High undercooling of bulk water during acoustic levitation

Transcription

1 Vol. 46 No. 3 SCIENCE IN CHINA (Series G) June 2003 High undercooling of bulk water during acoustic levitation LÜ Yongjun ( d ), CAO Chongde ( ) & WEI Bingbo ( ) Department of Applied Physics, Northwestern Polytechnical University, Xi an , China Correspondence should be addressed to Lü Yongjun ( lmss@nwpu.edu.cn) Received August 8, 2002 Abstract The experiments on undercooling of acoustically levitated water drops with the radius of 5 8 mm are carried out, and the maximum undercooling of 24 K is obtained in such a containerless state. Various factors influencing the undercoolability of water under acoustic levitation are synthetically analyzed. The experimental results indicate that impurities tend to decrease the undercooling level, whereas the dominant factor is the effect of ultrasound. The stirring and cavitation effects of ultrasound tend to stimulate the nucleation of water and prevent further bulk undercooling in experiments. The stirring effect provides some extra energy fluctuation to overcome the thermodynamic barrier for nucleation. The local high pressure caused by cavitation effect increases the local undercooling in water and stimulates nucleation before the achievement of a large bulk undercooling. According to the cooling curves, the dendrite growth velocity of ice is estimated, which is in good agreement with the theoretical prediction at the lower undercooling. The theoretical calculation predicts a dendrite growth velocity of 0.23 m/s corresponding to the maximum undercooling of 24 K, at which the rapid solidification of ice occurs. Keywords: acoustic levitation, undercooling, water, crystal growth. Water is the most common substance in nature and also a necessary substance for life. Although water has been intensively studied, there remain important and accessible fields of experimentation in which its properties are less known, such as the density maximum at 277 K at normal pressure, amorphous ice formation when undercooled to 228 K [1], and polyamorphic transition in solid water [2]. Therefore, the research on undercooled water plays an important role in understanding phase transformation characteristics of water and its thermodynamic properties. In 1973, Kanno et al. [3] undercooled liquid water to 181 K at 2.00 kilobars using a differential thermal analysis technique. Smith [4] indicated that liquid water can exist in the vicinity of 150 K at 1 atm on the basis of his investigation on the diffusivity of amorphous ice in films less than 500 nanometres thick. This provides another explanation for anomalous thermodynamic properties at low temperatures. Sassen [5] found that water droplets of Φ5 µm levitated in cloud can be undercooled approaching 233 K. Hare and Sorensen [6] measured the density of undercooled water in the temperature range of K using Mossop method, which provides accurate density data for further research. In 1886, acoustic levitation phenomenon was observed, which was not completely theoretically explained by King until Acoustic levitation can effectively avoid the heterogeneous nucleation from container walls, and consequently, tends to increase undercooling level, which is very beneficial for studying the metastable behavior of water. Trinh et al. [7,8]

2 260 SCIENCE IN CHINA (Series G) Vol. 46 successfully measured the surface tension and sound velocity of water in the range of 5 30 K by acoustic levitation. It is worthwhile to note that the volume of water in the above work is very small, involving water films less than a few hundred nm thick or small drops with diameter of the order of micron. This makes it difficult to study and measure phase transition process and thermodynamic properties. In this work a single-axis acoustic levitation apparatus was used to undercool bulk water and investigate the rapid solidification dynamics of bulk undercooled water. 1 Experimental procedure The experiments were performed with a single-axis acoustic levitator, as shown in fig. 1(a). The acoustic levitator consists of two elementary components: a vibrating device and a reflector. The vibration system involves a magnetostrictive transducer with a resonant frequency of 16.7 khz and an emitter. This acoustic levitator can stably levitate any substance whose density is less than tungsten (ρ = g/cm 3 ) by optimizing the reflector [9 12]. Fig. 1. Schematic of experimental apparatus. (a) Acoustic levitation experiment; (b) experiment for comparison. The experiment was carried out under the (0, 1, 3) mode [12]. The levitation region was enclosed by a PMMA cylindrical chamber to avoid violent air convection during cooling process. The distilled water was injected to the middle position of the levitation region by a syringe through a hole in the wall of resonant chamber. Water drops were cooled at 65 K/min indirectly by low temperature nitrogen flowing through the cooling tube, which was fixed at the top of chamber. In the experiments, the sample temperature was monitored by a NiCr-NiSi thermocouple with its tap 1 mm near the water drop. Cooling curves were recorded by a highly accurate recorder. To examine the effect of container walls on the undercooling behavior, comparative experiments were also performedinaglasstube,asshowninfig.1(b). 2 Results and discussions 2.1 Influence of impurities on the undercoolability of water The experiments were performed with distilled and tap water samples by acoustic levitation. Due to the statistical feature of nucleation, the experiments were repeated at least 10 times for each category. The distribution of undercooling is shown in fig. 2(a). Each circle represents the undercooling achieved by an individual experiment. The mean undercooling of the distilled water drops is 15 K, and that of tap water drops is 7.2 K. Obviously, the undercooling level of distilled water is much higher than that of tap water. Meanwhile, the undercooling of distilled water drops

