µm droplet and 52 kW/(m²·K) for the 0.5 µm droplet. Note that a non-linear scale is used (stretched over -2 °C). The freezing direction is always from bottom to top.



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Figure 7. Portion of ice in the droplet after dendritic freezing. a) for varied heat transfer coefficients at a freezing rate of 0.1 m/s (no convection, black squares), relative velocity 10 m/s (moderate convection, red dots); relative speed 100 m/s (high convection, blue triangles); and no convection, but within liquid oil (white diamonds). b) for varied freezing velocities and no convection: 0.02 m/s (magenta stars), 0.1 m/s (black squares), 0.5 m/s (green diamonds). A critical radius between 0.1 and 0.5 µm could be revealed dependent on the chosen parameters. The dotted lines describe the ice portion after stage-one freezing (SOF) calculated by equation 1a and 1b being in good agreement with the simulations.


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V. Discussion and Conclusions Fig. 3 summarizes the available experimental and computational data concerning freezing velocities. For supercooled droplets only experimental data comprised by Pruppacher

1

and ours are available

which both are in accordance with regard to an increasing freezing rate till 17 K supercooling while our data outreach the Pruppacher data till 24 K supercooling showing a decreasing tendency after a maximum at about 16 K supercooling. The latter is in accordance to the theoretical results of Rozmanov et al. who investigated the slow-growing basal plane and even better of Weiss et al. who studied the fast-growing secondary prismatic face.57,58 Both theoretical studies show a clear maximum for the freezing velocity for which the increasing viscosity with decreasing temperature (Wilson-Frenkel kinetics 61) should be crucial. However, one has to consider that the theoretical studies have general constraints as they both do not base on an adiabatic model which considers heat production during freezing but withdraw heat out of the whole drop volume evenly. Due to this and to the restriction that only very small drop volumes are manageable by simulations today, only a limited comparison with the experimental data is possible. For the future, higher computation power and improved modeling on the theoretical side and the investigation of higher supercoolings than -24°C on the experimental side would be advantageous. While in the presented experiments with acoustically levitated droplets the above mentioned > 90% nucleation rate via the surface is predominantly caused by heterogeneous nucleation due to small ice aerosol particles, a very similar result is reported, e.g., by the work of Djikaev et al. who investigated homogeneous nucleation and give thermodynamic explanations for "the experimental and molecular dynamics simulation evidence that ... the freezing of atmospheric aerosols occurs beginning at the droplet surface".63 In this context a transition is suggested by Djikaev which is similar but not equivalent to the presented one, where it was shown that at decreasing droplet size the onset of the crystallization changes from volume-based to the surface-stimulated mode at about 10 µm droplet radius.64 To understand the results depicted in Figs. 5, 6 and 7 one has to consider the interplay of two dynamic processes: heat production driven by the proceeding dendritic freezing front and heat conduction within the droplet and out of the droplet through the surface under consideration of the (strongly 17 ACS Paragon Plus Environment


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different) heat capacities of water and ice. For the bigger millimeter sized droplets the portion of heat which is released via the drop surface during the first-stage freezing phase can be neglected and equation 1a in its simplest form (without the ΔQ term) describes the portion of formed dendritic ice rather well. For the droplets with radii below 100 µm and increasingly below 10 µm the relative heat transport via the surface increases and the dendritic ice portion in the initially freezing drop region increases strongly towards 100 % in opposite to the finally freezing drop region where the ice portion even decreases below 27 % , compare most important middle row in Fig. 5. In some cases we observed droplets after the dendritic freezing phase with the initial part completely frozen and at the same time a nearly 100 % dense ice shell formed in the other drop part while the core part contains liquid water portions of 70 to 100 %, e.g., compare second half slice in the bottom row of Fig. 5. This means that the droplet could burst during the subsequent second freezing stage as the forming ice increases in volume and in turn liquid is squeezed out of the droplet, forming frozen noses which exactly describes the situation observed by Leisner in electrodynamical traps for supercooled freezing droplets.65,66 For smaller droplets than 1 to 10 µm, due to the relatively large surface and relatively short distances for heat conduction, the surface including most parts of the droplet, or even the whole droplet volume can completely freeze in one step, see e.g. first rows in Fig. 5 with 0.1 µm drop diameter. This marks the "bifurcation" range where the freezing process splits into two stages at which we aimed in this work. In summary, it can be stated that the simulation of the freezing process of supercooled droplets was successfully performed by our FEM model. In view of possible applications this rather small critical bifurcation size means that the two-stage freezing process is the prevailing process in nature although this is generally not considered in the atmospheric research.1,57,58,59,20,67 Finally, it will be interesting if our estimation of the critical radius in the range of 0.1 to 1 µm can be verified by molecular dynamics simulations or by experiments, especially using collisional cooling cells the future.


68,23

which both is planned for

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VI. Additional Materials The used FEM model is described here in detail. For all mesh zones the general heat balance equation is solved: 𝜕𝑇

𝜌𝑐𝑝 𝜕𝑡 + 𝜌𝑐𝑝 ∙ 𝑎 ∙ ∇𝑇 = ∇ ∙ (𝜆∇𝑇) + 𝑄

(4)

for the solver variable of Comsol a; with density ρ, heat capacity cp, temperature T, time t, nabla operator 𝛻, thermal conductivity λ and added or removed heat Q. The heat capacity and thermal conductivity are functions of the portion of ice in each element. Table 1 displays the used values.

