- Email: [email protected]
Liquid-to-crystal heterogeneous nucleation: bubble accelerated nucleation of pure supercooled water
Chemical Physics Letters 368 (2003) 177–182 www.elsevier.com/locate/cplett
Liquid-to-crystal heterogeneous nucleation: bubble accelerated nucleation of pure supercooled water A.F. Heneghan, A.D.J. Haymet
*
Department of Chemistry, University of Houston, Houston, TX 77204-5003, USA Received 4 October 2002; in final form 14 November 2002
Abstract Bubble formation in a single sample of pure water, held in the sample holder of our automated lag-time apparatus, is shown to lead to accelerated liquid-to-crystal heterogeneous nucleation. This phenomenon shifts the entire probability distribution for heterogeneous nucleation, together with the average lag-time hsi, to significantly warmer temperatures. In the linear supercooling experiment analyzed here, this is equivalent to reducing the average time to nucleation. Using a single assumption tested in previous work, the entire Ônucleation curveÕ is extracted from a single experiment. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Nucleation Bubble accelerated heterogeneous nucleation is studied here using a new version of our automated lag-time apparatus, denoted ALTA4, which we have designed, built and operated recently. Operated in a linear supercooling mode, this experiment determines the so-called Ônucleation curveÕ from a continuous, automated experiment on a single (unchanging) pure water sample for several hundred consecutive runs [1,2]. Here we quantify the accelerated rate of nucleation caused by air bubble formation in a sample of supercooled pure water used in one of our experiments. The entire survival probability distribution is shifted by approxi-
*
Corresponding author. Fax: +713-743-2709. E-mail address: [email protected] (A.D.J. Haymet).
mately 2.4 °C to warmer temperatures by the presence of the bubble in the sample. This is a significant and potentially useful effect. Much recent effort has been focused on understanding liquid-to-crystal nucleation [3–5], but challenges still remain. Important differences between liquid-to-crystal nucleation and gas-toliquid nucleation [6] have been exhibited [7,8]. It is perhaps not widely known that, in the liquid-tocrystal case, classical nucleation theory and its socalled Ôcapillarity approximationÕ break down [7]. A further challenge is to understand how to control the nucleation temperature of supercooled solutions. This is of importance in many fields including protein crystallization [9–11], where different products may be formed depending on the temperature of crystallization, artificial snow making [12,13], where conversion of water to snow is sought at as warm a temperature as possible,
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 8 3 5 - 3
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A.F. Heneghan, A.D.J. Haymet / Chemical Physics Letters 368 (2003) 177–182
and in the food industry [14–16] where the temperature of recrystallization directly affects food and food quality. Control of the nucleation temperature is important for so-called Ôcold energyÕ storage devices [17]. In these devices, a slurry of ice and supercooled water is used to help level electrical load discrepancies between day and night time caused by increasing air conditioning use. Our experiments, coupled with previous work [17,18], yield a data set which may be used to understand how to control the nucleation temperature in supercooled solutions, and evaluate candidate theories for crystal nucleation [19]. The introduction of bubbles into the supercooled solutions is a promising method for increasing the nucleation temperatures of these solutions. Elsewhere we exhibit the dependence of the nucleation temperature upon addition of crystals of (insoluble) AgI salt [1] and biological ÔantifreezeÕ proteins [20,21].
