Dynamics of Supercooled Droplets: Impacts, Jets, Explosions and More

Master thesis report

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

Defence committee: Author: Prof. Dr. Devaraj van der Meer Dr. Chao Sun Sebastian Sterl MSc. Sander Wildeman Dr. Stefan Kooij

May 20, 2015 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. Contents

Acknowledgments 3

1 Introduction 4 1.1 Supercooled droplets ...... 4 1.2 Experimental research on supercooled droplets ...... 5 1.3 Current study ...... 6

2 Motivation 7 2.1 Cloud physics and climate science ...... 7 2.1.1 Ice nucleation in experimental studies ...... 8 2.1.2 Ice nucleation in climate models ...... 9 2.2 Aircraft safety ...... 10 2.3 Geology and glaciology ...... 11

3 Experimental methods 13 3.1 Vacuum chamber ...... 13 3.2 Recording equipment ...... 15 3.3 Exploratory studies ...... 15 3.3.1 Single thermistor measurements ...... 16 3.3.2 Double thermistor measurements ...... 19 3.3.3 Maintaining constant droplet properties ...... 21 3.3.4 Controlled nucleation ...... 22

4 Impact of supercooled droplets 25 4.1 Parameters ...... 25 4.2 Experiments ...... 25 4.3 Scaling ...... 28 4.4 Modeling ...... 30 4.5 Summary ...... 31

5 Supercooled droplet trains 32 5.1 Experiments ...... 32 5.2 Printing structures ...... 36 5.3 Summary ...... 38

6 Shattering of supercooled droplets 39 6.1 Stages ...... 39 6.1.1 Icing ...... 39 6.1.2 Freezing ...... 39 6.1.3 Shattering ...... 40 6.1.4 Aftermath ...... 43

1 6.2 Modeling ...... 45 6.2.1 Frozen droplet lifetime ...... 45 6.2.2 Frozen droplet shattering ...... 47 6.2.3 Explosion ...... 53

7 Conclusion and discussion 56 7.1 Conclusion ...... 56 7.2 Challenges and recommendations ...... 57 7.2.1 Impact and jetting experiments ...... 57 7.2.2 Freezing experiments ...... 57

A Thermistor measurements 63

B Parameter estimation 64

C Modeling parameterizations 65

2 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.

3 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

ice nucleus Free in water energy

ice

water

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

4 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.

5 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.

6 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.

7 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 Ic (unstable), and ice with hexagonally arranged molecules Ih (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 Möhler[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

8 Figure 2.2: The phase diagram of water, including its known solid crystalline phases (indicated by Ic for cubic ice, Ih for 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. Kärcher 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

9 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

10 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

11 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.

12 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

13 K L

J G

D E I H A F

B C

A: Cooling bath & circulator G: Vertical nozzle positioning unit B: Cooled bottom plate H: Nozzle C: Pump I: Nozzle holder incl. piezo driver D: Pump hose J: Dispensing system E: Vacuum chamber K: Back pressure unit F: Deposition plate L: Syringe + pump

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 Tbottom as 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 underneath 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

14 −2

−4 C) ◦ ( −6 bottom T −8

−10 −15 −10 −5 ◦ Tcooling fluid ( 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

15 Front lighting Back lighting

M P M

vacuum vacuum N chamber N chamber Q

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

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 Tdrop could 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 Pnucl.

(a) (b) 3 10 10 10 C)

◦ liquid

( 0 5

drop −10 solid T 2 0 2 10 10 10

C) 0 P∞ (mbar) ◦ (

(mbar) −5 supercooled drop

∞ liquid 1 T P 10 Pnucl −10 Tnucl

0 −15 10 0 20 40 60 80 0 20 40 60 80 t (s) t (s)

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.

16 a t = 0 s b t = 23.5 ms

freezing front

c t = 63.5 ms d t = 177.5 ms

ejection of liquid

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 Tnucl and Pnucl. In Figure 3.6a, we show a scatterplot of the values (Tnucl,Pnucl) for 80 repetitions of the same experiment. The corresponding histograms of Tnucl and Pnucl 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 Pnucl with 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 Tnucl and 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 Pnucl, 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

17 (a) (b) 35

30 0.2

PDF 0.1 25

0 20 −20 −18 −16 −14 −12 −10 −8

(mbar) ◦ Tnucleation ( C) c 15 ( ) 0.2 nucleation P 10 0.1 5 PDF

0 0 −20 −18 −16 −14 −12 −10 −8 0 10 20 30 ◦ P Tnucleation ( C) nucleation (mbar)

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 Tnucl and 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 Pvap predicted 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 Ja. 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:

R Tf Cp(T, p)dT Ja = Tnucl , (3.1) Lf where Cp is the heat capacity at constant pressure, Lf is the latent heat of freezing, and Tf 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 Cv has 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 3 - we estimate Ja by using a constant Cp = 4.18 × 10 J/kg/K over the temperature range from ◦ ◦ 5 hTnucli = −15.4 C to 0 C, and assuming Lf = 3.34 × 10 J/kg. This results in a value of Ja ≈ 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 Ja < 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

18 (a) (b) 0 25

20 −5

C) 15 ◦ ( −10 PDF drop 10 T

−15 5

−20 0 0 20 40 60 80 0.2 0.25 0.3 0.35 0.4 ◦ t (s) −dTdrop/dt ( C/s)

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

19 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.

(a) (b) 6 liquid solid 10 solid 4 super-

C) cooled

◦ liquid ( 2

C) 0 i ◦ T

( supercooled liquid − T 0 o

T T −10 i To −2 liquid −20 −4 0 50 100 0 50 100 t (s) t (s)

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 To − Ti 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).

20 (a) Liquid and supercooled liquid phase (b) Supercooled liquid phase 0.8 1

0.6 0.8

0.6 0.4 PDF PDF 0.4 0.2 0.2

0 0 −4 −2 0 2 −4 −2 0 2 ◦ ◦ To − Ti ( C) To − Ti ( C)

Figure 3.10: Histogram of the temperature diﬀerence To − Ti during (a) the entire liquid phase, and (b) only the supercooled liquid phase, as obtained from repeated double-thermistor experiments.

3.3.3 Maintaining constant droplet properties The experimental methods used in the previous sections result in a continuously decreasing droplet temperature and chamber pressure during the evaporative cooling process. Since properties such as viscosity, surface tension, density, etc. are temperature-dependent, this means that the droplets in these experiments do not have properties that are constant with time. This is disadvantageous for investigating the properties of supercooled droplets. There is, however, a very simple method of obtaining approximately constant properties. In the experiments in the previous sections, the droplet was basically the only liquid present inside the chamber, with the cooled bottom plate serving to catch any vapour originating from the droplet during the cooling process. In theory, the ambient pressure during the evaporation process could be fixed to a constant value by depositing a “reservoir” of evaporable substance in the chamber. One could think of putting a container of water in the chamber. However, an even better method is depositing ice on the bottom plate of the chamber (by pouring some water on the plate and waiting for it to freeze), since this allows for temperature control over the reservoir by changing the bottom plate temperature. This ice starts to sublimate once the pressure in the chamber has become low enough. As long as the amount of ice is much larger than the amount of liquid in the chamber (assumed to be only the droplet), an equilibrium between sublimation and evacuation will be reached, and the pressure in the chamber will stay approximately constant near the vapour pressure of the ice for as long as there is a substantial amount of ice present. This provides the droplet with the opportunity to reach a temperature at which its vapour pressure corresponds to the ambient pressure, at which point net evaporation from its surface will cease. Its temperature will then stay nearly constant, too. The vapour pressure of the ice is a function of its temperature, which is determined by the temperature of the bottom plate itself. This means that by changing the temperature of the cooling fluid (and with it, that of the bottom plate and the ice on top of it), the vapour pressure of the ice can be tuned. A higher bottom plate temperature means a higher vapour pressure, and vice versa. Therefore, in an experiment with a reservoir of ice on the bottom plate, the constant pressure that will develop in the chamber, and the temperature of the droplet as it reaches equilibrium conditions, can be tuned simply by changing the temperature of the bottom plate. We have performed experiments to confirm the notion that keeping a reservoir of ice on the bottom plate can keep the chamber pressure and supercooled droplet temperature approximately constant for a substantial time span. In Figure 3.11, we plot an example time series of ambient pressure P∞ and droplet temperature Tdrop for such an experiment (in this case with the cooling fluid temperature set to −10◦C, corresponding to a bottom plate temperature of −6.5◦C), starting from

21 the initiation of evacuation. We can distinguish the following five phases in the time series: (a) the droplet is cooling while the pressure decreases, but still above melting point; (b) the supercooling phase starts, in which both P∞ and Tdrop decrease before reaching their respective equilibrium 2 values; (c) both P∞ and Tdrop remain constant for approximately 10 seconds; (d) the constant phase is over and both P∞ and T decrease further; (e) the droplet has frozen solid. The (nearly) constant values P∞,c and Tdrop,c have been indicated by red lines. Note the differences with the time series in Figure 3.4: the cooling of the droplet is basically “interrupted” from the moment when P∞ corresponds to the vapour pressure of the ice, and only when most of the ice has sublimated does the droplet continue its initial cooling. The time span occupied by phase (c) can, of course, be regulated by depositing more or less ice on the bottom plate, with a larger reservoir of ice corresponding to a longer phase (c). However, this depends, among other factors, on the shape and thickness of the ice reservoir. In Figure 3.12(a) we have plotted the equilibrium values P∞,c as a function of the bottom plate temperature Tbottom. The solid line in Figure 3.12(a) represents the vapour pressure of ice according to the parameterization by Murphy and Koop [48], given in full in Appendix C. It can be seen that the values of P∞,c closely correspond to the values of vapour pressure of ice evaluated at the relevant value of Tbottom, as should be expected. In Figure 3.12(b), we plot P∞,c versus its corresponding values of Tdrop,c in our experiments. We also plot the curve representing the vapour pressure of supercooled water, according to the parameterizations by Murphy and Koop [48] (see again Appendix C). The experimental points lie very near the vapour pressure curve, supporting the theoretical explanation provided above that the droplets adjust in temperature until their vapour 2 pressure corresponds to P∞,c, which in turn is fixed by the vapour pressure of ice at Tbot.

