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J. Phys. Chem. 1996, 100, 14276-14280
ARTICLES Isomeric Differences in the Nucleation of Crystalline Hydrocarbons from Their Melts Jinfan Huang, Wenqing Lu, and Lawrence S. Bartell* Department of Chemistry, UniVersity of Michigan, Ann Arbor, Michigan 48109 ReceiVed: April 4, 1996X
Large liquid clusters of the globular hydrocarbon 2,2,3-trimethylbutane were generated in supersonic expansions and frozen, in flight, by evaporative cooling while being monitored by electron diffraction. The nucleation rate rose to 8 × 1028 m-3 s-1 by the time half of the clusters were frozen. Nucleation properties are contrasted with those observed several decades ago by Turnbull and Cormia for three n-alkanes in emulsions. Despite the 15 order of magnitude difference between the rates of nucleation for the globular and linear hydrocarbons, it was possible to make valid comparisons and corroborate Turnbull’s inferences that the flexible, linear molecules nucleate by a rather different mechanism. Nucleation of the globular hydrocarbon was similar to that of other simple, rather rigid molecules studied in this laboratory and implied a solid-liquid interfacial free energy of about 4.6 mJ/m2, a value fairly typical for a substance of its (low) heat of fusion. On the other hand, the linear alkanes exhibit interfacial free energies that are twice as great and kinetic prefactors that are considerably greater, presumably because they nucleate in small molecular segments rather than by the association of entire molecules. Unlike several other simple substances which froze to metastable solids when cooled at extremely fast rates in this laboratory, 2,2,3-trimethylbutane froze to its thermodynamically stable form.
Introduction From time to time one reads in reviews of science how ignorant we are of the molecular behavior underlying one of the commonest transformations of matter, that of freezing.1 In order to draw interferences about mechanisms of processes, chemists characteristically turn to studies of kinetics. In research to find the relation between kinetics and structure, hydrocarbons have been particularly popular systems for investigation. They comprise a diverse class of compounds whose properties vary widely even among a family of isomers. This diversity might be expected to be reflected in the dynamics of homogeneous nucleation. As is well-known, normal hydrocarbons have comparatively low melting points and high entropies of fusion whereas the opposite is true of their highly branched, globular isomers.2,3 Because entropies of fusion play a major role in the theory of nucleation rates in freezing, it is of some interest to find the degree to which the globular and unbranched hydrocarbons exhibit significantly different behavior in crystallization from the melt. The present research is a step in this direction. What has impeded progress in nucleation research in the past has been the formidable difficulty of performing experiments for which the nucleation can be shown to be genuinely homogeneous. A half-century ago Vonnegut4 and Turnbull5-7 developed techniques that made such studies possible, although extremely tedious, and in 1960 Turnbull and Cormia8 carried out careful measurements of nucleation rates in the freezing of long-chain normal hydrocarbons. In the present research, measurements on a smaller, highly branched hydrocarbon, 2,2,3trimethylbutane (TMB), were made by a radically different technique.9 This new technique probes by electron diffraction the transformation of large molecular clusters cooling in X
Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)01014-3 CCC: $12.00
supersonic flow. Its virture is that homogeneous nucleation is assured in such experiments by the particularly deep undercooling attained and absence of contact of the system with foreign surfaces. A drawback is that it cannot be applied to arbitrary substances. Because the properties of TMB are favorable for the supersonic method, it seemed worthwhile to take advantage of the opportunity to compare it with the normally hydrocarbons analyzed previously. Results to be presented do bear out the differences implied by some of Turnbull’s hypotheses.5-8 Procedure Experimental Conditions. Vapor from a liquid sample of TMB (99.99%, Aldrich) was mixed with neon carrier gas in a stainless steel sample holder. Experiments were carried out with a subject mole fraction of 0.10 at a stagnation pressure of 2.48 bar and initial temperature of 313K. The vapor was expanded into a vacuum chamber (10-5 Torr) through a