Kinetics of Homogeneous Nucleation in the Freezing of Large Water

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J. Phys. Chem. 1995,99, 3924-3931

Kinetics of Homogeneous Nucleation in the Freezing of Large Water Clusters Jinfan Huang and Lawrence S. Bartell* Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 Received: August 8, 1994; In Final Form: November 15, 1994@

Water clusters of 4000-6000 molecules were produced by condensation of vapor in supersonic flow and cooled by evaporation until they froze at about 200 K. Rates of nucleation up to 1030m-3 s-’ were determined by electron diffraction measurements at microsecond intervals. Although nucleation rates were 20 orders of magnitude higher than in previous investigations of the freezing of water, this enormous disparity was accounted for naturally by the classical theory of nucleation. The free energy u,l of the solid-liquid interface implied by the results increases with temperature as F ,with n approximately 0.3-0.4, the same range of values as found for mercury in the only well-established trend known to the present investigators. The interfacial free energy of 21.6 mJ/m2 derived for clusters of water is virtually the same as that obtained for small water droplets by several workers but is substantially lower than the value inferred from the interfacial tension in the bulk system at 0 “C. This difference is a consequence of the different forms of ice encountered in the different experiments. Bulk water freezes to the thermodynamically stable hexagonal ice (Ih), whereas highly supercooled water freezes to the kinetically favored cubic ice (IC), a reaction product offering a lower free energy barrier. Anomalously, the ratio of uSlto the heat of fusion per unit area derived for supercooled water is only about two-thirds that suggested originally for water by Tumbull but since found to apply quite well to other nonmetallic substances. Two variants of the classical theory of homogeneous nucleation are compared, and some deficiencies of the theory are discussed.

Introduction The principal aim of the present research program is to determine how well current views of nucleation’ account for the kinetics of phase changes in condensed matter. Although the subject is fundamental and of considerable intrinsic importance in science and technology, it remains severely underinvestigated because the traditional techniques have been difficult to apply definitively. Our new method of studying nucleation by following the time evolution of phase transitions in beam of large molecular clusters formed in an expanding supersonic jet2-6 offers an alternative but incompletely evaluated approach. The present experiments offer the most detailed test of the method to date. A significant objective of our nucleation research is to provide new information about water, certainly the most important liquid by any standard of judgment. Frank’s pithy aphorism “Of all known liquids, water is probably the most studied and least understood” remains as true today as in 1972 when it was written.’ If anything, the enigma has deepened and the controversy surrounding it intensified in the intervening years.*-I6 It is premature to review here the anomalous behavior of supercooled water or the various hypotheses to account for it. We will examine just one aspect of the liquid but at a far deeper supercooling than had been attained in earlier investigations. What has been studied is the homogeneous nucleation of ice in very cold water and at an enormously higher rate than ever measured before. But the water was in the form of clusters of 4000-6000 molecules, immense by the standards of cluster research but submicroscopic and minuscule by the standards of previous research on nucleation. One of the points of contention about water concems whether the liquid, when confined to small dimensions, can be regarded as true liquid water.IJ-l7 It is of interest then to find to what degree the results for our minute water drops show commonality with the previous body of information about nucleation in supercooled water. Abstract published in Advance ACS Absrracts, March 1, 1995.

So far, among the several systems investigated in our research on the dynamics of f r e e ~ i n g , ~water , ~ . ~is the first example that had been studied definitively before. The greater body of early work on the freezing of highly supercooled water had been on fine mists formed in cloud chambers.8 Because little information about the duration of the freezing process was acquired, it was possible only to make rough guesses of nucleation rates. In these studies some of the nucleation was heterogeneous, especially in the case of the larger and warmer droplets. Fortunately, two careful and detailed studies of the freezing of droplets dispersed in emulsions have been In these investigations it was possible to discriminate between heterogeneous and homogeneous nucleation and to obtain what appear to be reliable homogeneous nucleation rates at well established temperatures. In neither these nor the earlier studies of nucleation in fine droplets had it been possible to determine the structure of the solid formed. It seems generally to have been assumed that the crystalline nuclei were of ordinary (hexagonal) ice. As will be shown presently, this assumption is almost certainly wrong. Two research groups have published results of electron diffraction analyses of the structures of water clusters formed (as in the present investigation) in supersonic flow.*O.*’ Both groups observed that their clusters were solid and of cubic ice. The clusters had not been examined early enough after their formation for possible liquid precursors of the crystals to have been seen, however. Therefore, no transitions were seen or suspected and no information about the kinetics of freezing was obtained. In the following we sketch the new experimental method and analyze the results it gives for water. We then reanalyze the results of prior researchIs.l9 using newer measurements of the properties of water to achieve a common basis of interpretation, thereby permitting as close a comparison with the present results as can be made.