3 No. 3 HIGH UNDERCOOLING OF BULK WATER DURING ACOUSTIC LEVITATION 261 mainly lies in the range of 8 24 K, whereas the undercooling of tap water drops is distributed mainly in the range of 5 7 K, as shown in fig. 2(b) and (c). This implies that the heterogeneous nucleation induced by impurities obviously influences the solidification process of water drops. Fig. 2. The undercooling of water drops under acoustic levitation and comparative experiment. (a) Distribution of undercooling; (b) statistical distribution of undercooling for tap water; (c) statistical distribution of undercooling for distilled water. On the other hand, the maximum undercooling of 24 K obtained in the experiments is quite smaller than the critical undercooling level of 0.2T m (54.6 K) for homogeneous nucleation, which suggests that there exist other factors influencing undercooling apart from impurities under acoustic levitation conditions. The sound wave is an important factor. It has been observed that a larger undercooling achieved in the experiment often corresponds to smaller deformation of water drop before the initiation of solidification. Since the acoustic radiation pressure on water drops is comparatively weak, the water drop becomes an approximate ellipsoid. Contrariwise, a smaller undercooling corresponds to larger deformation of water drops, and even oblate disk oscillating in the z axis, which is attributed to larger acoustic radiation force. That implies that the acoustic field is related to the undercooling level of water drops. In order to examine the heterogeneous nucleation caused by container walls, the undercooling of bulk water was measured in a glass tube. All of the obtained undercoolings were less than 1 K, indicating that the container walls remarkably facilitate nucleation. Therefore, acoustic levitation is an effective manner for studying the phase transition and thermodynamic properties of water. On the basis of the classical nucleation theory, variously sized crystal embryos firstly form in the melt before nucleation. With the structural and thermal fluctuations, some embryos continuously grow. Once the radius of an embryo becomes larger than the critical radius, r *, the embryo becomes a nucleus. The formation of nuclei is a result of the reduction in the total free energy, and it is the beginning of solidification process. The impurities provide probable heterogeneous nu-

4 262 SCIENCE IN CHINA (Series G) Vol. 46 cleation sites and lead to the initiation of solidification at small undercooling. The fact that the undercooling level of distilled water drops is higher than that of tap water drops elucidates that the impurities still influence the undercoolability of water drops, although acoustic levitation avoids the contamination of water drops from container walls. 2.2 The movement and cavitation effects induced by acoustic field The interaction of sound wave and water drops creates high potential energy at the surface of water drops during acoustic levitation. Considering the effect of surface tension, the impurities tend to gather at the surface of a drop. Hence, nucleation preferentially occurs from the drop surface, which is consistent with the fact that we observed in experiments. Another effect of acoustic field comes from the rotation of water drops, which may induce internal flow. The liquid flow might promote collisions of embryos or impurities, producing larger embryos or impurities cluster and facilitating the formation of nuclei. However, the probability of such a collision is insignificant with respect to the quite small size of embryos and impurities. Accordingly, the effect of this factor on undercooling can be negligible. On the other hand, the external energy is simultaneously introduced by the rotation and deformation of water drops. The external energy can overcome the activation energy G k that is necessary for nucleation, and thus, the enhancement of a higher undercooling level is hindered. The activation energy is calculated as follows: 4πσ 2σ G k =, 3 GLS where G LS is the Gibbs free energy difference between the liquid and solid per unit volume, Tm Tm CPL CPS GLS = Hm ( CPL CPS) dt T Sm dt, T T T 2 (1) (2) where H m is the latent heat of crystallization, S m the fusion entropy, C PS the specific heat of ice [13], C PL the specific heat of water. It can be seen that the activation energy decreases with the increase of undercooling (fig. 3). Accordingly, the influence of external disturbance on nucleation becomes more and more obvious with the increase of undercooling. In experiments, small undercooling is obtained at large acoustic radiation pressure and serious deformation, and large undercooling at small acoustic radiation pressure and small deformation. Therefore, it is important to carefully adjust the levitation state of water drops for achieving large undercoolings. Compared with the theoretical critical undercooling for homogeneous nucleation, the undercooling of 24 K is rather small. Besides the factors mentioned above, the cavitation in the undercooled liquid is another principal factor for heterogeneous nucleation. Although the water used in the experiments has been distilled, it is impossible to completely avoid gas microbubbles and solid impurities containing crevices in which gas pockets reside. A strong sound wave through the liquid can generate cavities by driving a gas pocket into oscillation until a small bubble is form. On the other hand, the unstable gas microbubbles tend to dissolve or gather at the surface of the drop, where they can more easily interact with ultrasound. If the bubble

