J. Phys. Chem. 1992, 96,8603-8610
8603
(7) Sloan, E. D. Clarhrare Hydrares of Narural Gases; Marcel Dekker:
surfaces, the thermodynamic properties of hydrates change the same way as those of ice. In fact, it is sufficient to know the change in the activity of water in order to predict how the hydrate behavior will change using the solid-solution model. In general, hydrates become less stable when present in confining geometries. However, the results are somewhat surprising in terms of the dissociation characteristics of the pore hydrates. On dissociation below the melting point of ice, the pores become plugged with ice, so,though less stable thermodynamically, the apparent stability of the hydrates is enhanced due to encapsulation among the pore walls and the ice cap. This also implies that when such a system is heated to the melting point of ice the hydrates in the interior of the pores will tend to dissociate explosively.
New York, 1990. (8) (9)
Englezos, P.; Bishnoi, P. R. AIChE J . 1988, 34, 1718. Robinson, D. B.; Ng, H.-J.;Chen, C.-J.Proc. Annu. Conv.GasprocCss.
Assoc. 1987, 66, 154. (10) van der Waals, J. H.; Platteeuw, J. C. A d a Chem. Phys. 1959,2, 1. (1 1) Handa, Y.P.; Zakrzewski, M.; Fairbridge, C. J . Phys. Chem., preceding paper in this issue. (12) Makogon, Y . F. Hydrafes of Narural Gas; Pennwell Books: Tulsa, AL, 1981. (13) Evrenos, A. I. J . Perr. Techno/. 1971, Sepr, 1059. (14) Cheng, W. K.;Pinder, K. L. Can. J . Chem. Eng. 1976, 54, 377. (15) Baker, P. E. In Natural Gases in Marine Sediments; Kaplan, I. R., Ed.; Plenum: New York, 1974; p 227. (16) Stoll, R. D. In Natural Gases in Marine Sediments; Kaplan, I. R., Ed.; Plenum: New York, 1974; p 235. (17) Stoll, R. D.; Bryan, G. M. J . Geophys. Res. 1979, 84, 1629. (18) Handa, P.; Stupin, D.; Zakrzewski, M. IEC Report No.EC-121991S, National Research Council of Canada. (19) Handa, Y.P. J. Chem. Thermodyn. 1986, 18, 891. 120) Handa. Y. P. J. Chem. Thermodvn. 1986. 18.915. (21) Robert$, 0. C.; Brownscombe, E.-R.; Howe, L. S.;Ramser, H.Per. Eng. 1941, 12, 56. ( 2 2 ) Deaton, W . M.; Frost, E. M. US.Bureau of Mines Monograph - . No. 8, 1946. (23) McLeod, H. 0.; Campbell, J. M. J . Per. Technol. 1961, 222, 590. (24) Marshall. D. R.: Saito. S.: Kobavashi. R. AIChE J . 1964. 10. 202. (25) de Roo, J. R.; Peters, C. J:; Lichienthaler,R. N.; Diepen, G. A. M. AIChE J . 1983.29, 651. (26) Frost, E. M.; Deaton, W. M. Oil Gas J . 1946, 45, 170. (27) Reamer, H. H.; Selleck, F. T.; Sage, B. H. Pet. Trans. AIME 1952, 195. 197. (28) Holder, G. D.; Kamath, V. A. J. Chem. Thermodyn. 1982,14, 1119. (29) Holder, G. D.;Godbole, S. P. AIChE J . 1982, 28, 930. (30) Dymond, J. H.; Smith, E. B. The Virial Coefficients ofpure Gases and Mixfures, Clarendon: London, 1980.
Acknowledgment Financial support for this work was provided, in part, by the Geological Survey of Canada under Gas Hydrate Project 870021. Registry No. Methane hydrate, 14476-19-8; propane hydrate, 14602-87-0.
References and Notes Judge, A. Proc. 4th Can. Permafrost Conf. 1982, 320. Kvenvolden, K. A. Chem. Geol. 1988, 71, 41. (3) Initial Reports of the Deep Sea Drilling Project. 1982, Leg 67; 1985,
(1) (2)
Leg 84.
(4) Brooks, J. M.; Kennicutt, M. C.; Fay, R. R.; McDonald, T. J. Science 1984,225,409. ( 5 ) Kvenvolden, K. A.; McDonald, T. J. Initial Reports of the Deep Sea Drilling Project. 1985, Leg 84, p 667. (6) Davidson, D.W. In Warm A Comprehensive Treafise;Franks, F., Ed.; Plenum: New York, 1973; Vol. 2, Chapter 3.
Electron Dlffractlon Studles of the Kinetics of Phase Changes in Molecular Clusters. 3. Solld-State Phase Transitions In SeF, and (CHs)SCCI Theodore S. Dibble and Lawrence S. Bartell*
9
Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 (Received: May 18, 1992; In Final Form: July 14, 1992)
Clusters of SeFd and of (CH3)$2C1 produced by the condensation of vapor in supersonic flow through a Lava1 nozzle were observed to undergo transitions from one crystalline phase to another. Selenium hexafluoride clusters transformed from the body-centered cubic (bbc) to the monoclinic phase with a nucleation rate of 5.5 X lo2*m-' s-l in the vicinity of 105 K whereas clusters of tert-butyl chloride underwent a transition from phase I11 (tetragonal) to phase IV at 6.4 X le7m-3 s-' at approximately 156 K. Results are interpreted in terms of the classical theory of homogeneous nucleation. In the kinetic prefactor, however, where the frequency of the primary molecular motions involved is conventionally ascribbd to translational jumps, we find it necessary to invoke reorientational jumps for the present transitions. Comparisons of heats of transition with interfacial free energies deduced from the nucleation rates indicate that Turnbull's empirical relation for interfacial tensions applies to boundaries between the two crystalline phases as well as to boundaries between solids and liquids. The extrapolation of nucleation rates to deep undercoolings via the formalism of nucleation theory leads to projections that rates as high as m-3 s-l are possible for the present solid-state transformations. Molecular dynamics simulations of phase transitions in clusters of SeFs corroborate this suggestion and help place constraints on aspects of the mechanism involved. The present experiments apparently constitute the fmt measurementsof nucleation rates and determinations of the cmeqonding interfacial free energies for transitions between two Crystalline phases in onecomponent systems and in systems of plyatomic molecules.