2. Supercooled liquid to crystal nucleation
possible on a single sample to generate the statistics of liquid-to-crystal nucleation. The data collected in this experiment are the lag-time to nucleation ti after cooling below the equilibrium freezing and melting temperature Tm , and the temperature of nucleation, Ti , as a function of run number for a single aqueous sample. Fig. 1 shows the measured lag-times as a function of run number for 444 consecutive runs on a single sample, which were collected uninterrupted over the course of 4 days on ALTA4. Note that the data are cleanly partitioned into two distinct sets, Set A with longer lag-times and set B with shorter lag-times. After run 294 the bubble was removed mechanically by shaking the NMR tube vigorously. The sample just prior to bubble removal is shown in Fig. 2. Another way to look at these data is shown in Fig. 3, where these data are transformed and plotted as a function of supercooled temperature. The time axis is readily transformed into temperature since in this experiment the supercooled temperature DT is just the cooling rate, a, times the
The apparatus and sample used in these experiments are identical to that described in reference [2] and the the basic operation of the instrument is described here. The sample studied here is 200 ll of commercial distilled water [22] which resides in a shortened NMR tube made of glass, and the measured linear cooling rate, a, for these experiments is 0:1131 K s1 or equivalently 6:786 K min1 . We show elsewhere [2] that this linear cooling rate may be varied by over an order of magnitude without affecting the conclusions. In this experiment, the temperature is decreased linearly and continuously by solid state Peltier devices which continue to decrease the sample temperature until the sample freezes. The scattering and interruption of a signal from a laser diode which shines through the NMR tube and onto a photodiode detector indicates when the sample is frozen. Upon detection of freezing, the lag-time is recorded and the apparatus reset for the next run by heating the sample with the Peltier devices to 283 K for four minutes ensuring all residual ice crystals have melted prior to commencing another run. This cycle is then repeated as many times as
Fig. 1. For pure water, the measured lag-times as a function of run number, denoted the ÔManhattanÕ. Two distinct regions are evident: the longer runs and lower temperatures of nucleation when no bubble is present (set A), and shorter runs and higher temperatures of nucleation when there is a bubble (set B).
A.F. Heneghan, A.D.J. Haymet / Chemical Physics Letters 368 (2003) 177–182
179
total time elapsed after passing through the equilibrium freezing temperature, Tf , DT ¼ a t:
Fig. 2. The NMR tube housing the sample of water used in these experiments. The bubble can clearly be seen at the bottom of the NMR tube.
Fig. 3. The temperature as a function of cooling rate time for the data, together with the temperature at nucleation with the bubble (filled squares, N0 ¼ 235) and without the bubble (open circles, N0 ¼ 209).
ð1Þ
The temperature at which nucleation occurs for the runs in Set A is denoted by a filled square, and for Set B is denoted by an open circle. The inset provides a closer look at the temperatures of nucleation for the sample with and without the bubble, and shows clearly the two temperature ranges where the samples freeze. Note that these two sets overlap slightly due to the inherent width of each of the survival curves. As a guide to the eye, the complete, measured thermal history of a single run, number 46, is shown as the solid line on this plot. One possible way to examine these data is to make a histogram of the number of samples which freeze in each time or temperature interval; however, the choice of the size for the bin width affects the final measured distribution. Instead, we calculate a survival probability [23,24], which is just the number of samples in the set unfrozen N ðtÞ after time ÔtÕ divided by the total number of samples in the data set N0 . Again, the time axis may be converted readily to temperature for the linear supercooling case, and these survival probabilities are shown in Fig. 4 for these two sets of data.
Fig. 4. The survival probability (top panel) for the sample with the bubble and without the bubble. The bubble shifts the survival probability to warmer temperatures. The companion nucleation curve (bottom panel) for the sample with the bubble and without the bubble.