glass miniature Laval nozzle with a throat diameter of 0.17 mm, exit diameter of 1.17 mm, and overall length of 15.15 mm. The gas flow was controlled by a pulsed valve operating at 19 Hz and pulse duration time of 0.4 ms. A 40 kV electron beam with the same frequency and duration, but with a delay of 0.6 ms, intersected the pulsed supersonic jet. The area of the cluster beam selected for analysis was transmitted through a Vee skimmer10 to the probing electron beam. Kodak medium slides recorded the resultant diffraction patterns, each representing the sum of 1140-3610 exposures. Clusters were monitored at distance intervals of 5.0 mm, starting 6.0 mm after clusters exited the nozzle. The average velocity of clusters was observed in a flight tube to be 630 m/s under the aforementioned flow conditions. Therefore, distance intervals of 5 mm correspond to time intervals of 7.9 µs. Analysis of Data. Because the procedure applied to the © 1996 American Chemical Society
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analysis and interpretation of the present experiments deviates from that used in our previous reports,9,11,12 we outline it in the following. As is universally done in treatments of nucleation the rate, J, of homogeneous nucleation is expressed by
J(T) ) A exp(-∆G*/kT)
(1)
where the free energy ∆G* of forming a spherical critical nucleus from the undercooled liquid is given in the capillary model by
∆G* ) 16πσsl3/[3(∆Gv + w′)2]
(2)
in which σsl is the principal unknown kinetic parameter supposed to correspond to the liquid-solid interfacial free energy per unit area. Although the capillary theory is known to have certain quantitative deficiencies, it is the only generally applicable theory available. Therefore, we shall apply it without further caveat. In the denominator of eq 2, ∆Gv represents the free energy of freezing per unit volume and w′ the work of changing the surface area of an outer liquid phase i due to the volume change accompanying the formation of the nucleus of phase j. It is ∆Gv which provides the driving force for freezing in the first place, and it is found by integrating the entropy of fusion over the range of undercooling. The contribution w′ for a nucleus growing in a spherical liquid droplet of radius r0 is (2σi/ r0)(Fi - Fj)/Fi, where 2σi/r0 is the Laplace pressure exerted by the outer phase on the inner phase and the F’s are densities of the phases.13 In the present treatment we neglect w′ because we do not know the relative densities at the deep undercooling attained but believe the difference is small. A number of different representations of prefactor A of eq 1 have appeared in the literature. In prior analyses of nucleation data carried out in this laboratory, the prefactor A was based either on a viscous flow model of molecular jump frequencies or, later, on a “free jump” model.9 The latter model incorporated jump frequencies taken from a treatment of crystal growth rates by Burke et al.14 after it was recognized that the viscous flow model excessively inhibited jumps at low temperatures. It now seems preferable to apply the prefactor of Grant and Gunton,15 a prefactor giving results close to those of our free jump model but one based upon a more rigorous derivation. Both the free jump and the Grant-Gunton prefactors were derived for monatomic systems and, hence, do not explicitly take into account the molecular reorientation encountered in polyatomic systems. Nevertheless, considering the present rather crude state of nucleation theory as it exists for general systems, this neglect is of minor consequence. In the Grant-Gunton theory for a system of volume Vs, the prefactor A is given by
A ) κΩ/Vs
(3)
where κ is the dynamic prefactor
κ ) 2λσslT/LR*3
(4)
with λ, L, and R* representing the thermal conductivity, heat of fusion per unit volume of solid, and radius of the critical nucleus. In eq 3 Ω is the dimensionless statistical prefactor
Ω ) (2/271/2)(Vs/ξ3)(σslξ2/kT)3/2(R*/ξ)4
(5)
with ξ representing a correlation length characterizing the thickness of the interface between the old and new phases. We arbitrarily take it to be about three molecular diameters. In the capillary approximation R* is given by
R* ) -2σsl/∆Gv
(6)
In our earlier measurements of nucleation we estimated the mean temperature at which nucleation took place and then compared the observed fraction F(t) of clusters frozen at time t with the value calculated by the theoretical expression, adjusting the interfacial free energy parameter σsl to achieve the best fit. Because the temperature of the evaporative ensemble of clusters corresponding to our system drops appreciably during the process of freezing, we now convert the rate J(T) to a function of time, J(t), by incorporating the temperature profile T(t) which we discuss in the Appendix. Then we numerically integrate the differential equation
