Dynamics of supercooled droplets: Impacts, jets, explosions and more

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Master thesis report

Dynamics of supercooled droplets:

Impacts, jets, explosions and more

Author:

Sebastian Sterl

Defence committee:

Prof. Dr.Devaraj van der Meer Dr.Chao Sun MSc.Sander Wildeman Dr.Stefan Kooij

May 20, 2015

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Abstract

When liquids are cooled below their freezing point without freezing, they are called “supercooled”. This counter-intuitive process occurs when an energy barrier exists for creating a nucleus of frozen substance in the liquid. A common occurrence of supercooled liquids is in the form of supercooled water droplets constituting atmospheric clouds at medium and high altitudes, and a common mechanism that causes such droplets to freeze is the presence of aerosols that serve as nucleation points. It is of high importance to our knowledge of cloud formation and the role of aerosols and ice clouds in the global climate system to understand the ways in which supercooled droplets behave, and the mechanisms by which they can freeze. Furthermore, it is highly relevant to the aircraft industry, since supercooled droplets impacting on airplane wings are one of the prime causes of aircraft icing.

In this thesis, we present experimental and numerical results on the freezing processes of supercooled droplets. We have used different experimental techniques to create supercooled macro- as well as microdrops, and performed experiments to vary their temperature and the influence of this temperature on a number of their properties. We have compared the characteristics of impacting supercooled droplets with those of non-supercooled droplets, and discovered the potential of using supercooled droplets for three-dimensional printing applications. Furthermore, we have used high- speed imaging to record the actual freezing process of supercooled droplets and infer the spatial and temporal scales involved herein, and shown how the freezing can cause droplets to explode. We have also constructed physical models to explain the dynamics of both the freezing process and of the explosion process.

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Acknowledgements

This thesis was written in a time when a lot of other things were happening. During the nine months which I spent on this project, I also managed to obtain my driving license, performed an internship as a technical consultant at African Energy & Consulting BV, regularly traveled up and down to Sweden, participated in the BestGraduates 2015 competition, co-wrote a paper based on an earlier internship, had a number of interviews for full-time positions, and accepted a job as a Junior Researcher at the New Climate Institute for Climate Policy and Global Sustainability in Cologne.

I would not have made it without the support of many people who deserve to be mentioned here.

On a professional level, I thank Sander Wildeman for his daily supervision and his continuous new ideas; Chao Sun for his constant support; Detlef Lohse for his input on the theoretical model on frozen droplet lifetime; Jacco Snoeijer for his input on the model on energy buildup inside a porous frozen droplet; Pascal Sleutel for help with the printing aspects of the setup; Devaraj van der Meer and Stefan Kooij for serving on the thesis committee; Gert-Wim Bruggert, Martin Bos and Bas Benschop for extensive technical help on our setup; and Joanita Leferink for administrative assistance.

On a personal level, I wish to thank all the friends I made in Enschede. Whether we met through our studies, through sports, through work or in other ways, my time in Enschede was shaped by you and I will not forget it. Here, I would specifically like to thank my office mates: Ignaas, Martin, Christa, Mirjam, Leonie, Pilar, Yaxing and Matthijs. I will remember the past months in Enschede with happiness, for which you deserve credit.

Furthermore, I thank my family for their support and help throughout my studies; I hope I made you proud. Lastly, my heartfelt thanks go to Chrissy, for her loving support, understanding, and help. I look forward to the next chapter with you.

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Chapter 1

Introduction

This section serves to give the reader a short introduction to supercooled liquids, the main subject of this thesis. We give a short overview of the physical principle behind supercooling, and provide an overview of the structure of this thesis.

1.1 Supercooled droplets

When a liquid is cooled to temperatures below its melting point, but remains liquid, it is termed

“supercooled”. This counter-intuitive process occurs when an energy barrier exists for creating a nucleus of frozen substance in the liquid. Even though being liquid is energetically unfavourable compared to being completely solid if the temperature is lower than melting point, the liquid phase can exist at such low temperatures if there is an energy barrier for ice nucleation. Such a barrier can exist if the local formation of an ice-liquid interface increases the free energy of the system.

As long as the energy for creating and maintaining such an interface is not available, there is an energy barrier to be overcome and freezing will not occur by itself. Only when a nucleus has attained a certain critical size, does it then become energetically favourable for the solid phase to keep growing [1]. A schematic display of the energy barrier concept is given in Figure 1.1, where the free energy of a supercooled droplet is plotted (qualitatively) against the volume fraction ice in the droplet.

When supercooled droplets do turn into ice, this happens through a process called nucleation [1, 2], in which the change of phase happens locally, and the new phase subsequently grows from that site. An ice nucleus can grow spontaneously, excited by thermal fluctuations; this process is called

Free energy

Vol % ice

ice nucleus in water

ice water

Figure 1.1: A schematic display of the energy barrier to be overcome for a supercooled water droplet to turn to ice.

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homogeneous nucleation. On the other hand, a nucleus can also grow from a particle - e.g. from an aerosol component, dust, soot, etc. - which forms a substrate onto which solid particles can adhere to form a crystalline structure; this process is called heterogeneous nucleation. A specific form of heterogeneous nucleation, termed contact nucleation, refers to the initiation of nucleation upon contact or impact of the supercooled liquid with some other matter.