0022-3654/95/2099-3924$09.00/0 0 1995 American Chemical Society

J. Phys. Chem., Vol. 99, No. 12, 1995 3925

The Freezing of Large Water Clusters

Procedure Brief Outline of Method. Neon carrier gas saturated with water vapor expands in flow through a miniature Laval (supersonic) nozzle and cools sufficiently in the process to condense the vapor into clusters of liquid water. As they pass through the nozzle surrounded by moist gas, the clusters remain substantially warmer than the gas because of the continuing condensation and evolution of heat. As they stream into the evacuated diffraction chamber beyond the nozzle at temperatures well above the so-called "evaporative cooling temperature," 22,23 they cool rapidly by evaporation until they freeze. The time evolution of cluster temperatures is inferred from the kinetics of evaporation calculated for a beam of microdrops in a rarefied, diverging jet of neon and water The transition from liquid to crystalline clusters is monitored by electron diffraction at time-of-flight intervals spaced closely enough together to define the rate of freezing. Experimental Details. Two sets of experiments were carried out, both at stagnation pressures of 4.4bar. Neon (99.999%, Air Products) was passed slowly through a cell containing water at 95 (run 1) or at 120 "C (run 2), to produce mole fractions of water vapor in the samples of approximately 0.2 and 0.4. The resultant mix was delivered through a pulsed valve27in 0.4 ms bursts at 10 Hz into the throat of a glass Laval nozzle (throat and exit diameters, 0.02 and 0.202 cm, respectively, and length 2.08 cm). An electron beam, pulsed in synchrony with the pulsed valve but with a duration time of 0.3 ms and delay of 0.6 ms, probed the steady-state plateaus of the supersonic pulses. Electron diffraction patterns were recorded on 4 x 5 in. Kodak medium slide photographic plates after being masked selectively by a rotating sector (sector figure, rl). Patterns were accumulated over 450- 1600 exposures. A vee skimmer2' selected the desired stream tubes of the cluster beam for diffraction analysis while screening the electron beam from the remaining output of the supersonic jet. Clusters were examined at spatial intervals of 0.3 cm, beginning 0.9 cm after their exit from the nozzle. Cluster velocities, measured in a flight were found to be 825 and 919 m/s in experiments with vapor initially at 95" and 120". The warmer sample had a greater initial enthalpy to be converted into mass flow. It is necessary to estimate a number of physical properties of supercooled water before an analysis of nucleation rates can be carried out. Values of the properties used and the methods for estimating them are found in the Appendix. Analysis of Data. From the development of crystalline peaks in the diffraction pattems can be derived the fractions, F(t), of clusters that have frozen. As discussed p r e v i ~ u s l y the , ~ ~time for clusters to freeze, once nucleated, is small compared with the statistical spread of nucleation times for the stochastic process. Therefore, the rate of freezing observed corresponds to the nucleation rate J rather than to the time for individual clusters to freeze. In principle, then, from each successive pair of values, F(tn)and F(tn+l),at known times of flight tfland tn+l could be calculated the nucleation rate at the mean temperature of the measurements. via

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s, A - ' Figure 1. Electron diffraction pattems of freezing water clusters at various times of flight beyond the nozzle, from top to bottom, in microseconds: (a) 10.9, (b) 14.5, (c) 18.2, (d) 21.8, (e) 25.5, (f) 29.1. I

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Figure 2. Electron diffraction pattems of freezing water clusters at various times of flight beyond the nozzle, from top to bottom, in microseconds: (a) 13.1, (b) 16.3, (c) 19.6, (d) 22.9, (e) 26.1, (029.4, (8) 32.6.

function J(t), whence F(t) is deduced by numerical integration of the expression

d ln[l - F(t)] = -J(t)Vc, dt

(2)

By adjusting the principal unknown of the classical theory, the parameter us,, it is possible to construct a curve F(t) passing smoothly through the experimental points. The kinetic parameter aslis usually referred to as the free energy per unit area of the interface between the old and new phases. More will be said about it presently. For the purposes of the analysis aSlis assumed to increase with temperature as OS1(T)=

a,,(T,)(T/T,)"

(3)

with n in the vicinity of 0.3 (see Discussion). The two variants of the classical theory of homogeneous nucleation adopted for the analysis are outlined in the appendix.