5 No. 3 HIGH UNDERCOOLING OF BULK WATER DURING ACOUSTIC LEVITATION 263 Fig. 3. The Gibbs free energy difference between ice and undercooled water and activation energy. (a) Gibbs free energy difference versus undercooling; (b) activation energy versus undercooling. has the proper size, it can be driven into resonant oscillation by sound wave. These small bubbles in resonance can continuously grow. In terms of the Crum s model [14], the threshold acoustic pressure amplitude for the growth of bubbles is given by: where 2σ Ci 1+ P R P C C = α Ci 2σ P (3 + 4 K) 1 + (4 3 η) K C0 R0P 0 1 4σ 2 α = 3η 1 + β, 3P0R0 (3η + 1 β ) 4σ + 4 3RP 0 0 K =, 4σ 1+ 3 RP ρω R0 0 1/2, (3) (4) (5) β =. (6) 3η P Here, C i is the concentration of dissolved gas in the liquid, C 0 the saturation concentration of the gas in unit volume, η a constant, σ the surface tension of water, R 0 the initial bubble radius, P 0 the ambient pressure. It is clear that the threshold acoustic pressure amplitude is a function of initial bubble radius and surface tension. On the basis of previous work [14], the threshold acoustic pressure amplitude increases monotonously with the increase of surface tension for a fixed R 0.Taking into account the fact that the surface tension of water increases with the decrease of temperature, a small threshold acoustic pressure amplitude is accessible for a small undercooling. The calculated threshold acoustic pressure amplitudes corresponding to undercoolings of 24 K and 5 K are shown in fig. 4. Assuming the initial bubble radius is 20 µm, the thresholds are 1.46D10 4 Pa and 1.37D10 4 Pa at 24 K and 5 K respectively, which demonstrates that the cavitation is easier to oc-

6 264 SCIENCE IN CHINA (Series G) Vol. 46 Fig. 4. The threshold acoustic pressure amplitude versus bubble radius. cur at a smaller undercooling. Meanwhile, the larger the initial radius is, the smaller the threshold acoustic pressure amplitude is. Once the bubble resonates, it could eventually collapse. Even sonoluminescence (SL) [14,15] is observed in some materials. SL is believed to be caused by the microshocks converging at the center of the gas bubble to create a very short duration ( 50 ps [15] ), high temperature compression. The collapse of gas bubble causes transient, high pressure in the water near the bubble wall. Investigations have shown that the pressure increase in this region is almost adiabatic and the pressure can reach 5 GPa or higher [15]. This adiabatic compression process of water can be described as follows [16] : p 0 p 0 1 T = C T [ C vα ( P P )], (7) where T 0 is the initial temperature, P 0 the atmospheric pressure, v the molar volume, α the thermal expansion coefficient and C p the specific heat. These parameters are constants without considering the changes of pressure. On the basis of the above conclusion that the cavitation occurs more possibly at the smaller undercooling of 5 K, the adiabatic compression is calculated by setting 268 K as the initial temperature, as shown in fig. 5. When the pressure exceeds 0.83 GPa, the temperature of the water near the bubble wall is actually below the melting point of high pressure ice, and the local undercooling increases with the increasing pressure. The change of melting point T m with pressure P is given by the Clausius-Clapeyron equation as Tn V Tm = Tn + ( P P0), Hm (8) where T n is the melting point at the atmospheric pressure and V the volume change. V 0 as P P 0 and V 0 asp P 0. H m is the fusion heat of water. According to eq. (8), the local undercooling of water near the bubble wall can be calculated. Assuming the final high pressure is up to 5 GPa before collapse, the local undercooling of 67 K can be predicted. On the other hand, the heterogeneous nucleation temperature Fig. 5. The adiabatic compression lines superposed on the change parallels the trend of the melting point phase diagram of water. change with pressure [18] and the undercooling corresponding to the heterogeneous nucleation temperature is smaller than 1 K. Therefore, water in the region nucleates firstly. It is inferred that