Iatraduction Transformations between crystalline phases have been studied by scientists in many disciplines and in widely varying materials.1*2 Most of the work in this field examines the overall rate of transformation' or its dependence on thermal history' rather than the nucleation rate itself. This is understandable in view of the experimental and theoretical difficulties surrounding nucleation, in general, and nucleation in crystalline phases, in particular. Additionally, analysis of experimental results requires extensive knowledge of the thermodynamic and physical properties of undercooled materials, information which is seldom available.
Experimental investigations into the nucleation of transitions between solid phases have been carried out for a number of metal alloy^,^*^^^ because of their obvious technological importance. We know of no previous determinations of the nucleation rate of transformations between crystalline phases in one-component systems or in polyatomic species. Neither are we aware of determinations of the interfacial free energy between two crystalline phases in such systems. Research a t the University of Michigan is currently focusing upon the structure and transformations of molecular clusters. Both experimental and computational techniques are being brought to
0022-3654/92/2096-8603S03.00/00 1992 American Chemical Society
8604 The Journal of Physical Chemistry, Vol. 96, No. ,21, 1992
Dibble and Bartell
bear in the study of nucleation in transitions between condensed phases. The experimental investigations detect changes in the internal organization of large molecular clusters formed in supersonic flow. In some cases liquid clusters have been observed to freeze in flight outside the nozzle, while in others solid clusters have been seen to undergo transformations between crystalline Analyses of the freezing of molecular clusters have already been p u b l i ~ h e d . ~We * ~ now report observations of the transition of SeF, from its bcc to its monoclinic phase and of the transition of (CH3)3CClfrom phase I11 to phase IV.
Background Clusters are formed in the present investigations by homogeneous nucleation of vapor, seeded into a rare gas carrier, during supersonic flow through a Laval nozzle. The expanding, cooling carrier gas removes the heat of condensation and allows clusters to accumulate thousands of molecules under mild expansion conditions. At comparatively high subject mole fractions and low carrier pressures, temperatures of clusters beyond the nozzle are governed by evaporative cooling. At higher concentrations of camer gas clusters are formed earlier in the flow and can be cooled by thermal accommodation before they exit the nozzle to well An electron beam below the evaporative cooling probes the clusters at some position beyond the nozzle. By translating the nozzle relative to the electron beam, we obtain a set of diffraction patterns over a range of times of flight of clusters after their formation. The patterns can be interpreted in terms of the size of the clusters and the fraction transformed at each time of flight. From these data we determine the nucleation rate, namely, the number of post-critical nuclei formed per unit volume per unit time. In the solid state, SeF6 appears to exhibit the same behavior as its homolog, SF6. Below their sublimation temperatures, both compounds exhibit bcc phases that are orientationally disordered (plastically crystalline). 2 ~ 1 Significant translational diffusion occurs in the solid state near the melting point of 239 K.I2-l4 Between 239 K and the transition temperature of 128 K, the average time between reorientational jumps for each molecule of SeF6 rises from about 2 to 8 ps.I5 In the low-temperature phase' of SeF6, molecular reorientation is significantly slower.I4J6 Michel et al.I3observed a single phase transition in crystalline J ~ ~by~ ~ SeF6 occurring at 173 f 15 K. Other s t u d i e ~ , ' ~some Michel's coauthors, also found only one transition, but that occurred near 128 K. Michel et al. indexed their powder diffraction pattern of the colder solid phase of SeF6 in terms of an orthorhombic cell of space group Pnma, but their assignments violated the reflection conditions for that space group. Moreover, their proposed indexing would lead to the counterintuitive implication that the lower temperature phase would be less dense than the bcc. In fact, the pattern they identified as orthorhombic, when combined with a pattern of the bcc phase, looks much like a pattern of the monoclinic phase of SeF, found in previous cluster studies.17 It appears that Michel et al. mistook some lines due to the monoclinic phase for lines arising from the bcc phase. We therefore conjecture that the phase identified by Michel et al. as orthorhombic was, in fact, monoclinic. Sulfur hexafluoride appears to transform from bcc to monoclinic through an intermediate trigonal structure.18-20Presumably SeF6 transforms the same way. Rotation of one-third of the molecules by 60' allows fluorine atoms to fit into the hollows of the nextnearest molecules, transforming the crystal to a trigonal structure. The molecules which have reoriented exhibit conspicuous orientational disorder in simulations,20but the other two-thirds of the molecules acquire an increased degree of order. Small displacements of the molecules complete the transition from trigonal to the monoclinic phase, in which all molecules are orientationally ordered. The phase transition is presumed to be driven by the decrease in potential energy accompanying the denser packing.21 terf-Butyl chloride exhibits four crystalline phases,22two orientationally disordered phases above 217 K and two ordered structures at lower temperatures. The two ordered phases were observed in previous cluster studies.23 These correspond to the
L
I
I
I
I
1
2
3
4
s
(A1)
Fignre 1. Electron diffraction patterns of clusters of SeF, in the supersonic jet. The uppermost pattern corresponds to a time of flight of clusters outside the nozzle of 7.6 ps; the next thra patterns were obtained at time-of-flight intervals differing by 7.6 1.1s. The lowest pattem, obtained with somewhat higher stagnation pressures, is of more completely transformed clusters. Intensities have been leveled and rescalcd as described in the Appendix of ref 8.
