J. Phys. Chem. 1995, 99, 11147- 11152
11147
Kinetics of Freezing of Dimethylacetylene. An Electron Diffraction Study Jinfan Huang, Wenqing Lu, and Lawrence S. Bartell" Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 Received: April 11, 1995@
The rate of nucleation of crystalline dimethylacetylene in the neat supercooled liquid has been measured in 130 8, clusters containing -1 1 OOO molecules. Clusters were generated by condensing the vapor in supersonic flow with neon carrier. At the rate of cooling to which they were subjected, the clusters dropped far below the bulk melting point of 241 K before they began to freeze at -155 K into a heretofore unknown solid phase. This solid had a density of packing close to that of the known tetragonal phase at a similar temperature. The maximum freezing rate was attained at 151 K where the nucleation rate reached 1.14 x critical nuclei m-3 s-I. Clusters froze to single crystals, consistent with Kashchiev's criterion for mononuclear freezing when it was applied with estimated rates of growth of postcritical nuclei. Because no thermodynamic information about the new solid phase was available, it was not possible to analyze the kinetic information in terms of the classical theory of homogeneous nucleation without introducing assumptions. However, plausible estimations of unknown quantities led to an interfacial free energy of the solid-liquid boundary of -20.0 mJlm2, a value in rough conformity with Tumbull's empirical rule. Differential scanning calorimetry carried out on a sample frozen by plunging the fluid into liquid nitrogen showed no signs of a transition at the normal tetragonal-to-tetragonal transition temperature. Whether the signal seen 41 deg higher involves the new solid phase produced in supersonic flow is not yet known.
Introduction Although the growth of crystals from their melts is commonplace and of considerable importance in science and technology, little is known about the molecular mechanism which initiates the transformation.' The kinetics of phase transitions in condensed matter is not as well understood as that of the kinetics of condensation of vapor,* partly because of the greater theoretical complexity and partly because of the greater difficulty in making experimental measurement^.^-^ A significant advance conceptually was made when the Becker-Doring classical theory of homogeneous nucleation' in the vapor phase was extended to freezing by Tumbull and Fisher.8 Recently, Oxtoby and others have begun to explore the magnitude of deviations from the classical theory that might be exhibited by a more realistic treatment.2.9.'o A new experimental technique" has also been devised to facilitate the measurement of rates of homogeneous nucleation in condensed systems, including solid state transformations12as well as freezing. This technique is based on the ability to produce large liquid clusters (or solid clusters, under certain conditions)by condensation of vapor in supersonic expansions and to supercool them rapidly. Phase changes are readily observed by methods already developed to monitor clusters. The information provided may well prove to be enlightening about the dynamics of condensed phase transitions if it can be conclusively demonstrated that matter as finely divided as it is in clusters is adequately representative of matter as it is normally encountered. A large step in this direction wasFecently made when it was shown that water clusters only 60 A in diameter and freezing 20 orders of magnitude faster than water droplets containing 10i2-fold more molecules nevertheless gave results that were entirely consistent with those of the larger droplets when interpreted in terms of the classical theory. l 3 Moreover, molecular dynamics simulations of considerably smaller clusters have shown that properties of cluster cores closely resemble those of bulk matter.I4 Be that as it may, @
Abstract published in Advance ACS Abstracts, June 15, 1995.
0022-365419512099-11147$09.00/0
too few results have been acquired to date to evaluate the new method as thoroughly as might be desired. Therefore, it is worthwhile to study a wider variety of systems. Systems already investigated by the new technique include liquid clusters of CC4I5 and CH3CC13I6 which froze to plastically crystalline solids, NH3I7 and H2013 which froze to ordered solids, and SeF6 and C(CH&C112 which underwent transitions from orientationally disordered to ordered crystalline phases. One interesting aspect of the studies is that the solid phase obtained in very fast transitions is determined by the kinetics rather than by the thermodynamics of the transition. In some cases the result is a phase not encountered in conventional studies of the bulk. While such a practical result has been known from antiquity (cf. the tempering of Damascus steel and other alloysI8), it is now possible to explore some of the kinetic aspects rather directly and in more elementary systems. At the present stage of development the new technique is not applicable to arbitrary systems. It can only be applied to substances that undergo extremely rapid transitions in the vicinity of the so-called evaporative cooling t e m ~ e r a t u r e ' ~ of. ~ ~ the material. Rules of thumb to help recognize such materials have been discussed elsewhere.*' One interesting possibility so identified is that of dimethylacetylene (DMA), the first quasilinear molecule to be examined by the new technique. It freezes at 240.8 K to a tetragonal solid which, in tum, transforms to another tetragonal solid phase at 154 K.22-24 In the present paper we report the rate of nucleation observed when cooling was fast enough to attain an extremely high degree of supercooling and comment upon the form of the solid that was produced. A brief study by differential scanning calorimetry will also be discussed.
