2317
J . Phys. Chem. 1992, 96, 2317-2322
Electron Diffraction Studies of the Kinetics of Phase Changes in Molecular Clusters. 2. Freezing of CH,CCI, Theodore S. Dibble and Lawrence S. Bartell* Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 (Received: September 19, 1991; In Final Form: November 14, 1991) The time evolution of the freezing of CH3CCl3has been monitored with the aid of a recently developed technique in which highly supercooled clusters, generated by condensation of vapor in supersonic flow through a Laval nozzle, are probed by electron diffraction. Clusters are observed to freeze into the stable high-temperature solid (Ib) at about 165 K with a nucleation rate of 1.9 X lo2, m-3 8. The time of flight of clusters outside the nozzle before the onset of freezing corresponds to the period of evaporative cooling required to achieve a nucleation rate sufficiently high for freezing to be observed in the supersonic jet, rather than the lag time for homogeneous nucleation, which is much shorter. If results are interpreted in terms of the classical nucleation theory of Turnbull and Fisher, and critical nuclei are assumed to be of phase Ib, the interfacial free energy between the liquid and the solid is found to be 4.7, mJ/m2. This result is in satisfactory agreement with estimates derived from Turnbull's empirical relation. The possibility that critical nuclei of the solid might be of the metastable fcc phase (Ia) is discussed, and arguments against this possibility are advanced. Estimates of the growth rate of postcritical nuclei imply that our observations correspond to mononuclear freezing of clusters into single crystals.
Introduction Freezing is almost always initiated heterogeneously, that is, by the catalytic effects of foreign particles.' In the absence of such effects freezing occurs when random fluctuations generate a nucleus of the solid which can spontaneously grow to bulk size.2 One goal of experiment is to determine the interfacial free energy between solid and liquid, usl,the critical parameter in nucleation theory. However, experimental studies of genuinely homogeneous nucleation are demanding because of the difficulty of entirely removing extraneous effects and verifying that they have been removed. A liquid to be studied must be broken up into small drops in order to isolate most of the sample from effects of p s i b l e catalytic impurities. This is often done by emulsifying the liquid3v4 or by covering drops with an inert layer.5 The emulsion or the oxide layer may introduce impurities and catalytic surface effects, however, and experiments with the same substance often obtain significantly different values of us1.6*7Unfortunately, ud is difficult to determine by other experimental methods and troublesome even to define the~retically.*-~ However, values of usl obtained from nucleation experiments and those found by other methods generally agree within the wide error limits common to all such determinations: In contrast with most work in this field,5-9our concern is with systems of plyatomic molecules. In this paper we describe a study of the rate of freezing of 100 A diameter clusters of liquid CH3CC13(l,l,l-trichloroethane) produced in a supersonic jet. Electron diffraction patterns of clusters, obtained at time-of-flight intervals of microseconds, reveal the extent of freezing and the cluster size. From these data are deduced a nucleation rate, from which is derived a solid-liquid interfacial free energy. Volumes of drops (clusters) produced in a supersonic jet are 6-12 orders of magnitude smaller than those in most studies of freezing. Also, the time during which clusters are normally observed is 4-8 orders of magnitude shorter than typical time ~ c a l e s . ~ J ~Consequently, J'
-
(1) Turnbull, D. J . Chem. Phys. 1950, 18, 198. (2) Fisher, J. C.; Hollomon, J. H.; Turnbull, D. J . Appl. Phys. 1948, 19,
775. (3) Turnbull, D. J . Chem. Phys. 1952, 20, 411. (4) Clausse, D. Rev. Phys. Appl. 1988, 23, 1767. (5) Devaud, G.; Turnbull, D. In Phase Transitions in Condensed Systems-Experiments and Theory; Cargill, G . S . 111, et al., Eds.; Materials Research Society: Pittsburgh, 1987. (6) Eustathopolous, N. Znf. Met. Rev. 1983, 28, 189. (7) Doremus, R. H. Rates of Phase Transformations; Academic Press: New York, 1985. (8) Defay, R.: Prigogine, I. Surface Tension and Adsorption; Longmans: London, 1966. (9) Woodruff, D. P. The Solid-Liquid Interface; Cambridge University Press: Cambridge, 1973. (10) Thomas, D. G.; Stavely, L. A. K. J . Chem. SOC.1952, 4569. (11) Turnbull, D.; Cech, R. E. J . Appl. Phys. 1950, 21, 804.