d ln[1 - F(t)] ) -J(t)Vcl dt
(7)
in which Vcl is the cluster volume, and adjust the parameter σsl to attain the optimum fit to the experimental data. The treatment as outlined applies to a system of clusters of uniform size. However, an analysis explicitly taking into account the characteristic distribution of cluster size yielded virtually identical results over the midrange of F(t) if Vcl were taken to be the mean cluster volume. It is our custom when assigning the cluster volume available for nucleation to subtract the volume occupied by the surface molecules9 where nucleation never occurs in systems for which the solid is wetted by the melt. In the present case the surface molecules make up about 25% of the cluster. It is worth mentioning that both the free jump and the Grant-Gunton prefactors have yielded theoretical F(t) curves in better agreement with our experiments than did the viscous flow prefactor. We take as our nucleation rate the value from the calculated curve best reproducing our experimental points. For purposes of analysis we assume that the parameter σsl varies with temperature as
σsl(T) ) σsl(T1)(T/T1)n
(8)
where n is a number presumably rather smaller than unity. Some theoretical treatments suggest that n is positive, but too little experimental information exists to make an intelligent choice. Data for the freezing of mercury yield n ≈ 0.3-0.4, whereas data for water suggest a negative value of about -0.07. For want of a better estimate, we shall taken n to be zero. This choice has little effect on the derivation of σsl from our data, but it does appreciably affect the extrapolation of our nucleation rates to higher and to lower temperatures. Results Because of the substantial undercooling attained before the sample freezes, it is worthwhile to know what phases are thermodynamically stable at low temperatures. In our supersonic jet clusters cool to 169 K (about 80 deg below the freezing point) before they begin to freeze. Huffman et al.3 reported no solid-state phase changes on cooling until the sample reached 121 K. To confirm this for our sample, we subjected it to differential scanning calorimetry and found no solid-state transitions between the melting point and 130 K. In addition, X-ray diffraction powder patterns taken at 236, 193, and 138 K all exhibited the same pattern, the pattern given also by our clusters after they froze as is illustrated in Figure 1. Clusters emerge from the nozzle as liquid microdrops which cool by evaporation as they fly into the evacuated diffraction chamber. Figure 2 depicts how their temperature falls with time. By the time clusters have begun to freeze their average diameter is ∼130 Å (∼5500 molecules). The evolution of their electron diffraction patterns as a greater and greater fraction of the
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Figure 1. Observed powder diffraction patterns of TMB: top, X-ray pattern at 138 K; bottom, electron diffraction pattern of clusters at ∼160 K. The greater breadth of the electron diffraction peaks is due to the small particle size. The differences in relative intensities originate largely from the greater contribution of the hydrogens in the electron patterns.
Figure 2. Thermal history of TMB clusters calculated as discussed in text. Elapsed times are reckoned from the emergence of the clusters from the nozzle.
Figure 3. Electron diffraction patterns of TMB clusters. Times of flight beyond the nozzle tip, from top to bottom, in µs: 9.5, 17.4, 25.4, 33.3, 41.2, 49.1.
Figure 4. Fraction of TMB clusters that have frozen as a function of the time after exiting the nozzle. Filled circles are experimental values. Curves, left to right, are calculated via eqs 1-8 with interfacial free energies of 4.45, 4.55, and 4.65 mJ/m2.
population crystallizes is shown in Figure 3. In Figure 4 the experimental values of F(ti) are compared with three curves calculated via eqs 1-7 for progressively increasing interfacial free energies, σsl, of 4.45, 4.55, and 4.65 mJ/m2. The onset of
Huang et al.
Figure 5. Projected temperature dependence of the nucleation rate for the freezing of TMB calculated to pass through the experimental point at 164 K, assuming that the interfacial free energy varies as T-0.3 (long dashes), T0.0 (solid), and T0.3 (short dashes).