Experimental Section Neon carrier gas was passed over a liquid sample of DMA (99.99%, Aldrich) in a small Pyrex tube to conduct the vapor through a mixing chamber into a stainless steel sample holder. The mixed gas 37 mol % in subject and at a stagnation pressure 0 1995 American Chemical Society
11148 J. Phys. Chem., Vol. 99, No. 28, 1995
of 2.0 bar and initial temperature of 293 K was admitted through a pulsed valve to a miniature glass Laval nozzle (0.2 mm entrance id., 2.02 mm exit i.d., overall length of 20.1 mm) where the flow became supersonic. During the cooling expansion toward the evacuated diffraction chamber the vapor condensed into liquid clusters whose properties were monitored after the pulsed jet emerged from the nozzle. Pulses were generated with a frequency of 19 Hz and a duration time of 0.4 ms and transmitted through a Vee skimmeS5 to select the desired stream tube of the cluster beam for diffraction analysis. Clusters were probed by a 40 kV electron beam pulsed at the same frequency and same duration time, but with a delay time of 0.6 ms. Kodak medium slide photographic plates recorded the diffraction patterns, which were accumulations of 1140-4940 exposures in the present experiments. Monitoring of clusters began 8.5 mm after clusters emerged from the nozzle and continued at intervals of 3 mm. The average cluster velocity was measured to be 703 d s in a flight tube under the aforementioned flow conditions. Therefore, spatial intervals of 3 mm correspond to time-of-flight intervals of 4.3 ps between successive observations. Measurements of transition temperatures of the solid were made with a Perkin-Elmer DSC 7-series thermal analysis system to investigate whether the temperature of phase transition in a bulk sample depended upon the rate at which a liquid was cooled during freezing. The instrument was calibrated with the solid state phase transition temperature of cyclohexane (-87 "C). To get a cooling rate for a bulk sample to approach in order of magnitude the rate attained by evaporating clusters, a sample of 4.4 mg of DMA in a sealed aluminum sample pan was thrust into liquid nitrogen. The frozen sample was then subjected to a heating rate of 10 " C h i n in the DSC scan.
Analysis of Data Computer modeling26-28 was carried out to obtain the temperature profile of the clusters during their growth inside the nozzle, first to c o n f i i that they emerged appreciably above their "evaporative cooling temperature" and then to establish the simpler but more important temperature profile of the evaporative ensemble of clusters during the period of flight while they were being monitored. As will be shown below, the structure of the solid that formed when the highly supercooled liquid clusters froze was not that of either of the known thermodynamically characterized stable phase^.?^,^^ Therefore, analyses of the kinetic data could not be carried out rigorously. Because such analyses provide useful perspectives into the transformations investigated, provisional analyses were performed based on the following approximations. We reckoned the degree of supercooling from the 240.8 K melting point of the known tetragonal form. Although the melting point of the present metastable solid must be somewhat lower, the error in the -90 deg supercooling is probably modest. Then, in view of the similarity of the packing densities of the new and the known phases at a given temperature, and for want of better evidence, we took the heat of fusion to be that of the stable form and assumed that the heat capacity of the new solid was a smooth curve, evenly cleaving those of the two tetragonal phases. Physical properties incorporated into the modeling are given in the Appendix. In applying the theories of homogeneous nucleation and of growth velocities of crystalline nuclei inside the supercooled liquid, we used two different models of molecular jump frequencies. Those applying to growth rates were outlined explicitly by or implied by the treatment of Burke et aL3I There is evidence that the model based on diffusiodviscous flow may
Huang et al. grossly exaggerate the inhibiting effect of viscosity on jumps at low temperatures, while the free jump model may err in the opposite direction. In the first model the jump frequency for nucleation is taken asI5
v = kT/v,q where v, represents the volume per molecule and 7 the viscosity, and in the second
v = @)(3RT/M)"*[ 1 - exp(AG/RT)]