There are different methods of generating supercooled droplets in the laboratory. Droplets can be supercooled in a controlled way by i.e. trapping them in a cold gaseous environment or a cold airflow; such trapping can be done through, for example, optical or acoustical means [3, 4].

Another method is to use the principle of evaporative cooling [5], by which droplets are placed in an environment with a pressure lower than the vapour pressure of the liquid phase; the droplets will tend to evaporate due to the underpressure. Since evaporation is endothermic, heat is removed from the droplets in the process; therefore, the droplets can be cooled to sub-freezing temperatures while evaporating. We have chosen to use the principle of evaporative cooling to create supercooled droplets in the current study.

1.2 Experimental research on supercooled droplets

Various groups of experimentalists have investigated several characteristics of supercooled droplets in the past. We discuss some of the relevant literature in this section, focusing on the techniques used to create supercooled droplets, as well as on the variety of measured parameters in studies on supercooled droplets.

Murray et al. [6] experimentally investigated the nucleation of ice from micrometer-sized droplets suspended on a glass substrate at temperatures around −37^◦C. They used their data to calculate the nucleation rate as a function of temperature, and to infer the ice-water interfacial energy using classical nucleation theory. Their data on nucleation rates are in good agreement with earlier results from (among others) Duft and Leisner [7], who conducted experiments using electromagnetically levitated supercooled microdroplets; as well as those from DeMott and Rogers [8], who studied nucleation inside a cloud chamber.

Observations of the freezing process have been performed by Ishizaka et al. [3] using laser- trapped supercooled aqueous ammonium sulfate droplets. They succeeded in directly filming the freezing of supercooled droplets at temperatures below −60^◦C with a CCD camera. Furthermore, Diehl et al. [9] observed the freezing process of sulfuric and nitric acid solution droplets levitated acoustically. They recorded the freezing temperature as a function of acid concentration, as well as a number of typical time scales of freezing.

Shin et al. [5] investigated the production of ice particles from supercooled droplets by spraying water droplets into a vacuum chamber, operated at pressures close to 1 mbar. A similar study on droplet freezing was conducted by Satoh et al. [10]; in this study, droplets were suspended on a thermocouple wire inside the vacuum chamber. In this way, the temperature of the droplets during supercooling and freezing could be recorded. A more recent paper by Sellberg et al. [11] describes a study in which the principle of evaporative cooling was used to probe the structure of micron-sized, supercooled water droplets at temperatures down to 227 K using X-ray scattering techniques. In this study, droplets were dispensed into a high-vacuum environment at a pressure of 10⁻³ mbar, far below the vapour pressure of water. It was found that metastable liquid water droplets could exist even at −46^◦C, although only on millisecond time scales.

Maitra et al. [12] have performed experiments to investigate the spreading of water droplets, supercooled in a sub-zero temperature environment, on superhydrophobic substrates. They identified several effects of the increased viscosity of supercooled droplets on spreading and penetration characteristics, among which decreased bubble entrapment and decelerated contact line motion in the recoiling stage. Using high-speed imaging techniques, Jung et al. [13] investigated the phase tran- sitions involved in the freezing of micrometer-sized supercooled droplets, in particular evaporation and re-condensation from their surface, on substrates with widely differing thermal conductivities.

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Inspired by the successes of these studies in creating and controlling supercooled droplets in the laboratory, we have set out to perform our own experimental study on supercooled droplets.

1.3 Current study

There are different aspects of interest in the mechanisms of supercooling and the physical behaviour of supercooled droplets. In this thesis, we distinguish between three different cases that are investigated experimentally and numerically:

(1) Impact. We have performed experiments to investigate the impact characteristics of supercooled droplets on solid substrates as compared to those of non-supercooled droplets. The results inform us about a number of differences in fluid properties between supercooled and non-supercooled liquids.

(2) Jetting. We have performed experiments in which liquid jets were used to create a stream of supercooled droplets with sizes in the order of tens of microns. It was found that these droplet streams could be used to print solid structures by controlling their impact direction and speed.

(3) Freezing. The physical process by which a supercooled droplet freezes is characterized by a wealth of behaviour in different stages. We have performed experiments in order to investigate the initiation of the freezing process of millimeter-sized droplets and its consequences.

This thesis is organized as follows. In chapter 2, we provide an overview of the motivations into research on supercooled liquids using literature review. In chapter 3, we describe in detail the experimental setup and methods that were used to perform our study. In chapter 4, we present an overview of the experimentally obtained results for the first case described above, the impact of supercooled droplets, as well as a short accompanying numerical study. In chapter 5, we present experimental results for the second case, experiments on jetting. In chapter 6, we present experimental as well as numerical results pertaining to the third case, freezing sessile droplets. Finally, chapter 7 provides the reader with a summary of the main findings of this study, as well as a number of recommendations for future studies.

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Chapter 2

Motivation

In this chapter, we describe the main motivation for research into the physical behaviour of supercooled liquids. The physics of supercooled water is extremely relevant in atmospheric physics and climate science, since supercooled droplets are the main constituents of certain types of clouds.

Furthermore, supercooled droplets are one of the chief causes of aircraft icing and are therefore important for safety reasons in the aircraft industry. Lastly, we shortly discuss the relevance of supercooled liquids in the geological concept of frost weathering.