nucleation occurs at about 169 K, and half the clusters are transformed by the time the temperature has dropped to 164 K where the nucleation rate has climbed to 7.9 × 1028 m-3 s-1. Despite the rather large scatter in the experimental data, it can be seen that the apparent precision in the measurement of σsl is better than 0.1 mJ/m2 (although the accuracy is substantially less because of the unresolved status of approximations in the theory applied). How the nucleation rate can be expected to vary with temperature, as projected from the present results by classical nucleation theory (incorporating the Grant-Gunton prefactor), is illustrated in Figure 5. In our earlier reports of nucleation we had presented detailed accounts of calculations of nucleation lag times and crystal growth rates to ascertain whether the lag times and growth times might interfere with our estimates of nucleation rates. We had concluded on the basis of the viscous jump model that neither source of delay would interfere, although some of the calculations gave only marginal assurance. Now that physical arguments and simulations tend to favor the growth law of Burke et al.14 and the molecular jump frequencies implied by the free jump model and Grant-Gunton prefactors,14,16-18 it turns out that the calculated nucleation lag and growth times are so much shorter than the time for an ensemble of clusters to freeze that they are unimportant for the present analysis. Therefore, calculations details will not be presented. One outcome of the fast growth rates is that, once a critical nucleus appears in it, a cluster freezes too quickly for another nucleus to appear and, therefore, freezes to a single crystal. Discussion One of the aims of this research, as mentioned in the Introduction, was to find whether the considerable differences between the thermodynamic properties of globular and straightchain hydrocarbons might lead to conspicuous differences in the nucleation phenomena. The answer is yes, and in several ways. In their emulsion experiments, Turnbull and Cormia8 noted that the long-chain hydrocarbons readily froze at relative undercoolings that were almost an order of magnitude smaller than those for simple molecular substances. By contrast, our globular hydrocarbon froze at a relative undercooling typical of those encountered in the clusters of simple molecules investigated by the supersonic technique. Turnbull and Cormia rationalized their results in terms of a previously proposed hypothesis whereby the nucleation process involves cooperative interactions between small segments of the n-alkane chains rather than between whole molecules. To obtain the correct order of magnitude for the prefactor A in eq 1, they found it necessary to assume that the molecular segment transferred in the elementary nucleation event included only about two carbon
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atoms. Although the ability of a fragment of a flexible molecule to undergo the required displacement independent of the rest of the molecule can be envisaged, a similar participation of a fragment of a more rigid globular molecule cannot. Other consequences of the structural differences are discussed below. A direct comparison between nucleation rates obtained in the emulsion and the cluster experiments is not germane because of the considerably different conditions applying to the two experiments. The different diameters of the droplets (∼ 3 × 10-6 m in emulsions vs 1.3 × 10-8 m in clusters), the extreme differences in the thermal histories (isothermal vs cooling rates in excess of 105 K/s), and consequent large differences in degree of undercooling led to enormously higher nucleation rates in the cluster experiments (by about 15 orders of magnitude). What can be compared meaningfully is the interfacial free energy parameter σsl derived. That such a comparison is legitimate in spite of the vastly different conditions of nucleation is suggested by another example. In the only available comparison of between results of the emulsion method and the cluster method, that of water,12 it was found that the derived interfacial free energies were virtually the same despite the 20 order of magnitude disparity between nucleation rates. In the present experiments σsl was found to be 4.6 mJ/m2, a value only half that reported by Turnbull and Cormia, who obtained 7.2, 9.6, and 9.3 mJ/m2 for C17H36, C18H38, and C24H50, respectively. The difference is too great to be ascribed to the differences in the techniques. We attribute it to differences in the molecular behavior. One way to gain perspective is in terms of an empirical relation between interfacial free energy and molar heat of fusion ∆Hfus found by Turnbull. For a variety of metalloids and simple nonmetallic substances he observed that
σsl ∼ 0.32∆Hfus/(V2NA)1/3
(9)
where V is the molar volume and NA is Avogadro’s number. We have also found the relation to hold reasonably well in our cluster experiments on systems of CCl4, CH3CCl3, CH3CtCCH3, and NH3.11 (In the case of water,12 for which ∆Hfus varies considerably with temperature, the agreement depends upon the value of the heat of fusion adopted.) There is no reason to believe that eq 9 should hold generally and, in fact, it does not come close to holding for the n-alkanes, yet the relation does give a plausible means of comparing related substances. It is instructive to consider the heptanes, all of which have comparable molar volumes. Heats of fusion decrease from 14.2 kJ/ mol for the unbranched, normal isomer to 2.2 kJ/mol for the present subject TMB. This greater than 6-fold difference implied by a blind application of eq 9 does not correspond to the observed difference between the n-alkanes of Turnbull and Cormia and TMB. In fact, if eq 9 were true, irrespective of molecular properties, the