Results

where VClis the mean cluster volume. In practice, the noise in the data makes it advantageous to analyze the entire collection of data from a run together. This is complicated by the fact that the cluster temperatures are slowly cooling during the run. Therefore, the profile T(t) experienced by the clusters is ~ a l c u l a t e d ~and ~ - used ~ ~ to convert the thermal function J(T) derived from classical nucleation t h e ~ r y ~ to , ~the ~ ?temporal ~'

Clusters were produced abundantly in both sets of supersonic expansions. A greater fraction of the water vapor condensed, and it condensed into larger clusters, in the run begun at the lower stagnation temperature (95 "C, run 1, as opposed to run 2, initially at 120 "C). This effect has less to do with the temperature itself than with the higher relative concentration of carrier gas in run 1 (5:l Ne to water, as compared with 1S:l in run 2). Reproduced in Figures 1 and 2 are the diffraction pattems of clusters superposed upon a background intensity

Huang and Bartell

3926 J. Phys. Chem., Vol. 99, No. 12, 1995 260 I

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Figure 3. Electrron diffraction pattems of ice clusters. Top: experimental pattern, including a small residual background of liquid water. Center: pattem calculated for cubic ice (IC).Bottom: pattem calculated for hexagonal ice (Ih).

arising from carrier gas and uncondensed water vapor. From the observed progression of the cluster intensities from those of a liquid to those of randomly oriented microcrystals can be derived the fraction of clusters that have frozen. Not only do the pattems yield the rate of the phase change from the time evolution of the intensities, they clearly identify the crystalline form produced as that of cubic ice ( 1 ~ ) from ~ ~ the s ~positions ~ of the Bragg reflections. Figure 3 compares a diffraction pattern of clusters obtained in run 1 (including some unfrozen material) with pattems calculated for perfect microcrystals of cubic and hexagonal ice. Moreover, breadths of the powder diffraction lines show that the cluster diameters were 71 8, (run 1) and 60 8, (run 2 ) after they froze and evaporatively dissipated the heat evolved during freezing. These values imply diameters of 72.5 and 61.5 A just before freezing (or 6300 and 3800 molecules per cluster) and 74 and 63 8, immediately after clusters left the nozzle, before they cooled to the freezing temperature by evaporation. See ref 6 for details of computing evaporation losses. Not all of the cluster volume implied by the diffraction pattems can be considered to be available for nucleation, however. In our molecular dynamics simulations of phase changes in cooling cluster^,^^.^^ critical nuclei have never been observed to form in the outermost, disordered layer of molecules in systems whose melt wets the solid. Accordingly, we have adopted the procedure of subtracting the outer molecular layer when calculating the effective volume of our clusters available for nucleation. By the conventions specified in ref 6, the effective cluster diameters were taken to be 66.3 and 55 8, in runs 1 and 2. Computations of the temperature profiles T(t) of the clusters in the two runs were carried out by numerically integrating the coupled differential equations governing the gas dynamics and mass and heat flux in the supersonic flow, i n ~ i d e ~and "~~ beyond26 the nozzle. Mass fractionation was taken into account approximately using experimental measurements of the fractionation in flow beyond the nozzle.37 Resultant temperatures are plotted in Figure 4. That the clusters of run 2 were hotter at the nozzle exit is due to the much higher concentration of uncondensed water vapor. Not only was the partial pressure of vapor initially higher, the fraction of vapor condensed during the expansion was lower because there was less carrier gas to act as a heat sink. The excess concentration of vapor surrounding the clusters initially kept them warm by continued, slow condensation and inhibited their evaporative cooling briefly after they left the nozzle. The resultant delay in the onset of freezing is evident in an examination of Figures 5 and 6. These figures compare the observed fractions frozen with those calculated

t, microsec Figure 4. Temperatures computed for water clusters as they cool by evaporation; elapsed times reckoned from time of exiting the nozzle. Ticks on the curves indicate the times at which the pattems of Figures 1 and 2 were recorded. I

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Figure 5. Fraction of clusters frozen as a function of the time after exiting the nozzle. Filled circles are experimental points for clusters condensed from vapor at an initial mole fraction of 0.2. Curves, left to right, calculated via eq 8 for interfacial free energies of 20.6, 21.6, and 22.6 mJ/m2.