tetragonal phase (111) and another form at lower temperature which appears to be the coldest phase (IV), the structure of which is not known. Each molecule reorients about its C-Cl axis about every 10 ps in phase 111; in phase IV the period is more than 200 ps.24 Translationaljumps and more complex reorientations occur rapidly in phase I but not in I11 or IV.24*25
Experimeatd Section Because the apparatus and procedures have been described elsewhere,"*26only a brief description will be given here. Gaseous selenium hexafluoride (99.5%, Noah Technologies) was seeded into neon (99.99996, Air Prcducts), making a mixture with an SeF6 mole fraction of 0.027. The total pressure was set to 4.8 bar. Neon carrier was passed through glass wool saturated with liquid fert-butyl chloride (9995, Aldrich) to make a mixture with a (CH3),CC1 mole fraction of 0.22 at a total pressure of 2.1 bar. Gas flow into a miniature glaas Laval nozzle (# 6)*' was controlled by means of a pulsed valve operating at 20 Hz. Electron beam pulses were timed to intersect a pulse of the supersonic jet at its earliest fully developed state before gas rebounding from the walls of the diffraction chamber could seriously degrade the jet. Photographic plates recorded diffraction patterns that were sums of hundreds to thousands of exposures (each of 0.3-msduration) depending upon flow and sampling conditions. Clusters of SeF, were monitored at time-of-flight intervals of 7.6 p s starting 7.6 ps after exiting the nozzle; (CH3)3CCIclusters were monitored at 1 4 - p intervals starting approximately 136 p s after clusters exited the nozzle. A Vee skimmer" was used in the (CH3)3CCl experiments to transmit only the central portion of the cluster jet. This was not necessary for SeF6 because the jet was probed closer to the nozzle before it diverged excessively. The conditions of the present experiments were chosen, based on previous experiments, to initiate the phase transition at a convenient distance beyond the nozzle. Changing conditions of the expansion to accelerate the cooling of clusters tends to make the transition occur earlier in the flow. Results Electron diffraction patterns of SeFs clusters are shown in Figure 1. Growth of rings due to the monoclinic phase can be observed. Ring breadths correspond to cluster diameters averaging 90A.28 Values of F(t), the volume fraction of clusters transformed at time t , are listed in the Appendix. From the values of F(r) and the mean cluster volume (derived from the cluster diameter), we
The Journal of Physical Chemistry, Vol. 96, No. 21, I992 8605
Solid-state Phase Transitions in SeF6 and (CH3),CC1
A = 2(a,/kT)1/Zv,-2/3[vo exp(-E/RT)]
I
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1
2
3
s
(A1)
Figure 2. Electron diffraction patterns of clusters of (CH3)3CCIin the supersonicjet. The upper four patterns were obtained at times of flight of clusters outside the nozzle of 60, 136, 150, and 164 ps. The bottom pattern was obtained by increasing the carrier gas pressure 3-fold, with a corresponding decrease in the concentration of (CH3)3CCI. The top and bottom patterns are taken as reference patterns of phase 111 and phase IV, respectively.
arrive at a nucleation rate, J, of 5.5 X loz8m-, s-l by applying eq 24 of ref 8. The diffraction patterns of clusters of (CH3),CC1 reproduced in Figure 2 exhibit a steady growth of Debye-Scherrer rings of phase IV at the expense of the tetragonal phase. For this transformation a nucleation rate of 6.4 X loz7m-3 s-I is derived from the F(t) values listed in Table I1 of the Appendix. Ring breadths of tetragonal clusters obtained under milder but almost identical conditions correspond to an average cluster diameter of 149 A.
Interpretation of Results According to the classical theory of homogeneous nucleation, the steady-state nucleation rate can be expressed asz9 J = A exp(-AG*/kT)
(1)
where AG* is the free energy barrier to formation of a critical nucleus of a new phase and A is a kinetic prefactor. In the classical theory of homogeneous nucleation, AG* is determined by the interplay among three terms: the Gibbs free energy of transformation, the energy required to create an interface between two phases, and the strain energy.1° Sources of strain energy include shear, mismatch of lattice parameters, and the difference between the specific volume of the transforming region and that of the surrounding matrix. In the case of freezing, strain energy does not arise because the liquid matrix flows, and the shape of the nucleus tends to be spherical to minimize the interfacial area. For many transformations in the solid state, however, strain energy is believed to dominate the free energy barrier and to govern the shape of the nuclei.’ To calculate the effect of strain energy, one must known the effect of lattice misfit and the value of such quantities as the bulk modulus of the matrix phase, the fractional volume change of transition at the nucleation temperature, and the shear angle of transition. Because this information is not available, and may not pertain to a crystal the size of a cluster, we will neglect the strain energy in the analysis. In the absence of strain energy, AG* is given by
AG* = 16rass3/( 3AGV2)
(2)
where a, is the interfacial free energy between two solid phases (considered an average over crystal faces) and AG, is the Gibbs free energy of transition per unit volume. The kinetic prefactor, A, is taken to bez9
(3)
where u, is the molecular volume. Nucleation theory postulates that molecules cross the interface between the matrix and the nascent nucleus with a frequency defined by the bracketed expression. For solid-state transitions in monatomic systems, the value of E is often considered to equal the activation barrier to d i f f u s i ~ n , ~and ~ * ~u0’is identified with the vibrational frequency of the crysta131 or is derived from experimental studies of translational diffusion.29 These definitions of u0 and E may be applicable in studies of polyatomic species, as well, at least for reconstructive transformations (transformations in which the primary coordination changes during the t r a n s i t i ~ n ) . ~For ~ *transitions ~~ from plastically crystalline to ordered crystalline forms of some substituted adamantanes, however, El Adib3’ has shown that the activation bamer in eq 3 corresponds to the bamer to molecular reorientation rather than translation. It is probable that El Adib’s interpretation applies to the bcc-temonoclinic transitions of SF6and SeF,, which are initiated by molecular rebrientation. In the case of (CH,),CCl, the mechanism of the transition from phase I11 to phase IV is not known, and phase I11 is not even a plastically crystalline phase according to Timmerman’s criterion that the entropy of fusion be less than 2.5R.” Nevertheless, it is plausible to apply El Adib’s interpretation again, as will be shown. Analogous considerationsmay apply to the nucleation lag time, e,,., which is a measure of the time to reach a steady-state nucleation rate.35 For freezing, 8,. can be e x p r e s ~ e das ~,~~ (4)
where D = DOexp(-EJRT) is the diffusion coefficient. For the transformations studied here, we substitute assfor asland a reorientational frequency, u0 exp(-EIRT), for the frequency of translational jumps, ~ D / V , ~ to / ~obtain ,
e,.
