Vitrification and Levitation of a Liquid Droplet on Liquid Nitrogen

Vitrification and levitation of a liquid droplet on liquid nitrogen

Young S. Songa, Douglas Adlera, Feng Xua, Emre Kayaalpb,2, Aida Nureddinc, Raymond M. Anchanc, Richard L. Maasd, and Utkan Demircia,e,1

aBio-Acoustic-Microelectromechanical Systems in Medicine Laboratory, Center for Bioengineering, Department of Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA 02115; bFaculty of Medicine, Yeditepe University, Istanbul, Turkey 34755; cCenter for Infertility and Reproductive Surgery, Obstetrics Gynecology and Reproductive Biology, and dDivision of Genetics, Department of Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA 02115; and eHarvard-Massachusetts Institute of Technology Health Sciences and Technology, Cambridge, MA 02139

Edited by Daniel D. Joseph, University of Minnesota, Minneapolis, MN, and approved January 13, 2010 (received for review December 4, 2009)

The vitrification of a liquid occurs when ice crystal formation is pre- study, we investigate the heat transfer and simultaneous phase vented in the cryogenic environment through ultrarapid cooling. In change of a nanoliter volume droplet levitating on liquid nitro- general, vitrification entails a large temperature difference be- gen. Understanding the vitrification and film boiling phenomena tween the liquid and its surrounding medium. In our droplet vitri- is also of great importance to other broad applications, such as fication experiments, we observed that such vitrification events metallurgy, nuclear reactors, turbine machineries, and biopreser- are accompanied by a Leidenfrost phenomenon, which impedes vation (13–16). In particular, vitrification is regarded as the only the heat transfer to cool the liquid, when the liquid droplet comes feasible way to successfully cryopreserve human cells and tissues into direct contact with liquid nitrogen. This is distinct from the such as oocytes and brain tissue (17, 18). more generally observed Leidenfrost phenomenon that occurs Upon ejecting a droplet into liquid nitrogen, as demonstrated when a liquid droplet is self-vaporized on a hot plate. In the case in Fig. 1A, the liquid nitrogen surrounding the droplet absorbs the of rapid cooling, the phase transition from liquid to vitrified solid heat from the liquid of relatively higher temperature and evapo- (i.e., vitrification) and the levitation of droplets on liquid nitrogen rates (14, 15, 19, 20). As soon as droplets are immersed in liquid

(i.e., Leidenfrost phenomenon) take place simultaneously. Here, we nitrogen, the evaporated nitrogen results in a buoyant force and a BIOPHYSICS AND