3.3.4 Controlled nucleation

In Figure 3.11, it can be seen that, even though the properties P∞ and Tdrop can be kept constant for a substantial time span, the eventual values of Pnucl and Tnucl are uncontrolled: the nucleation happens when the constant phase (c) is already over, and thus Pnucl and Tnucl cannot be set in the same way as P∞,c and Tdrop,c. For controlled investigation of the freezing characteristics of supercooled droplets, it would be beneﬁcial if the droplets in our setup could be “forced” to freeze during phase (c). We have performed various tests to investigate whether this can be achieved. The ﬁrst type of test was to check whether giving the droplet an external “kick” during phase (c) could induce nucleation. We have tried this by shooting a pulsed laser beam from a Litron 532 nm Nd-YAG 400 mJ laser into the droplet during phase (c). Such a laser pulse (if focused well) could induce a shock wave in the droplet. High-speed recordings revealed that droplets would generally only vibrate after receiving a shock from a laser pulse, without freezing. However, on occasion, freezing was observed a short while after such a shock – typically between 0.01 and 0.2 seconds later. In order to make sure that nucleation occurred, therefore, we only needed to pulse continuously during phase (c) and wait until such a freezing event happened. Even though it could take up to tens of laser pulses, this method was found to be very reliable in eventually inducing droplet freezing while phase (c) was still in progress. In Figure 3.13, we plot two example time series of P∞ and Tdrop in an experiment where laser pulses were shot at a supercooled droplet during phase (c) at a frequency of 2 Hz. Comparing these time series to Figure 3.11, it can be seen that the laser pulsing indeed forced the droplet to freeze during phase (c), basically skipping phase (d). In Figure 3.14, we present a series of frames from an example high-speed recording of the induction of nucleation by laser pulsing. Another, less violent but more tedious, method of inducing controlled nucleation that was tested, consisted of simply touching the supercooled droplet with a sharp tip of ice. This not only enabled control over the moment of nucleation, but also over the location from which it started. We present a few images from an example recording of such a controlled nucleation in Figure 3.15.

2The diﬀerence between the curves for the vapour pressures of ice and water is small. In the ranges displayed here, the vapour pressure of supercooled water is at most 9% higher than the vapour pressure of ice, at T ≈ −10◦C.

22 3 10

2 10 C)

◦ 0 ( (mbar) drop ∞

1 T P 10 −5 (e) P∞,c Tdrop,c

(a) (b) (c) (d) (e) −10 (a) (b) (c) (d) 0 10 0 100 200 300 400 0 100 200 300 400 t (s) t (s)

Figure 3.11: Time series of ambient pressure P∞ (left) and droplet temperature Tdrop (right) during an experiment with a reservoir of ice on the bottom plate (which had a temperature of −6.5◦C).

(a) (b) 5.5 5.5

5 5

4.5 4.5

(mbar) 4 (mbar) 4 ,c ,c ∞ ∞

P 3.5 P 3.5

3 3

2.5 2.5 −10 −8 −6 −4 −2 −10 −8 −6 −4 −2 ◦ ◦ Tbottom ( C) Tdrop,c ( C)

Figure 3.12: (a) Calibration curve displaying the constant values P∞,c as a function of the bottom plate temperature Tbot obtained in experiments with a reservoir of ice on the bottom plate. The solid line indicates the vapour pressure of ice. Errors in P∞,c correspond to the measurement uncertainty in the readings (5%), plus the standard deviation of values measured across repeated experiments. (b) A plot of P∞,c versus Tdrop,c. The dashed line indicates the vapour pressure of supercooled water. Errors in Tdrop,c correspond to the standard deviation of values measured across repeated experiments.

3 10 10

5 2 10 C) ◦ ( 0 (mbar) drop ∞

1 Laser pulsing starts T P 10 −5

(a) (b) (c) (d) (a) (b) (c) (d) 0 10 −10 0 50 100 150 200 0 50 100 150 200 t (s) t (s)

Figure 3.13: Time series of ambient pressure P∞ (left) and droplet temperature Tdrop (right) during an experiment with a reservoir of ice on the bottom plate (which had a temperature of −6.5◦C) and in which laser pulsing was used to control nucleation. The laser was pulsed at a frequency of 2 Hz.

23 (a) t =0 s (b) t =0.0012 s (c) t =0.008 s

(d) t =0.013 s (e) t =0.016 s (f) t =0.024 s

Figure 3.14: Pictures from an example recording in which nucleation is induced in a supercooled droplet by shooting it with a laser pulse. In frame (a), the droplet is hit by the laser pulse. In frame (b), it is vibrating from the shock wave that was induced by that pulse, as manifested in the slight motion blur. In frame (c), the ice front has just started to grow on the surface of the droplet; in the subsequent frames (d)-(f), it continues to envelop the entire droplet.

(a) t =0 s (b) t =0.003 s (c) t =0.006 s

(d) t =0.01 s (e) t =0.014 s (f) t =0.018 s

Figure 3.15: Pictures from an example recording in which a nucleation point is induced on the surface of a supercooled droplet by touching it with ice. Here, it is seen (a) how a pointy ice block touches a sessile droplet on a hydrophobic plate, and how (b)-(e) the freezing front spreads from the location where the droplet was touched across the entire droplet. In (f) the entire surface is covered by ice.

24 Chapter 4

Impact of supercooled droplets

In this chapter, we discuss the experimental results of the performed experiments and the analysis of the collected data regarding the impact of supercooled droplets on solid substrates, and provide a short discussion of a numerical model developed to explain some of the observed phenomena.

4.1 Parameters

The main relevant parameter in droplet impact experiments is the spreading diameter of the droplet, D, during its impact phase. As the droplet hits a surface, it will ﬂatten out and spread. The spreading process is inﬂuenced by certain parameters. These are mainly the impact speed, viscosity, and surface tension of the liquid. The two relevant dimensionless parameters in droplet impact experiments are, therefore, the Weber number W e, given by

ρU 2D W e = 0 0 , (4.1) σ where ρ denotes the liquid density, U0 its speed of impact, D0 the diameter of the impacting droplet (assumed spherical), and σ its surface tension; and the Reynolds number Re, given by ρU D Re = 0 0 , (4.2) η where η denotes the dynamic viscosity of the liquid. The most relevant output parameter of our droplet impact experiments was the maximum spreading diameter, Dmax (usually normalized by D0). In Figure 4.1, we show four stages of a droplet impact process: the arrival, impingement, spreading, and retraction of the droplet. The maximum spreading diameter, denoted Dmax, is always deﬁned as the maximum lateral spread of the droplet on the surface. In our experiments, we have attempted to investigate how the spreading process depends on the degree of supercooling of the droplets.

4.2 Experiments

In these experiments, the pressure in the test chamber was always lowered down to 4 mbar, before the syringe pump was used to pump degassed MilliQ water through the nozzles at a relatively slow speed, such that droplets formed and grew on the nozzle tip until they detached. Impacts of the droplets on glass surfaces were recorded using a high-speed camera. In order to make general statements about the inﬂuence of the degree of supercooling on the droplet impact characteristics, we must make the important note that the temperature of the droplet and its impact velocity U0 are not independent: the longer the distance the droplet has to fall

25 (1) arrival (2) impingement

(3) spreading (4) retraction

Dmax

Figure 4.1: Pictures from a high-speed recording of a droplet impact that represent four successive stages of impact: (1) arrival, (2) impingement, (3) spreading, and (4) retraction of the droplet. In this ﬁgure, the time between (1) and (2) and between (2) and (3) is 2.5 ms; the time between (3) and (4) is 18.75 ms. before impact, the more it cools down. Measuring the temperature of the droplets directly proved to be a challenging experiment. In order to get a rough idea of the temperature dependence on falling height, we performed the following test. First, we positioned the thermocouple wire directly underneath the nozzle. Subsequently, we allowed droplets from the nozzle to impact directly onto the tip of the wire. (Usually, though not always, the droplets would freeze upon impact on the wire.) Since the temperature readings could only be taken once per second, we repeated this procedure a number of times in order to get a range of observed temperatures. In Figure 4.2, we provide a plot of the observed temperatures (with error bars representing the range between the extremes of temperature readings observed) versus the impact velocity. The √ 2 latter was calculated as U0 = 2gd, with g = 9.81 m/s and d the vertical distance between the nozzle and the thermocouple wire. (Subsequent high-speed recording revealed this method to be accurate to approximately 0.01 m/s, whether the ambient pressure was atmospheric or 4 mbar, indicating that the droplets were approximately in free-fall.) We note here that, due to the fact that droplets sometimes froze on the thermocouple wire (a process which takes much less than a second), this adds extra uncertainty to these measurements, since there is a near-instantaneous increase in temperature when the freezing happens. Therefore, whether a recorded value indicated a liquid or solid temperature can only be said with certainty in the cases where the droplet did not

26 U0 (m/s) 0.0 1.4 2.0 2.4 Model Experiment −2

−4 C) ◦ (

T −6

−8

−10 0 100 200 300 d (mm)

Figure 4.2: The measured temperature of falling, supercooled droplets of 3 mm diameter in a 4 mbar environment versus their distance from impact (lower axis) and their speed at impact (top axis). The error bars denote the ranges of observed temperatures for each impact speed. The solid line represents the solution to equation (4.8) under the conditions mentioned in section 4.4.

freeze upon impact. Nevertheless, a cooling trend with increased U0, as is to be expected, seems to be clear from Figure 4.2. The measured temperatures for the lowest two velocities are nearly identical. The reason for this is probably that the formation of the droplet at the nozzle tip takes some time, during which the growing droplet already has the chance to cool down. For short falling distances, then, the temperature is likely to be determined mostly by this cooling down on the nozzle tip, rather than the cooling during free-fall (which takes a relatively short time compared to the formation time on the nozzle tip). The time it takes a droplet to fall a distance d is given by t = p2d/g. The average cooling rate that follows from the data points in Figure 4.2 can thus be calculated to be approximately −25◦C/s, much larger than the rates displayed in Figure 3.7. This difference is unsurprising, given that the data come from different experimental conditions; the rates in Figure 3.7 correspond to cooling during a continuous evacuation starting from atmospheric pressure, whereas the data in Figure 3.7 were recorded in a setting where the ambient pressure was already as low as 4 mbar from the start. We have performed droplet impact experiments for different U0 at both atmospheric pressure (i.e. with non-supercooled droplets, at T ≈ 12◦C) and at 4 mbar (supercooled) with a high-speed camera. From the recordings, we tracked the spreading of the droplet during impact as a function of time, enabling us to identify Dmax. In Figure 4.3, we show the evolution of D(t) for a supercooled and a non-supercooled droplet, both with an impact speed of 2.3 m/s. It can be clearly seen that the supercooled droplet spread less far: Dmax = 3.3D0 for the supercooled droplet as compared to 4D0 for the normal one.