=
3kTu, AGvzv0exp(-E / R T)v,~/~
If, in the analysis of our results for SeF6,we had assumed that the jump frequency in eq 3 corresponded to a frequency of translational jumps,I2 we would have obtained a physically meaningless result making prefactor A less than the observed nucleation rate for all temperatures below the bulk transition temperature. In fact, the average time between translationaljumps in the bcc phase is more than 1 s at the transition temperature. Therefore, translational jumps do not occur on the time scale of our experiment, a fact which is reflected in values of 9,. calculated on this basis (eq 4, with a, substituted for as,)that are on the order of seconds. Consequently, in the present study, we are compelled to interpret eq 3 as referring to reorientational jumps. The frequency of translational jumps in the tetragonal phase of (CH3)$CI is not known. We obtain a crude estimate by extrapolating the Arrhenius temperature dependence in phase I to temperatures at which the tetragonal phase (111) is supercooled. This extrapolation certainly overestimates the frequency of translational jumps in phase I11 because the frequency of translational j u m p is expected to decrease discontinuouslyat the phase transitions between phase I and phase 111. Moreover, translational jumps almost certainly have a higher activation barrier in phase I11 than in phase I. If, nevertheless, we were to employ this extrapolation in our analyses, we would obtain values of A significantly greater than the nucleation rate. On the other hand, values of 8,. would be of the order of 0.01 s, far longer than the time scale of the experiment. Therefore, in the case of (CH3)3CCIwe again interpret eq 3 in terms of reorientational jumps. Unfortunately, it is not known with any precision what values of u0 and E, should be adopted for either of the two compounds studied herein or even if molecular reorientation follows Arrhenius behavior in the supercooled matrix phases. It is thought that the dominant reorientational motion observed in bcc SF6corresponds
Dibble and Bartell
8606 The Journal of Physical Chemistry, Vol. 96, No. 21. 1992
to 90° jumps between equivalent orientation^.^' The frequency of the 60° rotation about a C3molecular axis, which presumably initiates the bcc-monoclinic phase transition, is not known. We will neglect the distinction. In the case of (CH,),CCl, it is uncertain what reorientational motion initiates the phase transition. One Clue is that, as in the bcc-to-monoclinic transition in SeF,, the frequency of molecular reorientationaljumps about the C-CI axis in (CH3)3CCldecreases by an order of magnitude as a result of the 111-to-IV t r a n ~ i t i o n . ~ ~For * *want ~ of more definitive information, we attribute the reorientational jumps of (CH3)3CCl to motions about this axis and infer vo and E from the temperature dependence measured for phase III.24 In the absence of more definitive data, we will follow Angell's suggestion38that rotations in supercooled molecular crystals exhibit Arrhenius behavior, and represent the frequencies of molecular reorientations in !kF6 and (CH3)3CClby Arrhenius rate laws, and will extrapolate from the measured range into the supercooled regime. Derived results are insensitive to details of the extrapolations. Experimental and estimated values of physical properties incorporated into our analyses are given in the Appendix.
Discussion Identification of the rate of transformation of clusters with the nucleation rate requires that growth of post-critical nuclei is fast on the time scale of the experiment, i.e., that nucleation rather than propagation of the phase transition (growth) determines the rate of transformation. K a ~ h c h i e vhas ~ ~shown that the linear growth rate, G, required to fulfill this condition can be expressed as G >> J p t 3 a ' 1 3
(6)
where Vis the volume of the sample and a is the ratio of the volume in which the transformation is detected to the sample volume. Using the cluster volumes and nucleation rates determined previously and setting a equal to unity, we calculate that a growth rate of at least lo4 m/s is needed for both (CH3)3CCland SeF,. No experimental determinations of the rate of growth appear to be available for either SeF, or (CH3)3CCI. From the previous section it is clear that translational jumps are too slow to participate in the phase transition, but we are unaware of a theory of growth based on molecular reorientations which could be used to quantify growth rates. If we assume that the rate of radial growth is of the magnitude of the product of the molecular diameter and the frequency of reorientation, as might be reasonable at the significant supercoolings attained by the clusters, we obtain growth rates of the order of 50 m/s for both compounds. In molecular dynamics simulationsof SeF6,40941 the bcc-to-monoclinic phase transition occurs in clusters of a few hundred molecules in 10-50 ps, implying a growth rate of at least 10 m/s. These values are still 5 orders of magnitude higher than required by Kashchiev's criterion for mononuclear transformation. Therefore, we conclude that the observed rates of transformation are not limited by the rates of propagation of the phase changes and that the transition in each cluster is the result of the formation of a single critical nucleus. Inference of u, from a measured nucleation rate requires a reasonably accuate knowledge of the cluster temperature. Cluster temperatures cannot be determined from our diffraction data because neither the coefficients of thermal expansion nor accurate lattice constants are known for (CH3)3CC1or SeF6.42The high mole fraction and low total pressure used in the (CH3)3CCIexperiments assure that the clusters of (CH3)3CCIexit the nozzle at temperatures well above the evaporative cooling temperature and subsequently cool by evaporation. Estimates of the evaporative cooling of (CH3),CC1 based on two empirical rules8 are 170 and 160 K. Because the single temperatures suggested by such rules do not describe the changing temperatures of clusters in flight outside the nozzle, we calculated the evaporative cooling along the cluster trajectories. The results of these computations, the principles of which are described in detail e l ~ e w h e r e , 4are ~ - ~depicted in Figure 3. They indicate that (CH3)3CClclusters formed under the experimentalconditions have
180 1-
k5
\
170
I
100
10
200
Time (ps) Figure 3. Calculated thermal history of clusters of (CH3),CC1,cooling by evaporation in flight outside the nozzle. 4.00
I
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3.15 3.50 3.25
3.00 2.15 2.50 2.25 I 145
I 150
155
160
165
T (K) Figure 4. Interfacial free energy, u,,, in mJ/m2, between phase I11 and phase IV of (CH3)3CCI,calculated from the experimental nucleation rate. Bccausc the nucleation temperature is not well known, u,, is calculated to reproduce the single observed rate over a range of aaumed cluster temperatures. The curve does not represent the (unknown) temperature dependence of u".