investigate these two simultaneous physical events by using a the- pressure differential pushing droplets to the liquid nitrogen sur- COMPUTATIONAL BIOLOGY oretical model containing three dimensionless parameters (i.e., Ste- face (i.e., levitation phenomenon; Fig. S1 and Video S1). This fan, Biot, and Fourier numbers). We explain theoretically and levitation continues until the droplet temperature reaches the observe experimentally a threshold droplet radius during the vitri- Leidenfrost temperature (Fig. 1D; Video S1). Assuming that fication of a cryoprotectant droplet in the presence of the Leiden- the static feature of the droplet primarily governs the entire sys- frost effect. tem [as first reported by Frederking and Clark (21)], the flow behavior of nitrogen gas environing a droplet and following PHYSICS cryopreservation ∣ leidenfrost ∣ film boiling ∣ phase change ∣ crystallization pressure development can be analytically modeled as an incom- pressible flow in cavities, Fig. 1B and C (also see SI Text). The ilm boiling is a phenomenon of great interest that has applica- vapor blanket acts as a heat-insulating layer, thereby hindering Ftions in a variety of fields such as aerospace, cryogenics, and heat transfer for cooling. electronics. In addition, a similar phenomenon can be easily observed in daily life, e.g., the “dancing droplets” on a frying pan Results and Discussion when the droplets come into contact with the pan that is much Once the temperature of a liquid approaches its freezing point, hotter than the boiling temperature of the droplets (1, 2). The solidification of the liquid (crystallization or vitrification) starts droplets are heated and evaporate on the pan. As a result, they depending on cooling conditions such as cooling rate, nucleation are provided with a levitation force by the generated vapor layer. sites, and pressure (22–24). To describe such a solidification pro- This solid–liquid film boiling (or liquid–liquid film boiling) is cess, two models, i.e., the Stefan and zone models, have been em- usually caused by a huge temperature distinction between adja- ployed (25). Unlike the Stefan model, where the entire domain is cent substances (3–7). Film boiling, also referred to as the Lei- sharply divided into solid and liquid subdomains by a moving denfrost effect, generally impedes the heat transfer between phase interface, the zone model benefits from more efficiency neighboring materials due to the big thermal resistance induced in characterizing the crystallization process using a propagating by evaporated vapor layer (8, 9). A great amount of effort has zone. In the current study, the zone model is adopted by using been spent to understand the film boiling behavior of plates or the following nonisothermal kinetic equation proposed by Bou- spheres occurring at extremely high temperatures with strong tron and Mehl (13): evaporation of a liquid (10–12). On the other hand, when a liquid dχ 2 − ∕ ¼ χ3ð1 − χÞð − Þ Q RT [1] droplet falls into an extremely cold medium such as a cryogenic ka Tf T e fluid (e.g., liquid nitrogen), the relatively hot droplet boils the dt surrounding medium. Unlike a droplet on the pan, the vapor from the medium, not from the droplet itself, levitates the dro- Author contributions: U.D. designed research; Y.S.S. and D.A. performed research; F.X., plet. Furthermore, abrupt phase change from liquid to solid oc- E.K., A.N., R.M.A., R.L.M., and U.D. analyzed data; and Y.S.S., F.X., and U.D. wrote curs along with film boiling during cooling down of the droplets. the paper. Here, we investigate the vitrification of a droplet via rapid The authors declare no conflict of interest. freezing and its levitation by the Leidenfrost effect. In principle, This article is a PNAS Direct Submission. vitrification implies the phase transition of a liquid to a glass Freely available online through the PNAS open access option. (amorphous ice) with a very low degree of crystallization. To 1To whom correspondence should be addressed. E-mail: [email protected]. the best of our knowledge, there have been no prior attempts 2Present address: Jamaica Hospital Medical Center, 8900 Van Wyck Expressway, Jamaica, to examine the vitrification behavior of droplets that are ejected NY 11418. directly into liquid nitrogen and the accompanying Leidenfrost This article contains supporting information online at www.pnas.org/cgi/content/full/ phenomenon on the surface of liquid nitrogen. In the current 0914059107/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.0914059107 PNAS Early Edition ∣ 1of5 Downloaded by guest on September 30, 2021 χ 0 χ 1 ¼ 2∕α where is the degree of ice crystallization ( < < ), t (s) is diffusion time) tc R with unit of second, dimensionless ra- ∕ time, Q (J mol) is the activation energy, Tf (K) is the final dius r ¼ r∕R, where R is the radius of the droplet and r is the temperature of the freezing process, R (J∕mol K) is the gas con- coordinate in the radius direction, dimensionless time (Fourier ¼ ∕ ð¼ Þ stant, and ka is a characteristic constant. In the case of a spherical- number) t t tc Fo indicating the ratio of the heat conduc- ¼ L ∕ δ shaped droplet, ka πδ2νT r , where L (J kg) is the latent heat, tion rate to the rate of thermal energy storage, and dimensionless f f T−T∞ temperature T ¼ − , in which T (K) is the initial temperature (m) is the thickness of the transition layer between liquid phase Ti T∞ i ν 2∕ of droplets, and T∞ (K) is the temperature of liquid nitrogen. The and solid phase, rf (m) is the radius of ice region, and (m s) is the kinematic viscosity. This zone model is based on the hypoth- resulting heat transfer equation is expressed as esis that the sum of specific and latent heats can be given by an ∂ ∂χ enthalpy function (13, 23). T 2 ¼ ∇ T þ St [2] The primary mode of heat transfer in the droplet at cryogenic ∂t ∂t temperatures is heat conduction rather than heat convection, as in which ∇ is a differential vector operator and St denotes the L the Peclet number with physical meaning of the ratio of convec- Stefan number St ¼ ð − Þ [L (kJ∕kg) is the heat of fusion] in- ¼ ∕α ≪ 1 cp Ti T∞ tion to conduction in the convecting region, Pe DV in [D dicating the ratio of sensible heat to latent heat. The imposed ∂T is characteristic length, i.e., the diameter for spherical droplets, boundary conditions are as follows: ¼ 0,atr ¼ 0 (at the cen- λ ∂r α ¼ ∂T V in is the internal flow velocity of liquid within droplets, ρ is ¼ − ¼ 1 cp ter of droplets) and ∂r BiT at r (at the surface of dro- 2 3 hR the thermal diffusivity with unit of m ∕s, ρ (kg∕m ) is density, c plets). Here Bi ¼ λ , where h denotes the convective heat transfer (J∕kg K) is specific heat, and λ (W∕m K) is thermal conductivity]. coefficient (W∕m2 K) indicating the ratio of the heat transfer re- Dimensional analysis is carried out using corresponding charac- sistances inside and at the surface of a body. The initial condition ¼ ¼ 0 teristic quantities as explained below: characteristic time (heat of T Ti at t is applied. The distinguishing characteristic of a