27 4 not supercooled supercooled

3 0

D/D 2

1 impact

0 0 0.02 0.04 0.06 0.08 t (s)

Figure 4.3: The measured spreading diameter D(t), normalized by the droplet diameter D0, of a supercooled and a non-supercooled droplet, for U0 = 2.3 m/s.

We hypothesize that the difference in maximum spreading diameter between supercooled and non-supercooled droplets could originate from the temperature-dependence of either the surface tension, or the viscosity of the liquid. It is known from literature that the surface tension [50] and the viscosity [51] of supercooled water are different from non-supercooled water: in the ranges of temperatures observed in our experiments, both quantities increase with decreasing temperature. Therefore, experiments conducted at the same U0 at atmospheric pressure and low-pressure, respectively, are not conducted with the same W e and Re, since both of these quantities are temperature-dependent. (We note here that the density, which enters both W e and Re, is also temperature-dependent; however, in the range of temperatures relevant for the current experiments - say, from 10◦C to −10◦C - the variation in density is close to 0.1% [49], whereas the variation in surface tension lies around 5% [50], and that of the viscosity is close to 50% [51].) We have repeated the experiments for different dropping heights, i.e. for different U0. In Fig- ure 4.4a, we show the measured values of Dmax/D0 versus the impact velocity, for supercooled (red) and non-supercooled (blue) droplets. The latter consistently show a higher spreading diameter than the former.

4.3 Scaling

In Eggers et al. [52] two scaling laws for Dmax are derived on the basis of the dominance of viscous eﬀects and surface tension, respectively. In the former case, it is assumed that the initial kinetic energy of the impacting droplet is completely dissipated by viscosity. The kinetic energy of the 3 2 impacting droplet scales as ρD0U0 , whereas the energy dissipation during the impact scales as 3 η(U0/h)Dmax, where h is the thickness of the ﬂattened drop as it reaches maximum diameter 2 3 Dmax. Along with volume conservation, which implies hDmax ∼ D0, this means that

28 (a) (b) (c) 4 4 4 0 0 3.5 0 3.5 3.5 /D /D /D 3 3 3 max max max D D D 2.5 2.5 2.5

2 2 2 0 1 2 3 5 10 15 4 5 1/5 U0 (m/s) W e1/2 Re

Figure 4.4: The measured maximum spreading diameter Dmax, normalized by the droplet diameter D0, of super- 1/2 1/5 cooled (red) and non-supercooled (blue) droplets, against U0 (a), W e (b) and Re (c). Circles indicate data for non-supercooled droplets; triangles indicate data for supercooled droplets.

D ρD U 1/5 max ∼ 0 0 = Re1/5. (4.3) D0 η Similarly, when we assume that the kinetic energy is purely transformed into surface energy, which 2 scales as σDmax, we get

D ρD U 2 1/2 max ∼ 0 0 = W e1/2. (4.4) D0 σ 1/2 1/5 We can thus also plot the values Dmax/D0 versus W e and Re , to correct for the effects of the temperature-dependent surface tension and viscosity, respectively. The results are plotted in Figures 4.4b and 4.4c. Here, we have used the “calibration plot” in Figure 4.2 to estimate the temperature at which to evaluate σ and η. For the values that we have used, the reader is referred to Appendix B. However, no clear trends emerge from either of these plots. This is likely to reflect the fact that the viscous and surface tension effects do not dominate over one another, but are rather simultaneously at play in determining the dynamics of impact. As argued by Eggers et al. [52] and Laan et al. [53], in practice the dependence of Dmax/D0 on Re and W e has to be gathered into a single scaling law, since in many practical situations, both dimensionless groups W e and Re are in similar orders of magnitude. The approach is then to interpolate between both equations (4.3) and (4.4), which results in the following functionality: D max ∼ Re1/5f(W eRe−2/5), (4.5) D0 where f(x) is a function with the following limits:

lim f(x) ∼ x1/2 (4.6) x→0 in the capillary regime (where W eRe−2/5 goes to zero, because W e Re), and

lim f(x) ∼ 1 (4.7) x→∞ in the viscous regime (where W eRe−2/5 goes to infinity, because Re W e). −1/5 −2/5 Arguing from this position, it makes sense to plot (Dmax/D0)Re against W eRe , as has been done in Figure 4.5. The points should then fall approximately into the same trend as −1/5 1/2 1/2 reported by Laan et al. [53], who fit the function (Dmax/D0)Re ≈ P /(1.24 + P ) with P = W eRe−2/5 to their data points. This fit has been plotted as a dotted line in Figure 4.5. It is seen that our data points are indeed very close to this fit.

29 0.8

0.7 5 / 1 − Re ) 0 0.6 /D max D ( 0.5

0 1 10 10 W eRe−2/5

1/5 −2/5 Figure 4.5: The quantity Dmax/D0 divided by Re is a function of the parameter W eRe , according to equation (4.5). The experimental data for supercooled (red) and non-supercooled (blue) droplets are close to falling onto the ﬁtted trend from Laan et al. [53] (dashed line).

4.4 Modeling

In this section, we give an overview of a simple modeling study used to numerically estimate some of the relevant quantities in droplet evaporation experiments. This approach is based mostly on the modeling by Sellberg et al. [11]. The model is based on Knudsen’s theory of evaporation in a low-pressure environment. In our approach, we model a droplet in an environment with pressure P∞ and at temperature T∞. The droplet itself is modeled as consisting of a single evaporating entity with temperature Ts(t) and radius R(t). The temperature change of the droplet is then given by

dTs(t) (Pvap(Ts) − P∞) As(t)∆Hvap(Ts) = − p . (4.8) dt 2πmkBTs(t) Cp(Ts)ρ(Ts)Vs(t) 2 3 Here, As(t) = 4πR(t) is the surface area of the droplet, and Vs(t) = (4/3)πR(t) is its volume; furthermore, Pvap is the saturation vapour pressure, m is the mass of a molecule of droplet substance (assumed pure), ∆Hvap is the enthalpy of vapourization per molecule, Cp is the speciﬁc heat capacity of droplet substance, and ρ is its density. 1 If appropriate parameterizations for substance properties – namely, ∆Hvap, Cp, Pvap and ρ – are known as a function of temperature, this very simple model can be run to provide an estimation for droplet temperature in a low-pressure environment, such as during evaporative cooling. We take the enthalpy of vapourization from [54]; the speciﬁc heat capacity from [55]; the saturation vapour

1 p The prefactor (Pvap(T ) − P∞)/ 2πmkB Ts(t) represents the flux of molecules of evaporating substance per unit area. This can be seen from dimensional analysis as follows. The average kinetic energy of such a molecule, mv2 with 2 v denoting velocity, is due to thermal activation and therefore assumed to be of the order kB T . Thus, v ∼ kB T/m. The molar flux J through a surface with area A, assuming uniform velocity and normal direction everywhere along the surface, is then proportional to vAn/V , with n/V denoting the density of molecules. Using the ideal gas law, 2 √ V/n = kbT/P , we find J ∼ vAn/V = vAP/(kB T ). Since v ∼ kB T/m, this gives J/A ∼ P/ kB T m.

30 4 x 10 (a) (b) 4.65 4500

4.6 4400 4.55 (J/kg) 4300 (J/kg/K) vap 4.5 p H

C 4200 ∆ 4.45 4.4 4100 −20 0 20 −20 0 20 T (◦C) T (◦C) (c) (d) 30 1000 ) 20 3 998 (mbar) (kg/m

vap 10 ρ 996 P

0 994 −20 0 20 −20 0 20 T (◦C) T (◦C)

Figure 4.6: The representations as used in our modeling approach (equation (4.8)) of a number of properties of water taking the supercooled range into account. Plotted are (a) enthalpy of vaporization; (b) isobaric heat capacity; (c) vapour pressure; (d) density. pressure from [48]; and the density from [56]. Overviews of the parameterizations in numerical form are presented in Appendix C. We plot these parameterizations graphically in Figure 4.6. We can use this model to estimate the temperature of falling supercooled droplets versus their in-flight time. We have run a simulation according to equation (4.8) using a first-order accurate Euler time-stepping scheme. We have used P∞ = 4 mbar (the approximate ambient pressure ◦ during the experiments), Ts(t = 0) = −2 C (the estimated initial temperature of the droplets at the moment of pinch-off, inferred from the temperatures at the lowest values of d in Figure 4.2), an initial droplet diameter of D(t = 0) = D0 ≈ 3 mm (as inferred from camera recordings of falling droplets). The resulting curve Ts(t) is plotted in Figure 4.2 as a solid line. It can be seen that, in terms of the temperature difference between droplets at the lowest and highest investigated impact speeds, the agreement between model and measurements is quite close.

4.5 Summary

We have used our setup to perform experiments on the impact and spreading characteristics of supercooled droplets on a solid surface, and compared these to non-supercooled droplets. We have shown that, when accounting for the temperature dependency of viscosity and surface tension, the impacts of supercooled droplets can be modeled well with the scaling approach by Laan et al. [53].

31 Chapter 5

Supercooled droplet trains

Another interesting application of our experimental setup is the creation of supercooled liquid jets. These could be easily formed by increasing the pumping rate of the syringe pump up to a point where the ﬂow rate was too fast for a sequence of suspended macrodrops to grow on the tip of the nozzle. Instead of macrodrops, a very thin liquid jet was then ejected out of the nozzle opening. Such a jet breaks up into micro-sized droplets after having fallen a certain distance, essentially turning into a rapid vertical stream of supercooled microdrops. Upon impact on a solid surface, it was found that these streams can form vertical pillars (stalagmites) of ice. For this, the surface need not be supercooled; the initial impact on a surface can provide an opportunity for ice nucleation by itself. Once ice has formed at the initial point of impact on the surface, the rest of the jet arriving will impinge on this ice and can easily freeze by itself. The result is a rapidly growing stalagmite of ice. One could call this an “ice printing” process. In this section, we provide the results of experiments that were conducted to investigate some properties of these growing stalagmites, and some factors that inﬂuence their growth.