(2)
in whichfrepresents the fraction of the jumps to the new phase which are successful (taken to be 0.27), 13. is the mean free path, M the molecular weight, and AG the (bulk) molar free energy change from the old phase to the new. Nucleation rates were assumed to be represented
J = A exp( -AG*lkT)
(3)
where AG* is given by the standard capillary formula and the preexponential, by (4) with frequency v expressed by either eq 1 or 2. Similarly, the crystal growth velocity v3' was taken to be one mediated by diffusion, namely v = (foa/A2)exp(ASIR>[1 - exp(AG/RT)]
(5)
or one based on free jumps (eq 2) with
v = av
(6)
where D is the coefficient of diffusion, a the crystal layer spacing, A the mean free path in the liquid (taken to be 0.5 A), and AS the entropy of fusion. In our computations the coefficient of diffusion was evaluated by the approximation of Li and C h a r ~ g , ~ ~
D = kT/2zqvm113
(7)
Results Figure 1 plots electron diffraction patterns of DMA clusters at successively greater distances from the nozzle in which they were formed. The early patterns are of liquid clusters. A steady growth of Debye-Schemer rings can be seen as the clusters undergo continued evaporative cooling, accelerating the process of freezing. Ring breadths indicate that the clusters were 120 8, in diameter when they fully froze and, hence, -130 8, in diameter as liquid clusters before they dissipated the heat of crystallization by e ~ a p o r a t i o n .The ~ ~ diffraction pattern reveals that the solid formed did not correspond to either of the two crystal structures known from previous X-ray studies. This is shown in Figure 2 by the comparison of the observed diffraction pattern with those calculated from the two known ~ t r u c t u r e s . ~ ~ . ~ ~ The new solid form yields a much simpler diffraction pattern, and one which can be indexed as cubic, with intensities approximately following those for a bcc lattice. That the measured lattice constant of 4.63 A corresponds to just one molecule per unit cell shows the impossibility of such an assignment unless molecules are statistically disordered with an average density resembling a bcc distribution of matter. How the molecules pack to achieve such a diffraction pattern has not yet been determined. Because the solid produced by the rapid freezing of the highly supercooled liquid was a new phase for which no thermo-
-
Kinetics of Freezing of Dimethylacetylene
J. Phys. Chem., Vol. 99, No. 28, 1995 11149
0
10
20
30
40
50
70
60
80
time, microsec Figure 3. Temperature of liquid clusters of dimethylacetylene beyond the nozzle, calculated from the kinetics of condensation and evaporation of clusters in a divergent supersonic flow. I , 0.5
1
1.5
2
2.5
3
3.5
1
I
-
0.8
4
I
s,A '
0.6
Figure 1. Electron diffraction patterns of freezing clusters of di-
0.4
methylacetylene at various times of flight beyond nozzle: From top to bottom: 12.1, 16.4,20.6,24.9,29.2, 37.3, and46.2ps. and fully frozen, denser clusters produced under colder flow conditions.
-
0.2
-
0 -
I
I
I
1
I
I
0
10
20
30
40
50
-0.2
time. microsec Figure 4. Volume fraction F(r) of clusters of dimethylacetylene frozen as a function of time elapsed since exiting the nozzle: points, experimental; curves, left to right, calculated according to eqs 2-4 with the data of Figure 3, respectively, adopting interfacial free energies of 19.5, 20.0, and 20.5 mJ/m?. I
I
I
I
I
1
I
I
100
3 0.5
1
1.5
2
2.5
3
3.5
4
s,A '
dynamic information is available, it is not possible to analyze the kinetic data as rigorously as has been done in prior studies by the present method. For sake of illustration, we introduced the assumptions described in the previous section. From the time evolution of the diffraction intensities can be derived the fractions, F(t,), of clusters that have frozen at times t,. In our treatment these were related to the nucleation rate J( r ) by first calculating the thermal profile T(t) as mentioned in the previous section, then expressing the classical nucleation rate as the temporal function J ( t ) of eq 3, and numerically integrating the differential equation d ln[l - F(t)] = -J(t)Vc, dt
90
d
80
4-
70
-
I 60
-
iL
Figure 2. Electron diffraction patterns of crystalline clusters of dimethylacetylene: top, experimental intensities: middle, intensities calculated for high-temperature tetragonal phase; bottom, intensities calculated for low-temperature tetragonal phase.