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Figure 6. Fraction of clusters frozen as a function of the time after exiting the nozzle. Filled circles are experimental points for clusters condensed from vapor at an initial mole fraction of 0.4. Curves, left to right, calculated via eq 8 for interfacial free energies of 20.7, 21.7, and 22.7 mJ/m2.

from the classical nucleation theory using three different assumed interfacial free energies, usl. Results give a quick, visual impression of the sensitivity of F(t) to changes in u s ] . Nucleation rates for the two runs were inferred from the slopes of the curves in Figures 5 and 6 best representing the experimental points. Their values at 200 K are plotted and compared in Figure 7 with the results of Wood and WaltonI8

J. Phys. Chem., Vol. 99, No. 12, 1995 3927

The Freezing of Large Water Clusters

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T, K Figure 7. Experimental nucleation rates (m-3 s-I) for the freezing of water (expressed in terms of log[J(T)]): clusters, circles; water-in-oil emulsions, squares, ref 18, and solid diamonds, ref 19. Calculated temperature dependence according to eq 7 with prefactor of eq 8 (solid curve, us[ = P 3 0 ) or of eq 9 (dashes, usl =

36 32 h

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T, K Figure 8. Interfacial free energies (mJ/m2) for the boundary between ice and water. Values derived from nucleation rates to cubic ice, prefactor of eq 8: clusters, circles; water-in-oil emulsions, squares, ref 18, and solid diamonds (superposed on each other), ref 19. Dashed line, calculated with u,l= Value at 273 K with error bar computed from equilibrium contact angles between water and two crystals of hexagonal ice sharing a grain boundary.38

and Butorin and S k r i p ~ v . ’Free ~ energies of the water-ice interface derived from the nucleation rates (eq 2) are compared in Figure 8 with the thermodynamic water-ice value determined from interfacial tension measurements at the freezing point.38

Discussion Experimental versus Calculated Nucleation Rates. A remarkable feature of our experiments is the prodigious rate at which our clusters froze once the onset of the phase change was reached. Our nucleation rates climbed to 1030 m-3 s-I before a significant fraction of the clusters were able to freeze on the time scale of the experiment. These rates are 20 orders of magnitude higher than the typical rates measured by Wood and Walton’* and Butorin and S k r i p ~ v in ’ ~ microemulsions. Nevertheless, as shown in Figure 7, this enormous disparity is readily rationalized in terms of the classical theory of nucleation without assuming that the confined water in clusters differs in any essential way from water in vastly larger drops. Of course, such an agreement is in no way a proof that confined water is altogether “normal” in behavior. Whether eq 8 or eq 9 of the Appendix is employed, the function calculated via classical nucleation theory passes equally well through the points for the clusters and the microemulsions-provided that the exponent n adopted for the dependence of asl on temperature (eq 3) is adjusted suitably.

Values of n required to force J(T) to pass through the experimental points are 0.30 for the “viscous jump” model of eq 8 and 0.39 for the “free jump” model of eq 9. Nothing is known about the quantitative temperature dependence of the kinetic parameter usl for substances like water. Only for the freezing of mercury has the dependence been found with acceptable reliability according to a recent review of the matter.39 The classic study of nucleation in mercury by Tumbullm implied a value in the vicinity of 0.3-0.4, Le., a value of the magnitude found in the present investigation. Tumbull’s prefactor A resembled eq 8 somewhat more than eq 9 but tended to be intermediate (because the glass divergence of viscosity at low temperature was not built into his model). Of course, mercury is not sufficiently similar to water for the present agreement to be persuasive, but it is the only comparison available. As suggested by Tumbul14’and expanded upon by Spaepen,42n is expected to be positive rather than negative; that is, the free energy of the interface is expected to increase as temperature rises because of the interfacial entropy. The entropy tends to be negative because liquid in contact with the crystal is forced into a structure more ordered than that of the bulk. Another recent observation of the freezing at very deep s u p e r ~ o o l i n gsuggests ~~ a nucleation rate about 10 orders of magnitude higher than those in the microemulsions. The temperature of the water vapor at the onset of nucleation of liquid droplets in a cloud chamber was very well known (217 K). Unfortunately, the temperature of the droplets, themselves, when they froze, was not. It seems to have been intermediate between ours and those of the microemulsions. It is worth pointing out that our observed nucleation rates are many orders of magnitude higher than the maximum rate for water predicted by FukutaU in his calculations to illustrate the inhibiting effect of the work w’ (eq 11 in the Appendix) incurred when the droplet surface expands as a solid nucleus is formed in the interior. This discrepancy in rate arose because Fukuta took the difference in density between the liquid and solid to be that of hexagonal ice and water at the freezing point. Such an assumption would normally be an entirely reasonable one. But supercooled water is anomalous and its liquid density quickly decreases toward that of ice as the substance cools. At temperatures lower than about 225 K the density of l i q ~ i d ’ ~ . ~ ~ is very close to the density of hexagonal i ~ e and ~ ~ equally s ~ ~ close to the density of cubic ice inferred from our experimental lattice constant. Therefore, the w” correction is minor at high supercoolings. Interfacial Free Energies. Values of the interfacial free energies at 200 K derived from runs 1 and 2 were 21.55 and 21.7, d / m 2 , based of the prefactor of eq 8. The exact numerical values of us’derived depend, of course, upon the kinetic model used, being higher by about 8% if eq 9 is applied instead of eq 8. This difference can hardly be regarded as serious in view of the unsettled status of nucleation theory. Values of as’from various experiments at various temperatures are plotted in Figure 8. The points for the supercooled systems were all calculated (or recalculated) on the basis of eq 8 to be consistent. It can be seen in Figure 8 that the interfacial free energies derived from nucleation rates of clusters at 200 K are very close to those derived from the 20 order-of-magnitude lower rates in microemulsions. What appears at first glance to be serious is the large discrepancy between the values for highly supercooled droplets and the thermodynamic value for the bulk system at the freezing point. If one is tempted to attribute this discrepancy to the fact that asl determined from cluster data is a kinetic rather than a true thermodynamic variable, it should be remembered that the bulk value pertains to hexagonal ice,