5.1 Experiments

The components of the setup for jetting experiments were precisely the same as those used for the droplet impact experiments. We typically used pumping rates of 50 mL/hr or 100 mL/hr. With the nozzle opening being 70 µm wide, this resulted in jet ejection speeds of 3.6 m/s and 7.2 m/s, respectively. In Figure 5.1, we show four pictures out of an example recording (125 fps, shutter speed 1/125 s) of an ice stalagmite growing upwards while being fed by an incoming supercooled droplet train ejected at 3.6 m/s. The time between successive pictures is 0.8 seconds. From the distance calibration, we can estimate the growth speed Ustalagmite ≈ 9.2 mm/s. If we now assume the stalagmite, in a simpliﬁed version, to be approximately cylindrical in shape, we can also estimate its average volu- −8 3 3 metric growth to be V˙stalagmite ≈ 1.4 × 10 m /s. Assuming the density of ice to be 916.7 kg/m , −6 this corresponds to a change in mass M˙ stalagmite = 9.5 × 10 kg/s. This mass change is a way of quantifying the eﬃciency of the printing process. To clearly image what exactly happens at the stalagmite tip, where it is met by the incoming jet, more stringent recording settings are needed. We have performed a high-speed recording at 75000 frames per second, with a shutter time of 10−6 seconds, to record the process of impingement of jet droplets on the stalagmite tip. In Figure 5.2, we show four pictures from such a recording that show in more detail what happens. From inspection of the recording, we can state the following. First of all, the microdroplets that arrive at the stalagmite are already spread out quite substantially,

32 1 mm

Figure 5.1: Four example pictures, 0.8 seconds apart in time, showing the growth of an ice stalagmite through continuous emission of a microjet from the nozzle at 50 mL/hr. The broken-up jet itself cannot be distinguished in these pictures. The distance from the nozzle was approximately 50 mm for these pictures.

33 1 2

3 4

Figure 5.2: Four example pictures, 13.3 microseconds apart in time, showing the impact process of a broken-up jet onto the tip of the ice stalagmite that it is forming. In pictures (1) and (2), the circle shows an emitted droplet that “missed” the stalagmite. In picture (2), the arrow shows the beginning splash of a droplet, indicating the presence of a liquid puddle at the very tip of the stalagmite. with some droplets in fact “missing” the stalagmite and ending up somewhere else. However, the majority of droplets do arrive within a spatially confined region that is narrow enough to support the growth of a stalagmite. Furthermore, the droplets do not freeze directly upon impact on the stalagmite; rather, a small puddle exists on top of the stalagmite that freezes at the bottom and receives new liquid at the top from arriving droplets. This results in small-scale splashes that can be distinguished in the high-speed recording. Measuring the thermodynamic characteristics of the jet proved to be a very challenging experiment. In order to probe the temperature of the impinging jet, we performed the following test. First, the syringe pump (set to a pumping rate of 50 mL/hr) was started and jet ejection initiated. Subsequently, the thermocouple wire was moved into the jet, such that an ice stalagmite started being printed onto its tip. This experiment was repeated a number of times; each time, the temperature reading at the moment when stalagmite formation started was recorded. This experiment was done for different distances between thermocouple wire and nozzle, to probe the effect of jetting distance on the temperature of the (broken-up) jet at the point of stalagmite formation. In Figure 5.3, we plot the results as obtained from a number of such experiments, for the two different jetting rates, at different vertical nozzle-wire distances d. Although the spread in the data is quite substantial - up to ±2◦C at the same d - there seems to be a clear cooling trend as d increases, as would be expected. We note, again, that the fact that the thermocouple had a time resolution of one second leads to a definite uncertainly in the measurements, since the process of freezing upon impact happens on much shorter timescales than this. We also note that the temperature plots can not be said to represent the temperature of the stalagmites in any way, since these will continuously cool through sublimation in the low-pressure environment. We can only speak of the droplet temperature upon impact; there is no “stalagmite temperature”. The reason why no larger values of d have been included in this plot is a practical problem. As the distance from the nozzle to the point of jet impact increases (past the point of jet breakup), the jet fans out more and more in the lateral dimensions. The consequence of this is that, if d is too large, it becomes impossible to create narrow stalagmites due to the droplets constituting the broken-up jet no longer being confined to a small lateral extent. Instead of a stalagmite, the result

34 (a) 50 mL/hr (b) 100 mL/hr 0 0 −2 −2 −4 −4 C) C)

◦ −6 ◦

( ( −6

T −8 T −8 −10 −10 −12 −12 0 50 100 150 0 100 200 300 d (mm) d (mm)

Figure 5.3: The measured temperature of impacting, supercooled jets, emitted at 3.6 m/s (a) and 7.2 m/s (b) from a 70 µm diameter nozzle, at different distances from the nozzle. is a spread-out snowy layer. Thus, with the current type of setup, this printing technique can only be employed in a certain vertical distance range. We have made an attempt to measure the influence of d on the printing speed Ustalagmite and printed mass M˙ stalagmite. This is interesting to investigate since there seem to be three factors influencing the efficiency of printing. Firstly, as mentioned, there is the lateral spreading of the jet, leading to lower printed mass for larger d. Secondly, the temperature measurements (Figure 5.3) indicate that a larger distance d means that the droplets are colder upon impact, which increases the chance that they will nucleate and freeze on top of the pillar. Thirdly and lastly, as seen in the high-speed recordings of Figure 5.2, the impact of droplets on the stalagmite tip can happen in the form of splashes. In that case, a larger distance would imply a tendency to more violent splashing, leading to a higher loss of liquid. Thus, the effect of increasing d is likely to be determined by a combination of these three factors. Furthermore, we also expect the jet ejection speed to play a nontrivial role, since a higher speed will both mean a different lateral spreading process, a different average droplet temperature for the same d (since the droplets have had less time to cool in the underpressure environment), and a changed effect of splashing intensity on d. Thus, all three factors mentioned above are also affected by the pumping speed with which the printing is performed. In Figure 5.4 we plot the vertical growth speed of stalagmites for the two different jet ejection rates at 50 mL/hr (a) and 100 mL/hr (b). In both cases, it can be clearly seen that the growth speed decreases sharply with increasing d. The dotted vertical lines in the plot indicate the minimum distance required for a stalagmite to grow. For smaller d, it appears from our recordings that the jet is not yet broken-up, meaning impingement of a continuous jet, which does not support the growth of a stalagmite. Note also that using the higher jet ejection rate, pillars can be grown at larger distances from the nozzle. In Figure 5.5, we plot the measured average diameter Dstalagmite as a function of d for the two different jet ejection rates. It appears that the stalagmites become thicker for increasing d, probably due to an increased spreading of the jet, although this trend is much more pronounced for the higher ejection rate. The question now is how the decrease of growth speed, and increase of diameter, with d in combination reflect the “efficiency” of printing. We plot the amount of printed substance M˙ stalagmite for both jet ejection rates in Figure 5.6. Interestingly, it appears as if there is an optimum distance from the nozzle at which most liquid is captured by the growing stalagmite. This trend is much more pronounced for the higher ejection

35 (a) 50 mL/hr (b) 100 mL/hr 20 15

15 10 (mm/s) (mm/s) 10

5 stalagmite stalagmite 5 U U

0 0 0 50 100 150 0 100 200 300 d (mm) d (mm)

Figure 5.4: The measured growth speed of the stalagmites created at a jet ejection speed of (a) 50 mL/hr and (b) 100 mL/hr, for diﬀerent nozzle-stalagmite distances d.

(a) 50 mL/hr (b) 100 mL/hr 2.5 2.5

2 2 (mm) (mm) 1.5

1.5 stalagmite stalagmite

D 1 D

0.5 1 0 50 100 150 0 100 200 300 d (mm) d (mm)

Figure 5.5: The measured average diameter of the stalagmites created at a jet ejection speed of (a) 50 mL/hr and (b) 100 mL/hr, for diﬀerent nozzle-stalagmite distances d. rate. This means that, with this method of printing, one could speak of an “optimal distance” at which to print. We assume that this is the case for a wide range of jet ejection speeds.

5.2 Printing structures

By manually moving the bottom surface while the jetting continues, one can “print” curved ice stalagmites, a collection of short ice pillars, or potentially even pre-deﬁned patterns or shapes. An example of a collection of ice pillars created in such a way is displayed in the photo in Figure 5.7.

36 −5 −5 x 10 (a) 50 mL/hr x 10 (b) 100 mL/hr 2.5 5

2 4 (kg/s) (kg/s) 1.5 3 stalagmite stalagmite

˙ 1 ˙ 2 M M

0.5 1 0 100 200 0 100 200 300 d (mm) d (mm)

Figure 5.6: The measured average amount of printed ice per unit time (in kg/s) of the stalagmites created at a jet ejection speed of (a) 50 mL/hr and (b) 100 mL/hr, for diﬀerent nozzle-stalagmite distances d.

Figure 5.7: By moving the bottom surface during the ejection of a supercooled jet, structures such as this one can be made.

37 5.3 Summary

We have performed exploratory experiments that are basically a variation of those on droplet impact described in chapter 4. Using a slightly higher pumping rate, the same setup can be used to create jets which break up into micrometer-sized droplets that supercool during their fall. Such supercooled droplet streams can be used to print ice pillars. Our investigations indicate that several factors inﬂuence the total amount of ice that can be printed, with an optimum distance (in terms of total printed mass per unit time) from nozzle to printing site emerging. Potentially, techniques such as this could have more advanced applications in the printing of solid 3D structures in general.

38 Chapter 6

Shattering of supercooled droplets

As explained in section 3.3.1, it is the surface of supercooled droplets that freezes ﬁrst. We have conducted experiments to investigate in more detail what happens after the surface of a supercooled droplet freezes over and forms a solid shell. The experiments in section 3.3.1 were conducted with suspended droplets. The suspension of the droplets aﬀects their shape and therefore also likely the exact way and shape in which they freeze over. In order to mimic conditions in nature (i.e. in clouds) in which droplets undergo supercooling, we have performed experiments in which millimeter-sized droplets on a hydrophobic substrate are supercooled inside the vacuum chamber. The hydrophobicity of the substrate ensured that the droplets had a large contact angle and would freeze over in a near-spherical shape. We used the same hydrophobic substrates with candle soot as described in chapter 3.

6.1 Stages

We found that, when supercooling a droplet on such a hydrophobic surface, three processes always happened in short succession. Firstly, the droplets would ice over in a near-spherical shape and form a frozen “marble”. Subsequently, the bulk of the droplet would start to freeze, and a time span in the order of seconds to tens of seconds afterwards, the droplets were usually seen to shatter into two or more pieces. In the following paragraphs, we will give a more detailed analysis of these observed processes.