cooled by evaporation to about 156 K during the comparatively long time of flight before they are probed by the electron beam. Values of u, for the interface between phases I11 and IV of (CH3)3CClshow a strong dependence on the assumed cluster temperature. Therefore, in Figure 4 we plot over a range of assumed cluster temperature values of a, that would reproduce the single measured nucleation rate J. At the estimated cluster temperature of 156 K, the value for a, is 3.2 mJ/m2. The evaporative cooling temperature of SeF6 is about 120 K according to conventional rules of t h ~ m b . ~However, , ~ ~ . ~ at the relatively low mole fraction and high total pressure used in the SeF6experiments, condensation occurred early in the flow and clusters were cooled strongly, before they left the nozzle, by thermal accommodation with expanding carrier gas. It is likely that they cooled significantly below the evaporative cooling temperature. As in the case of (CH3)3CCl,we attempted to calculate the thermal history of clusters of SeF,. Unfortunately, unlike the calculations for (CH3)3CCl,which were extremely insensitive to poorly known condensation parameters, calculations for SeF6 with a high concentration of carrier gas were very unstable. Cluster temperatures ranging from 90 to 120 K in the region probed by the electrons could not be ruled out by the computations when tenable variations were made in the parameters governing the rate of condensation and cluster growth. Other considerations narrow the b i t s . Experimental studies of clusters of sF6obtained under the same conditions"+*' suggest that SeFs clusters produced in this experiment are appreciably warmer than the lower limit of 90 K calculated above. Cluster temperatures were known to be approximately 100 K for SF6 by virtue of the appearance of the 95 K transition induced by slightly more extreme conditions. Presumably clusters of SeF, would be somewhat warmer. Moreover, because the heat of transition48warms transforming clusters by about 20 deg, clusters that are initially bcc cannot transform completely unless they are colder than about 110 K. Under similar conditions (mole fraction 0.03 rather than 0.027, neon carrier, total pressure 4.8 bar), clusters of SeF6 appear to have transformed quite completely at times of flight longer than
Solid-state Phase Transitions in SeF6 and (CH&CCI
-
5.5
-
5.0 4.5
-
4.0
-
\
\ - I
3.0 90
95
100 105 110 115 120
T (K) Figure 5. Calculated value of u,, for the bcc-monoclinic interface of ScF6: dependence on assumed cluster temperature. See caption to Figure 4.
those examined here. Because the temperature at which clusters of SeF6 transform is not well known, values of usscalculated to reproduce the single measured nucleation rate are shown in Figure 5 for a range of assumed cluster temperatures. Calculated values of us fall monotonically from 5.1 to 3.8 mJ/m2 as the assumed cluster temperature rises from 100 to 110 K. Whether the time elapsed between the formation of clusters and the onset of their transformation provides information of kinetic significance needs to be examined. This time of flight might conceivably be related to the time to develop a steady-state nucleation rate, a time thought to be about 38,..35 Such an interpretation appears unlikely, because lag times calculated from eq 5 are several orders of magnitude shorter than the observed delays, being approximately 0.1 ns for SeF6 and 1 ns for (CH3)3CCl. In the case of (CH&2Cl the delay is readily attributable to the time required for the warm clusters emerging from the nozzle to cool by evaporation to a sufficiently low temperature. That is, clusters will not be able to transform until they reach an undercooling at which the nucleation rate is high enough to induce the phase transition during the characteristic time of observation. The extreme temperature dependence of the nucleation rate to be derived below together with the cooling rate of Figure 3 makes this account reasonable. However, it also implies that the calculated nucleation rate corresponds to an average over a range of temperatures. In the case of SeF6,the argument presented above concerning the degree of undercooling of SeF, clusters implies that cluster temperatures have fallen appreciably below their evaporative temperature in the cooling expansion of the carrier gas. If clusters had cooled far below their evaporative cooling temperature before they exited the nozzle, their vapor pressure would have become so low that further evaporative cooling in flight beyond the nozzle would have been negligible. If they were unable to cool further, there would seem to be no reason for the clusters to have waited to transform until 10 ps after they had exited the nozzle. Therefore, it is reasonable to conclude that the SeF6clusters leave the nozzle cooler than but only moderately cooler than the swalled 'evaporative cooling temperature". They continue to cool another degree or two in the next 10 ps until they reach a nucleation rate high enough for the transition to occur on the time scale of the observations. Just how rapidly the nucleation rate varies with temperature is discussed below. To the best of our knowledge, no prior measurements of free energies of interfaces between two crystalline phases of the same compound exist. Those derived from our nucleation rates, namely os = 3.2 mJ/m2 (tert-butyl chloride) and 4.4 mJ/m2 (selenium hexafluoride), are of the same magnitude as solid-liquid interfacial free energies for comparable substances. Values of us have been deduced for metal alloys and for grain boundaries between metals, h0wever.4~These values are typically much higher than the above values, as would be expected in view of the higher cohesive forces in metallic systems exemplified by large surface tensions and heats of transition. Such considerations invite a question. Might Turnbull's empirical ruleSorelating solid-liquid interfacial free energies per unit area to heats of melting, namely
The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 8607
be fruitfully extended to interfaces between two different solid phases to yield us = k$hHtr/(V2Na)'13