2.1 Cloud physics and climate science

The physics of clouds forms one of the major uncertain components in current efforts to model and predict climate change [14]. Clouds play a part in the climate system in different ways: they influence precipitation, albedo effects and greenhouse warming, among other things. The formation mechanism of clouds and in particular the role of natural and anthropogenic aerosol therein remains a knowledge gap in climate science. Despite the importance of ice in cloud physics, the understanding of the processes governing the formation of ice in clouds under different atmospheric conditions, and the influence of anthropogenic activity through the release of aerosols thereon, remains incomplete [14, 15].

Cloud physics act on many scales. While the physical size of a cloud system can easily span distances in the order of 10³ km, the formation of individual droplets and crystals that make up a cloud is a microscale phenomenon. The micro- and macroscales of clouds are intimately coupled:

the macrophysics determine the environmental background (temperature, pressure, wind) in which the microphysics take place, while the microphysics of the cloud determine, for example, its global release of heat through condensation and freezing [2, 16].

The physics of clouds involves the three phases of matter: gas, liquid and solid. Most of the condensed water in the troposphere exists in the liquid phase; the rest is in the solid phase [1].

Ice is mainly prevalent in clouds that reach large altitudes. The most important type are cirrus clouds, which consist exclusively of ice crystals [17]; other types are cirrostratus and cirrocumulus clouds [1]. In Figure 2.1, an illustration is given of common cloud types and the typical altitudes above ground level that they reach (here in feet); ice clouds are typically prevalent at altitudes between 8 and 17 kilometers [17]. Cirrus clouds play an important role in the Earth’s heat budget.

They cover up to 30% of the surface of the planet, and exert a net heating effect due to a high efficiency in absorbing heat radiated away from Earth’s surface [18]. As such, cirrus clouds exert a considerable influence on the climate of the Earth, and it is thus of great importance for global climate models to accurately represent and parameterize the mechanisms by which they are formed.

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Figure 2.1: An illustration of the altitudes that different cloud types typically reach (numbers in feet). From http: // images. intellicast. com/ App_ Images/ Resources/ Clouds/ NIMB500x246. jpg .

In the atmosphere, homogeneous nucleation of supercooled droplets typically takes place at temperatures below −30^◦C to −40^◦C, whereas heterogeneous nucleation is the dominant form of nucleation at higher temperatures [1].

2.1.1 Ice nucleation in experimental studies

There have been numerous experimental attempts to quantify a number of important parameters relating to the freezing of supercooled water droplets in atmospheric conditions.

One question of importance is which crystalline phase of ice is dominant in clouds. There are many known crystalline phases of ice, dependent on temperature and pressure (see Figure 2.2), but under atmospheric conditions, only two of these are relevant: ice with cubically arranged molecules I_c (unstable), and ice with hexagonally arranged molecules I_h(stable). Since cubic and hexagonal ice have different thermodynamical properties, the mechanisms and rates of conversion from water to cubic ice and hexagonal ice are of importance for the rates of dehydration of clouds [1]. Evidence has been found in a number of studies (as summarized i.e. in [6]) that ice crystals nucleated from water droplets over the temperature range 200 K to 240 K, and over a droplet size range from nanometers to micrometers, are always initially in the unstable arrangement Ic. If the temperature after crystallization is high enough, they may relax to the more stable hexagonal arrangement Ih. Evidence for the existence of cubic ice has also been found in the atmosphere by a number of field studies, as summarized in [1].

Heterogeneous nucleation in clouds has also been a topic of experimental and field investigations.

Summaries of previous investigations is given in the review by Cantrell and Heymsfield [1] and the more extended one by Hoose and M¨ohler [20]. The main questions on a microphysics level addressed by such research are typically: firstly, what kinds of particles constitute the most common nucleation sites for heterogeneous ice formation; and secondly, what are the atmospheric conditions that favour heterogeneous nucleation (regarding temperature, humidity, et cetera). The main candidates for nucleation sites inside atmospheric water droplets have been suggested to be mineral dust and desert sand particles, metallic particles, as well as soot from aircraft [1]. Even macromolecules, such as from pollen, may play a role [21]. Heterogeneous nucleation is thus a natural process which can be enhanced by human activity.

Haag et al. [22] report on in situ aircraft measurements of cirrus cloud composition in the northern and southern midlatitudes, and claim that their analysis shows homogeneous freezing to be the dominant mechanism in the formation of such clouds. However, Cziczo et al. [17] report on in situ aircraft measurements in North and Central America. They claim that heterogeneous

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Figure 2.2: The phase diagram of water, including its known solid crystalline phases (indicated by Icfor cubic ice, Ihfor hexagonal ice, and with roman numerals for various other phases) versus pressure as well as temperature. L indicates the liquid phase. Picture taken from [19].

nucleation is the dominant freezing mechanism of high-altitude cirrus clouds, with mineral dust and metallic particles being the main drivers (i.e. providers of nucleation sites) for this process.

Contact nucleation, which falls under heterogeneous nucleation, is specifically relevant in atmospheric science. In this context, it mainly refers to the initiation of droplet freezing upon external contact of a water surface with an aerosol particle [1, 2]. Experimental research by Durant and Shaw [23] has shown that the nucleation temperature of water in surface contact with volcanic ash particles is higher than that of water droplets in which such a particle is completely immersed. This would imply that such contact freezing is not dependent on the impingement of aerosol on droplet, but purely determined by whether the aerosol particle is partly or completely immersed. Causes for this phenomenon are currently unknown.