6.1.1 Icing In Figure 6.1, we plot a number of frames from a high-speed recording of the icing process of a suspended drop. In (a), the droplet is shown shortly before the freezing sets in. In (b) to (d), the droplet surface is in the process of freezing over; in (e) the freezing is near-complete; and in (f) a small amount of liquid has been ejected from the interior of the droplet, as indicated by the arrow (similar to the process seen in Figure 3.5). Most of the droplets that we froze in this way exhibited a similar kind of extremity, caused by the extrusion of liquid shortly after the surface freezing completed.

6.1.2 Freezing The process by which the bulk of the iced-over droplet freezes can take a time in the order of seconds. As mentioned, we observed that the frozen droplets would shatter into pieces after a couple of seconds. We assume that the physics behind this process is that the gradual freezing over of the droplet from the outside creates large inner stresses due to the density of ice being lower than that of water, eventually causing the droplet to crack open and blow apart.

39 We denote the time between the freezing over of the surface of the droplet and the moment of shattering as the lifetime τlife of the droplet. This lifetime was found to depend on the initial size of the droplet. We have performed a number of experiments to investigate the dependence of τlife on the initial radius Ro of the droplet. Results are given in Figure 6.2. This data shows an approximately quadratic trend of τlife with respect to Ro in the investigated range. We need to take into account the fact that larger droplets tend to sink in under their own weight more than small droplets. This means that larger droplets will have a flatter initial shape than smaller droplets, which is likely to affect the freezing process and thus τlife . The shape of the droplet can be quantified through the aspect ratio H/D, where H is the height of the droplet and D its diameter (measured in the horizontal direction). As can be seen in Figure 6.3, while the small droplets are approximately spherical (aspect ratio of close to 1.00), the large droplets are considerably flatter and it is unlikely that one can model them as “spherical”.

6.1.3 Shattering In all observed cases, the frozen droplet shattered, cracked open, or exploded after the expiration of τlife . In most cases, the frozen droplet was seen to shatter into two large pieces (as well as a number of shards flying in all directions, usually). In Figure 6.4, we show a number of frames from a high-speed recording of such a shattering process. We can use such a recording to make an estimate of the energy released during such an explosion. While such a calculation will be very rough, it can give an indication of the order of magnitude of the energies involved. The estimation is done as follows. We see in Figure 6.4 that, after the shattering, the two largest pieces undergo both translation and rotation. Therefore, the energy imparted to those pieces by the explosion is divided between kinetic energy and rotational energy. We have made a schematic drawing in Figure 6.5 to indicate the movements involved in a simplified manner. In the figure, m1 and m2 denote the mass of the two large pieces, v1 and v2 their translational velocities, and ω1 and ω2 their rotation speeds. We denote the mass of the frozen droplet (before shattering) by M = m1 + m2. From the recording in Figure 6.4, we estimate the velocity v1 ≈ 0.5 m/s and v2 ≈ 0.8 m/s. From momentum conservation m1v1 + m2v2 ≈ 0, we can estimate m1 ≈ M/(1 + v1/v2) and m2 ≈ M/(1 + v2/v1). Thus, we can estimate the total kinetic energy

(a) t = 0 s (b) t = 0.005 s (c) t = 0.01 s

(d) t = 0.02 s (e) t = 0.03 s (f) t = 0.04 s

Figure 6.1: A sequence of pictures showing the icing over of a supercooled droplet. The droplet in the liquid state had a radius of 2.0 mm.

40 30

20 (s) 15 life τ

0 0 1 2 3 4 5 6 Ro (mm)

Figure 6.2: The experimentally observed lifetime τlife of frozen supercooled droplets versus their initial radius Ro. The blue line corresponds to a quadratic ﬁt through the data points.

1 D 0.9 0.8 H 0.7 H/D 0.6 0.5 2 4 6 D/2 (mm)

Figure 6.3: The aspect ratio of droplets on the hydrophobic substrate versus their radius. The picture on the right illustrates the calculation of the aspect ratio.

41 (a) t = 0 s (b) t = 0.002 s

(e) t = 0.008 s (f) t = 0.01 s

Figure 6.4: A sequence of pictures showing the shattering of a frozen supercooled droplet into two large pieces. The initial droplet radius was 1.2 mm.

2 ω1 ω v1 v2

m2 m1

Figure 6.5: A schematic illustrating the parameters used to calculate the translational and rotational energies of the shattered droplet pieces.

42 1 E ≈ m v2 + m v2 kin 2 1 1 2 2 1 M 2 M 2 ≈ v1 + v2 2 1 + v1/v2 1 + v2/v1 M = v v , 2 1 2 from which it follows that Ekin/M ≈ 0.2 J/kg. 2 2 Furthermore, we can estimate the rotational speed ω1 ≈ 2 × 10 rad/s and ω2 ≈ 5 × 10 rad/s. Inferring the moments of inertia I1 and I2 about the respective axes of rotation is not easy from 2 2 the recording, so we make the order-of-magnitude estimate I1 ≈ m1R and I2 ≈ m2R . Then, the total rotational energy is estimated as

1 E ≈ I ω2 + I ω2 rot 2 1 1 2 2 1 M 2 2 M 2 2 ≈ R ω1 + R ω2 2 1 + v1/v2 1 + v2/v1 MR2 v ω2 + v ω2 = 1 2 2 1 , 2 v1 + v2 from which it follows that Erot/M ≈ 0.1 J/kg. Thus, kinetic and rotational energies are in the same order of magnitude, and the total energy released is also in the order of tenths of a Joule per kilogram.

6.1.4 Aftermath By observing the pieces of the shattered frozen droplet, we have tried to infer small amounts of additional information about the freezing and explosion processes. In particular, since droplets were usually seen to explode into two large pieces, each near half the original droplet in size, we took camera pictures of the cross-section of such pieces to obtain some insight on how the inward freezing process happened. We used a Nikon D300 digital camera in order to take such close-up pictures through the glass in the sidewalls of the vacuum chamber, where necessary via a mirror mounted on the movable deposition plate with double-sided tape. In Figure 6.6, we present a number of example pictures of the cross-section of such halves of frozen droplets, revealing a range of both similarities and differences. In (a) and (c), it appears that a substantial chuck in the middle of the drop is missing, thus essentially leaving a thick hemispherical shell. In (b), we see what looks like a pattern of concentric circles with different radii on the cross-section. In (d), we see a piece that seem to have a rather smooth ice surface at large radii, but a very rough one nearer the center. In (e) and (f), we see two pieces of the same droplet where, next to a similar rough patch near the center as in (d), a crack had permeated across the entire droplet diameter. We theorize that these rough “inclusions” observed in the photos might be caused by the droplets exploding before the freezing process was entirely completed. In that case, as soon as an explosion is initiated, the remaining liquid would very quickly freeze upon being exposed, and either be blown away in one or more separate pieces (as might have happened in the case of (a) or (c)) or freeze in a much rougher structure as part of the larger pieces (as might have happened in the cases of (d), (e) and (f)). However, as the “onion-ring” structure in (b) would seem to suggest, this need not necessarily happen in all cases. In section 6.2.2, we present two theoretical approaches to find an explanation for why a frozen droplet might crack open and explode before having frozen completely to the core.

43 (a) (b)

(e) (f)

Figure 6.6: Photos of the cross-sections of several pieces of shattered supercooled droplets.

44 Ro

Tice

Twater

Ri(t) ΔR(t)

Figure 6.7: A schematic illustrating the model described in section 6.2.1.

6.2 Modeling

In this section, we present a numerical model of the freezing process of a sessile supercooled droplet and discuss its results in comparison to the experimental data. We also present two hypotheses on the causes of the shattering of the frozen droplets, as well as another model for the initial stages of the explosion itself.

6.2.1 Frozen droplet lifetime Here, we develop a theoretical framework to model the freezing process of a supercooled droplet. We represent the droplet as a spherical shell of outer radius Ro and inner radius Ri consisting of ice, filled by water. We assume that the temperature difference ∆T = Twater − Tice > 0 between the inner and outer side of the shell drives a heat flux from the liquid inside the droplet ◦ to the outside. The temperature Tice is assumed to be a number of degrees below 0 C (see, for example, Figure 3.4). The temperature of the water in contact with the ice shell is assumed to be ◦ Twater = 0 C. A schematic drawing of this model setup is given in Figure 6.7. The temperature gradient in the ice shell is then approximately dT ∆T ≈ , (6.1) dr ∆R(t) with ∆R(t) = Ro − Ri(t). The associated thermal energy change, according to Fourier’s law of conduction applied to a spherical shell, is then given by dE ∆T ≈ 4πR (t)R κ , (6.2) dt i o ∆R(t) where κ is the thermal conductivity of ice. Now, the amount of energy loss needed to freeze enough supercooled water to decrease the inner radius of the ice shell at a rate dRi/dt is given by dE 2 dRi = Lf ρice4πRi , (6.3) dt phase dt where Lf is the latent heat of freezing. Assuming that all lost energy contributes to the freezing of supercooled water inside the shell, it follows from equations (6.2) and (6.3) that dR κ R ∆T i = − o . (6.4) dt Lf ρice Ri(t) Ro − Ri(t)

45 −3 x 10 4 Ro = 1 mm Ro = 2 mm R 3 o = 3 mm Ro = 4 mm

(m) 2 i R

0 0 10 20 30 40 t (s)

Figure 6.8: Numerical solution of equation (6.4) for the material values indicated in the text, for a number of diﬀerent droplet radii Ro.

3 We have numerically solved equation (6.4) using κ = 2.22 W/m/K, ρice = 916.7 kg/m and 5 ◦ Lf = 3.34 × 10 J/kg, for a number of different values of Ro and for ∆T = 10 C. Results are plotted in Figure 6.8. We define the lifetime τlife of a freezing droplet as the time until the inner radius reaches zero, when the built-up internal strains and stresses are assumed to lead to shattering of the droplet, as experimentally observed. In Figure 6.9, we plot τlife against the radius Ro for a number of different values of ∆T on a double logarithmic scale. 2 According to our model, τlife scales as Ro, which can be shown as follows. We nondimensionalize ρ L equation (6.4) by defining the time scale T = ice f R2, and the dimensionless parameters t0 = t/T κ∆T o 0 and R = Ri/Ro. Rewriting equation (6.4) results in dR0 1 = − , dt0 R0(1 − R0) which can be solved analytically by seperation of variables, subsequent integration, and using the initial condition R0(0) = 1. This yields the implicit relationship 1 1 1 R03 − R02 + = t, 3 2 6 from which it follows that R0(t0 = 1/6) = 0. Therefore, 1 ρ L τ = ice f R2, (6.5) life 6 κ∆T o proving the quadratic dependence of droplet lifetime on initial radius in this model.