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A.F. Heneghan, A.D.J. Haymet / Chemical Physics Letters 368 (2003) 177–182
The dramatic acceleration of nucleation when the bubble is present is shown clearly in this probability. In some alternate analyses [25], the maximum of the first derivative of this probability is determined and used in further analysis. We have shown that the temperature which 50% of the samples survive T50% is a good approximation to the maximum of the first derivative [2]. Elsewhere [1] we show this quantity is an excellent definition for the so-called Ôsupercooling pointÕ of biological solutions. Here, we simply use the temperature at which 50% of the samples are frozen as a guide to differentiate between the data sets with the bubble and without the bubble. The supercooled temperature at which half the samples freeze is )13.1 °C without the bubble, and )10.7 °C with the bubble present. Hence, the presence of the bubble induces nucleation at 2.4 °C warmer temperatures than the sample without the bubble, which is a significant effect. In this sample, the bubble is a surface on which nucleation may occur, by reducing the effective free energy barrier to nucleation. The acceleration of the rate of nucleation, due to the lowering of the free energy barrier, is similar to the effect obtained by adding an (insoluble) AgI crystal into the solution. The AgI is thought to provide a surface where the crystal phase may form more readily. AgI is known to be an extremely good ice nucleator [26]. We show elsewhere [1] the presence of an AgI crystal shifts the entire survival probability to temperatures 7.65 °C warmer than for pure water without the AgI present. Hence, it is clear that the bubble acts as a nucleator, albeit not as efficient as an AgI crystal. The actual bubble formed and present during all indicated runs is shown above in the photograph of Fig. 2. Even after many thermal cycles, the bubble size and position within the NMR tube remained constant, positioned at the bottom of the NMR tube. The bubble was removed mechanically after run 294 by vigorous tapping on the bottom of the NMR tube. The approximate diameter of the NMR tube is 5000 lm and the approximate bubble diameter is 2000 lm. The effect of different bubble sizes or different bubble positions within the NMR tube has not yet been measured in our apparatus. In the absence of the bubble (and any
other added nucleators), the supercooled water nucleates heterogeneously on the surface of the NMR tube, and there can be some variation (unimportant for this study) from tube to tube. For this reason we record a unique identification number of each tube used in our laboratory, and tube 20010701 is used here. This is the same tube used in supercooled water experiments analyzed in reference [2]. 3. Discussion The survival probabilities shown in Fig. 4(top panel) may be analyzed according to the method described earlier by us [1] to obtain the so called Ônucleation curveÕ, by which we denote the average lag-time, hsi as a function of supercooled temperature DT . For this sample with and without the bubble, this analysis is presented in Fig. 4(bottom panel) and outlined here. Our simple analysis of these data draws upon ideas from macroscopic chemical kinetics. We consider an initial state A, the ÔreactantÕ (the supercooled liquid) which evolves to the equilibrium state B the ÔproductÕ (the crystal) with ÔrateÕ k, k
A ! B:
ð2Þ
At equilibrium the system is all ÔproductÕ. We may now examine various assumptions for the macroscopic kinetics of this ÔreactionÕ, as a test of whether or not our data are consistent with the given assumption [23]. Under this assumption, the nucleation event occurs in a single step, evolving directly from the supercooled state A to the final crystal state B. Hence, the survival probability is F ðtÞ ¼ N ðtÞ=N0 ¼ exp½kt ;
ð3Þ
and the average lag time is hsi ¼ k 1 ¼ ln 2 half-life. Further, we assume a functional form for the dependence of the average lag-time, hsi, as a function of the fixed supercooled temperature, DT . One such functional form arises from the formalism of classical nucleation theory [7], " # Wb3 hsi ¼ P exp ; ð4Þ 2 T ðDT Þ
A.F. Heneghan, A.D.J. Haymet / Chemical Physics Letters 368 (2003) 177–182
where P and Wb3 are parameters which are fit to the data. In this work, for the sample without the bubble P ¼ 6:47 1010 and Wb1=3 ¼ 34:37 K, while for the sample with the bubble P ¼ 5:80 109 and Wb1=3 ¼ 19:4 K. Thus from this single experiment and subsequent fit to the data, we have obtained direct fits for the kinetic prefactor parameter P and the parameter Wb3 . These data are further used to plot the functional for the average lag-time, hsi, as a function of the fixed supercooled temperature, DT for the sample both with and without the bubble present. Again, this function demonstrates the dramatic acceleration of the bubble-induced heterogeneous nucleation. As mentioned above, this first derivative is the instantaneous nucleation probability at any given supercooled temperature. Fig. 5 shows this first derivative for the samples with and without the bubble. Although there is some overlap between the nucleation probabilities, Fig. 5 shows the very low probability of nucleation at cooler temperatures for the sample with the bubble. Conversely, it is not very probable for the sample without the
Fig. 5. The first derivative of the fit to the function for the survival probability for the sample with the bubble (left) and without the bubble (right). The presence of the bubble shifts the first derivatives or instantaneous nucleation probability to warmer temperatures.