(8)
where kT and k$ are universal constants? Analogous relations have been noted between the liquid-vapor surface tension and heat of e~aporization.~~ In eqs 7 and 8 the terms A&/(pNa)It3 can be regarded as enthalpies of transition per unit area, and the physical constants to evaluate them are tabulated in the Appendix. Turnbull found that kT tended to be approximately 0.45 for metals and 0.32 for certain metalloids and nonmetallic compounds. Our studies of the freezing of CC14*and CH3CC139agreed with the latter value. Results of the present investigation make k$ 1 0 . 3 for (CHJ3CC1 and 0.2-0.3 for SeF6 (where the estimated temperature and heat of transition are more uncertain). It appears, then, that interfacial free energy and enthalpy are related in much the same way for solidsolid interfaces as they are for solid-liquid interfaces. Although Turnbull's relation can be of considerable value when educated guesses have to be made when no experimental information exists (see, for example, ref 52), it cannot be supposed that the relation holds rigidly. Because the temperature dependences of usland AHfusare quite different, Turnbull's kT of eq 7 cannot be truly constant over a range of undercoolings. For us and k $ the temperature dependence is unknown. The extraordinarily large rates observed in the highly undercooled clusters are of interest in themselves. Moreover, once the kinetically relevant interfacial free energy becomes available, it is possible to estimate what rates can be attained at even deeper undercoolings. That these model computations lead to some remarkable projections for the present solid-state transitions is discussed next. In prior studies of the kinetics of freezing of liquid clusters, it was argued that the temperatures at which the clusters were observed to freeze in supersonic flow were close to the temperatures corresponding to the maximum nucleation rates theoretically possible for the substances investigated. The present treatment implies that the solid-state transitions of (CHJ3CC1 and SeF6 are entirely different. Subjective notions about molecular mobilities in liquids and solids might lead one to believe that transformations between solid phases tend to be more sluggish than the freezing of liquids. On the other hand, the solid-state nucleation rates measured in the present study are of the same magnitude as the rates previously found for the freezing of clusters. What is crucial to note is the difference in mechanism believed to operate. Translational jumps of molecules become rate limiting in the freezing of very cold, viscous liquids where the jump rate deviates greatly from Arrhenius behavior as the glass transition is approached. In the two solid-state transitions reported here, it is inferred that it is molecular reorientational jumps rather than translational jumps that usher in the transformations. Because reorientational rates are believed to conform more nearly to an Arrhenius temperature dependence than does viscous flow, it is quite possible that nonreconstructive solid-state transitions can achieve considerably higher nucleation rates at great undercoolings than can freezing. Evidence supporting this conjecture is given below. It is instructive to apply the present treatment to extrapolate the nucleation rate to lower temperatures. This can be done by inserting into eqs 1-3 the interfacial free energies a, derived above along with the physical properties listed in the Appendix. For illustrative purposes we take u,(T) to be anstant. Surface tensions are known to decrease with temperature, and solid-liquid interfacial tensions are believed to increase.s3 As mentioned before, the behavior of us( r ) is unknown. If a, were strongly temperature dependent, the slope of J(7') calculated from eqs 1-3 could be shifted substantially. We will show, however, that the order of magnitude of the estimated temperature dependence of J(7') is entirely reasonable if u, is fmed. Figures 6 and 7 give the results of our calculations of J( 7') for (CH3)$C1 and SeF6, where J( r ) is constructed to pass through the single measured rate for each
8608 The Journal of Physical Chemistry, Vol. 96, No. 21, 1992
c,
M
3
TABLE I: T b e ~ ~ t ~ o d yand ~ mPhysical i~ Properties” quantity SeFn ref (CH3,CCI 128 12 183.1 T,,, K AH,.. J/mol (2180) 48 1860 c,(i’j-‘cp(2)*,(5.54 1 0 . 1 5 2 5 ~ ) 48 (68.7 - 0 . 4 6 ~ ) J/(mol K) V. cm3/mol 63.7 17 92.7 1.4 X 10l2 15 7.6 X 10l2 uo) HZ E: kJ/mol 2.4 15 7.5 Do,”cm2/s 0.1 12 0.025 E,,” kJ/mol 38 12 15.4
35t \ 1
30 25
~~
-
20-
15 120 130 140 150 160 170
Figure 6. Temperature dependence of nucleation rate for the phase 111-phase IV transformation of (CH,)$Cl, constructed to pass through the experimental nucleation rate at 156 K. The interfacial free energy
is assumed to be independent of temperature. I
-3
I
25
20 15 10
70
80
90
100
110
Dibble and Bartell
120
T (K) Figure 7. Two constructionsof the temperaturedependence of nucleation
rate for the bcc-to-monoclinictransition of SeF+ These two constructions were carried out to illustrate the range of uncertainty corresponding to the 10-deg uncertainty in the inferred nucleation temperature. The solid and dashed curves are constrained to pass through the experimental nucleation rate at 110 and 100 K, respectively. The circles are rates derived from MD simulations of the phase changes in 150-molecule clusters. substance. In the case of SeF, two curves are plotted. They span the estimated range of uncertainty in the temperature of the transforming clusters. The predicted temperature dependences are quite different from those calculated for the freezing of clusters in refs 8 and 9. Unlike the latter, for which the J( 7‘) curves soon peak and then decline as the temperature falls below the evaporative cooling temperature, the present J(7‘) curves continue to rise with further cooling. If these projections are correct, the nucleation rates, which at loz8 m-3 s-’ in the present experiments are already enormous, can climb well over a millionfold higher with further decrease in temperature. Such a prediction is based, of