long-chain n-alkanes (C18-C24) would have interfacial free energies in excess of 40 mJ/m2 and TMB only about 3.4 mJ/m2. Only the latter value is of the correct magnitude. Since eq 9 was formulated to convert from the heat of fusion per mole to the heat of fusion per unit area (of a layer one molecule thick), it was not intended to apply to long, flexible molecules. It is probably significant that if one selects a twocarbon segment rather than a whole n-alkane molecule, the disparity between the observed value and the value calculated via eq 9 is considerably reduced. We conclude, therefore, that the present globular hydrocarbon acts much like other simple molecules in nucleation events. By contrast, the n-alkanes behave very differently, perhaps because nucleation can be initiated by small segments of the flexible hydrocarbon chains. An opportunity to investigate the mech-
anism of nucleation in more detail has been offered by a massive set of molecular dynamics simulations recently carried out on the melting and freezing of n-alkanes up to dodecane.20 The authors reported trends in melting and freezing points with chain length but did not comment upon the molecular mechanisms involved. Even though the stage of the computations was preliminary and the approximations adopted did not reproduce the odd-even melting differences, it is probable that the simulations contain information that could test the hypothesis of Turnbull and Cormia. It would be of great interest to follow the molecular behavior during nucleation to find how the individual segments of the hydrocarbon chains interact in MD simulations. Until that is done, we must be content with the observation that nucleation rates confirm that the n-alkanes do indeed behave quite differently from simple, rather rigid molecules. Acknowledgment. This research was sponsored by a grant from the National Science Foundation. We thank Mr. Paul Lennon for expert assistance in carrying out the supersonic experiments and Prof. Anding Jin for taking the X-ray diffraction patterns at Nanjing Normal University. Appendix Auxiliary Apparatus. X-ray powder diffraction patterns were taken with a Rigaku D/max-RC rotaflex using Cu KR radiation. A Perkin-Elmer DSC 7-series thermal analysis system was used to investigate possible phase transitions in our sample at low temperatures. Estimate of Temperature. Although the temperatures of the frozen clusters can sometimes be deduced from their lattice constants, the temperatures during the freezing process in our experiments can only be inferred indirectly. Under experimental conditions comparable to those of the present experiments, the growing clusters generated are comparatively warm and bathed in vapor that continues to condense upon them, keeping them warm until they exit the nozzle. After their entry into the vacuum chamber they begin to cool rapidly by evaporation until they reach their “evaporative cooling temperature” some dozens of microseconds later. Two rules of thumb for estimating the evaporative cooling temperature, Tevp, are commonly applied. The first proposes for a normal liquid that21
Tevp ) 0.04∆Evap/R
(10)
where we use the value of ∆Evap at Tevp. An alternative approximation11 makes Tevp the temperature at which the bulk vapor pressure is 0.4 Pa. These approximations suggest that Tvap is 170 K, or 172 K, according to the ∆Hvap(T) listed below or to the vapor pressure equation of ref 22. Because there cannot be an absolute evaporative temperature independent of time, the numbers suggested can at best be rough estimates. A more reliable guide is to integrate the coupled differential equations that govern nucleation, growth, evaporation, condensation, and thermal accommodation in the expanding medium of vapor and rarefying carrier gas that the clusters find themselves in.23,24 If clusters exit the nozzle at temperatures well over Tevp, as they do in the present experiments, the temperature profile T(t) of clusters in flight beyond the nozzle depends mildly upon cluster diameter but very little upon details of processes inside the nozzle. Experimental results are based upon the time evolution T(t) for liquid clusters computed in this manner. Over the time span of concern in the present research, T(t) is probably correct to perhaps a degree or two. As shown in Figure 2, the temperature at the time of onset of nucleation is about 169 K,
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close to the empirically estimated evaporated temperatures, but the additional cooling during the freezing is appreciable. Estimation of Thermodynamic Data. The free energy of freezing per unit volume is calculated from the entropy of transition in the standard way, making use of the estimated difference in heat capacities of the liquid and solid. The heat capacity of solid phase I in J mol-1 K-1 is represented in the region of interest by
Cp ) 99.25 + 0.2071T + 0.000659T2
(11)
and that for the supercooled liquid via an extrapolation of the heat capacity for the normal liquid by
Cp ) 82.25 + 0.4462T - 0.000020T2
(12)
using the experimental data from ref 3. The heat of vaporization was simplified for use in the program to model nucleation, growth, and evaporation of the liquid clusters inside and beyond the nozzle. It was estimated from the enthalpies and heat capacities from refs 3 and 25 to be
∆Hv(T) ) 42.95 - 0.0370T kJ/mol in the mean over the range of temperature of interest. The liquid density was extrapolated from the data from ref 22 and taken as 0.786 g/cm3 at 168 K. The thermal conductivity was assumed to be the same as that for 2,2-dimethylpentane whose temperature dependence was given in ref 26. References and Notes (1) See for example: McBride, J. M. Science 1993, 256, 814. Berry, S. R.; Rice, S. A.; Ross, J. Physical Chemistry; Wiley: New York, 1980; p 860.