181
bubble to nucleate at warmer temperatures. Fig. 5 shows clearly the two distinct ranges of nucleation temperatures for the sample with and without the bubble. These data may be compared to the work by Inada et al. [17] in which the effect of ultrasonic vibration upon the nucleation temperature is examined. In that study, the frequency and the power output of an ultrasonic vibrator were chosen such that cavitation occurred within the water sample. They were able to induce nucleation at temperatures approximately 8 °C warmer by using ultrasonic vibration. In our study, we are able to induce nucleation with a single bubble at temperatures which are approximately 3 °C warmer. There are a couple of sources for the difference. The first is the sample volume. We use an NMR tube with an inner diameter of 5 mm while they use a vessel with an inner diameter of 140 mm. There is a greater probability for heterogenous nucleation to occur on larger surface areas, and bubble volumes. A second source of difference is the method of formation of the bubbles. Our bubble is stationary, and it is not formed as the result of a sono-mechanical disturbance to the system. The disruption to the system by ultrasonic vibration is greater than the disruption to the system by a stationary air bubble. We have shown the presence of a bubble within our water sample leads to accelerated nucleation. The entire nucleation probability distribution is shifted to warmer temperatures. This bubble provides an additional surface on which nucleation may take place, lowering the effective free energy barrier to nucleation. This bubble serves as a catalyst to the nucleation. This process is similar to introducing an impurity such as an (insoluble) crystal of AgI into the sample holder. Elsewhere [1] we have shown the effect of introducing an AgI crystal into the water solution. For comparison, the temperature where 50% of the samples survive when an AgI crystal is introduced is )6.1 °C while the bubble only shifts the temperature where 50% survive to )10.7 °C. Although the air bubble induces nucleation at warmer temperatures than without the air bubble, the air bubble does not appear to be as good a nucleator as a crystal of AgI. The challenge to theory is to explain the
182
A.F. Heneghan, A.D.J. Haymet / Chemical Physics Letters 368 (2003) 177–182
molecular origin of such shifts in the Ônucleation curveÕ by external nucleators such as AgI, bubbles, and biological ÔantifreezeÕ molecules.
Acknowledgements The authors acknowledge gratefully support from ACS-PRF under grant number AC 33707AC9, and Texas ARP grant 003652-0303-1999 supporting ice/water interface studies.
References [1] A.F. Heneghan, P. Wilson, A.D.J. Haymet, PNAS 99 (2002) 9631. [2] A.F. Heneghan, A.D.J. Haymet, J. Chem. Phys 117 (2002) 5319. [3] J. Huaug, L.S. Bartell, J. Phys. Chem. 99 (1995) 3924. [4] D.W. Oxtoby, Nature 347 (1990) 725. [5] V. Talanquer, D.W. Oxtoby, J. Chem. Phys. 109 (1998) 223. [6] V. Talanquer, D.W. Oxtoby, J. Phys. Chem. 99 (1995) 2865. [7] A. Laaksonen, V. Talanquer, D.W. Oxtoby, Ann. Rev. Phys. Chem. 46 (1995) 489. [8] A. Laaksonen, I. Napari, J. Phys. Chem. B 105 (2001) 11678.