Huang and Bartell

3928 J. Phys. Chem., Vol. 99, No. 12, 1995 whereas the cluster values are for cubic ice. In prior evaluations of kinetic versus thermodynamic values it was concluded that the two quantities seem to be the same, within the rather large scatter of experimental values.48 Nevertheless, it is fair to mention that Oxtoby, among others, considers the variable ~ $ 1 to have a valid, physical meaning only at the equilibrium freezing point.49 The similarity between the cluster values of o,l and those from the microemulsions leaves little doubt that the water droplets in the emulsions froze to cubic ice rather than to hexagonal ice. Indeed, why cubic ice nucleates in the first place is accounted for very well by the results in Figure 8. The lower interfacial free energy associated with cubic ice means a smaller barrier, AG*, to the formation of a cubic nucleus from the liquid. Therefore, the reason cubic ice is formed lies in the kinetics of the process. In experiments carried out on short time scales, the cubic form simply outcompetes the hexagonal. This experimental finding qualitatively corroborates theoretical modeling by T a k a h a ~ h i who , ~ ~ concluded on the basis of broken hydrogen bonds in surfaces that AG* is lower for cubic than for hexagonal nuclei. Quantitatively, our implied difference is greater. We know of no evidence to suggest a thermodynamic bias favoring cubic ice at any temperature. As far as we are aware, no direct transformation from the hexagonal form to the cubic has ever been seen. It is possible to obtain cubic ice by means other than the rapid freezing of small droplets. One way has been to warm the vitrified, amorphous solid (formed by subjecting liquid microdrops or vapor to very cold surfaces) until the solid melts at a temperature far below those of the present experiment. Once molten in its extremely supercooled condition, it quickly freezes to cubic ice.5' Another way to form it is from one of the high pressure forms of crystalline water.52 The two-component model of liquid water formulated by Robinson and co-workers" accounts for the unusual dependence of density on temperature by attributing the decreasing density upon cooling below 4 "C to an increasing concentration of an icelike component. It appears from the interfacial free energies that cold water fits more comfortably against cubic than hexagonal ice. This suggests the possibility that Robinson's low temperature liquid component may resemble cubic ice more than hexagonal. Some references have been made39to the claim of Wood and Walton that they successfully established the temperature dependence of ~ $ for 1 water. This claim was disputed by Butorin and S k r i p ~ v because '~ of the near impossibility in kinetic measurements on emulsions of attaining the requisite precision when experiments cover only a small span of temperature. In fact, in our own analyses of the raw data published by Wood and Walton, we obtained a scatter too large to yield a meaningful value for do,l/dT. Curiously, the slope do, I/dT reported by Wood and Walton made their interfacial free energies extrapolate to the thermodynamic value at the melting point.38 From the evidence cited above it can be seen that such an extrapolation implies far too steep a slope and an endpoint at the wrong type of interface. Turnbull Relation. In 1950 TumbullS3 noticed that the interfacial free energies found from the kinetics of freezing were closely related to the heat of fusion per unit area, Ahf,,,, defined by