0022-3654/92/2096-2317$03.00/0
freezing cannot be seen unless clusters are sufficiently undercooled for nucleation rates to be about 15 orders of magnitude higher than those typically citede5
Background A. Cluster Formation and History. Molecular clusters are formed by expansion of vapor, seeded into a rare gas carrier, through a nozzle into a vacuum. In the expanding flow, the vapor cools and condenses into clusters which grow until the rate of condensation of the rarefied vapor no longer exceeds the rate of evaporation. Within the confines of a Laval nozzle, under some expansion conditions, the vapor density remains high long enough for clusters to accumulate thousands to tens of thousands of m o l e c ~ l e s . ~ Thermal ~J~ accommodation by the carrier removes the heat of condensation from the growing clusters. After leaving the nozzle, clusters cool quickly by evaporation, and their temperatures may fall far below the melting point. Because the vapor pressure, and hence the evaporation rate, fall rapidly as the temperature falls, cluster temperatures soon begin to level off.l4 Two rules of thumb have been proposed to characterize the temperature of clusters after evaporation on a 50-ps time scale. One15J6gives the evaporative cooling temperature, Tevp,of a normal liquid as
where hevap( TeV)is the molar energy of vaporization at Tw. A possibly more general rule, which may be more appropriate for compounds not obeying Trouton's rule, expresses the evaporative cooling temperature as17
Tevpi= T(for which vp is 0.4 Pa)
(1b)
that is, temperature at which the (bulk) vapor pressure is 0.4 Pa. Because the cooling rate of clusters quickly becomes slow on the time scale of the experiment, such rules of thumb are not unreasonable under conditions where evaporation governs the cluster temperature. An alternative approach, and one which is more helpful in interpreting the present experiments, is based on com(12) Wegener, P. P. In Nonequilibrium Flows; Wegener, P. P., Ed.; Marcel Dekker: New York, 1969. (13) Abraham, 0.; et al. Phys. Fluids 1981, 24, 1017. (14) Bartell, L. S.; Dibble, T. S. Z . Phys. D 1991, 20, 255. (1 5) Gspann, J. In Physics of Electronic and Atomic Collisions; Datz, S., Ed.; North-Holland: New York, 1982. (16) Klots, C. J. J. Phys. Chem. 1988, 92, 5864. (1 7) Bartell, L. S.; Dibble, T. S. J . Phys. Chem. 1991, 95, 1159. The value of us,obtained for CCI, in this reference is revised in Appendix C of the present work.
0 1992 American Chemical Society
2318 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992
Dibble and Bartell
puter modeling of cluster nucleation and growth. Such calculations can now offer a plausible account of cluster temperatures along their trajectories. Detailed discussions of the governing relations may be found e l s e ~ h e r e . ' Computations ~~~~ of the thermal history of clusters inside the nozzle have been extended to the fractionated flow beyond the nozzle exit.20 B. Nucleation Theory in Condensed Phases. The theory of homogeneous nucleation in liquids adopted for the present investigation was formulated by Turnbull and Fisher21 and by Buckle.22 The nucleation rate, J , is defined as the number of postcritical nuclei (those which may grow spontaneously) formed per unit time per unit volume of the melt.23 If viscous flow is used to model the activation barrier for molecular jumps across the solid-liquid interface, the nucleation rate at steady state in an isothermal system is expressed as17,22
J = f ( ~ ~ ~ k ~ T ) ~ / ~exp(-AG*/kBT) /(~,~/~q)
C
(2) 1
where usl is the interfacial free energy between solid and liquid, u, is the molecular volume, and 9 is the liquid viscosity. The free energy barrier, AG*, to the formation of a critical nucleus from the liquid is given by AG* = 1 6 m ~ , ~ ~ / ( 3 A G , ~ )
(3)
where AG, is the Gibbs free energy of freezing per unit volume a t the temperature of the supercooled liquid. K a ~ h c h i e vhas ~ ~examined the nucleation rate in the transient regime, Le., before the nucleation rate reaches steady state. Extrapolating the number of postcritical nuclei formed in the steady-state regime to zero, one obtains an effective nucleation lag time, On*.The time for the nucleation rate to reach steady state is 3 e,,..Kashchiev derived an equation for 8,. which can be expressed in terms of the quantities employed above as
8,. = 2nuS1q/(AG:u,~/~)
(4)
C. Turnbull Reaction for Interfacial Free Energy. Values of uSIobtained for water and about 20 monatomic metals and me-
talloids are in approximate accord with an empirical relation described by T ~ r n b u l l .H~e~found that us, =
k T m f U s / (p/3Na1/3)
(5)
where kT is a parameter ranging from 0.32 to 0.55,P a n d MfU are the molar volume and enthalpy of fusion, and N, is Avogadro's number. Water (near 0 "C)and metalloids seem to have kTvalues near 0.32, although usl for water appears to be appreciably lower near -36 OC,the temperature at which water freezes homogeMetals typically exhibit kT values of 0.45-0.55.6 D. Electron Diffraction Technique. Because clusters are randomly oriented with respect to the electron beam, diffraction patterns of crystalline clusters correspond to powder patterns. Cluster diameters can be deduced from the breadths of well-resolved Debye-Scherrer rings,28and the extent of freezing at each time of flight can be inferred from comparisons of diffraction patterns with reference patterns of solid and 1 i q ~ i d . lIf~ clusters freeze rapidly enough, once nucleated, to make a second nucleation event unlikely, then the rate of freezing equals the rate of nucleation. It is reasonable to identify the observed extent of freezing at each time, t , with the volume fraction frozen, F(t), rather than the fraction of clusters frozen. It has been shown," for a typical (18) Bartell, L. S. J . Phys. Chem. 1990, 94, 5102. (19) Bartell, L. S.; Machonkin, R. A. J . Phys. Chem. 1990, 94, 6468. (20) Bartell, L. S. Unpublished research. (21) Turnbull, D.; Fisher, J. C . J . Chem. Phys. 1949, 17, 71. (22) Buckle, E. R. Proc. R . SOC.London 1961, A261, 189. (23) Turnbull, D. J . Appl. Phys. 1950, 21, 1022. (24) Kashchiev, D. Surf. Sci. 1969, 14, 209. (25) Bigg, E. K. Proc. Phys. Soc. 1953, 668, 688. (26) Wood, G. R.; Walton, A. G. J . Appl. Phys. 1970, 41, 3027. (27) Butorin, G. T.; Scripov, V. P. Sou. Phys.-Crysrallogr. (Engl. Transl.) 322. (28) Bartell, L. S. Chem. Reu. 1986, 86,491.