(8)
where Vci is the cluster volume minus the surface layer.34 By adjusting the principal unknown of classical theory, the parameter u,l (usually referred to as the interfacial free energy), it is possible to construct a curve passing smoothly through the experimental points. We assume that this curve is our best representation of the nucleation rate. For sake of computation the kinetic parameter us[ is assumed to increase with temperature as F,j,the factor by which the only two well-studied interfaces
-
E
(21 Q)
I -200
-150
-100
-50
0
-I
I 50
TI "C Figure 5. Differential scanning calorimetry curve of dimethylacetylene cooled by immersion in liquid nitrogen.
known to us have been determined to increase, namely, those of mercury3 and water.I3 Results are insensitive to the exact exponent. Figure 3 portrays the profile T(t)used in the analysis, and Figure 4 illustrates the experimental and calculated fractions F(t) of clusters frozen. Freezing began to be apparent at -155 K, and the maximum conversion rate was reached at -151 K at a nucleation rate of 1.14 x m-3 s-I. The curve best representing the experimental points was that based on the free jump model of eq 2 calculated with an interfacial free energy of 20.2 mJ/m2. An interfacial free energy 11% lower resulted if eq 1 were used, instead. Reproduced in Figure 5 is the DSC scan of the bulk sample that was prepared from the liquid by plunging the small aluminum sample holder filled with DMA into liquid nitrogen. No sign of the known solid state transition at 154K was22.23 observed. A small, reproducible peak at 195 K was seen, as well as one at the normal melting temperature. Whether the
11150 J. Phys. Chem., Vol. 99, No. 28, 1995 signal represents a conversion of the new phase into the stable high-temperature solid phase is not yet known.
Discussion
40
,
Huang et al. I
I
3 5 1 30
\.
I
We base our derivation of nucleation rate on our inference that the rate at which clusters freeze is governed by the rate of nucleation of the solid phase, not by the rate at which clusters solidify, once nucleated. For one thing, as discussed below, the calculated velocity of the advancing solid-liquid interface is fast enough to freeze a cluster in a fraction of the time period over which an ensemble of clusters is observed to freeze, and the evaporative mechanism for dissipating the heat of crystallization is more than adequate to keep the freezing cluster For another, according to the theory of Wu3s and others,36the time lag for the buildup of a steady state of production of precritical nuclei is calculated to be a small fraction of a microsecond and, hence, too short to have a discernible effect on the rate of freezing. It is of some interest to compare the interfacial free energy estimated from the nucleation rate with the expected value according to a relation proposed by Tumb~ll.~’Tumbull found that interfacial free energies per unit area derived from the kinetics of freezing tend to be a constant fraction of Ahm,the heat of fusion of a molecular layer per unit area, where Ah,,, is defined in terms of the molar enthalpy and volume as AHm/ (V2N~)”3.That fraction kT, tended to be about 0.32 for metalloids and nonmetals. Results for clusters of CCl4I5 and CH3CC1jl6 were consistent with this fraction, but H20L3and NH3” gave somewhat lower fractions. Dimethylacetylene yielded the ratio 0.29, suggesting that the order of magnitude of the derived interfacial free energy is reasonable, despite the fact that the result is based on the melting point and heat of fusion of the stable phase, not the actual phase that was generated. Of course, an error in heat of fusion in the derivation of o,r cancels to a certain extent in the Tumbull ratio. It is to be noted that the barrier to nucleation is proportional in the classical theory to the cube of the interfacial free energy. Therefore, since the form of the solid frozen was mediated by kinetics, it is to be expected that the interfacial free energy of the new solid obtained is lower than that of the thermodynamically stable phase. Estimates were made of the velocity of crystal growth, partly to find whether its slowness might be a contributing factor in the speed with which freezing could take place (see above) and partly to apply Kashchiev’s criterion for mononuclear freezing.j* To this end, two expressions for the speed were invoked. As sketched in the section on analysis of results, one expression (eq 5) can be expected to yield a speed that tends to be too slow at low temperatures, and the other (eqs 2 and 6) must err, if at all, in the opposite direction. In a molecular dynamics simulation, eqs 2 and 6 accurately represented the growth rate of argon 100 faces in a highly supercooled system.31 In some treatments of crystal growth a key factor is the rate of dissipation of heat. In the present case this rate is not a problem because, as mentioned above, the rapid evaporation accompanying a slight warming keeps the cluster cool, and the thermal conductivity is high enough to prevent an appreciable thermal gradient across a cluster. To provide a lower limit for the growth rate with eq 5, we estimated the viscosity of cold DMA assuming that the glass temperature may be as high as 105 K (see Appendix). This gave a velocity of -0.04 d s . The other expression gave a speed 4000-fold faster. Either speed could freeze a cluster with a radius 60 8, in a fraction of a microsecond.