2.1.2 Ice nucleation in climate models

While this thesis is primarily concerned with the microphysics of nucleation in supercooled droplets, we briefly mention some examples of research on the aspects of large-scale physics of clouds associated with the nucleation of ice. We do this to highlight the importance of understanding these freezing processes in research efforts to improve global climate predictions, as well as the main challenges that remain in understanding the coupling between microphysics and macrophysics of ice clouds.

In order to realistically represent the effects of homogeneous and heterogeneous nucleation in the atmosphere for application in General Circulation Models (GCMs), realistic parametizations of cloud formation, based on the physics of nucleation, are necessary. K¨archer and Lohmann [24]

developed a parameterization of homogeneous freezing of supercooled droplets, and used it to perform the first interactive simulations of cirrus clouds in a GCM [25]. The same authors also

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developed a parameterization of cirrus cloud formation due to heterogeneous nucleation [26]. Similar efforts were made by, for example, Liu et al. [27]. Barahona and Nehes [28] developed an analytical parameterization for cirrus cloud genesis which explicitly incorporates the competition between the two types of nucleation, homo- and heterogenous.

Gettelman et al. [29] incorporated the ice schemes described in [27] into the Community At- mospheric Model (CAM) GCM. Results from their simulations imply that homogeneous freezing dominates ice formation in tropical cirrus clouds, but that heterogeneous nucleation processes can dominate in dust-rich upper-tropospheric regions and in the Arctic. Jensen and Ackerman [30]

numerically studied the influence of homogeneous aerosol freezing on ice formation in rising high- altitude tropical cumulonimbus clouds. Their simulations suggested that the homogeneous freezing process is the main mechanism for generation of tropical tropopause layer (TTL) cirrus clouds.

The cited examples show that, both from experimental and numerical investigations, there is a high amount of uncertainty regarding the relative importance of homo- and heterogeneous nucleation processes in cloud ice formation.

2.2 Aircraft safety

Apart from the interest in the fields of meteorology and climatology, the study of supercooled droplets and ice nucleation is important for understanding the processes behind ice accretion on structures such as aircraft and ships [31]. Ice accumulation on aircraft, mainly on aircraft wings, has been the cause of a number of serious accidents in aviation [32]. Such accumulation can occur when the aircraft is flying through clouds consisting of supercooled droplets. Aircraft icing is hazardous because it increases the weight of the aircraft, decreases its lift, and increases its drag; see also Figure 2.3.

Other processes in which icing plays a fundamental role are marine icing (i.e. ice accretion on ship structures), ice accretion on wind turbine blades leading to reduced power generation, and ice deposition on power cables during, for example, snow storms [33].

In the context of aircraft icing, two types of icing can be distinguished: glaze icing, in which supercooled droplets arrive and form a layer of water before freezing, and rime icing, in which supercooled droplets directly freeze upon impact on a solid surface, trapping air along with them and giving the ice a white appearance. This is indicated schematically in Figure 2.4. The effects of ice accretion on aircraft also depend on a number of other factors, including the liquid water content (LWC) of the clouds the aircraft is passing through, the ambient temperature, the average cloud droplet size, the geometry of the iced object (usually the airfoil), and the aircraft speed [34].

In order to increase flight safety by reducing the amount of ice accretion on aircraft structures, fundamental understanding of the processes of ice nucleation upon impact are required [35]. For an insightful review on aircraft icing, including approaches to calculate droplet trajectories and icing rates, and an overview of ice protection systems, the reader is referred to the article by Gent et al. [36].

A numerical study into the accretion of ice from supercooled droplets impacting on a surface is described by Myers & Hammond [35] and Brakel et al. [31], modeling the impact of incoming supercooled drops onto a solid surface. Their one-dimensional model can describe both rime and glaze icing processes. Another insightful numerical study has been performed by Wang et al. [37], who demonstrate a combined droplet tracking and splashing model to illustrate the effect of supercooled droplet impact on airfoil-shaped objects. A detailed account of a numerical study into aircraft icing can be found in the PhD thesis by Hospers [38], which describes the development and validation of a Eulerian method that includes the effects of disperse droplet sizes, droplet splashing, droplet deformation, and rebounding after impact.

Insightful experimental studies into the interaction of supercooled droplets with solid surfaces in the context of applications for aircraft icing have been performed, among others, by Fumoto &

Kawanima [39], who visualized the impact of supercooled droplets on metal substrates; by Jung

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Figure 2.3: A cartoon illustrating the combined effects that icing has on aircraft performance. From http: // vortex. accuweather. com/ adc2004/ pub/ includes/ columns/ newsstory/ 2011/ 400x266_ 12101129_

icing-affects-aircraft. jpg .

Figure 2.4: Schematic illustration of the difference between rime and glaze icing on an airfoil structure. Picture taken from http: // www. newscientist. com/ data/ images/ archive/ 2810/ 28106101. jpg .

et al. [40], who investigated the freezing of supercooled droplets on a solid surface under different shearing conditions; by Jung et al. [41], who focused on how the roughness and wettability of the impact surface can be changed to influence the freezing delay for impacting supercooled droplets; and by Yang et al. [42], who examined a number of conditions for instantaneous and non-instantaneous freezing of supercooled droplet impact on metal surfaces.