15
2
25
1 Figure 1. Electron diffraction patterns of clusters of CH3CCI3formed in a supersonic jet: (a) time of flight 60 ps beyond the nozzle tip; (b) 63.8 ps; (c) 67.5 ps; (d) 75 ps; (e) 78.8 ps; (f) 100 ps, obtained under similar but not identical conditions. Intensities have been leveled and resealed as described in the Appendix of ref 17. s ('4.1
distribution of cluster sizes, that the steady-state nucleation rate can be derived from F ( t ) via the relation J = 3[(1 - F)-II4 - 1]/[( V ) ( t - to)] (6) where (V) is the mean cluster volume and to is the time at which freezing begins. From the slope of the plot of 3[(1 - F)-'/4l]/( V) versus t - to,one obtains a nucleation rate. The distance from the nozzle to the electron beam is converted to a time of flight of clusters as discussed in Appendix B. E. Freezing of CH3CCI3.The behavior of CH3CC13upon freezing is only poorly understood. It typically freezes into the face-centered cubic phase (Ia), which, upon standing or cooling, usually transforms to phase (Ib).ze34 Phase Ib (Z= 21) thought to be cubic due to a near absence of birefringence and, like phase Ia, is plastically crystalline. An ordered orthorhombic crystalline phase is stable below about 224 K.'* Because the orthorhombic phase always transforms to phase Ib rather than Ia and because Ib has a higher melting point than Ia?1933phase Ia is thought to be thermodynamically metastable a t all temperatures. Experimental Section The apparatus and procedures have been described elsehere;^^^^^ only a brief description will be given here. Neon (99.999%, Air Products) is passed through glass wool saturated with liquid CH3CC13(99+%, Aldrich) to make a gaseous mixture with CH3CC13mole fraction 0.088 at 2.1-bar total pressure. Gas flow into a miniature glass Lava1 nozzle (No. 6)" was controlled by means of a pulsed valve operating at 20 Hz. Electron beam pulses were timed to intersect a steady-state section of the pulse of the supersonic jet. Photographic plates record diffraction patterns that are sums of between 3000 and 6000 expasure, each of 0.3-ms duration. Use of a Vee skimmer36 ensured that only the central section of the supersonic jet was probed by the electron beam. Clusters were monitored at time-of-flight intervals of 3.8 or 7.5 ks, starting 60 ps after exiting the nozzle. The conditions of the present experiment were selected to induce freezing to occur at a convenient distance outside the nozzle. When a stagnation pressure of 2.9 bar (CH3CC13mole fraction (29) Rudman, R. Mol. Cryst. Liq. Cryst. 1970, 6, 427. (30) Saint-Guirons, H.; Xans, P.J . Phys. C 1980, 13, LS35. (31) Figuire, P.; Guillaume, R.; Swarc, H. J. Chim. Phys. 1971,68, 124. (32) Rudman, R.; Post, B. Mol. Crysr. 1968, 5 , 95. (33) Silver, L.; Rudman, R. J . Phys. Chem. 1970, 74, 3134. (34) Morrison, J. A.; Richards, E. L.; Sakon, M. Mol. Crysr. Liq. Crysr. 1977, 43, 59. (35) Bartell, L. S.; Heenan, R. K.; Nagashima, M.J . Chem. Phys. 1983, 78, 236. (36) Bartell, L. S.; French, R. J . Rev. Sci. Insrrum. 1989, 60, 1223. (37) Valente, E. J.; Bartell, L. S. J . Chem. Phys. 1983, 79, 2683.
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2319
Kinetics of Phase Changes in Molecular Clusters 0.5
7
A
-
55
60
65
70
75
80
time (ps) Figure 2. Volume fraction, F(t), of clusters that have frozen at time t after exiting the nozzle: points, experimental; solid line, calculated from eq 6 with to = 61 ps, J = 1.9 X m-3 s-I, and (V) corresponding to a sphere of diameter 134 A. Curve f of Figure 1 was used as the reference corresponding to complete freezing.