Ah,, = AHf,,,/(V2NA)2'3

(4)

where AHfu5and V represent the molar heat and molar volume, and N A is Avogadro's number. Tumbull found that

where k T was about 0.45 for metals and 0.32 for water and several metalloids. At the time no very precise measurements of nucleation rates had been carried out on water, but Tumbull applied certain empirical rules of thumb relating nucleation rate to the degree of supercooling when freezing took place (which rules of thumb do not work for our molecular clusters). Nineteen years later when the thermodynamic value of ~ $ was 1 measured,38 it was found to agree very well with Tumbull's estimated value. It is an ironic twist, then, to find that the values of os[ derived from accurately measured nucleation rates of clusters and emulsions disagree markedly with Tumbull's projection, giving k T a value of only 0.22 for water droplets if the heat of fusion for hexagonal ice is used for Ahf,,,. That, of course, is the heat originally adopted. To be consistent in the application of Tumbull's relation, the appropriate heat of fusion to be used for nucleation in highly supercooled droplets should be that for cubic ice. That value has been reported54to be only 160 J/mol less at 160 K than the heat for hexagonal ice at the same temperature (2.6% less than AHf,, at 0 "C). Even though the value of k T originally proposed for water and other nonmetallic substances seems cot to apply accurately to water, it has worked quite well for the other clusters we have studied, including CC4 CH3CCb4 and slightly less well for NH3.h Critical Nuclei. According to the classical (capillary) nucleation t h e ~ r y , ~the ~ , number ~' of molecules contained in critical nuclei is

n* = ( 3 2 / 3 ) ~ [ ~ , , / ( A G ~V,W , ')]~V*N,~ where is the free energy of fusion and w' is a minor correction defined in the appendix (eq 11). Application of eq 6 to our results implies that there are 52 (eq 8) or 65 (eq 9) molecules in the critical nuclei of clusters at 200 K. Little significance should be attached to the exact values of n* because the theory completely ignores the thickness of the transition region between the old and new phases. Transition layers are believed to be several molecules t h i ~ k and, ~ ~ . ~ ~ therefore, could contain more molecules than computed above for n*. What effect such transition layers have on the nucleation rate and how the contents should be apportioned to the two phases are questions that have not been resolved in the capillary theory. Therefore, the application of the classical theory in the present treatment can be considered to be little more than an empirical scheme (with a certain historically significant rationale) to relate highly disparate rates on a common basis. Crystalline Form. Diffraction intensities of our clusters deviate somewhat from those calculated for perfect, syherical microcrystals of cubic ice, as illustrated in Figure 3. Some of the discrepancies in the figure are due to an unfrozen portion of the clusters. The rest may be due, in part, to imperfections in the longer-range internal order and, in part, to surface disorder. Somewhat similar discrepancies were also seen in our clusters of cubic ammonia,6 another system with networks of hydrogen bonds. Diffraction pattems of crystalline clusters of molecules such as SiF4 and SF6, whose intermolecular forces are less specific and less inclined to lead to networks, have tended to agree more closely with pattems calculated from ideal structures. Nevertheless, it is clear that our clusters :re cubic, not hexagonal, ice. The lattice constant, a = 6.36(2) A, implies a density of 0.93 g/cm3 at approximately 206 K. Among the forms of solid produced by the freezing of small water droplets and their subsequent growth in a moist atmosphere, the most commonly seen, of course, is snow. Viewed under a magnifying glass, the beautiful flakes of snow display a dazzling variety of aspects, but many exhibit a decidedly hexagonal external shape betraying their crystal structure (Ih).

J. Phys. Chem., Vol. 99, No. 12, 1995 3929

The Freezing of Large Water Clusters On the other hand, measurements of the angles between the c-axes in snow polycrystals led Kobayashi et a1.57,58to propose a mechanism in which microdrops in the atmosphere freeze to cubic crystals that act as nuclei for condensation of water vapor. The epitaxial growth of hexagonal ice upon the 111 planes of the cubic cores accounts for the observed directions of the hexagonal axes. Such a mechanism is consistent with the present results.

and Harrowellm found that jump rates based on diffusion models do indeed considerably exaggerate the inhibiting effect of viscosity, while jump rates implied by the BBG model tend to be too high. For purposes of illustration we show the results of adopting each of the expressions for the prefactor A. The two treatments can reasonably be expected to yield results bracketing the range to be expected for a capillary theory. The free energy barrier, AG*, to the formation of a spherical nucleus is given by