Fig. 1. Generation and levitation of droplets on top of liquid nitrogen. (A) A schematic description of the experimental setup and of the behavior of droplets over time. It is a cross-sectional image of the ejector that is driven by an electric pulse. (B) Schematic demonstration of a droplet in the liquid nitrogen. The droplet in dark blue color is surrounded by the liquid nitrogen vapor as shown in the schematic drawing. (C) Schematic illustration of a droplet floating around on top of liquid nitrogen. (D) Photograph of a droplet levitating on liquid nitrogen. The droplet stained with a color dye is partially submerged in the liquid nitrogen as illustrated in C. (Scale bar, 1 mm.)

2of5 ∣ www.pnas.org/cgi/doi/10.1073/pnas.0914059107 Song et al. Downloaded by guest on September 30, 2021 result, it has a small degree of crystallization (χ ≈ 0.023). In con- trast, the temperature plot of a droplet with a dimensionless radius R ¼ 0.2 (R ¼ 200 μm) has a plateau regime due to the latent heat caused by crystallization of liquid (Fig. 2B). As seen in Fig. 2A and B, the ice crystal formation in the two droplets stops at a dimensionless temperature of approximately T ¼ 0.56 (i.e., T ≈ 200 K) during cooling. It is known that the total degree of crystallization depends on how quickly the droplets pass the dangerous temperature region (DTR) (13). Therefore, shorten- ing the time duration passing through the DTR is critical to minimize ice crystal formation and improve vitrification during rapid cooling. Because the surface of a droplet comes in contact with liquid nitrogen first, it cools down more quickly than the inner regions (Fig. 2A and especially Fig. 2B). However, the surface (or outer region) takes more time to pass through the DTR than the inner regions due to the latent heat flowing radially outward from the inner region to the outer region of the droplet, thereby generating more ice crystals in the outer region (Fig. 2B) (27). Fig. 2C and D present the profiles of temperature and crystallinity across the 200 μm droplet at Fo of 1.3, which was chosen to clearly see the difference of the crystallinity arbitrarily. Because the internal thermal resistance of the droplet is smaller than the external thermal resistance between droplet surface and liquid BIOPHYSICS AND COMPUTATIONAL BIOLOGY PHYSICS

Fig. 2. Change in temperature and degree of crystallization as a function of time. (A) Plot of the dimensionless temperature and crystallinity for a droplet with a dimensionless radius of 0.02. The left-hand and right-hand axes of the graph denote the dimensionless temperature and degree of crystallinity, respectively. (B) Change in the dimensionless temperature and degree of crystallization for a droplet with a dimensionless radius of 0.2 with respect to the dimensionless time, Fo.(C) Profile of the dimensionless temperature at Fo ¼ 1.3.(D) Contour of the degree of crystallization at Fo ¼ 1.3.

phase transition is an abrupt change in thermophysical properties, in particular the thermal conductivity (k), heat capacity ρ (cp), and density ( ), with respect to a thermodynamic variable such as temperature. In our calculations, these temperature- dependent properties are considered (13, 26). The predicted temperature variations and degree of crystallization are demonstrated at three locations inside the droplet (center, middle, and surface) as shown in Fig. 2A and B. The Fig. 3. Variation in the degree of crystallization and dimensionless Leiden- ¼ ∕ frost time as a function of the radius. (A) Plot of the crystallinity vs. the di- radius of droplets is normalized as R R Rm, where Rm is 1 × 10−3 mensionless radius. (B–D) Images of droplets in liquid nitrogen. (Scale bar, the maximum radius ( m) considered in this study. A μ 100 m.) (E) Graph of the dimensionless Leidenfrost time vs. the dimension- ¼ 0 02 ¼ 20 μ t ∕t small droplet with a dimensionless radius R . (R m) less radius. The dimensionless Leidenfrost time is defined as LF c , in which A t cools down quickly due to its low heat capacity (Fig. 2 ). As a LF is the Leidenfrost time.