course, on the presumed adherence of the rates of the proposed molecular jumps to an Arrhenius law. That this remarkable rate of increase of J( 7‘) is real is corroborated by molecular dynamics simulations of the bcc-monoclinic phase change in clusters of SeF6.40s41 In these simulations clusters containing 150 molecules were cooled in 10-deg steps every 50, 100, or 200 ps, starting with the bcc structure. Altogether 15 independent rum were made. Although this number is insufficient to generate precise statistics of the stochastic process of nucleation, it is enough to yield quite reasonable orders of magnitude. Most clusters spontaneously transformed at various times in the range 80-90 K. Nucleation always began in the interior of the cluster. This and related evidences4 raises questions about what fraction of the total volume of a cluster can be considered to be the ‘effective” volume for nucleation. In our experimental clusters of 3600 molecules 28% of the molecules are in the outer layer, but in 150-molecule clusters, fully two-thirds of them are. For the present order-of-magnitude purposes we shall assume that the effective volume for nucleation is that of the interior, or one-third that of the 150-molecule clusters. The nucleation rates estimated for the times and temperatures in the MD simulations are 2 X m-3 s-I at 90 K and 2 X m-3 s-’ at 80 K. They can be
-
ref 55 55 55 56
24 24 57 57
Numbers in parentheses correspond to unavailable information estimated as described in the Appendix. bIndices 1 and 2 refer to the phases stable at high and low temperature, respectively. CQuantitiesu0 and E refer to molecular reorientation. “The frequency of translational jumps is taken as 600 exp(-E,/RT)/vm2/’. seen to be of the magnitude predicted (Figure 7) from the experimental rate. The investigation of nucleation rates in extremely small, highly undercooled aggregates of matter is too new to have been fashioned into a fully developed technique. Nevertheless, the opportunities it offers for studying processes which have heretofore eluded observation make it a promising avenue of approach to pursue. In the present analyses simplifying assumptions were made, some of whose consequences can be checked. Underlying the entire treatment is the classical theory of homogeneous nucleation, a theory which invokes critical nuclei with bulklike chemical potentials and surfaces with bulklike interfacial free energies. The molecular motions associated with the propagation of the transformations studied were taken to be reorientational jumps adhering to the Arrhenius rate law. The assumption of translational jumps had led to contradictions. Also,the strain energy of one solid phase formed in another was neglected on the basis of the belief that the strain would be minor in particles only a few unit cells across. The availability of the molecular dynamics (MD) simulations for SeF6makes it feasible to carry out a preliminary assessment of some of the assumptions. As stated above, the nucleation rate J(7‘) determined from the M D results was in conformity with extrapolations of the experimental rate to large undercoolings. Additional checks are suggested by the simulations. Nucleation rates developed after the end of the induction period for forming nuclei are only part of the story. In simulations the time lag before steady-state nucleation is reached ought to be compatible with the kinetics of buildup of precritical nuclei implied by the assumed rate law for molecular reorientations. This law was taken to correspond to a mild, Arrhenius temperature dependence. Consequences of assuming that the frequency of the proposed molecular jumps falls off more rapidly with temperature than described by the Arrhenius rate law are straightforward. If the frequency is considered to fall off much more rapidly, as it is believed to in the case of freezing: then the calculated lag time for the bcemonoclinic transition of SeF6 would be longer at large undercoolings, according to eq 4, than the lag times found in the MD simulations. Some interferences can be drawn about the strain energy as well. If strain energy made a significant contribution to the free energy barrier in the formation of a critical nucleus, it would presumably be far more important for the large clusters produced in experiments than for the 150-molecule clusters in the simulations. Therefore, if the strain energy had been influential in the experiments, the value of a,, derived ignoring strain would have been artificially large. Its adoption in calculating nucleation rates at deep undercoolings would have yielded values of J( r ) much lower than those seen in the simulations. Moreover, the observation that interfacial free energies for boundaries between different solid phases are essentially the same fraction of the heat of transition per unit area as is found in boundaries between other condensed phases (solid and liquid, in freezing) lends some plausibility to the values of a, derived. It cannot be claimed that the assumptions introduced in the interpretation of the present results have been rigorously tested. Overall, however, the picture
Solid-state Phase Transitions in SeF6 and (CH3)$C1
TABLE II: Extent of Transformationa (CH3)3CCl
SeF6 f
(as) 7.6 15 23 31
F(t)
0 0.25 0.37 0.35
t (PSI 60 136 150 164
F(t)
0 0.44 0.53 0.60
,I F ( f ) corresponds to the volume fraction of clusters transformed a t time of flight, 1, of clusters outside the nozzle. F(r) was determined as described in the Appendix of ref 8, based on the diffraction range of 1.5 < s (A-I) < 2.5 for (CH&CCl and 2.7 < s (A-I) < 3.1 for SeF6. that emerges seems to be internally consistent and pleasingly coherent.
Acknowledgment. This research was supported by a grant from the National Science Foundation. We thank Mr. Shimin Xu for permission to use his unpublished results for simulations of SeF6 clusters. T.S.D. acknowledges the receipt of a Hermann and Margaret Sokol Fellowship and a PreDoctoral Fellowship from the Rackham Graduate School. We thank Mr. Paul Lennon for his invaluable assistance in operating the electron diffraction unit.