Final Comments When research on this system was first begun, it was not fully appreciated how instructive the results would be, or how ramified their analysis, owing to water’s unusual properties. Therefore, since there is much still to be learned, we intend to extend this research to a wider range of cluster sizes and a higher degree of statistical precision. Current results already crudely discem the variation of usl over the range of temperatures covered. A more precise determination might help to discriminate between the models of molecular jumps discussed above. To what degree small molecular clusters clusters of highly confined water molecules can be considered to act as valid model systems for drops containing lo1’- lOI3-fold more molecules and nucleating 1020-foldmore slowly remains to be seen. The preliminary hint that the water in clusters is little perturbed by its confinement is based solely on the success of classical theory in connecting the points in Figures 7 and 8 by adopting the same thermal dependence of u,l that applies to mercury. Neither the classical theory nor the comparison are compelling. Nevertheless, results offer fresh and thoughtprovoking evidence about a well-studied but incompletely understood system.

Acknowledgment. This research was supported by a grant from the National Science Foundation. We thank Mr. Paul Lennon for his valuable contributions to the experiments. Appendix Expressions Adopted for Analyzing Nucleation Rates. As formulated by Turnbull and FisheIjO and Buckle3I for the process of freezing, the steady-state rate J for the formation of critical nuclei per unit volume per unit time can be expressed as

J = A exp( -AG*/kZ‘)

(7)

where AG* is the free energy barrier to the formation of the critical nucleus believed to initiate the transition. A current variant of the theory using viscous flow to model the molecular jumps across the solid-liquid interface takes the prefactor to be3

A = 2(a,,k~1/2/(v;/3v)

(8)

where usl is the interfacial free energy per unit area of the boundary between solid and liquid, vm is the molecular volume, and v(T) is the liquid viscosity. This expression appears to make the jump rate excessively small at low temperatures where the viscosity increases steeply, and the expression

A = (5f/avm2/3 ) ( 3 ~ , , / m ) ” ~ [-l exp(-AGfu,/RT)]

(9)

suggested by the heuristic treatment of jump frequencies of Burke, Broughton, and Gilmer (BBG),59has been proposed as a plausible a l t e m a t i ~ e ? Parameter ~ f is the fraction of jumps that are successful (commonly taken as ~ 0 . 2 7 ) ,a~ is~ a characteristic intermolecular spacing, and m is the molecular mass. In a fundamental analysis of crystal growth rates, Oxtoby

AG* = 16na,3/[3(AGv

+w’)~]

(10)

as shown by Gibbs over a century ago,61where AGv 0 is the Gibbs free energy of the transition from the old to new phase per unit volume at the standard pressure and supercooled temperature as outlined in the next section. The term w’ expresses a correction for the work of changing the surface area of the outer phase i due to the volume change as a nucleus of phase j forms in the interior. For a nucleus growing in a spherical particle of radius ro

where 2u,lrois the Laplace pressure exerted by the outer phase on the inner phase and the e’s are densities of the phases. In interpreting the time of development of the solid phase in terms of the nucleation rate, it is necessary to consider the development of transient populations of embryos leading ultimately to critical nuclei,62 and the time of growth of the solid phase in the liquid, once nucleation has taken p l a ~ e . ~ ~ . ~ ~ In all studies of phase changes in clusters encountered in this research program, including the freezing of water, the times to develop steady-state populations and to freeze a nucleated cluster have been small compared with the statistical spread of nucleation times.29 Estimation of Properties of Supercooled Water. It is necessary to be able to calculate evaporation rates of clusters as a part of the calculation of their temperature profile T(t). In addition, estimates are needed for the values of a number of thermodynamic properties entering the two representations of the nucleation theory investigated in the present work. For these purposes are required extrapolations of known information about supercooled water into a range of temperature that has been inaccessible to experiment. Even if such measurements were to exist for bulk water down to 200 K and below (a seemingly impossible eventuality) it is not clear that they would apply to our clusters. As mentioned in the Introduction, some workers believe that properties of water in confined spaces are dtered.I4-l7 The structure and behavior of water near a surface may differ from that of the bulk. The diverging thermodynamic and transport properties proposed to occur as supercoolingdeepens8-I2 might be associated with spatial and temporal fluctuations of density that would be frustrated by nanometer-sized clusters probed on a microsecond time scale. Be that as it may, we will adopt the following compromises. One of the most important quantities in applications of the classical nucleation theory for freezing is AGv, the free energy of freezing per unit volume of liquid. This quantity can be derived from the heat of fusion and ACp, the difference in heat capacity of the liquid and the solid, via AGv = -AGf,,IV, where