2.3 Geology and glaciology

Freezing of supercooled water is also of relevance in the fields of geology and glaciology, as it can exist inside porous media such as rocks, influencing a number of processes that are driven by the freezing of liquid in such media.

When water freezes in a porous medium such as rock, it creates large internal pressures that can cause a rock to develop fractures and potentially crack apart. This is a process known as

“frost weathering”, whose consequences can be observed in nature in conditions ranging from single cracked rocks to pattern formations in entire landscapes, as can be seen in Figure 2.5.

Apart from fundamental relevance in fracture mechanics in geology, frost weathering thus plays an important role in the geomorphological development of (in particular) glacial and periglacial

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Figure 2.5: (Left) An example of a rock cracked open during freezing conditions. Picture from http: // teach.

albion. edu/ jjn10/ files/ 2010/ 10/ frost-wedging-example3. jpg . (Right) An interesting example of a large-scale consequence of frost weathering: the creation of “patterned ground” in an area with permafrost. Patterned ground is created when the freezing of groundwater causes an upwards swelling of soil [43]. Picture from http: // blogs.

agu. org/ martianchronicles/ files/ 2011/ 03/ Blog2_ 6. jpg .

regions [44, 45]. One would tend to think that the principle of rock fracturing in freezing conditions is rather trivial; as water inside rock pores freezes into ice, it expands, thereby creating large internal pressures that can create fractures and cracks and eventually cause the rock to break apart.

However, the claim that volumetric expansion of water upon freezing is the main, or sole, driver of frost weathering has been refuted by several studies. In fact, several investigations have shown that supercooled water plays a key role in frost weathering processes. Walder & Hallet [44]

developed a mathematical model to describe frost weathering of porous rock. Their results imply that the migration of unfrozen, supercooled water towards freezing centers within porous rock plays a critical role, and that this depends on the temperature of the supercooled water as well as its cooling rate. A modeling study by Rempel et al. [46] suggests that one of the prime cause of pressures large enough to crack open porous rocks is the existence of supercooled interfacial liquid films separating ice from rock (a process also called “premelting”). Furthermore, a modeling study by Vlahou & Worster [47] suggests that only in extremely impermeable rocks could volumetric expansion be the sole driver of fracturing, and confirms the notion from [46] that the presence of liquid films at sub-melting temperatures between ice and rock, and the associated pressure across such films, is one of the chief drivers of rock fracturings in freezing conditions.

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Chapter 3

Experimental methods

In this section, we describe in detail the experimental setup and methods that were used to perform this study, and the relevant parameters and settings that were used throughout the experiments.

The aim of our experiments was mainly to investigate impact and freezing processes of supercooled droplets. Thus, we needed a setup that could accommodate visual recordings of supercooled droplets, preferably at various temperatures, and that was high enough to perform impact experiments.

3.1 Vacuum chamber

A schematic of the setup is given in Figure 3.1, indicating the principal hardware components of the experiment. The principal component of the setup is a stainless-steel vacuum chamber with dimensions 325 x 140 x 195 mm. The vacuum chamber is connected to a Brook Compton BS2208 pump via a 12 mm diameter tube connected to an opening in one of the chamber sidewalls.

The pressure in the chamber could be measured via a MicroPirani pressure transducer that was connected to the pump hose. On two sides, transparent plates were installed in the chamber walls so that its contents could be accessed visually.

Inkjet nozzles of type MD-K-501 (microdrop Technologies GmbH), held in place by an MD- H-501 piezo-electrically driven holding unit, could be inserted into an opening in the top of the chamber. In this way, water jets or droplets could be deposited into the chamber on command.

The holding unit was connected by tubing to regular lab syringes that contained the working fluid, and by electric wiring to a Microdrop MD-E-3000 dispensing unit. The dispensing of fluid could thus be driven piezoelectrically as well as manually, the latter being done using a syringe pump to directly apply a force on the syringes and push the working fluid through the nozzle and into the vacuum chamber. In order to control the position of the fluid meniscus on the tip of the nozzle before ejection of drops/jets, a Microdrop AD-E-130 back pressure unit could be used to suck excess fluid back in or eject remaining air inside the nozzle. In all our experiments, we used MilliQ water degassed in a separate vacuum bell-jar as the working fluid.

Using the vacuum pump, the pressure in the chamber could be brought down to as low as 10⁻² mbar if no fluid was injected via the nozzle or otherwise present in the chamber. However, the presence of a to-be-supercooled liquid in the chamber presented a challenge to controlling the pressure. This is because evaporation from a droplet surface, caused by evacuation of the chamber, will produce vapour that in turn works to increase the chamber pressure. To ensure that the evaporative cooling process did not counteract itself in this way and the pressure could be controlled to some extent, the bottom plate of the vacuum chamber was designed to contain channels through which cooling fluid could be pumped. This was done using a Julabo F26 Cooling Circulator with a 50 vol% mixture of glycerol and water serving as the cooling fluid. By cooling the bottom plate

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A

A: Cooling bath & circulator B: Cooled bottom plate C: Pump

D: Pump hose E: Vacuum chamber F: Deposition plate

B C

E

F

D

G

G: Vertical nozzle positioning unit H: Nozzle

I: Nozzle holder incl. piezo driver J: Dispensing system

K: Back pressure unit L: Syringe + pump H

I

J

K _L

Figure 3.1: A schematic of the experimental setup used in this study.

to sub-zero temperatures, water vapour and excess fluid in the chamber could condense and freeze on the bottom plate, thus leaving the partial pressure of water in the chamber nearly unaffected by the evaporative cooling from the droplet. In section 3.3.3, we describe a method by which the bottom plate temperature can be used to accurately tune the ambient pressure in the chamber.