00
I
I
100
120
I
I
1
,
I
I
140
160
\
180
200
T, K Figure 6. Dependence of nucleation rates (m-3 s-I) for the freezing of dimethylacetylene. Solid point: experimental value for 130 A clusters. Curves, calculated to pass through the experimental point: solid line, according to the free jump model; dashed line, according to the viscous flow model.
Kashchiev’s criterionj8 for a material to freeze into a single crystal over a volume V is given by
G >>
Jv’3a‘‘3
(9)
where G is the linear growth rate of the solid into the liquid and a is the ratio of the volume in which the transformation is detected to the sample volume. If the mean cluster volume and nucleation rate determined previously are inserted into eq 9 with a taken to be unity, the requirement for mononuclear freezing of DMA is G >> 0.0008 d s . This criterion is satisfied even if the lower limit of growth velocity is adopted. It is worth extrapolating our nucleation rate to much higher and lower temperatures according to the classical theory of nucleation. Even though classical nucleation theory is known to have only qualitative validity, such extrapolations have proven to be helpful in the past by indicating whether the nucleation rate might rise sufficiently at low temperatures to make the phase change occur on a time scale accessible to molecular dynamics simulations. Projections so carried out were successful in identifying several systems appropriate for investigation. In the present case, stronger caveats are offered because the ingredients entering the computations are more speculative than usual, as mentioned above. As shown in Figure 6, the result of the free jump, but not the viscous jump, model projects a rate just accessible to simulations of freezing carried out with current computational power. Whether the free jump model errs too much on the positive side remains to be established. The fact that a new phase of DMA was generated when the clusters froze is not a particularly rare phenomenon in cluster studies,*’ but it warrants a few comments. We do not believe the metastable phase occurred merely because the clusters were extremely small compared with normal samples. In our experience, clusters 2 orders of magnitude smaller yet in volume nevertheless spontaneously adopt the bulk phase in molecular dynamics simulations when the thermodynamically favored form is not in conflict with the kinetics.*’ Neither do we attribute the formation of the metastable phase directly to the rapid cooling, itself. What a combination of the small volume for nucleation and the rapid cooling are responsible for is the attainment of a very high degree of supercooling. This, in tum, leads to a high nucleation rate in which the metastable form simply outcompetes the stable form. In bulk systems what is normally seen on the enormously longer time scales of the experiments is the result of a long period of equilibration. Nevertheless, certain methods of quenching may make it possible to obtain even in bulk systems those phases which are readily produced in cluster systems. Clusters, then, may prove
J. Phys. Chem., Vol. 99, No. 28, 1995 11151
Kinetics of Freezing of Dimethylacetylene to be useful in forecasting materials with novel properties. In any event, they provide convenient model systems for investigating transformations induced under unusual conditions.