-1 I
-2 -3 -4
oc
v
M
0 3
I
I
I
1
I
150
160
170
180
190
-
-5 -6 -7
140
T (K) Figure 4. Nucleation time lag, €In., for freezing of CH3CC13in an isothermal system. Calculation based on eq 4 using the temperature-dependent viscosity and AGv. Upper curve, results if critical nuclei are of phase Ia; lower curve, results if critical nuclei are of phase Ib. l
8
O
7
I 7 T 7 170
1
Ei
2
150 1
1
3t
I
1
10
100
1000
time (ps) Figure 5. Thermal history of liquid clusters of CH3CC13outside the nozzle, calculated from modeling of the condensation and evaporation of clusters formed in supersonic flow. Based on a cluster diameter of 134
A. 140 1 5 0 160 1 7 0 1 8 0 190 2 0 0
T (K) Figure 3. Solid-liquid interfacial free energy, uSlrof CH3CC13calculated from the single measured nucleation rate for a range of assumed cluster temperatures: upper curve, results if critical nuclei are of phase Ib; lower curve, results if critical nuclei are of phase la. Note that the curves signify nothing about the temperature dependence of uSl.
F(t) was already greater than 0.5 by the time 0.063) was clusters had exited the nozzle. RHults Electron diffraction patterns of clusters are displayed in Figure 1. LkbyeScherrer rings of phase Ib evolve from the broad band of the liquid at s = 1.3 A-l. Due to the orientational disorder in the plastically crystalline solid, only the rings of low order are observed. Ring breadths of more fully frozen clusters roduced under similar conditions imply cluster diameters of 134 Figure 2 depicts the volume fraction, F(t), of clusters frozen at time 1. Clusters formed under the conditions described above do not begin to freeze until they are well beyond the nozzle. Applying eq 6 and the values of F(t) corresponding to patterns c, d, and e of Figure 1, we obtain a nucleation rate of 1.9 X lo2*m-3 s-l. If the temperature of freezing were known, the interfacial free energy, u,,, could be inferred from the single measured nucleation rate by invoking eqs 2 and 3. Because this temperature is only approximately known, we plot over a range of assumed temperatures in Figure 3 the value of ud which reproduces that nucleation rate. Even though the solid phase observed is phase Ib, it is conceivable that the critical nuclei might possess the structure of phase Ia, instead, and both possibilities are examined. Values
1.
(38) Bartell, L.S.;Valcnte, E.J.; Dibble, T.S.J . Phys. Chem. 1990, 91, 14S2.
of thermodynamic and physical properties used to calculate usl are discussed in Appendix A. At 165 K, our best estimate of the cluster temperature upon freezing (see Discussion), usl is 4.78 mJ/m2 if the nucleus is of phase Ib or 1.9 mJ/m2 if of phase Ia. The calculated temperature dependence of the nucleation lag time, en.,is shown in Figure 4. Values of 0.8 or 5 MS are obtained for 8,. at 165 K if critical nuclei are of phase Ib or Ia, respectively.
Discussion A satisfactory analysis of results requires a knowledge of the temperature at which the clusters freeze. Estimates of the evaporative cooling temperature for CH3CC13clusters based on eqs la and l b are 187 and 176 K, respectively. Figure 5 illustrates a more detailed construction of the thermal history of liquid clusters in supersonic flow outside the nozzle, obtained by numerically integrating the relevant rate Such computations provide information that is more realistic than the single temperatures suggested by empirical rules. According to the computations, clusters exit the nozzle, approximately 50 p after they are fully grown, at about 177 K and cool to 165 K by the time they begin to freeze, -60 p later. Figure 5 also implies that liquid clusters continue to cool during the period over which clusters are observed to freeze. Therefore, the nucleation rate derived represents an average over a range of temperatures. Curves of calculated interfacial tension, shown in Figure 3, display broad maxima near the freezing temperature of 165 K whether critical nuclei are of phase Ia or Ib, 50 the uncertainty in the cluster temperature contributes little to the uncertainty in uSI. This is due to the interplay between the increase in AG, and the decrease in fluidity as the temperature decreases. The sharp rise in viscosity as the temperature falls below 170 K causes a large increase in, and correspondingly increased uncertainty in, the calculated lag time. Despite this uncertainty, a large lag time cannot explain why clusters do not begin to freeze
2320 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 30
28 26 24 22 20
18
-
16
I
145
I
IS5
I
I
165
I
I
175
I
after the formation of a postcritical nucleus. If growth of postcritical nuclei is fast on the time scale of the experiment, the first postcritical nucleus forming in each cluster forestalls the appearance of a second, the observed rate of freezing corresponds to the rate of nucleation, and the freezing is said to be mononuclear. If, however, growth of postcritical nuclei were to take place over the time scale of the experiment, the probability that several nuclei form per cluster would be appreciable (polynuclear freezing). Kashchiev has determined that, for freezing to be mononuclear, the linear growth rate, G, of the solid into the liquid must fulfill the requirement40
I
G >> JV'/3a'/3
1x5
T (K) Figure 6. Temperature dependence of nucleation rate for freezing of CHICCIS,calculated to pass through the observed rate at 165 K. Critical nuclei are taken to be of phase Ib. Solid line, constant osl;dashed line, us,= dashed and dotted line, usI = c2T; dotted line, usl= c,P.