46 2 10 ∆T = 10 ◦C ∆T = 15 ◦C ∆T = 20 ◦C Experimental

1 10 (s) life τ

0 10

0.3 0.5 1 2 3 5 Ro (mm)

Figure 6.9: The droplet lifetime τlife versus droplet radius Ro, for three different temperature differences ∆T (corresponding to different outer shell temperatures, since the temperature at the inner shell is assumed to be zero degrees Celsius), on a log-log scale. The dots with error bars are the experimental data points already displayed in Figure 6.2.

It can be observed in Figure 6.9 that, while the experimental and numerical data predict the same order of magnitude of τlife , the experimental data do not follow a power-law scaling with exponent 2. Instead, a power-law ﬁt to the experimental data would give an exponent of approximately 1.35.

6.2.2 Frozen droplet shattering The model setup described in section 6.2.1 implies that internal strains are developing in the freezing droplet. In this section, we present two hypotheses on how such strains are developed and how they could be connected to droplet shattering.

Ice contraction model One hypothesis is based on the fact that the density of ice is lower than that of water (at 0◦C). Assuming that the outer ice shell on the droplet is rigid, this means that the freezing ice will tend to compress the leftover water by expanding inwards. In this section, we model how this can cause strains and a buildup of deformation energy inside the droplet, and why this would result in mechanical means of energy release (i.e. through cracks) in the droplet during the freezing process. The buildup of internal stress is due to the bulk compressibility of water. This is quantified by the bulk modulus K, which for water is approximately 2.2 × 109 N/m2. We assume that the ice exerts a normal (radially inward) force on the water. The bulk strain V is defined as the relative ∂V change in volume due to the compression of water: V = V , where ∂V denotes an infinitesimal 3 change in volume, and V = (4/3)πRi is the initial volume of the water. We can calculate the bulk strain as follows. Assume that an infinitesimal amount of water inside 2 the shell of ice, with mass ∂m = 4πρwaterRi ∂Ri and taking up a volume of ∂Vwater = ∂m/ρwater, freezes onto the shell. After freezing, it will take up a volume of ∂Vice = ∂m/ρice. Thus, the net change in volume ∂V of the amount of substance with mass ∂m is then

47 1 1 ∂V = ∂Vice − ∂Vwater = ∂m − , ρice ρwater which means that the bulk strain can be calculated as ∂Vice − ∂Vwater ρwater ∂Ri ∂Ri V = = 3 − 1 ≡ 3β . V ρice Ri Ri 3 3 3 In this case, ρwater ≈ 10 kg/m and ρice ≈ 916.2 kg/m , giving β ≈ 0.09. The pressure change ∂P on the remaining water inside the shell due to this compression eﬀect is then given by ∂P = KV . Through partial integration from Ro to Ri, it is readily seen that the pressure as function of the inner radius Ri is then given by

Ro 8 Ro P (Ro) − P (Ri) ≈ −P (Ri) = 3Kβ ln ≈ 6 × 10 ln . (6.6) Ri Ri

The compression force on the water is then given by F (Ri) = −P (Ri)A(Ri) (directed radially inwards), where A(Ri) denotes the surface area of the water in contact with the ice shell, 2 A(Ri) = 4πRi . The total work done through compression Wtot, assuming the droplet freezes completely, is equal to the total stored compression energy, and is given by

Z 0 Z Ro 2 Ro Wtot = F (Ri)∂Ri = 12πKβRi ln ∂Ri Ro 0 Ri 4 R Ri=Ro = πKβR3 3 ln o + 1 3 i R i Ri=0 4 = πKβR3. 3 o In Figure 6.10, we show the total compression energy versus the initial radius of the droplet according to this model. 3 The energy density u in a completely frozen droplet, equal to Wtot/M (with M = (4/3)πRoρwater), is then independent of the droplet’s initial radius, and is given by

Kβ 1 1 u = = K − ≈ 2 × 105 J/kg. ρwater ρice ρwater By comparison, the energy density of TNT is only about a factor 10 higher. No frozen droplet shell can withstand a stress of close to 6×108 Pa (as in the beginning stages of freezing according to equation (6.6), when Ri ≈ Ro). The tensile strength of ice, according to data available in the literature, is in the order of 1 MPa [57]; therefore, already in the beginning stages of the freezing (when the shell is very thin), we assume that the high stress results in cracks and faults in the ice and in local expulsions of liquid through the frozen shell. This process continues as the freezing proceeds. Thus, we assume that the energy does not build up as the droplet freezes, but is instead continuously released through mechanical faults and liquid extrusions. This explains why the kinetic energies calculated in section 6.1.3 are orders of magnitude lower than the value of 2 ×105 J/kg calculated in this section.1 Why do the frozen droplets still explode, as experimentally observed, if the energy is continuously released? Our freezing model predicts that the decrease of the inner radius happens fastest during the end of the freezing process (see Figure 6.8). Could it be that the freezing in this ﬁnal stage

1In hindsight, this justiﬁes the fact that the freezing temperature of water was not adapted according to its pressure, using the Clausius-Clapeyron relation, in the model in section 6.2.1.

48 2 10

1 10

(J) 0 10 tot W

−1 10

−2 10 0 1 2 3 4 5 Ro (mm)

Figure 6.10: The work done by compression, equal to the total compressional energy stored in the ice, versus the initial radius of the droplet Ro, according to equation (6.7). happens so fast that the compression energy cannot be released fast enough mechanically compared to the speed at which it builds up, resulting in a ﬁnal explosion? Let us assume that this “fast freezing” starts when the inner radius has reached a value denoted Rf , with Rf Ro. We assume that all energy resulting from prior compression was already mechanically released. In that case, analogously to equation (6.7), the energy built up during the ﬁnal freezing stage is given by 4 W = πKβR3 , f 3 f meaning that the energy density is given by

W R 3 1 1 u = f = f K − . M Ro ρice ρwater Assuming that all the compression energy is converted into rotational and translational kinetic energy after the explosion, with u ∼ 0.1 J/kg as inferred in section 6.1.3, this implies Rf ∼ 10 µm for Ro ∼ 1 mm. The energy leading to explosion would therefore be built up during the freezing of a ﬁnal liquid core in the frozen droplet, with a radius in the order of micrometers. However, this order of magnitude is so small that it could not explain the sizes of the rough inclusions in the droplet pieces after shattering, observed in the photos in Figure 6.6.

Porosity model Another hypothesis is based on the assumption that the ice shell is porous. It has been repeatedly observed how, after the initial freezing of the surface of a supercooled droplet, liquid from the inside is extruded through the shell and freezes in the outside environment (see, for example, Figure 6.1). This could imply that the ice is permeable to some degree. A pressure drop over the ice shell would then be the cause of such an extrusion of liquid. We attempt here to estimate such a pressure drop, starting from Darcy’s law Π dp q” = − , (6.7) µ dr

49 where Π is the permeability of the ice, dp/dr is the pressure gradient across the ice shell, and q” is the ﬂux, or net discharge per unit area (units of m/s), of liquid from the inside of the droplet through the porous shell. The total discharge of liquid through the shell Q (in m3/s) should be independent of the radius; therefore, the ﬂux will be given by Q q” = . (6.8) 4πr2 Equating (6.7) and (6.8), we obtain the pressure gradient dp Qµ 1 = − . dr 4πΠ r2 Integrating from the inner to the outer radius, this leads to the total pressure drop Qµ R − R ∆p = o i . (6.9) 4πΠ RoRi Now, the total liquid discharge Q is equal to the temporal change of volume of the liquid core with radius Ri(t):

dV 2 dRi Q = = 4πRi(t) . dt dt Using the differential equation (6.4) obtained from our previous model on the droplet lifetime, we can express dRi/dt in terms of Ri and Ro, and obtain R R (t) κ∆T Q = 4π o i . Ro − Ri(t) Lf ρice Inserting this expression back into (6.9), we obtain κ∆T µ ∆p = , (6.10) Lf ρiceΠ implying that the excess pressure in the liquid core is independent of Ri. Using the values mentioned in section 6.2.1 and η = 1.3 × 10−3 kg/(ms), we find 10−10 N ∆p ≈ . Π The fact that frozen droplets develop cracks during their freezing indicates that the pressure on the ice shell is larger than the tensile stress of ice, which is in the order of 1 MPa, as mentioned. In order for ∆p to be in the order of this tensile stress, the permeability would have to be at most Π ≈ 10−16 m2. This is in the same order as the typical permeability of limestone [58]. Whether or not this represents a realistic value for the ice droplets with which we are concerned here is unknown. Measurements of the permeability of sea ice have yielded values of Π that are orders of magnitude higher [59]; however, sea ice permeability is likely affected greatly by the presence of brine channels, in view of which the large discrepancy between the permeability of sea ice and that of supercooled ice inferred from our simple model may be unsurprising. In the following, we estimate how this pressure would induce elastic deformation of the porous ice shell and what elastic energies are thereby stored. We start by using the elastic constitutive equations and stress-strain relations in a solid shell. We assume that the displacement field is radial, ~u = urˆr, and that the stresses only depend on the radial coordinate r. The stresses σrr and σθθ = σφφ, according to the stress-strain relations, are given by

E σ = [(1 − ν) + ν + ν ]; rr (1 + ν)(1 − 2ν) rr θθ φφ E σ = [ν + ] , θθ (1 + ν)(1 − 2ν) rr θθ

50 where E and ν are the Young’s modulus and Poisson’s ratio, respectively. The strains rr and θθ = φφ are given by ∂u = r ; (6.11) rr ∂r u = r . (6.12) θθ r The stresses can then be rewritten as follows: ∂u u σ = (2˜µ + λ) r + 2λ r ; (6.13) rr ∂r r ∂u u σ = λ r + 2(λ +µ ˜) r . (6.14) θθ ∂r r Here,µ ˜ and λ are the ﬁrst and second Lam´econstants, respectively, given by E µ˜ = ; 2(1 + ν) νE λ = . (1 + ν)(1 − 2ν)

Now, from the equilibrium equations, it is given that ∂σ 2 rr + (σ − σ ) = 0. ∂r r rr θθ Inserting equations (6.13) and (6.14), this results in

∂2u 2 ∂u 2 ∂ 1 ∂(r2u ) r + r − u = r = 0. ∂r2 r ∂r r2 r ∂r r2 ∂r The solution for the displacement ﬁeld is then B u (r) = Ar + , (6.15) r r2 where A and B are constants to be determined from the boundary conditions. Inserting equation (6.15) back into equation (6.13) results in

2B σ = 3λA + 2˜µ A − . (6.16) rr r3

The boundary conditions at the inner and outer radius of the shell are σrr(Ri) = −∆p and σrr(Ro) = 0, respectively. Then, A and B are found to be as follows:

3 ∆p Ri A = 3 3 ; 3λ + 2˜µ Ro − Ri 3 3 ∆p RoRi B = 3 3 . 4˜µ Ro − Ri Similarly, inserting equation (6.15) back into (6.14) results in

2˜µB σ = σ = (3λ + 2˜µ)A + . (6.17) θθ φφ r3

51 5 10

0 10 (J/kg) u

−5 10 0 0.5 1 Ri/Ro

3 Figure 6.11: The elastic energy per unit mass (u = Eel/M, with M = (4/3)πRoρwater) versus Ri/Ro according to equation (6.18). The horizontal lines denote the range 0.1 J/kg – 1 J/kg.