do the approximate rules so far proposed, we include values of the latter for purposes of comparison. According to one of the rules, the evaporative temperature for a normal liquid (obeying Trouton’s rule) is20
Acknowledgment. This research was supported by a grant from the National Science Foundation. Appendix Estimation of AG,( )‘2 for Supercooled Dimethylacetylene. The free energy of freezing per unit volume, AG,(T), is a quantity used in the derivation of the interfacial free energy, nucleation lag time, and crystal growth rate of supercooled clusters. In order to construct a plausible value for this variable from the entropy of fusion at the melting point, it was necessary to estimate the difference in heat capacity between the liquid and solid phases. For this purpose we subjectively extended the experimental curve for the l i q ~ i d ~into ~ . ’ the ~ supercooled region, obtaining the representation C&T) = - 14.89
+ 0.8063T - 0.00111415
(10)
in J K-’ mol-’ over the range from -145 to 240 K. For the unknown solid we passed a smooth curve to cleave more or less evenly through the segments of the heat capacities of the two known tetragonal phase^^','^ to get C&T) = 11.74
+ 0.5866T - 0.0011472
(11)
over the same temperature range. The above functions are not to be considered as our recommended, optimized approximations. They simply document what was used in the analysis of the present data. The heat capacity of the vapor was also determined. It was computed from the vibrational f r e q u e n c i e ~ ~ ~ to help construct the temperature dependence of the heat of vaporization. Over the range of validity of the above expressions for the liquid and solid, we obtained for the vapor C&T) = 37.94
+ 0.11105T+ 7.84 x
(12)
Temperature Profile and Evaporative Cooling Temperature. The temperature profile of liquid clusters as they fly supersonically into the evacuated diffraction chamber, surrounded by a rapidly rarefying atmosphere of vapor and carrier gas, was determined by integrating the coupled differential equations goveming condensation, evaporation, thermal accommodation, and the dynamics of expanding gas.2s A similar but more complex computer m ~ d e l i n g had ~ ~ . determined ~~ the temperature profile inside the nozzle and established that the temperatures of clusters as they exited the nozzle (at -182 K) were appreciably higher than the so-called evaporative cooling temperature19to be discussed below. Because the initial cooling rate beyond the nozzle is of the order of IO7 IUS,the temperature some microseconds downstream of the nozzle where the cooling rate has fallen by several orders of magnitude is not particularly sensitive to the exact exit temperature. As cooling takes place by evaporation, the vapor pressure drops rapidly, progressively inhibiting further evaporation and cooling as shown in Figure 3. As a consequence, the temperature drop is more nearly linear in the logarithm of the time than in the time itself. Therefore, the temperature soon ceases to change by more than a few degrees during the time of typical experiments on clusters, and its characteristic value in this range is commonly referred to as the “evaporative cooling temperature”. Although we believe that our explicit computation of the temperature profile gives a more accurate determination of the cluster’s temperature than
where AEvapis the change in internal energy on vaporization at Tevp. An alternative proposal compensating to some extent for deviations from Trouton’ s rule simply adopts the temperature at which the vapor pressure has fallen to -0.4 Pa (at which point the evaporation rate has become very low). From the estimated difference in heat capacity of the liquid and vapor can be constructed a good representation of AHvap(T)and, via the Clapeyron equation, the temperature dependence of the vapor pressure extrapolated beyond the range of existing experiments!O Estimates of the evaporative cooling temperature for liquid DMA clusters from these two approaches are 151 and 149 K, respectively. According to our computed profile, the cluster’s maximum rate of freezing occurred at 151 K. Commonly, there is a greater disparity than this between the various estimates of the temperature. Glass Transition and Viscosity at Low Temperature. The temperature dependent viscosity ~ ( rfor ) liquid DMA has been determined experimentally from extrapolations of results for solutions4’ Results for the solvents CC4 and C6D6 are consistent (once an obvious typographical error is corrected) but somewhat different from those for CS’. Therefore, the latter results were discounted and the expression log v(T) = -1.9087
+ 383.67/T
presumed to be accurate to temperatures down to about 240 K was adopted. In order to extrapolate the viscosity down to temperatures far beyond the range of validity of an equation such as (14), a stratagem outlined e l ~ e w h e r ewas ’ ~ implemented. Expressing the viscosity by the form
logv(T)=A+BI(T-To)+CT+D?
(15)
it yielded a good representation of the data in the original range of validity but forced the viscosity to diverge to lo1*Pa s at the glass transition temperature, T,. Although the glass temperature is unknown for DMA, Angell has suggested guidel i n e ~ for ~ ~its. estimation. ~ ~ According to one of these$2 the “ideal glass transition temperature” TOis the lowest temperature at which a glass can exist if the Kauzmann paradoxu is to be avoided. Angell states that the measured glass transition temperature T, is typically 10-20 “C above For dimethylacetylene we find that To is approximately 85 K. Since our objective is not to establish the true glass temperature but, rather, to find the limiting “worst case” inhibition of nucleation and growth rate (Le.,the lowest molecular jump rate according to eq I), we adopt 105 K for the glass temperature.