until 60 ps after exiting the nozzle. If this 60 ps of flight were associated with the time to reach a steady-state nucleation rate (-38,.) it would imply a cluster temperature of 153 K (according to Figure 4), more than 10 deg colder than our best estimate of cluster temperature. The time for clusters to reach 153 K by evaporative cooling would be about 600 ps according to our computations (Figure 5). Calculated values of 8,. at 165 K imply a time to develop steady-state nucleation in an isothermal system of only -2 ps if nuclei are of phase Ib. Furthermore, the distribution of subcritical nuclei formed at higher temperatures would lower the time required to attain a steady-state nucleation rate at 165 K. Therefore, the only reasonable interpretation of the delay is to attribute it to the time required for continued evaporative cooling to lower the temperature of the initially warm clusters until the nucleation rate has become high enough to induce freezing. Unfortunately, the temperature dependence of nucleation rate, J( T), for CH3CC13cannot be deduced reliably from the available information. For one thing, the nucleation rate is extraordinarily sensitive to changes in the value of the interfacial free energy between solid and liquid, and hence, J( T ) depends critically upon how usIvaries with temperature. A 20% change in usIcan alter J by 5 orders of magnitude. It is instructive to adopt various power laws to express the unknown temperature dependence of the interfacial free energy and to calculate the corresponding functions, J( r). Theoretical considerations of the solid-liquid interface Results suggest that usItends to increase with temperat~re.63~~ for several representations of us](7') are displayed in Figure 6. At small undercoolings J( T ) rises rapidly as the temperature falls. As the temperature approaches the glass transition temperature, however, the viscosity rises steeply, depressing the nucleation rate. A temperature-independent us]would generate a peak nucleation rate at a temperature higher than that at which freezing was observed, an unphysical result in view of the short lag time expected at such temperatures. Curves for other power laws display peaks at or just below the estimated freezing temperature, implying that clusters formed under our experimental conditions freeze when their temperatures approach the maximum nucleation rate theoretically possible for CH3CCl,. Using the J( T ) functions corresponding to the various power laws together with the thermal history of the clusters, we have calculated the fraction of (134-A diameter) clusters frozen in flight outside the nozzle. All of the J( Tj functions tested imply that freezing occurs significantly earlier than observed in our experiment. The best agreement is with an improbably rapid increase of usl with temperature, suggesting that systematic errors exist in some of the quantities constructed, such as AG,( T), v( r), and T(t),or in our implementation of nucleation theory. In calculating the nucleation rate as described in the previous section, it was implicitly assumed that clusters froze immediately (39) Spaepen, F. Acta Merall. 1975, 23, 729,
Dibble and Bartell
(7)
where Vis the volume of the liquid sample (cluster volume) and a is the ratio of volume of solid detected to the volume of liquid. For a nucleation rate equal to 1.9 X lo2*m-3 SKI, a volume equal that of a sphere of diameter 134 A, and a equal to unity, the requirement for mononuclear freezing of CH3CC13is G >> 0.3 mm/s. Crystal growth velocity may be calculated by the WilsonFrenkel theory of diffusion limited growth.41 As described by Burke et this theory expresses the velocity, u, at which a planar solid-liquid interface propagates into the liquid, by the relation
where D is the diffusion coefficient, a is the site-site distance in the crystal, A is the mean free path in the liquid,& is an empirical constant equal to the fraction of molecular jumps which succeed in sticking to the solid, and a ( T , , , ) and Ab(r) are the molar entropy and temperature-dependent Gibbs free energy of fusion, respectively. Burke et al., in their molecular dynamics simulation of argon, appeared to use a value of A close to one-half the root-mean-square width of the first peak in the pair correlation function of the they found& = 0.27 gave the best fit to the theory. An estimation of the velocity of propagation of freezing in CH3CC13can be made by calculating the diffusion coefficient from the viscosity via the Stokes-Einstein equation44and evaluating AG( r ) as in Appendix A. Parameter a was estimated to be 6.7 and A was estimated from the pair correlation function of liquid C C l t 5 to be 0.5 A. The resultant growth rate at 165 K was -5 mm/s, a value larger than that required for mononuclear freezing according to Kashchiev's criterion. The formulation of Kelton et al.46 is probably more relevant to our problem, that of radial growth, than the Wilson-Frenkel model of linear growth. Kelton et al. used numerical integration to solve the coupled kinetic equations for nucleation and growth of solid from a viscous liquid in an isothermal medium. Our previous analysis" extrapolated the results of Kelton et al. and estimated that the time required for the adiabatic freezing of a CCl, cluster is roughly 3 times the nucleation lag time. For phase Ib of CH3CC13,the corresponding time to freeze the entire cluster would be about 2-3 p s . Such a time corresponds to an average (radial) growth velocity of about 2 mm/s, again a value larger than that required for mononuclear freezing. It is reasonable to conclude, therefore, that our observations reflect the kinetics of nucleation rather than growth of postcritical nuclei. For substantially larger drops polynuclear freezing would have occurred. Even though we observe only phase Ib, the possibility that the critical nuclei might have possessed the structure of phase Ia should be considered. If this had been true, and if the growing crystalline (40) Kashchiev, D.; Verdoes, D.; van Rosmalen, G. M. J . Cryst. Growth 1991, 110, 373.