[9] O. Galkin, P.G. Vekilov, Proc. Natl. Acad. Sci. 97 (2000) 6277. [10] D.M. Blow, N. Chayen, L.F. Floyd, E. Saridakis, Protein Sci. 3 (1994) 1638. [11] N.E.C. Ane, J. Radcliffe, D.M. Blow, Protein Sci. 2 (1993) 113. [12] B.J. Mason, Contemp. Phys. 33 (1992) 227. [13] J. Nelson, C. Knight, J. Atmos. Sci. 55 (1998) 1452. [14] E.K. Harper, C.F. Shoemaker, J. Food Sci. 48 (1983) 1801. [15] W.L. Kerr, D.S. Reid, Lebens. Wiss. u Technol. 27 (1994) 225. [16] C.A. Knight, J. Hallett, A.L. DeVriess, Cryobiology 25 (1988) 55. [17] T. Inada, X. Zhang, A. Yabe, Y. Kozawa, Int. J. Heat Mass Tran. 44 (2001) 4523. [18] X. Zhang et al., Int. J. Heat Mass Tran. 44 (2001) 4533. [19] L. Granasy, T. Borzsonyi, T. Pusztai, P.F. James, European Space Agency. [Special Publication] SP SP-454 (2001) 629. [20] K. Fairley et al., J. Biol. Chem. 277 (2002) 24073. [21] P.W. Wilson, A.F. Heneghan, A.D.J. Haymet, Cryobiology June (2002), in press. [22] The water is OmniSolv reagent grade distilled water which was filtered using a 0.2 lM filter. The water was purchased from EM Science. [23] T. Barlow, A. Haymet, Rev. Sci. Instruments 66 (1995) 2996. [24] A.F. Haneghan, P.W. Wilson, G. Wang, A.D.J. Haymet, J. Chem. Phys. 115 (2001) 7599. [25] V.P. Skripov, Curr. Top. Mater. Res. 2 (1976) 327. [26] B. Vonnegut, J. Appl. Phys. 18 (1947) 593.
Liquid-to-crystal heterogeneous nucleation: bubble accelerated nucleation of pure supercooled water A.F. Heneghan, A.D.J. Haymet
*
Department of Chemistry, University of Houston, Houston, TX 77204-5003, USA Received 4 October 2002; in final form 14 November 2002
Abstract Bubble formation in a single sample of pure water, held in the sample holder of our automated lag-time apparatus, is shown to lead to accelerated liquid-to-crystal heterogeneous nucleation. This phenomenon shifts the entire probability distribution for heterogeneous nucleation, together with the average lag-time hsi, to significantly warmer temperatures. In the linear supercooling experiment analyzed here, this is equivalent to reducing the average time to nucleation. Using a single assumption tested in previous work, the entire Ônucleation curveÕ is extracted from a single experiment. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Nucleation Bubble accelerated heterogeneous nucleation is studied here using a new version of our automated lag-time apparatus, denoted ALTA4, which we have designed, built and operated recently. Operated in a linear supercooling mode, this experiment determines the so-called Ônucleation curveÕ from a continuous, automated experiment on a single (unchanging) pure water sample for several hundred consecutive runs [1,2]. Here we quantify the accelerated rate of nucleation caused by air bubble formation in a sample of supercooled pure water used in one of our experiments. The entire survival probability distribution is shifted by approxi-
*
Corresponding author. Fax: +713-743-2709. E-mail address: [email protected] (A.D.J. Haymet).
mately 2.4 °C to warmer temperatures by the presence of the bubble in the sample. This is a significant and potentially useful effect. Much recent effort has been focused on understanding liquid-to-crystal nucleation [3–5], but challenges still remain. Important differences between liquid-to-crystal nucleation and gas-toliquid nucleation [6] have been exhibited [7,8]. It is perhaps not widely known that, in the liquid-tocrystal case, classical nucleation theory and its socalled Ôcapillarity approximationÕ break down [7]. A further challenge is to understand how to control the nucleation temperature of supercooled solutions. This is of importance in many fields including protein crystallization [9–11], where different products may be formed depending on the temperature of crystallization, artificial snow making [12,13], where conversion of water to snow is sought at as warm a temperature as possible,
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 8 3 5 - 3
178
A.F. Heneghan, A.D.J. Haymet / Chemical Physics Letters 368 (2003) 177–182
and in the food industry [14–16] where the temperature of recrystallization directly affects food and food quality. Control of the nucleation temperature is important for so-called Ôcold energyÕ storage devices [17]. In these devices, a slurry of ice and supercooled water is used to help level electrical load discrepancies between day and night time caused by increasing air conditioning use. Our experiments, coupled with previous work [17,18], yield a data set which may be used to understand how to control the nucleation temperature in supercooled solutions, and evaluate candidate theories for crystal nucleation [19]. The introduction of bubbles into the supercooled solutions is a promising method for increasing the nucleation temperatures of these solutions. Elsewhere we exhibit the dependence of the nucleation temperature upon addition of crystals of (insoluble) AgI salt [1] and biological ÔantifreezeÕ proteins [20,21].