Table 1. Physical values used in FEM model. 50 cp,s

cp,l

λs

λl

ΔH

ρ

2000 [J/kg/K]

4400 [J/kg/K]

2.25 [W/m/K]

0.5 [W/m/K]

334000 [J/kg]

1000 [kg/m³]

At the outer boundary to the surrounding medium (air or oil) the following boundary condition is set: −𝑎 ∙ (−𝜆∇𝑇) = 𝛼 ∙ ∆𝑇

(5)

where α is the heat transfer coefficient, which is a function of the radius (see eqs. 2 and 3). Caused by the latent heat ΔHm of water the following power per volume PV is released:

𝑃𝑉 =

∆𝐻𝑚 ∙𝜌∙𝑢

(6)

𝑑

with freezing speed u and drop diameter d. The maximum portion of ice x in each zone is calculated in layers in the following way: 1) portion of ice by heat storage xs: 𝑥𝑠 =

𝑐𝑝,𝑤𝑎𝑡𝑒𝑟 ∙∆𝑇

(7)

∆𝐻𝑚 −𝑐𝑝,𝑖𝑐𝑒 ∙∆𝑇+𝑐𝑝,𝑤𝑎𝑡𝑒𝑟 ∙∆𝑇

2) portion of ice by heat transfer x*, which strongly depends on the freezing rate u: 19

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With the heat transfer equation 𝑄 =

𝜆∙𝐴∙∆𝑇 𝑑

, and 𝑃𝑉∗ = 𝑡

𝑄

1 ∙𝑉

and 𝑃𝑉 =

Page 20 of 27

∆𝐻𝑚 ∙𝜌 𝑡1

(eq. 6) the portion of ice

due to heat transport x* can be predicted, where t1 = d/u is the time for one step and V the volume of the element. The heat transfer splits into two parts, into the still liquid part (𝑥𝑙∗ ) and in opposite direction into the stage one ice (𝑥𝑠∗ ): 𝑥 ∗ = 𝑥𝑙∗ + 𝑥𝑠∗ 𝑥𝑙∗ = 𝑥𝑠∗ =

1/2∙(𝜆𝑠 +𝜆𝑙 )∙∆𝑇𝑖∗ ∙𝐴𝑖,𝑙 𝑉𝑖 ∙𝑑1

∗ 2/3∙(𝜆𝑠 +𝜆𝑙 )∙∆𝑇𝑖−1 ∙𝐴𝑖,𝑠

𝑉𝑖 ∙𝑢∙∆𝐻𝑚 ∙𝜌

1

∙𝑃 = 𝑉

(8)

1/2∙(𝜆𝑠 +𝜆𝑙 )∙∆𝑇𝑖∗ ∙𝐴𝑖,𝑙

(9)

𝑉𝑖 ∙𝑢∙∆𝐻𝑚 ∙𝜌

,

(10)

where λl and λs are the thermal conductivities of the liquid and solid water, ΔTi* the supercooling at the interface between active zone and water and ΔTi-1* the temperature in the ice zone behind the active zone, Ai,l and Ai,s the surface area between the active zone and the liquid or ice, Vi the volume of the zone and d1 the thickness of the layer. Knowing the maximum possible portion of ice x, the right amount of produced heat results as 𝑃𝑣 ∙ 𝑥.

The heat flux 𝑞̇ to the environment is determined by the temperature difference at the surface and the heat transfer coefficient is:

(11)

𝑞̇ = 𝛼 ∙ (𝑇0 − 𝑇𝐴 ).

To increase the approximation to the real freezing, in each freezing step of the 30 layers additional iterations within the preceding layers have been performed which however cost considerably more calculation time. So we found as a compromise between accuracy and manageable computation costs to carry out four iterations within the preceding freezing volume when the dendritic freezing front reaches layer 8, 15, 22 and 30. The physical reason for the iterations is that during the time span in which the freezing front advances and locally heats up the droplet towards 0°C, the preceding regions of the droplet can proceed cooling so that the phase transition can continue there under the boundary condition that the temperature locally cannot exceed 0°C. 20 ACS Paragon Plus Environment

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List of symbols and acronyms Α

contact area

a

solver variable

cp

heat capacity

fps

frames per second

d

diameter

ΔH

latent heat

Nu

Nusselt number

Pr

Prandtl number

PV

Power density

ΔQ

heat

𝑞̇

heat flux

r

radius

Re

Reynolds number

SOF

stage one freezing

t1

time for one step

T0

temperature at beginning

T

temperature

u

freezing speed

V

volume of zone

x*

portion of ice due heat transfer

x

portion of ice

Greek symbols: 21 ACS Paragon Plus Environment


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α

heat transfer coefficient

λ

thermal conductivity

ν

viscosity

ρ

density

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Subscripts: air

air

crit

critical

exp

experimental

i

counter

l

liquid

s

solid

theo

theoretic

Acknowledgement We gratefully thank Volker C. Weiss for supporting us with newly computed data which extend the picture of the freezing rate dependence of droplets on the supercooling. This research was supported by the Deutsche Forschungsgemeinschaft (Grant BA 2176/3-2 and Grants BA 2176/4-1, BA 2176/4-2 and BA 2176/5-1). The authors declare no competing financial interest.

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