Song et al. PNAS Early Edition ∣ 3of5 Downloaded by guest on September 30, 2021 nitrogen (Bi < 1, i.e., heat transfer occurs faster inside the droplet), the range of the temperature variation across the droplet is narrow (Fig. 2C). After calculating the final degree of crystallization of a droplet, we averaged the value using the relevant volume. The calculated degree of crystallization of droplets is found to be a strong function of the droplet size (Fig. 3A). As the droplet radius increases, the degree of crystallization also increases, especially between 0 and 0.2 dimensionless radii. The droplets with a low degree of crystallization after cooling can be regarded as vitrified droplets. During cooling, homogeneous ice nucleation first happens between homogeneous nucleation temperature Th and the melt- −42 ∼ 0 ing temperature Tm (e.g., °C for water), followed by het- erogeneous ice nucleation in the range of the glass transition temperature (Tg)toTh (28). In the nucleation regimes, including the DTR as shown in Fig. 2A and B, the probability for a volume of ice crystal (V) can be given by JðT;PÞ × V × Δt, where JðT;PÞ is the nucleation rate as a function of temperature T and pressure P (29). Rapid cooling can shorten the time duration in the nuclea- Δ tion regime ( t), thus decrease the probability of ice crystal nu- St Bi ∼108 ∕ Fig. 4. Plot of the degree of crystallization in the domain of the and cleation. For instance, a cooling rate of °C min allows the numbers. The degree of crystallization increases with an increase in the St vitrification of pure water (30). It is noted that there exists a crit- and Bi numbers. In particular, the degree of crystallization drastically rises ical droplet size beyond which incipient nuclei grow leading to the with respect to the Bi number. formation of ice crystals. Otherwise, incipient nuclei collapse and an ice crystal is not created. The thermodynamic characteristic tion, the experimental and numerical results show the same temperatures (T , T , and T ) change along with the composi- g h m trends (Fig. 3E). tion and concentration of solutes. In the current study, we took By using two governing dimensionless parameters (i.e., Bi and into account of cryoprotective agents (CPAs), which are chemi- St) in our system, we analyzed the characteristics of crystallization cals used to minimize damage of cells or tissues during freezing. of droplets (i.e., χ). That is, the degree of crystallization increases T and T decrease, while T increases, with increasing CPA con- h m g with an increase in either St or Bi (Fig. 4). In particular, we find centration. By increasing CPA concentration, T and T become g m that the crystallinity is a stronger function of Bi than St. This is closer and eventually meet. As a result, a vitrified solid can be attributed to the fact that the heat transfer at the interface be- generated even at a slow cooling rate in the case of ∼60% wt/ tween the droplets and the liquid nitrogen is more critical in de- wt of CPAs (31). Fig. 3B–D shows the frozen droplets in the liquid termining the crystallization of droplets than the internal heat nitrogen. As the droplet size increases, morphological changes capacity of the droplets. Here, it is noted that, in the case of are observed: Vitrified droplets are translucent with bright cores St ¼ 0, it is identical to the uncoupled approach, where the en- (Fig. 3B), but crystallized droplets contain granular particles in- ergy equation [2] is decoupled from the kinetics of ice crystal for- side (ice crystals) (Fig. 3C and D). In addition, it is shown that the mation (Eq. 1) (27). large droplets have grain-like crystal particles due to the longer In summary, we have theoretically analyzed the vitrification cooling time. The experimental results shown in Fig. 3B–D are and Leidenfrost phenomena of droplets in liquid nitrogen using found to be consistent with the numerical findings. dimensionless physical parameters and compared the theoretical Fig. 3E demonstrates the hovering time of droplets on liquid results with experimental data. This dimensionless analysis ex- nitrogen. When droplets plummet into liquid nitrogen, three re- plains the simultaneous levitation and phase change of droplets. gimes are defined: a film boiling regime, a