In the two previous analyses of nucleation rates in water, two different simplifications for AGf,,(T) were applied. In one caseI9 it was assumed that ACp is negligible and, in the other,I8

3930 J. Phys. Chem., Vol. 99, No. 12, 1995

Huang and Bartell

that ASfu,(T) obeys an empirical relation suggested by Hoffman.@ Unacceptably different results are given by the two methods and neither is optimal. Our scheme is based on a proposed model for supercooled water dividing C, into a “normal” component and an ‘‘anomalous’’ ~ o m p o n e n t .Ac~~ cording to some hypotheses, the anomalous component diverges as water approaches the critical temperature Ts believed to be -226 K. We fitted the observed heat capacity of supercooled water (measured down to 236 K@) with the function (in J mol-’ K-1)

C, = 72.92

+ 0.01896(T - 226) + 41.7/[1 -t 0.0072(T - 226)*] (13)

for T > 226 K. This function peaks at 226 K, constraining the heat capacity at 226 K to go no higher than 114.4 J K-’ mol-’. Below 226 we adopted the polynomial

C, = 3.990 + 0.3220T - 0.0003608?

(14)

to represent the normal component suggested by Oguni and Ange1165 in their Figure 5. Because the heat capacity of supercooled water is substantially greater than that of the solid, the approximation for AGfus(T)that neglects AC, makes the free energy of fusion too large and, hence, the derived free energy for the interface between the liquid and solid phases, too large. In addition to the above considerations is the complication that our solid and, as shown above, the solid encountered in the previous measurements of nucleation rate were not ordinary hexagonal ice (Ih) but cubic ice (IC)whose heat capacity and enthalpy in the region of concem are unknown. M e a ~ u r e m e n t s ~ ~ at lower temperatures yield values of C, that are slightly higher than for ordinary ice67and indicate that the enthalpy of normal ice is -160 J/mol lower. To account for these differences in a crude way, we subtracted 80 J/mol from the molar free energy of fusion of ice Ih, to obtain an approximation for the molar free energy of fusion of ice IC. Adequate analytical representations of our results in the range of concem are (in J/mol) 1139.5

+ 13.016T-

0.06499P

(15)

AG,,(T) zz -2007

+ 37.1632‘-

0.11022

(16)

AG,,(T)

%

below 226 K, and

above 226 K. If we had used only the “normal” component of C, for the liquid, which would certainly have made the heat capacity too small, our values of AGf,,(T) would have been augmented by -60 J/mol, or 5% at 200 K. If AC, had been neglected altogether, AGf,, at 200 K would have been higher by 477 J/mol, or 42%. In the case of water, the Hoffman approximation is better than the neglect of AC,, (contrary to the assertion in ref 19) but gives values of AGf,, somewhat smaller than our approximation. Even if our approximations are imperfect, they are probably better that those of the previous analyses and, applied systematically to the present and previous experimental rates, they allow results to be compared on the same basis. Liquid densities were taken from the model of Robinson and co-workers14 which gave an excellent representation of experimental data over the measured range. They were taken to be 0.93 at 200 K, the temperature of freezing. Liquid viscosities were estimated via eq 17 of ref 3, using the coefficients of Yaws6* for viscosity in the normal range of temperature and the glass transition temperature of 136 K tabulated by Angell

et al.69 Surface tensions of water in the range of interest were taken to be = 111.63 - 0.13167T

(17)

based on a linear representation of the results of Flonano and AngelL70 Computations modeling the formation of and the thermal history of clusters in nozzle flow and beyond require analytical approximations for the heat of vaporization AHv(7‘) and the vapor pressure ~ ( 7 ‘ ) .The former quantity can, of course, be computed from AHb(373.15)and AC,(T) for vaporization, For our purposes it was sufficient to use the linear approximation in (J/mol)

AH,(T)

%

56317 - 41.84T

(18)

and to express the vapor pressure, via the Clapeyron equation, as

lnp(7‘) = l n p , - 6773.3(1/T - UTi) - 5.032 ln(T/T,)

where pl = 2.149 torr and T I = 263.15 K.’* Possible Interference by Clathrate Formation. The question has been raised whether clathrates of neon in water might have been formed and altered experimental results. Such an eventuality could not have happened because “neon atoms are too small and volatile to be retained in [water] cavities.”72 By the time our clusters are cool enough to be remotely attractive hosts for neon to dissolve in, the neon has become highly rarefied.

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