2. Supercooled liquid to crystal nucleation
possible on a single sample to generate the statistics of liquid-to-crystal nucleation. The data collected in this experiment are the lag-time to nucleation ti after cooling below the equilibrium freezing and melting temperature Tm , and the temperature of nucleation, Ti , as a function of run number for a single aqueous sample. Fig. 1 shows the measured lag-times as a function of run number for 444 consecutive runs on a single sample, which were collected uninterrupted over the course of 4 days on ALTA4. Note that the data are cleanly partitioned into two distinct sets, Set A with longer lag-times and set B with shorter lag-times. After run 294 the bubble was removed mechanically by shaking the NMR tube vigorously. The sample just prior to bubble removal is shown in Fig. 2. Another way to look at these data is shown in Fig. 3, where these data are transformed and plotted as a function of supercooled temperature. The time axis is readily transformed into temperature since in this experiment the supercooled temperature DT is just the cooling rate, a, times the
The apparatus and sample used in these experiments are identical to that described in reference [2] and the the basic operation of the instrument is described here. The sample studied here is 200 ll of commercial distilled water [22] which resides in a shortened NMR tube made of glass, and the measured linear cooling rate, a, for these experiments is 0:1131 K s1 or equivalently 6:786 K min1 . We show elsewhere [2] that this linear cooling rate may be varied by over an order of magnitude without affecting the conclusions. In this experiment, the temperature is decreased linearly and continuously by solid state Peltier devices which continue to decrease the sample temperature until the sample freezes. The scattering and interruption of a signal from a laser diode which shines through the NMR tube and onto a photodiode detector indicates when the sample is frozen. Upon detection of freezing, the lag-time is recorded and the apparatus reset for the next run by heating the sample with the Peltier devices to 283 K for four minutes ensuring all residual ice crystals have melted prior to commencing another run. This cycle is then repeated as many times as
Fig. 1. For pure water, the measured lag-times as a function of run number, denoted the ÔManhattanÕ. Two distinct regions are evident: the longer runs and lower temperatures of nucleation when no bubble is present (set A), and shorter runs and higher temperatures of nucleation when there is a bubble (set B).
A.F. Heneghan, A.D.J. Haymet / Chemical Physics Letters 368 (2003) 177–182
179
total time elapsed after passing through the equilibrium freezing temperature, Tf , DT ¼ a t:
Fig. 2. The NMR tube housing the sample of water used in these experiments. The bubble can clearly be seen at the bottom of the NMR tube.
Fig. 3. The temperature as a function of cooling rate time for the data, together with the temperature at nucleation with the bubble (filled squares, N0 ¼ 235) and without the bubble (open circles, N0 ¼ 209).
ð1Þ
The temperature at which nucleation occurs for the runs in Set A is denoted by a filled square, and for Set B is denoted by an open circle. The inset provides a closer look at the temperatures of nucleation for the sample with and without the bubble, and shows clearly the two temperature ranges where the samples freeze. Note that these two sets overlap slightly due to the inherent width of each of the survival curves. As a guide to the eye, the complete, measured thermal history of a single run, number 46, is shown as the solid line on this plot. One possible way to examine these data is to make a histogram of the number of samples which freeze in each time or temperature interval; however, the choice of the size for the bin width affects the final measured distribution. Instead, we calculate a survival probability [23,24], which is just the number of samples in the set unfrozen N ðtÞ after time ÔtÕ divided by the total number of samples in the data set N0 . Again, the time axis may be converted readily to temperature for the linear supercooling case, and these survival probabilities are shown in Fig. 4 for these two sets of data.