transition boiling regime, and a nucleate boiling regime. The evaporation rate of film Methods boiling is known to be lower than that of peak nucleate boiling by Experimental Methods. A valve-based droplet ejector with nozzle diameter more than one order of magnitude (14). During the transition 150 μm (SMLD-5b, TechElan) was connected to a syringe through a needle from film boiling to nucleate boiling, the vapor blanket surround- and tubing, and the liquid was then loaded into the syringe. The air line ing the droplet disappears. Subsequently, the droplet sinks be- was connected to the syringe to provide a pressure of 34 kPa to overcome neath the liquid nitrogen. The temperature at the moment the surface tension of a droplet at the orifice of the ejector (Fig. 1A). The that film boiling changes into nucleate boiling corresponds to the ejector was controlled by a pulse generator (8112A 50 MHz, Hewlett-Pack- Leidenfrost temperature. The hovering time is the elapsed time ard). Droplets are generated to implement rapid cooling by using an air-pres- when the calculated temperature of droplets reaches the Leiden- sure, pulse-controlled solenoid valve. Larger droplets were generated using a micropipette. Cryoprotectant, 3 M 1,2-propanediol (PrOH, EMD Chemicals), frost temperature. The Leidenfrost time normalized by the was employed to vitrify droplets. Blue food dye was used to stain the dro- diffusive characteristic time (tc) declines dramatically and ap- plets. The generated droplets were plunged into a liquid nitrogen bath (glass proaches an asymptotic value (∼1.5). As the size of the droplet Petri dishes, Pyrex). The liquid nitrogen container was covered with alumi- increases, the influence of heat diffusion in the droplet strength- num foil in a bid to prevent possible electrostatic interactions between ens. It is interpreted that the smaller droplets are more easily af- the container and droplets. Right after the immersion of droplets, the levita- fected by external heat transfer than by internal heat diffusion tion time of the droplets was measured until they sank into the liquid nitro- because the Biot number (Bi) gets smaller (≪1). Similar to gen bath. The frozen droplets were taken for imaging (Spot Advanced, the change in crystallinity (Fig. 3A), the dimensionless Leiden- Diagnostic Instruments, Inc.) under a microscope (Eclipse Ti-S, Nikon) using frost time drastically alters between 0 and ∼0.2 (Fig. 3E). Please a polarized filter. Droplet sizes were measured based on recorded images. note that thermodynamics and heat transfer between the droplet Numerical Simulations. Finite element simulation was carried out to model and liquid nitrogen modeled in this study play a more important the heat transfer, phase change, and Leidenfrost phenomena of droplets. role in determining the vitrification behavior and hovering time To predict the temperature and degree of crystallization of droplets during of the droplet than relevant hydrodynamics. The hydrodynamics cooling, we simultaneously calculated Eqs. 1 and 2 because they are coupled of the convection film boiling is explained in SI Text. In addi- to each other. For the calculation, we assumed a spherical symmetry based on

4of5 ∣ www.pnas.org/cgi/doi/10.1073/pnas.0914059107 Song et al. Downloaded by guest on September 30, 2021 the analytical solutions of fluid dynamics of the nitrogen vapor (SI Text). The Hasan Keles, and Dr. Ragip Akbas for feedback and discussions. This fourth-order Runge–Kutta algorithm and the backward Euler method, which work was supported by National Institutes of Health Grant R21 EB007707. is an implicit method solving an equation set to move forward to a next time U.D. was partially supported by R01-AI081534. R.L.M. and R.M.A. were supported by 1RL1 DE019021. This work was performed at the Bio-Acoustic step during calculation, were employed for the numerical simulations. Microelectromechanical Systems in Medicine Laboratories, Center for Bioengineering, Brigham and Women’s Hospital, Harvard Medical School ACKNOWLEDGMENTS. We would like to thank Dr. Can Erdonmez, and Harvard-Massachusetts Institute of Technology Health Sciences Dr. Edward Haeggstrom, Dr. Sangjun Moon, Naside Gozde Durmus, Onur and Technology.

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