References and Notes (1) McBride, M. Science 1992, 256, 814. (2) Laaksonen, A.; Talanquer, V.; Oxtoby, D. W. Annu. Rev. Phys. Chem., in press. (3) Tumbull, D. J. Chem. Phys. 1952, 20, 411. (4)Tumbull, D.; Cormia, R. L. J . Chem. Phys. 1961. 34, 820. ( 5 ) Cormia, R. L.: hice, F. P.; Tumbull, D. J. Chem. Phys. 1962.37,
1333. (6) Miyazawa, T.: Pound, G. M. J . C p ~ s t Growth . 1974, 23, 45. (7) Becker, R.; Doring, W. Ann. Phys. 1935, 24, 719. (8) Turnbull, D.; Fisher, J. C. J. Chem. Phys. 1949, 17, 71. (9) Oxtoby, D. W. Adv. Chem. Phys. 1988, 70, 263. (10) Oxtoby, D. W. J . Phys.: Condens. Mutter 1992, 4 , 7627.
Huang et al.
11152 J. Phys. Chem., Vol. 99, No. 28, 1995 (11) Bartell, L. S.; Dibble, T. S. J . Am. Chem. SOC. 1990, 112, 890. (12) Dibble, T. S.; Bartell, L. S. J . Phys. Chem. 1992, 96, 8603. (13) Huang, J.; Bartell, L. S. J . Phys. Chem. 1995, 99, 3924.
(14) Bartell, L. S.; Xu, S . J . Phys. Chem., in press. (15) Bartell, L. S.: Dibble, T. S. J . Phys. Chem. 19511, 95, 1159. (16) Dibble. T. S.; Bartell, L. S. J. Phys. Chem. 1992, 96, 2317. (17) Huang, J.; Bartell, L. S. J . Phvs. Chem. 1994. 98, 4543. (18) Sherby, 0. D.; Wadsworth, J. Sci. Am. 1985, 252, 112. (19) Gspann, J. Physics of Elecrronic and Aromic Collisions; Datz, S., Ed.; North-Holland: New York, 1982. (20) Klots, C. J. J . Phys. Chem. 1988, 92. 5864. 121) Bartell, L. S. J . Phys. Chem. 1995, 99, 1080. (22) Yost. D. M.: Osbome. D. W.: Gamer. C. S. J . Am. Chem. Soc. 1941, 63, 3492. (23) Osbome, D. W.; Gamer, C. S.; Yost, D. M. J . Chem. Phvs. 1940. 8. 131. (24) Givens, F. L.; McCormick, W. D. J . Chem. Phvs. 1977, 66, 5829. (25) Bartell, L. S.; French. R. J. J . Rev. Sci. Instrum. 1989, 60. 1223. (26) Bartell, L. S. J . Phys. Chem. 1990, 94, 5120. (27) Bartell. L. S.; Machonlun, R. A. J . Phys. Chem. 1990, 94, 6468. (28) Bartell, L. S. Unpublished research. (29) Pignataro. E.: Post, B. Acta Cystullogr. 1955, 8, 672. (30) Mike, M. G.;Segerman, E.; Post, B. Acta Cystallogr. 1959, 12, 390.
(31) Burke, E.; Broughton. Q.;Gilmer, G.H. J . Chem. Phys. 1988.89, 1030. (32) Li, J. C. M.; Chang, P. J . Chem. P h y . 1955, 23, 519. (33) See, for example, ref 17. (34) Clusters such as those of dimethylacetylene for which the solid is wetted by the liquid melt first in the surface layer but never freeze first in this layer. Therefore, we do not include the surface layer of molecules as contributing to the volume available for nucleation. See ref 21. (35) Wu, D. J . Chem. Phys. 1992, 97, 9622. (36) Kashchiev, D. Surf. Sci. 1969, 14, 209. (37) Tumbull, D. J . Appl. Phys. 1950, 21, 1022. (38) Kashchiev, D.: Verdoes. D.; van Rosmalen, G.M. J . C r y . Growth 1991, I 1 0, 373. (39) Herzberg, G. Infrared and Raman Spectra of Polyatomic Molecules: D. Van Nostrand: Princeton, NJ, 1945. (40) Hesig, G. B.; Davis. H. M. J . Am. Chem. SOC.1935, 57. 339. (41) Parker, R. G.;Jonas, J. J . Chem. Eng. Data 1972, 17, 300. (42) Angell, C. A.; Sare. E. J. J . Chem. Phys. 1970, 52, 1058. (43) Angell, C. A.: Sare, J. M.; Sare, E. J. J . Phys. Chem. 1978. 82, 2622. (44) Kauzmann, W. Chem. Rev. 1948, 43. 219.
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