The total elastic energy Eel is then given by [60, 61] 1 Z Eel = (σrrrr + σθθθθ + σφφφφ) ∂V 2 V Z 1 ∂u ur = σrr + 2σθθ ∂V. 2 V ∂r r

Filling in the expressions for σrr (6.16), σθθ (6.17), and ur (6.15), and using the expression in equation (6.10) resulting from the porous-ice model for the pressure drop ∆p, we ﬁnally arrive at the following expression:

2 3 3 3 κ∆T µ Ri Ro Ri Eel = 2π + 3 3 . (6.18) Lf ρiceΠ 3λ + 2˜µ 4˜µ (Ro − Ri )

In Figure 6.11, we plot the energy per unit mass u = Eel/M according to equation (6.18) as a −16 2 ◦ 6 function of the normalized inner radius Ri/Ro (using Π = 10 m , ∆T = 10 C,µ ˜ = 6.57×10 Pa, and λ = 3.38×106 Pa). The horizontal lines in the ﬁgure denote the range 0.1 J/kg – 1 J/kg, which is the order of magnitude of the energy densities that were estimated from the camera recordings in section 6.1.3. Assuming that the droplet explosion is related to the release of the built-up elastic energy, this would imply that the explosion happens when the droplet diameter is approximately 20% of Ro. This is much closer in order of magnitude to the sizes of the rough inclusions seen in Figure 6.6. It could be that at higher Ri/Ro, during the initial stages of the freezing, the energy is released manifested in cracks appearing across the surface of the frozen droplet. Possibly, at some point during the freezing process, the cracking is no longer suﬃcient to release surplus elastic energy, thus leaving an explosion as the next option. At which Ri/Ro this happens probably depends on how exactly the droplet froze over (e.g. where did the freezing start, how symmetrically did the droplet freeze, etc.), and is thus determined to a large extent by factors uncontrollable by the experimentator. However, typical sizes of the rough inclusions in Figure 6.6 are at least in the same order (10% of Ro) as would be predicted by equation (6.18) assuming that the explosion happens through a conversion of elastic energy to kinetic energy, with an order of magnitude inferred from the camera recordings in section 6.1.3.

52 6.2.3 Explosion In this section, we present a mathematical model aiming to provide insight into the energy of the actual explosion. We start by assuming that every explosion is initiated by the formation of a single crack across the entire diameter of a droplet with a thick solid shell and a liquid inclusion in its core. The two halves are assumed to be subsequently blown to either side. In this model, we only consider the dynamics on extremely short timescales after the separation of the two halves. A schematic picture of the model settings is given in Figure 6.12. We denote the separation between the two halves by ξ(t) = 2x(t), as indicated. Assuming that the halves separate symmet- 4 3 rically and that the liquid inclusion was spherical (with radius Ri and volume V = 3 πRi ), the 2 separation opens up an additional cylindrical volume of ∆V (t) = ξ(t)πRi for the liquid to take up. The cross-section of this volume is indicated as the shaded region in Figure 6.12. We assume that the pressure P (t) inside the liquid inclusion in the frozen droplet is given by P (t) = P0 − Pdrop(t), with P0 the pressure just before explosion, and Pdrop(t) the drop in pressure due to the expansion of the liquid into the additional volume ∆V . The drop in pressure is then given by

2 ∆V (t) ξ(t)πRi Pdrop(t) = K = K 3 V (4/3)πRi 3 x(t) = K, 2 Ri with K the bulk modulus of water. Then, Newton’s second law (force equals mass times accelera- tion) gives

M ∂2x P (t)A = , 2 ∂t2 2 with the surface area A = πRi and M/2 the mass of each half of the droplet. Inserting P (t) leads to the following diﬀerential equation:

∂2x 3πR K 2πR2P + i x(t) − i 0 = 0. (6.19) ∂t2 M M With initial conditions x(0) = 0 and x0(0) = v(0) = 0, the solution to equation (6.19) is

x(t)

M M 2 2

(t)

Figure 6.12: A schematic picture of the assumed initial stage of explosion used in the model in section 6.2.3. The droplet is assumed to break into two equal halves and separate symmetrically.

53 0.06

0.04 (J/kg) /M kin

E 0.02 = u

0 0 0.5 1 Ri/Ro

Figure 6.13: A plot of the maximal kinetic energy per unit mass as a function of Ri/Ro according to equation (6.22).

" r !# 2 R P 3πR K x(t) = i o 1 − cos i t . 3 K M

From this, we ﬁnd the maximum velocity vexplosion ≡ max[v] to be r 2 R P 3πR K v = i 0 i . (6.20) explosion 3 K M Thus, the associated maximum kinetic energy of the explosion (i.e. of both halves) is given by 1 E = Mv2 kin 2 explosion 2 P 2R3 = π 0 i 3 K 1 VP 2 = 0 . 2 K The total kinetic energy per unit mass of the initial droplet is then given by

E E 1 P 2 R 3 u = kin = kin = 0 i . (6.21) 4 3 M 3 πRoρwater 2 Kρwater Ro We can slightly adapt the results from the previously-used porosity model (section 6.2.2) to estimate P0 as a function of Ri. We make the assumption that, at the moment of explosion, the azimuthal 2 stress σθθ is equal to the tensile strength of ice , denoted σf . Solving equation (6.17) for ∆p then results in 2Note the diﬀerences between these two models: in the porosity model, the assumption was that the cracking would be caused by the total pressure drop over the shell being equal to the tensile strength of ice. Thus, in that model, ∆p was assumed ﬁxed and equal to the tensile strength of ice, determining σθθ, whereas we now assume σθθ to be equal to the tensile strength, determining P0.

54  R 3  1 − i  Ro  ∆p(Ri) = σf   .  1 R 3   + i  2 Ro

We use this ∆p(Ri) for P0 in equation (6.21); the expression becomes

 3 2 2 3 1 − Ri 1 σ R Ro u = f i   . (6.22) 2 Kρ R  3  water o 1 + Ri 2 Ro With K = 2.2 × 109 Pa, the results from equation (6.22) are as displayed in Figure 6.13. Although the quantitative values of u are typically an order of magnitude smaller than those inferred from the recordings in section 6.1.3, there is a peak in the energy density (rather close to u√∼ 0.1 J/kg) as 1/3 Ri decreases from Ro to 0, at a value of Ri/Ro ≈ 0.6 (the exact value is Ri/Ro = [( 33 − 5)/4] ). If the peak in the energy density is a trigger for the explosion, this model would explain why the radii of the inclusions observed in Figure 6.6 are also in the order of 10% of Ro. We shortly synthesize the results from this section with the hypothesis presented in section 6.2.2. The “big picture” would be that a buildup of strain inside the droplet shell causes the buildup of elastic energy in the shell, as well as the formation of small-scale cracks in the shell, eventually followed by a crack permeating across the entire droplet diameter. The elastic energy is converted into kinetic energy in the explosion, which gives the unfrozen liquid in the core the chance to expand. The energy calculated in equation 6.22 thus represents the (negative) work done by the liquid upon expansion. At its maximum, when Ri/Ro ≈ 0.6, this work is not far in magnitude from the experimentally observed kinetic energies with which the droplet halves ﬂy apart after an explosion.

55 Chapter 7

Conclusion and discussion

In this chapter, we summarize the most important results and insights obtained from this study into the behaviour of supercooled droplets, and provide a number of recommendations for future research in the same area.

7.1 Conclusion

In this thesis, I present experimental and numerical results from various studies on the behaviour of supercooled droplets. Diﬀerent experimental techniques have been used to create supercooled macro- as well as microdrops using the principle of evaporative cooling, and experiments have been performed in which their size, impact speed and temperature could be varied and measured. The main ﬁndings presented in this thesis can be summarized as follows:

We have designed an experimental setup that allows for the controlled creation of supercooled droplets with temperatures of as low as ≈ −18◦C, supports experiments on impact and jetting of supercooled droplets, is compatible with high-speed recording equipment, and allows for a variety of experimental investigations on supercooled droplets in a ﬂexible way. The setup can be used to create supercooled droplets with approximately constant temperatures for times in the order of minutes or longer.

We have compared the characteristics of impacting supercooled droplets with those of non- supercooled droplets. We have conﬁrmed that, when correcting for the temperature dependency of viscosity (and, to a lesser degree, surface tension), the maximum diameter during impact of supercooled droplets in the temperature range −2◦C to −8◦C follows closely the same scaling with W e and Re as previously presented for diﬀerent liquids in [53].