Fig. 4. The survival probability (top panel) for the sample with the bubble and without the bubble. The bubble shifts the survival probability to warmer temperatures. The companion nucleation curve (bottom panel) for the sample with the bubble and without the bubble.
180
A.F. Heneghan, A.D.J. Haymet / Chemical Physics Letters 368 (2003) 177–182
The dramatic acceleration of nucleation when the bubble is present is shown clearly in this probability. In some alternate analyses [25], the maximum of the first derivative of this probability is determined and used in further analysis. We have shown that the temperature which 50% of the samples survive T50% is a good approximation to the maximum of the first derivative [2]. Elsewhere [1] we show this quantity is an excellent definition for the so-called Ôsupercooling pointÕ of biological solutions. Here, we simply use the temperature at which 50% of the samples are frozen as a guide to differentiate between the data sets with the bubble and without the bubble. The supercooled temperature at which half the samples freeze is )13.1 °C without the bubble, and )10.7 °C with the bubble present. Hence, the presence of the bubble induces nucleation at 2.4 °C warmer temperatures than the sample without the bubble, which is a significant effect. In this sample, the bubble is a surface on which nucleation may occur, by reducing the effective free energy barrier to nucleation. The acceleration of the rate of nucleation, due to the lowering of the free energy barrier, is similar to the effect obtained by adding an (insoluble) AgI crystal into the solution. The AgI is thought to provide a surface where the crystal phase may form more readily. AgI is known to be an extremely good ice nucleator [26]. We show elsewhere [1] the presence of an AgI crystal shifts the entire survival probability to temperatures 7.65 °C warmer than for pure water without the AgI present. Hence, it is clear that the bubble acts as a nucleator, albeit not as efficient as an AgI crystal. The actual bubble formed and present during all indicated runs is shown above in the photograph of Fig. 2. Even after many thermal cycles, the bubble size and position within the NMR tube remained constant, positioned at the bottom of the NMR tube. The bubble was removed mechanically after run 294 by vigorous tapping on the bottom of the NMR tube. The approximate diameter of the NMR tube is 5000 lm and the approximate bubble diameter is 2000 lm. The effect of different bubble sizes or different bubble positions within the NMR tube has not yet been measured in our apparatus. In the absence of the bubble (and any
other added nucleators), the supercooled water nucleates heterogeneously on the surface of the NMR tube, and there can be some variation (unimportant for this study) from tube to tube. For this reason we record a unique identification number of each tube used in our laboratory, and tube 20010701 is used here. This is the same tube used in supercooled water experiments analyzed in reference [2]. 3. Discussion The survival probabilities shown in Fig. 4(top panel) may be analyzed according to the method described earlier by us [1] to obtain the so called Ônucleation curveÕ, by which we denote the average lag-time, hsi as a function of supercooled temperature DT . For this sample with and without the bubble, this analysis is presented in Fig. 4(bottom panel) and outlined here. Our simple analysis of these data draws upon ideas from macroscopic chemical kinetics. We consider an initial state A, the ÔreactantÕ (the supercooled liquid) which evolves to the equilibrium state B the ÔproductÕ (the crystal) with ÔrateÕ k, k
A ! B:
ð2Þ
At equilibrium the system is all ÔproductÕ. We may now examine various assumptions for the macroscopic kinetics of this ÔreactionÕ, as a test of whether or not our data are consistent with the given assumption [23]. Under this assumption, the nucleation event occurs in a single step, evolving directly from the supercooled state A to the final crystal state B. Hence, the survival probability is F ðtÞ ¼ N ðtÞ=N0 ¼ exp½kt ;
ð3Þ
and the average lag time is hsi ¼ k 1 ¼ ln 2 half-life. Further, we assume a functional form for the dependence of the average lag-time, hsi, as a function of the fixed supercooled temperature, DT . One such functional form arises from the formalism of classical nucleation theory [7], " # Wb3 hsi ¼ P exp ; ð4Þ 2 T ðDT Þ