7 No. 3 HIGH UNDERCOOLING OF BULK WATER DURING ACOUSTIC LEVITATION 265 the heat produced during bubble collapse cannot cause obvious temperature rise in the water during such short period of time because the density and specific heat of water are much larger than those of gas. The high pressure ice film remains and serves as a heterogeneous site for further nucleation. Thus, cavitation decreases undercooling lever of water and promotes nucleation. Consequently, it is presumed that a degassing process before levitation conduces to the achievement of a large undercooling. 2.3 Rapid growth of ice dendrite in undercooled water Fig. 6 displays the cooling curves corresponding to the cases of solidification by acoustic levitation in a glass tube. The undercooling of water obtained in the tube is only 0.4 K and the cooling curve does not show evident recalescence. The solidification time of a sample can be measured from the cooling curve and the growth velocity of ice dendrite can be determined by the ratio of the diameter of the ellipsoid sample to the solidification time. In the case of undercooling of 0.4 K, the solidification length is about 7 mm and the solidification time is 108 s. Thus the growth velocity of ice dendrite is estimated to be 6.4D10 5 mcs 1.Although the estimation is relatively rough, it is reliable at least in the order of magnitude [19].For samples undercooled at 12 K and 24 K by acoustic levitation, their cooling curves are characterized by evident recalescence, which is the thermal effect of rapid solidification [20]. Based on the dendrite growth theory [21,22], the growth velocity and tip radius of ice dendrite versus undercooling are given by: 2α Cp 2 V = T, H m Γ 1 R =, σ * T Fig. 6. Cooling curves of water at different undercooling. (9) (10) where V is the growth velocity of ice dendrite, R the dendrite tip radius, Γ the Gibbs-Thomson parameter, α is the thermal diffusion coefficient, α =2P t /VR, P t the thermal Peclet number. σ * the stability parameter (σ * = 1/4π 2 ). The experimentally estimated and theoretically calculated growth velocities versus undercooling are shown in fig. 7 and the thermophysical parameters used for calculations are listed in table 1. Apparently, the growth velocity increases with the increase of undercooling in the cases of both theoretical calculation and experimental measurements. When undercooling is smaller than 20 K, the theoretical predictions of the growth velocity of ice dendrite agree well with the experimental measurements. Once the water drop is undercooled by larger than 20 K, the differences between them are visible due to the sharp recalescence on the cool-

8 266 SCIENCE IN CHINA (Series G) Vol. 46 ing curves which make accurate measurements difficult. The predicted growth velocity at the undercooling of 24 K is 0.23 m/s and the measured value is about 0.17 m/s. For both the theoretical predictions and the experimental measurements, the growth velocity is larger than the empirical critical value of 0.01 m/s for rapid solidification. Therefore, rapid solidification occurs at large undercooling in this work. The calculated dendrite tip radius drastically decreases with the increase of undercooling and attains 0.11 µm at the undercooling of 24 K. Table 1 Thermophysical parameters used for calculations C p/jc(kgck) H f /JCkg D10 4 α /m 2 Cs D10 8 Γ /KCm D Conclusion (1) The undercooling and rapid solidification of bulk water are investigated by acoustic levitation. The maximum undercooling of 24 K is achieved. (2) Various factors influencing the undercoolability of water are analyzed. The container walls can drastically catalyze nucleation in water and the containerless state provided by Fig. 7. Dendrite growth velocity and dendrite tip radius versus undercooling. acoustic levitation is effective to undercool bulk water. The stirring induced by acoustic wave provides extra external energy to overcome the energy barrier and decreases the undercooling level. The cavitation effect tends to increase the local undercooling and simulate nucleation. (3) The growth velocity and tip radius of ice dendrite as functions of undercooling are calculated on the basis of LKT dendrite growth theory. The theoretically predicted and the experimentally measured growth velocities at the undercooling of 24 K are 0.23 m/s and 0.17 m/s respectively, which indicates that rapid solidification occurs. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos & ) and Fok Ying Tung Education Foundation (Grant No ). The authors are grateful to Dr. W. J. Xie, X. J. Han, W. J. Yao and Mr. J. Xu for helpful discussions. References 1. Mishima, O., Stanley, H. E., The relationship between liquid, supercooling and glass water, Nature, 1998, 396: Mishima, O., Clvert, L. D., Walley, E., Melting ice I at 77 K and 10 Kbar: a new method of making amorphous solids, Nature, 1984, 310: Kanno, H., Speedy, R. J., Angell, C. A., Supercooling of water to 92k under pressure, Science, 1975, 189: Smith, R. S., Bruce, D. K., The existence of supercooled liquid water at 150 K, Nature, 1999, 398:

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