In Figure 3.2, we provide a calibration plot of the temperature at the surface of the bottom plate Tbottomas a function of the temperature of cooling fluid circulating through the bottom plate.

Typically, the temperature of the cooling fluid was set to be between −5^◦C and −13.5^◦C, resulting in a bottom plate surface temperature between −2.5^◦C and −8.7^◦C. Unless mentioned otherwise, the cooling fluid temperature was set to −10^◦C in all of our experiments.

The sidewall of the vacuum chamber contained a hole through which necessary units could be inserted, mounted on a stainless-steel stick. A rectangular aluminum plate screwed onto this stick was used as a support for the deposition of supercooled droplets. With the cooling fluid temperature set to −10^◦C, the temperature of this suspended plate was close to +10^◦C. Droplets from the nozzle were found to be able to nucleate into ice on this above-zero temperature surface, thus providing proof that they became supercooled in the vacuum chamber.

Instead of the depositioning unit, ferrite-zinc core thermocouple wires, connected to a HH506RA Multilogger Thermometer, could be inserted through the hole in the sidewall and positioned un- derneath the nozzle. In this way, the temperature of droplets ejected from the nozzle could be measured. A description of a short experiment to test the accuracy of such temperature measurements is given in Appendix A. An identical thermocouple wire could also be positioned vertically, by inserting it through the opening that normally accommodated the nozzle. The thermocouple wires used in our setup were 2 mm in diameter.

In the experiments on droplet freezing, the droplets were positioned on hydrophobic surfaces to ensure they retained an approximately spherical shape. The hydrophobic surface had to be selected based on the thermal conductivity of its material. Many hydrophobic surfaces are made out of metallic micropillars; however, this kind of surface is unsuitable for the present experiments, since the large thermal conductivity of the metal would tend to raise the temperature of a droplet and prevent it from supercooling. Thus, a hydrophobic surface with low thermal conductivity was

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−15 −10 −5

−10

−8

−6

−4

−2

T

cooling fluid

(

^◦

C) T

bottom

(

◦

C )

Figure 3.2: Calibration plot showing the surface temperature of the bottom plate Tbottom versus the temperature of the cooling fluid

needed. We decided to perform our experiments using a glass surface that had been held above a candle flame for a couple of seconds. The layer of soot deposited on the glass surface was enough to render it hydrophobic, with a contact angle of about 160^◦for room-temperature water.

3.2 Recording equipment

In order to visualize the processes of ice formation from supercooled droplets, we used cameras from the Photron series (SA2 or SA1.1) on a movable stage to take high-speed recordings of freezing suspended or sessile droplets and of the impacts of droplets and jets ejected from the nozzle onto the deposition plate. In general, we used an Olympus ILP light source to provide illumination. We used both front-lighting and back-lighting configurations, depending on the specific experiment.

Front-lighting was used in cases where the observation of freezing fronts was necessary. The light guide was then positioned such that the light was directed first through a beam expander, which could be used to widen or narrow the illuminated area, and then through the same chamber window in front of which the camera was positioned. Back lighting was used in those cases where freezing fronts need not be tracked. It was implemented by placing the light source on the opposite side of the vacuum chamber as the camera, and directing the light first through a diffuser plate and then into the chamber. These methods are schematically displayed in Figure 3.3.

All high-speed recordings were processed using image analysis tools in MATLAB.

3.3 Exploratory studies

In this section, we describe the results obtained from a number of exploratory experiments designed to test the supercooling capabilities of the current setup, and explore the controllability of the conditions under which droplets could supercool. We first describe experiments in which a single

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N

M M

N P

Q

vacuum chamber

M: Light source and guide P: Beam expander N: Camera Q: Diffuser plate

vacuum chamber

Front lighting Back lighting

Figure 3.3: A schematic of the lighting methods used in this study.

thermistor was used to probe the temperature of a droplet during continuous evacuation of the chamber. Subsequently, we describe experiments in which two different thermistors were used to probe the surface and interior temperature of supercooled droplets during evacuation, respectively.

Lastly, we describe a method of using the current setup to create supercooled droplets with a constant temperature in an environment at constant pressure.

3.3.1 Single thermistor measurements

In these experiments, macro-sized droplets of degassed MilliQ water with a diameter of ≈ 3 mm were deposited on the tip of the thermocouple wire before the vacuum pump was switched on.

The initial temperature (i.e. before evacuation) of the droplets was close to 12^◦C in all cases. The pressure in the champer P_∞and the temperature of the droplet surface Tdropcould be logged as a function of time after evacuation. In Figure 3.4, we show an example of the time series P_∞(t) and Tdrop(t) of one such experiment.

It can be clearly seen how the pressure decreases in a near-exponential manner. Futhermore, the temperature initially decreases continuously after evacuation, eventually reaching a minimum - denoted Tnucl - at which the freezing suddenly sets in. At this point, the temperature of the droplet jumps back up to near the freezing point, due to the release of latent heat of freezing.