(41) Wilson, H. A. Philos. Mag.1900, 50, 238 and Frenkel, J. Phys. Z . Sowjefunion 1932, 1, 498 are cited by other authors. (42) Burke, E.; Broughton, J. Q.; Gilmer, G. H. J . Chem. Phys. 1988.89, 1030. (43) Verlet, L. Phys. Reu. 1968, 165, 201. (44) Einstein, A. Ann. Phys. 1905, 17, 549; 1906, 19, 289. (45) Sharkey, L. R.; Bartell, L. S. Unpublished research. (46) Kelton, K. F.; Greer, A. L.; Thompson, C. V. J . Chem. Phys. 1983, 79, 6261.
The Journal of Physical Chemistry, Vol. 96, No. 5. 1992 2321
Kinetics of Phase Changes in Molecular Clusters
Foundation and a Hermann and Margaret Sokol Fellowship.
TABLE I: Physical Properties" quantity
T,,,(Ib), K T,,,(Ia), K M d I b ) , J/mol h R d I a ) , J/mol tJ1) - c,(s), J/(mol K) U,cm3/mol
T K T(i65
K),Pa.s
value
ref
243.1 232 2350 910 (-42.5 0.536T - 0 . 0 0 1 3 7 ~ ) * 89.4 (1 30)c (0.72)c
49 33 49 33 49 32
+
Parentheses indicate nonexistent information, estimated as deare listed separately for solid phases scribed. Values of T, and Ia and Ib. *Heat capacity data that was extrapolated from 243 to 145 K. We are not aware of any experimental determination of T, or of viscosity for CH3CC13 below the melting point. T, was obtained as described in Appendix A using eq 15 of ref 17; liquid viscosity at any temperature above Tg may be deduced from that equation and T,.
mfU,
nuclei had quickly transformed to phase Ib while they were still small, we would not have detected phase Ia. What we would have observed, had this been the case, would have corresponded to the kinetics of nucleation of the phase Ia, not of Ib. If such a second nucleation event had taken place, however, this time from solid phase Ia to solid phase Ib in an extremely small volume, it would have required an unrealistically high nucleation rate. SaintGuirons and Xans'O studied phase transitions in emulsions containing 1-pm droplets of CH3CC13. During slow cooling to 123 K, most of the droplets froze into phase Ia and few transformed to phase Ib or phase 11, implying that the rate of homogeneous nucleation of Ib from Ia is low. Moreover, Rudman has observed that CH3CC13tends to freeze directly to phase Ib upon fast cooling, while more gentle cooling produces phase IaF9 Therefore, because our cooling rates are several orders of magnitude faster than even those used in Rudman's study, we infer that phase Ib was nucleated in our experiments. Finally, what can be said about our value of a,,? To the best of our knowledge, their exists no other determination of gSlfor CH3CC13. Evidence supports the assumption that critical nuclei were of the same phase, Ib, as the fully frozen clusters. However, the calculated interfacial tension of CH3CC13is in reasonable agreement with Tumbull's relation (eq 5) at the value of kT (0.32) for water (near 0 "C) and metalloids whether the nuclei were of phase Ib or Ia (0.34 if Ib and 0.36 if Ia). Does a relation seeming to hold primarily for metalloids and water have a very general applicability to nonmetallic substances? If so, since our fractional supercoolings are larger than those obtained n most experiments, agreement with the Turnbull relation might imply that interfacial free energies are relatively temperature independent, in disagreement with our rough modeling of the freezing rate from various representations of J( T ) and the calculated thermal history of clusters.
Conclusions We have observed the freezing of highly supercooled clusters of CH3CC13into the solid phase Ib. Because clusters in a supersonic jet are isolated from each other and are not in contact with solids which might act as catalysts for freezing, we can be confident that nucleation is homogeneous rather than heterogeneous. According to two independent estimates of the rate of propagation of freezing, the observed freezing rate corresponds to the rate of nucleation rather than the rate of growth of postcritical nuclei. From the measured nucleation rate, we obtain a value of 4.7, mJ/m2 for the solid-liquid interfacial free energy, us], and find that it is consistent with the value suggested by Turnbull's relation for nonmetals. Acknowledgment. This research was supported by a grant from the National Science Foundation. We thank Paul Lennon for assistance in operating the electron diffraction unit and in scanning photographic plates and Frederick Dulles for sharing his knowledge of computer systems and software. We gratefully acknowledge helpful comments by Dr. D. Kashchiev. T.S.D. acknowledges the receipt of a Dow-Britton Fellowship from the Dow Chemical Co.