A potential application of using supercooled droplets for three-dimensional printing has been identified. This method can be used for printing stalagmites of ice. We have investigated the influence of the distance of the printing nozzle to the deposition location on the printing speed and the amount of deposited mass. High-speed imaging techniques have been used to record the actual freezing process of supercooled droplets and infer the spatial and temporal scales involved herein. The same techniques have been used to show how the freezing can cause supercooled droplets to shatter. Different physical models have been constructed to explain the dynamics of the freezing process of a supercooled droplet, and the causes of the shattering of supercooled droplets after freezing. To do this, we have inferred principles from heat transfer and solid mechanics (elas- ticity theory). Our models provide insight into how heat transfer through the spherical ice

56 shell that appears at the start of the inward freezing process of supercooled droplets constrains the lifetime of such droplets; how the buildup of elastic energy through elastic deformation of the ice during the freezing could explain why such droplets tend to develop cracks and explode; and how an actual explosion might be initiated.

7.2 Challenges and recommendations

In this section, we present a number of challenges that were encountered during the current experiments and suggest a number of recommendations for future studies.

7.2.1 Impact and jetting experiments One of the main difficulties we encountered in the experiments on supercooled droplet impact (chapter 4) and on supercooled jets (chapter 5) was that the nozzles used were very prone to clogging. A number of causes can be responsible for this, among which entrapment of air bubbles in the nozzle tip (if the water had not been sufficiently degassed, for example). However, in the current setup, one specific reason for clogging was in fact the nucleation of ice on the nozzle tip. Since conditions for heterogeneous nucleation are near-uncontrollable, it was basically down to chance whether or not an experiment had to be stopped due to ice clogging of the nozzle or not. This significantly increased the time it took to perform the experiments and made them generally tedious. Apart from inconvenience for the impact experiments, it had another consequence in the printing experiments: the printing could sometimes stop abruptly as the nozzle clogged during the jetting itself. In order to develop any ice printing technique into something more practical, a solution to this problem would have to be found. This is likely to represent a significant challenge. Another challenge in these experiments was measuring the temperature of impacting droplets and jets. While we believe our measuring accuracy is reasonable, we are convinced that much more accurate information could be obtained if a higher temporal resolution were available for the temperature measurements. As has been shown, surface freezing processes (whether of suspended or impacting droplets) take place within tens of milliseconds, timescales which have not been captured at all in the temperature measurements done in this study (which had a resulution of one second). A large amount of information on the temperature evolution on short timescales is therefore lost, i.e. we have not been able to measure how much the temperature of a supercooled droplet changes during impact, for example, while this might have potentially important implications for studies on aircraft icing.

7.2.2 Freezing experiments In order to obtain more information about the inwardly-directed freezing process of supercooled droplets, described in chapter 6, we tried using a class 3B laser module ZM18B (sold by Z-Laser) with 40 mW output power in different configurations to illuminate a supercooled droplet about to freeze. We tried out the following exploratory tests: We directed the laser beam through a window in the vacuum chamber and let it illuminate the droplet from the side (from the camera’s point of view), using a mirror. The idea was to use the much higher albedo of ice as compared to water to obtain information about the ice content of the droplet. The light would (1) mostly pass through the droplet while it was liquid, (2) illuminate the iced outer shell of the droplet as its surface froze over (enabling accurate tracking of the freezing front), and (3) subsequently, through changes in the total reflected intensity, provide information about the percentage of ice in the inwardly-freezing droplet. While the idea of

57 Figure 7.1: Two pictures of a droplet, 2.6 mm in radius, freezing over while being illuminated by a laser from the right. The time between the two pictures corresponds to 21 ms. One can see a diﬀerent intensity in the light reﬂected from the ice front (coming from below) and the part of the droplet surface that is still liquid.

using this technique to track the freezing front as it spreads across the droplet worked to some extent (see Figure 7.1), the images during the inward freezing process became so noisy with speckles as to provide little quantitative or qualitative information of value. Instead of pure MilliQ water, we also tried using a 0.0008 vol% solution of FluoroMaxTM red fluorescent microspheres of 0.52 µm diameter to create the droplets. These microspheres absorb green light with a wavelength of 532 nm and emit red light with a mean wavelength close to 630 nm. We illuminated the droplet with the aforementioned laser module and put a dichroic mirror in front of the camera to capture only the emitted light. The idea was that the concentration of microspheres in ice might differ from that in water, thus providing information about the total ice content of a supercooled freezing droplet. However, this turned out not to be the case, at least not to an observable degree, although the principle of using microspheres itself proved to work very well in illuminating the droplet (in liquid as well as in solid form) very sharply. Therefore, this setup might be a useful alternative for imaging freezing droplets if front illumination with a light source is not available or not feasible in a setup. One of the major difficulties in carrying out the experiments on sessile droplet freezing was related to the preparation of the hydrophobic surface. Since it was clear that the surface on which the droplets were to be deposited had to have a low thermal conductivity in order for supercooling to occur, it was not possible to use surfaces with metallic micropillars. We prepared our own hydrophobic substrate with candle soot. However, this resulted in a number of problems of its own, chief among which were the contamination of the droplets with pieces of soot, and the low controllability of the small-scale surface structure, mainly in terms of roughness. Another problem was that, due to the flatness of the used surfaces, the droplets would often tend to move slighty out of focus during experiments, thus making it tedious and time-inefficient to obtain recordings such as those displayed in Figure 6.4. To remedy this, we designed a glass plate with a small concave patch to act as a potential minimum, in which a deposited droplet would have little to no propensity for moving.

It would be interesting to investigate whether the dependence of τlife on ∆T as predicted by equation (6.5) could be experimentally investigated. In our current experiments, we have found that the standard deviations in observed values of τlife are about as large as the diﬀer- ◦ ences between theoretical curves for τlife with ∆T varying by 10 C (see Figure 6.9). However, one could imagine performing experiments in which the supercooled droplet temperature and ambient pressure are kept constant and the droplets are then forced to turn into ice (as explained in section 3.3.3 and 3.3.4). One could then investigate whether the droplet lifetime depends on the temperature at which the droplet froze. This might give qualitative insights into whether equation (6.5) correctly models the dependence of τlife on ∆T .

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62 Appendix A

Thermistor measurements

As described in chapter 3, a HH506RA Multilogger Thermometer with a ferrite-zinc core thermocouple wire was used to determine the temperature of supercooled droplets in the vacuum chamber. Here, we provide a short discussion to show the accuracy of this method of measuring the temperature. A demonstration of the functioning of this measuring method can be done in three stages, as described by the following description of a short experiment.

Firstly, the vacuum pump and the circulating cooler were both put into operation. Once the ambient pressure was lower than the vapour pressure of water, and the bottom plate temperature was suﬃciently low, a number of droplets were ejected from a height of ≈ 130 mm from the thermocouple wire. At impact on the wire, the thermometer reading showed a temperature of ≈ -7◦C, and the droplets froze on the wire.

Subsequently, no further droplets were ejected. As evaporative cooling caused the ice on the wire tip to cool down further, the temperature reading decreased, reaching as far down as ≈ -12◦C within a matter of seconds. Once the ejection of droplets was started again, the temperature reading jumped back up to ≈ -7◦C as soon as the droplets hit the ice on the wire tip and froze thereon.

Lastly, if the vacuum pump was shut oﬀ, the pressure in the vacuum chamber rapidly rose up to the vapour pressure (and eventually beyond, due to minute leaks in the chamber that could only be observed when the pressure was brought down to millibar-level and the pump was subsequently shut oﬀ). When this happened, evaporative cooling ceased, and the temperature reading jumped back up to 0◦C, representative of the melting block of ice at the wire tip.

63 Appendix B

Parameter estimation

◦ For the impact speed U0 = 0.87 m/s, we estimate Tdroplet ≈ −2.5 C; for U0 ≈ 1.31 m/s, we estimate ◦ ◦ Tdroplet ≈ −6.0 C; and for U0 = 2.30 m/s, we estimate Tdroplet ≈ −7.5 C. We then use the following values: from [50], we estimate

σ(T = −2.5◦C) ≈ 1.045 × 0.07274 kg/s2; σ(T = −6.0◦C) ≈ 1.052 × 0.07274 kg/s2; σ(T = −7.5◦C) ≈ 1.055 × 0.07274 kg/s2.

Similarly, from [51], we estimate

η(T = −2.5◦C) ≈ 1.97 × 10−3 kg/(ms); η(T = −6.0◦C) ≈ 2.25 × 10−3 kg/(ms); η(T = −7.5◦C) ≈ 2.40 × 10−3 kg/(ms).

According to [50], the surface tension of water is not aﬀected in a signiﬁcant way by the presence of air; therefore, we assume the values of σ for supercooled droplets to depend only on the temperature of the droplets, not on the ambient pressure (which is after all lower in our experiments for supercooled droplets as compared to normal droplets). For non-supercooled droplets, we use σ = 0.07274 kg/s2 and η = 1.3 × 10−3 kg/(ms).

64 Appendix C

Modeling parameterizations

The parameterizations used in equation (4.8) are as follows. We take the enthalpy of vapourization from [54]; the speciﬁc heat capacity from [55]; the saturation vapour pressure from [48]; and the density from [56]. For the enthalpy of vapourization (in J/kg), we use

3/8 11/8 19/8 27/8 ∆Hvap(T ) = 1000 × (44.46 × Tc + 14.64 × Tc − 27.95 × Tc + 13.99 × Tc ), (C.1) with 647.126 − T Tc = 647.126 − Tf and Tf = 273.15 K the freezing temperature of water at standard atmospheric conditions. The isobaric heat capacity (in J/kg/K) is represented by

5.26 ! T − Tf + 100 C (T ) = 4.1855 × 103 0.996185 + 0.0002874 + 0.011160 × 10−0.036(T −Tf ) . p 100 (C.2) For the vapour pressure of water (in mbar), we have

6763.22 P (T ) = 0.01 × exp [54.842763 − − 4.210 log (T ) + 0.000367 × T + vap T 1331.22 arctan (0.0415(T − 218.8))(53.878 − − 9.44523 log (T ) + 0.014025 × T )]. (C.3) T Finally, the density (in kg/m3) is represented by

−3 2 −6 3 ρ(T ) = [999.8395+16.945176(T −Tf )−7.9870401×10 (T −Tf ) −46.170461×10 (T −Tf ) + −9 4 −12 5 −3 105.5630 × 10 (T − Tf ) − 280.5425 × 10 (T − Tf ) ]/[1 + 16.87985 × 10 (T − Tf )]. (C.4)

The vapour pressure of ice has not been used in equation (4.8), but is referred to in section 3.3.3. It is parameterized as follows (in mbar):

5723.265 P (T ) = 0.01 × exp 9.550426 − + 3.53068 log (T ) − 0.00728332 × T . (C.5) vap,ice T