Subsequently, the (now-frozen) droplet again decreases in temperature as evaporative cooling at its surface continues. The pressure at which the droplet froze solid on the wire is denoted P_nucl.

0 20 40 60 80

10⁰ 10¹ 10² 10³

t (s) P∞(mbar)

(a)

0 20 40 60 80

−15

−10

−5 0 5 10

t (s) Tdrop(◦C)

(b)

10⁰ 10²

−10 0 10

P∞(mbar) Tdrop(◦C)

liquid

solid

supercooled liquid

Tnucl

P_nucl

Figure 3.4: An example of the pressure (a) and droplet temperature (b) evolution after starting the vacuum pump in the current setup. The inset in (a) displays the droplet temperature versus the pressure.

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t = 0 s t = 23.5 ms

t = 63.5 ms t = 177.5 ms

freezing front

ejection of liquid

a b

c d

Figure 3.5: Four pictures out of a 2000-frame-per-second recording of the freezing of a suspended supercooled droplet. (a) Just before freezing sets in; (b) During the propagation of the freezing front along the surface of the droplet; (c) Ejection of core liquid after completion of the surface freezing; and (d) The shape of the eventually completely frozen droplet.

In Figure 3.5, we show four example pictures, taken from a high-speed camera recording, that show the droplet before, during and after freezing. These recordings show clearly that it is not the entire droplet that freezes initially, but that the freezing starts on the surface of the droplet. It is seen that, after the icing front has spread over the entire droplet surface, this surface bursts and liquid from the core is released into the surroundings, quickly freezing in its turn. The propagation of the freezing front over the entire surface takes approximately 50 ms.

Through repetition of the experiment it was found that there existed a quite substantial spread in the values of T_nucl and P_nucl. In Figure 3.6a, we show a scatterplot of the values (T_nucl,P_nucl) for 80 repetitions of the same experiment. The corresponding histograms of T_nucl and P_nucl are shown in Figures 3.6b and 3.6c, respectively.

It can be seen that most of the freezing events happened in a range of Tnucl with a mean value of −15.4^◦C and a standard deviation of 1.6^◦C, and a range of Pnuclwith a mean of 6.7 mbar and a standard deviation of 2.3 mbar. However, a very small number of freezing events were also observed to take place at approximately 30 mbar and −9^◦C.

No clear dependence between Tnucl and Pnucl is evident here. This is likely to underline the nature of heterogeneous freezing: nucleation sites can come from small impurities inside the liquid, or (in this case) the contact of the droplet with the thermocouple wire. Thus, a substantial spread in the exact moment of nucleation - and with that, in Tnucland Pnucl- is anticipated. It must also be noted here, however, that the thermometer we used had a temporal resolution of one second;

since the freezing itself happens within approximately 50 ms, an error in each measurement of Tnucl

of up to approximately 0.5^◦C is anticipated (estimated from the largest values of dT /dt observed in the few seconds before nucleation). Similarly, an error of approximately 0.3 mbar is to be expected in each reading of P_nucl, in addition to the inaccuracy of approximately 5% of each reading that is to be expected according to the specifications of the pressure transducer¹.

In Figure 3.6a, we also show a line corresponding to the vapour pressure of supercooled water, according to the fit by Murphy & Koop [48] (equation (10) in their paper) that is adequate to cover

1To be found on http://www.mksinst.com/docs/UR/925DS.pdf

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−200 −18 −16 −14 −12 −10 −8 5

10 15 20 25 30 35

Tnucleation(^◦C) Pnucleation(mbar)

(a)

−200 −18 −16 −14 −12 −10 −8 0.1

0.2

T_nucleation (^◦C)

PDF

(b)

0 10 20 30

0 0.1 0.2

P_nucleation(mbar)

PDF

(c)

Figure 3.6: (a) A scatterplot of the values (Tnucl,Pnucl) for 80 repetitions of the same suspended-drop experiment;

(b) and (c) The PDFs of Tnucland Pnucl, respectively.

the range of supercooled water temperatures encountered in the current experiments. It can be seen that all of the freezing events happened at a pressure above the vapour pressure Pvappredicted by this fit.

Why are the points not closer to the predicted values of vapour pressure? One would expect that, if P > Pvap, net condensation would occur on the droplet, and it would grow in size and its temperature would rise, moving its position in a (P, T )-plot closer to the line P = Pvap. The fact that this is not reflected by the data is indicative of the cooled bottom plate acting as a water vapour sink. The partial pressure of water vapour in the chamber is therefore extremely low, and the ambient pressures of Pnucl = 6.7 mbar ± 2.3 mbar are due mainly to gases other than water vapour.

Can we explain the fact that it is only the surface of the droplet that freezes initially, and not the entire droplet? We can find a clue by looking at a dimensionless group called the Jakob number, denoted J a. In the context of supercooled droplets, this number represents the ratio of sensible energy lost during the supercooling to the latent energy lost during freezing. It is therefore defined as follows:

J a = RT_f

TnuclCp(T, p)dT Lf

, (3.1)

where C_p is the heat capacity at constant pressure, L_f is the latent heat of freezing, and T_f is the freezing point of water, 0^◦C. Technically, in the current experiments, in which the pressure decreases with time, one ought to rather use Cv, the heat capacity at constant volume. However, since Cvhas not been experimentally measured for supercooled droplets, but is assumed to deviate by only a few per cent from Cp in the range of temperatures recorded in Figure 3.6a [49] - which, in turn, deviates only by a few per cent as a function of temperature in the range relevant here - we estimate J a by using a constant Cp = 4.18 × 10³ J/kg/K over the temperature range from hTnucli = −15.4^◦C to 0^◦C, and assuming Lf = 3.34 × 10⁵J/kg. This results in a value of J a ≈ 0.2.