A.F. Heneghan, A.D.J. Haymet / Chemical Physics Letters 368 (2003) 177–182
where P and Wb3 are parameters which are fit to the data. In this work, for the sample without the bubble P ¼ 6:47 1010 and Wb1=3 ¼ 34:37 K, while for the sample with the bubble P ¼ 5:80 109 and Wb1=3 ¼ 19:4 K. Thus from this single experiment and subsequent fit to the data, we have obtained direct fits for the kinetic prefactor parameter P and the parameter Wb3 . These data are further used to plot the functional for the average lag-time, hsi, as a function of the fixed supercooled temperature, DT for the sample both with and without the bubble present. Again, this function demonstrates the dramatic acceleration of the bubble-induced heterogeneous nucleation. As mentioned above, this first derivative is the instantaneous nucleation probability at any given supercooled temperature. Fig. 5 shows this first derivative for the samples with and without the bubble. Although there is some overlap between the nucleation probabilities, Fig. 5 shows the very low probability of nucleation at cooler temperatures for the sample with the bubble. Conversely, it is not very probable for the sample without the
Fig. 5. The first derivative of the fit to the function for the survival probability for the sample with the bubble (left) and without the bubble (right). The presence of the bubble shifts the first derivatives or instantaneous nucleation probability to warmer temperatures.
181
bubble to nucleate at warmer temperatures. Fig. 5 shows clearly the two distinct ranges of nucleation temperatures for the sample with and without the bubble. These data may be compared to the work by Inada et al. [17] in which the effect of ultrasonic vibration upon the nucleation temperature is examined. In that study, the frequency and the power output of an ultrasonic vibrator were chosen such that cavitation occurred within the water sample. They were able to induce nucleation at temperatures approximately 8 °C warmer by using ultrasonic vibration. In our study, we are able to induce nucleation with a single bubble at temperatures which are approximately 3 °C warmer. There are a couple of sources for the difference. The first is the sample volume. We use an NMR tube with an inner diameter of 5 mm while they use a vessel with an inner diameter of 140 mm. There is a greater probability for heterogenous nucleation to occur on larger surface areas, and bubble volumes. A second source of difference is the method of formation of the bubbles. Our bubble is stationary, and it is not formed as the result of a sono-mechanical disturbance to the system. The disruption to the system by ultrasonic vibration is greater than the disruption to the system by a stationary air bubble. We have shown the presence of a bubble within our water sample leads to accelerated nucleation. The entire nucleation probability distribution is shifted to warmer temperatures. This bubble provides an additional surface on which nucleation may take place, lowering the effective free energy barrier to nucleation. This bubble serves as a catalyst to the nucleation. This process is similar to introducing an impurity such as an (insoluble) crystal of AgI into the sample holder. Elsewhere [1] we have shown the effect of introducing an AgI crystal into the water solution. For comparison, the temperature where 50% of the samples survive when an AgI crystal is introduced is )6.1 °C while the bubble only shifts the temperature where 50% survive to )10.7 °C. Although the air bubble induces nucleation at warmer temperatures than without the air bubble, the air bubble does not appear to be as good a nucleator as a crystal of AgI. The challenge to theory is to explain the
182
A.F. Heneghan, A.D.J. Haymet / Chemical Physics Letters 368 (2003) 177–182
molecular origin of such shifts in the Ônucleation curveÕ by external nucleators such as AgI, bubbles, and biological ÔantifreezeÕ molecules.
Acknowledgements The authors acknowledge gratefully support from ACS-PRF under grant number AC 33707AC9, and Texas ARP grant 003652-0303-1999 supporting ice/water interface studies.
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