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- ∕ Ã 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 Ã 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 Ã 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 ∂ Ã ∂χ enthalpy function (13, 23). T 2 Ã Ã ¼ ∇ T þ St Ã [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 ∂TÃ Ã is characteristic length, i.e., the diameter for spherical droplets, boundary conditions are as follows: Ã ¼ 0,atr ¼ 0 (at the cen- λ Ã ∂r α ¼ ∂T Ã Ã V in is the internal flow velocity of liquid within droplets, ρ is Ã 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.

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