Appendix A. Physical Properties of CH3CCI3. In order to derive the solid-liquid interfacial tension of CH3CC13from the nucleation rate, one must know the thermodynamic and physical properties a t the cluster temperature, well below the range of stability of either the solid form detected or the liquid. Such nonexistent information must be estimated, somewhat subjectively, from extrapolations of the data available from the literature. The values we adopted are summarized in Table I. In the absence of experimental data on temperature dependence, the molar volume of the highly supercooled liquid was assumed to be the same as that of the solid at 235 K.3z The uncertainty in the molar volume is too small to affect results significantly. Viscosities of fragile liquids like CH3CC13are believed to follow a universal curve when plotted against Tg/T, where Tg is the glass transition temperat ~ r e . ~ 'Tgwas selected to adjust viscosities, calculated in accordance with an expression discussed elsewhere,17 to the known experimental values near room t e m p e r a t ~ r e . ~ ~ The Gibbs free energy of freezing per unit volume, AGv, is obtained from the relation AG"(T) = (1 /
nJ TAsfus(T ) d T Tnl
(All
where Asf,, is the molar heat of fusion, the temperature dependence of which is calculated from the difference between the somewhat uncertain extrapolations of the heat capacities of the liquid and phase Ib?9 The heat capacity of phase Ia was assumed to be equal that of phase Ib for want of better information. The heat of fusion of phase Ia has been characterized in a differential scanning calorimetry study by Silver and R ~ d m a n . ~ ~ Liquid vapor pressure is the most important factor governing the rate at which liquid clusters cool by evaporation. Available data on the vapor pressureMand AHwP49of CH3CC13do not extend below 267 K. Rather than extrapolating the published values of AHvap100 deg below observations, we determined the temperature dependence of AHvapfrom
AKap(T) = AHvap(T1) + JTACP(T) d T TI
(A2)
where AHvap(Tl)is the experimentally determined heat of vaporization at temperature Ti, and ACpis the difference between the heat capacity of the vapor and that of the liquid. The heat capacity of the vapor was calculated from the vibrational frequencies using a harmonic oscillator approximati~n,~' and the experimental heat capacity of the was subjectively extrapolated below the melting point, guided by the measurements on the solid. We conclude that ACp = 39.9 - 0.698T + 0.00121P J/(mol K). Vapor pressure is then extrapolated from the value of 5 Torr at 241.2 K by integrating the Clapeyron equation. At such low vapor pressures the vapor can safely be taken as ideal. B. Calculation of Cluster Velocity. No direct measurements of the velocities of CH3CC13clusters have been made, but velocities can be estimated by combining the theoretical work of Schwartz and Andress* with experimental determinations of the velocities of SF, clusters formed in supersonic flow through the Lava1 nozzle (No. 6 ) . Schwartz and Andres studied the terminal velocity of small particles accelerated by a carrier gas in a free jet expansion. As described in our previous work,I7the experimentally determined ratio of the velocity of clusters of SF,, formed in a nozzle ex(47) Angell, C. A.; et a]. J . Chim. Phys. 1985, 82, 773. (48) Timmermans, J. Physico-Chemical Constants of Pure Organic Compounds; Elsevier: New York, 1950. (49) Andon, R. J . L.; Counsell, J. F.; Lee, D. A.; Martin, J. F. J . Chem. Soc., Faraday Trans. 1 1973,69, 1721 and Suppl. No. 20745 (not 20785 as cited). (50) Ambrose, D.; Sprake, C. H. S.;Townsend, D. J . Chem. Soc., Faruday Trans. 1 1973, 69, 839. (51) Pitzer, K. S.; Hollenberg, J. L. J . A m . Chem. SOC.1953, 75, 2219. (52) Schwartz, M. H.; Andres, R. P. J . Aerosol Sci. 1976, 7 , 281.
J . Phys. Chem. 1992,96, 2322-2325
2322
pansion, to that of the neon carrier gasS3can be used to estimate terminal velocity ratios of clusters formed in the nozzle under different expansion conditions. Following this method, we estimated that the velocity ratio for 134-A CH3CCI3clusters formed under the conditions of this experiment was approximately 0.84. Using this ratio in computations of the history of clusters formed in supersonic flow, we obtained cluster velocities of 800 m/s. Therefore, a 3.0-mm distance interval of sampling corresponds to a 3.7,-ps time-of-flight interval. C. Note on Previous Results for C q . Values of the heat capacity of liquid and solid CCl, used in a previous study1’ had (53) French, R. J.; Bartell, L. S. Unpublished research.
been linear extrapolations of the experimental data of Hicks, Hooley, and Stephenson.” These extrapolations gave unphysically high values of ACpand hence low values of ud. A more plausible estimate, constructed as described above for CH3CC1,, gives CP(liq) - CP(s)= -24.1 + 0.268T- 6.66 X 10-T (J/(mol K)and leads to a corrected value of for CCl, at 175 K that is 5.46 mJ/m2 if critical nuclei are of phase Ib (kT = 0.35) and 4.0 mJ/m2 if of phase Ia (kT = 0.38). Agreement with Turnbull’s empirical relation is still satisfactory. Phase Ib remains the probable form of the critical nuclei. (54) Hicks, J. F. G.; Hooley, J. G.; Stephenson, C. C. J . Am. Chem. Soc. 1944,66, 1064.