(2) Handbook of Physics and Chemistry, 72nd ed.; Lide, D. L., Ed.; CRC Press: Boca Raton, FL, 1991. (3) Huffman, H. M.; Gross, M. E.; Scott, D. W.; McCullough, J. P. J. Phys. Chem. 1961, 65, 495. (4) Vonnegut, B. J. Colloid Sci. 1948, 3, 563. (5) Turnbull, D. J. Appl. Phys. 1949, 20, 817. (6) Turnbull, D. J. Chem. Phys. 1952, 20, 411. (7) Turnbull, D. Solid State Phys. 1956, 3, 225. (8) Turnbull, D.; Cormia, R. L. J. Chem. Phys. 1961, 34, 820. (9) Bartell, L. S. J. Phys. Chem. 1995, 99, 1080. (10) Bartell, L. S.; French, R. J. J. ReV. Sci. Instrum. 1989, 60, 1223. (11) Bartell, L. S.; Dibble, T. S. J. Phys. Chem. 1991, 95, 1159. Dibble, T. S.; Bartell, L. S. J. Phys. Chem. 1992, 92, 2317. Bartell, L. S. J. Phys. Chem. 1994, 98, 4543. Huang, J.; Lu, W.; Bartell, L. S. J. Phys. Chem. 1995, 99, 11147. (12) Huang, J.; Bartell, L. S. J. Phys. Chem. 1995, 99, 3924. (13) Fukuta, N. In Lecture Notes in Physics, Atmospheric Aerosols and Nucleation; Wagner, P. E., Vali, G., Eds.; Springer-Verlag: Berlin, 1989; p 504. (14) Burke, E.; Broughton, Q.; Gilmer, G. H. J. Chem. Phys. 1988, 89, 1030. (15) Grant, M.; Gunton, J. D. Phys. ReV. 1985, B32, 7299. (16) Harrowell, P.; Oxtoby, D. W. J. Chem. Phys. 1984, 80, 1639. (17) Oxtoby, D. W. AdV. Chem. Phys. 1988, 70, 263; Phys. Condens. Matter 1992, 4, 7627. (18) Oxtoby, D. W. Personal communication. (19) Turnbull, D. J. Appl. Phys. 1950, 21, 1022. (20) Esselink, K.; Hilbers, P. A. J.; van Beest, B. W. H. J. Chem. Phys. 1994, 101, 9033. (21) Klots, C. E. J. Phys. Chem. 1988, 92, 5824. (22) Smith, B. D.; R. Srivatava, R. Thermodynamic Data for Pure Compounds, Part B; Elsevier: Amsterdam, 1988; p 136. (23) Bartell, L. S. J. Phys. Chem. 1990, 94, 5120; unpublished work (flow beyond nozzle). (24) Bartell, L. S.; Machonkin, R. A. J. Phys. Chem. 1990, 94, 6468. (25) Waddington, G.; Todd, S. S.; Huffman, H. M. J. Am. Chem. Soc. 1947, 69, 32. (26) Vargaftik, N. B.; Filippi, L. P.; Tarzmanov, A. A.; Evgenii, E. T. Handbook of Thermal ConductiVity of Liquids and Gases; CRC Press: Boca Raton, FL, 1994.
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