What does this mean in physical terms? Freezing can only happen if enough energy (Lf) is released at the same time. The fact that J a < 1 here indicates that the energy lost through supercooling is not enough to lead to complete freezing. In other words: the energy deficit of a

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(a)

t (s) Tdrop(◦C)

0.2 0.25 0.3 0.35 0.4 0

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−dTdrop/dt (^◦C/s)

PDF

(b)

Figure 3.7: (a) Ensemble of the supercooling phase of all suspended-drop experiments; (b) Histogram of the rate of supercooling during the initial phase of supercooling.

supercooled droplet compared to a droplet at 0^◦C is not as much as the energy release required for complete freezing to occur. This might be a reason why it is only the surface of the droplet that initially freezes: the energy deficit from supercooling the liquid droplet can support the freezing of a shell of the droplet, but not the entire droplet at once.

What about the rates of supercooling? From the temperature histories as in Figure 3.4, we can extract the temperature readings from the point at which supercooling starts up to the point of freezing. An ensemble of these temperature histories between T = 0^◦C and T = Tnucl is given in Figure 3.7. What can be seen clearly is that during the initial phase of supercooling (lasting approximately 30 seconds in the current experimental conditions, see the dotted line in Figure 3.7a), the rate of supercooling seems to be well-defined. During this first phase, the average rate of supercooling is dT /dt = −0.34 ± 0.03^◦C/s; see also the histogram in Figure 3.7.

However, the first phase of well-defined cooling seems to be followed by a second phase in which the rate of cooling can decrease, and sometimes even reverse slightly, before the freezing eventually occurs, with large differences between individual experiments. The last phase of supercooling is therefore badly reproducible with the current experimental conditions, resulting in the large scatter of points in Figure 3.6a.

3.3.2 Double thermistor measurements

We have also performed experiments in order to measure any potential temperature difference between the surface of the droplet and its bulk liquid. Considering that the supercooling happens through evaporation at the droplet surface, one would assume that the surface temperature of a supercooled droplet (as measured in section 3.3.1) must tend to be lower than the temperature of the bulk liquid inside the droplet. In these experiments, two separate thermistors were used simultaneously in order to measure any potential differences between the temperature at the surface and in the bulk, respectively.

The experiments were conducted as follows. Since it was found to be practically impossible to immerse a second thermistor into a droplet suspended on a first one, we could not simply replicate the experimental conditions from section 3.3.1. Instead, our approach was to deposit a droplet on a hydrophobic substrate. We then used a vertically-placed thermistor wire to hold it in place and

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Outer thermistor Inner thermistor

Figure 3.8: Schematic of the experiments performed with two separate thermistors to simultaneously measure the bulk and surface temperature of a supercooled droplet.

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−10 0 10

t(s) T(◦C)

(a)

T_i T_o

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−4

−2 0 2 4 6

t(s) To−Ti(◦C)

(b) liquid

liquid supercooled

liquid

solid

supercooled liquid

Figure 3.9: A representative example result of one of the experiments in which both the surface and bulk temperature of a supercooled droplet were probed.

record its surface temperature, and horizontally inserted a second thermistor wire into the interior of the droplet to infer a “bulk” temperature of the liquid. A schematic drawing of this setup is given in Figure 3.8.

We note here that the values measured by the second thermistor should be seen to represent an “average” bulk temperature, since we assume that a radially-dependent temperature profile exists inside a droplet supercooled through evaporation, which would be impossible to probe with thermistor wires of the sizes we used. We also note that, in order to fit both thermistor wires, we had to perform these experiments with droplets of ≈ 6 mm diameter, twice as large as those used in section 3.3.1 (conversely, we could not have performed the experiments in that section with droplets of such a large diameter, since they could not have stayed suspended).

In Figure 3.9 we plot a representative example result of one of these experiments. Figure 3.9(a) displays two temperature curves, denoted To(“outer”, the surface temperature) and Ti(“inner”, the interior temperature), plotted from the moment evaporative cooling was initiated. Also indicated are the three regimes in which the droplet was a non-supercooled liquid, supercooled liquid, and frozen solid, respectively. In Figure 3.9(b), we plot the difference To− Ti as a function of time; it is clear that the surface temperature is lower than the bulk temperature while the droplet is liquid, as expected.

We have repeated the double-thermistor experiments a number of times in order to obtain statis- tical information on the temperature difference between the surface and the bulk of the supercooled droplet. In Figure 3.10(a), we plot a histogram of the temperature difference T_o− T_i from the moment of cooling initiation up to the point of freezing, from all experiments conducted with this setup. The mean temperature difference is approximately −1.1^◦C. Figure 3.10(b) shows the same histogram using only the temperature values in the times when the droplet is supercooled (i.e. when its surface temperature is lower than 0^◦C).

Dynamics of supercooled droplets: Impacts, jets, explosions and more