Evidence for a New Phase of Water: Water I1 Robin J. Speedy Chemistry Department, Victoria University, Wellington, New Zealand (Received: September 30, 1991)
Unstable glassy solid forms of water can be made by vapor deposition or by splat quenching the liquid. When they are annealed at about 130 K, both of the amorphous solids evidently relax into the same metastable amorphous state, which behaves reproducibly on thermal cycling and displays an apparent glass transition at 136 K to a liquid which freezes (irreversibly) to ice (lc) at 150 K. The apparently liquid water at 150 K cannot be a metastable extension of normal liquid water unless its entropy is implausibly small. I conclude that it is a distinct metastable amorphous phase of water, which I call water 11. Entropy estimates indicate that water I1 would be, thermodynamically, the most stable form of water at some temperature above 150 K unless it becomes absolutely unstable at some upper temperature limit of stability.
Introduction Angell and Sare’ pointed out that the entropy of liquid water tends to become less than that of ice when it is extrapolated down to the temperature Tg = 136 K at which vapor-deposited amorphous-solid water is reported to have a glass-to-liquid transition.2 From an analysis of the entropy and heat capacity2 of amorphous-solid water samples prepared by vapor deposition, Johari3 argued that it was impossible to have a continuity of metastable states connecting normal water and the glass. Subsequent studies have changed that view.+lI Studies of both vapour deposited samples and samples prepared by hyperquenching small liquid droplets show that, after suitable annealing, the two methods yield essentially the same produ~t.~,*J’ That product evidently has a very weak, reversible, glass transition at 136 K and it freezes, irreversibly, to ice (IC) at about 150 KSS4 Improved thermal studies of the product show that its heat capacity above the glass transitionS-l0 is much less than the value reported earlier by Sugisaki et ala2 Because of that change, Johari’s argument no longer holds watere10 and recent seem to take the view that continuity is possible. This paper shows, by a different thermodynamic argument, that it is not possible. (1) Angell, C. A.; Sare, E. J. J . Chem. Phys. 1970,52, 1058. (2) Sugisaki, M.; S u p , H.; Seki, S.Bull. Chem. Soc. Jpn. 196%,41,2591. (3) Johari, G. P. Philos. Mug.1977, 35, 1077. (4) Angell, C. A.; Tucker, J. C. J. Phys. Chem. 1980.84, 268. ( 5 ) Hallbrucker, A.; Mayer, E. J . Phys. Chem. 1987, 91, 503. (6) Johari, G. P.; Hallbrucker, A.; Mayer, E. Nature 1987, 330, 552. (7) Hallbrucker, A.; Mayer, E.; Johari, G. P.Philas. Mug.B 1989,60, 179. (8) Hallbrucker, A.; Mayer, E.; Johari, G. P. J . Phys. Chem. 1989, 93, 4986. (9) Johari, G. P.; Ram, S.;Astl, G.; Mayer, E. J. Non-Crysr. Sol. 1990, 116, 282. (IO) MacFarlane, D. R.; Angell, C. A. J . Phys. Chem. 1984, 88, 759. (11) Dubochet, J.; Adrian, M.; Vogel, R. H. Cryolerters 1983, 4, 233. (12) Rice, S.A,; Bergren, M. S.;Swingle, L. Chem. Phys. Leu. 1978,59, 14.
For brevity, I use the label “normal water” to refer to the stable liquid between 273 and 373 K, at 1 atm, and any stable or metastable extensions of the same phase which are connected to it in the sense that they can be reached from it reversibly, without any first-order discontinuity, by varying the temperature or pressure. For the other kind of water I use the label =water 11”. Water I1 refers to the thoroughly annealed amorphous material which behaves reproducibily when the temperature is cycled up and down below 150 K,displays an apparent glass transition at 136 K,and freezes (irreversibly) to ice (IC) at 150 K with the release of 1330 f 20 J m0l-l,~9~ and to states which are connected to it reversibly without any first-order discontinuity. For the purposes of the thermodynamic argument it is not necessary to specify whether water I1 is liquid or solid, although the apparent existence of a glass transition suggests that it is a liquid between 136 and 150 K. The hypothesis to be tested is that in principle (that is, if freezing could be avoided) normal water could be cooled reversibly from its freezing temperature T , = 273 K to water I1 at its glass transition temperature,+’ Tg = 136 K,without any discontinuity along the way. If that is possible, in principle, then normal water and water I1 belong to the same phase, and water I1 is a metastable continuation of normal water. If it is not possible, then water 11, if it is a metastable phase, is a distinct new phase of water.
Entropy of Water I1 The argument hinges on the magnitude of the entropy of the water 11. The thermodynamic properties of water I1 are best established at the temperature T I = 150 K,where it freezes to ice (IC). At TI the material is well above its apparent glass transition temperature T = 136 K, it behaves reproducibly, without hysteresis, when t i e temperature is cycled up and down and its excess enthalpy at TI, AH(Tl), has been below T1,S-9 measured precisely.’V8 AH( TI) is the heat released when water I1 freezes to ice (IC) at T , , plus the heat released when ice (IC)
0022-365419212096-2322$03.00/0 0 1992 American Chemical Society