Appendix Physical Properties of SeF6 and (CH3)$CI. Inference of us6 from the nucleation rate requires knowledge of the physical and thermodynamic properties of the supercooled parent phase, information which is not directly available. Also,the heat capacity and heat of transition of SeF6 are not known. It was therefore necessary to make subjective estimates of these properties. The molar enthalpy change, A&, of transformation for SeF6 was derived by assuming that the entropy change is the same as that for SF6.48 We assumed that lattice and rotational contributions to the heat capacity of SeF6 are comparable to those of SF6in the same phase at the same temperature. Accordingly, the free energy of transition of SeF, was constructed from the entropy of transition of SF6and the heat capacity of monoclinic SF6 and extrapolations of the heat capacity of bcc SF6!,@ The reorientation frequency in the supermled bcc phase of SeF6 was extrapolated from the data of Gilbert and Drifford.15 The molar volume was taken to be that of the bcc phase.” Heat capacities and the heat of transition of (CH3)$C1 have been determined The heat capacity of the tetragonal phase was extrapolated into the supercooled regime. The molar volume of (CH3)3CClwas taken from powder diffraction data of the tetragonal phases5, We estimated the temperature dependence of the rate of reorientation in the tetragonal phase of (CH3)$CI from a study employing quasi-elastic neutron scatt e r i ~ ~ gNMR . ~ ~ studies of the bcc phase determined the temperature dependence of the frequency of translational diffusi0n,2~,~’ which was extrapolated into the supercooled regime. Values of the estimated and observed physical properties incorporated into our analyses for both compounds are listed in Table I. clmter Velocities. Velocities of clusters of SeF6and (CH3)3CCl in the supersonicjet were inferred from a treatment by Schwartz and and re^.^* These authors have shown at what point in the expanding, accelerating carrier gas in a free jet the momentum transfer to seeded particles becomes ineffective. For a given oritice shape and carrier gas, the ratio of the terminal velocity of seeded particles to that of the carrier gas is determined by the slip number, a dimensionless quantity which depends upon the pressure of carrier gas and the size and density of the particles, etc. The results of Schwartz and Andres cannot be used directly to determine the terminal velocity ratio in an expansion through a Lava1 nozzle. The velocity of SF, clusters formed in supersonic flow through nozzle #6 and the velocity of its neon carrier gas have been determined,59however, and as described elsewhere: we can use the results of Schwartz and Andres to estimate how the velocity ratios in our expansions vary with changes in the pressure of the stagnant gas, the cluster size, and the cluster density.8 The observed velocity ratio of 1Wll SF6 clusters to carrier gas was 0.87 when the stagnation pressure was 3.9 bar. The cluster sizes,
The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 8609 densities, and stagnation pressures of the present experiments imply slip numbers corresponding to velocity ratios of 0.87for SeF6 and 0.88 for (CH3)3CCl. These velocity ratios were used in computations of the formation and evolution of clusters to estimate the location of the region inside the nozzle where the carrier ceased to accelerate the clusters. From these computations we deduce terminal cluster velocities of about 650 m/s for SeF6 and 720 m/s for (CH3)$C1. Trigonal SeF6. Separate experiments carried out in this laboratory suggest that SeF6, like SF,, may possess a trigonal phase. Electron diffraction patterns were obtained from an expansion of stagnant gas mixtures of SeF, in neon with an SeF, mole fraction of 0.027and total pressure 5.2-5.7bar. These patterns exhibited a weak Debydcherrer ring at s = 3.38 A-’ not assignable to the bcc or monoclinic phase. This is consistent with previous observations, in clusters, of the trigonal phase of SF6.Il If we assume that the SeF6 and SF6 trigonal structures are isostructural, the anomalous ring would be the 3 1 1 reflection. If the ax ratio is assumed to be 1.78 as in SF6, the cell constants of trigonal SeF6 are given approximately by a = b = 8.41 A, c = 4.73 A, a = B = 90°,and y = 120”. No sign of the trigonal phase appears in the kinetic experiments. However, if the ratelimiting step in the transformation of SeF6 is in reality the formation of the trigonal intermediate from the bcc, then the use of thermodynamic properties of the monoclinic phase in the calculation of us would be flawed to some extent. Such a reaction pathway would, however, require an extraordinarily fast transition from trigonal to monoclinic at an early stage of growth of the nuclei. Registry No. SeF6, 7783-79-1; (CH3)3CCI,507-20-0.
References and Notes (1) Rao, C. N. R.; Rao, K. J. Phase Transitions in Solids; McGraw-Hill: New York. 1978. (2) Phase Transformations in Solids; Smoluchowski, R., Mayer, J. E., Weyl, W. A., Us.; Wiley: New York, 1951. (3) Suga, H. J . Chim. Phys. 1985, 82, 275. (4) Fisher, J. C.; Holloman, J. H.; Turnbull, D. Met. Trans. AIME 1949, 185. 691. ( 5 ) Knapp, H.; Dehlinger, U. Acta Metall. 1956, 4, 289. (6) Bartell, L. S.; Dibble, T. S. J . Am. Chem. SOC.1990, 112, 890. (7) Beck, R.; Nibler, J. W. Chem. Phys. Lett. 1988, 148, 271. (8) Bartell, L. S.; Dibble, T. S. J . Phys. Chem. 1991, 95, 1159. (9) Dibble, T. S.; Bartell, L. S. J. Phys. Chem. 1992, 96, 2317. (10) Bartell, L. S. Chem. Rev. 1986, 86, 492. (11) Bartell, L. S.;French, R. J. Rev. Sci. Instrum. 1989, 60, 1223. (12) Virlet, J.; Rigny, P. Chem. Phys. Lett. 1970, 6, 377. (13) Michel, J.; Drifford, M.; Rigny, P. J . Chim. Phys. 1970, 67, 31. (14) Garg, S. H. J. Chem. Phys. 1977,66, 2517. (15) Gilbert, M.; Drifford, M . In Advances in Ramon Spectroscopy; Mathieu, J. P., Ed.; Heyden and Sons: London, 1972; Vol. 1. (16) Blinc, R.; Lahajnar, G. Phys. Rev. Lett. 1967, 19, 685. (17) Bartell, L. S.; Valente, E. J.; Caillat, J. C. J . Phys. Chem. 1987, 91, 2498. Note that the phases identified in this reference as triclinic can be described more simply in terms of a monoclinic unit cell. For details, see: Cockcroft, J. K.; Fitch, A. N. Z . Kristallogr. 1988, 184, 123. (18) Raynerd, G.; Tatlock, G. J.; Venables, J. A. Acta Crystallogr. 1982, 838, 1896. (19) Powell, B. M.; Dove, M.T.; Pawley, G. S.; Bartell, L.S. Mol. Phys. 1987, 62, 1127. (20) Pawley, G. S.; Dove, M. T. Chem. Phys. Lett. 1983, 99,45. (21) Bartell, L. S.; Xu,S. J . Phys. Chem. 1991, 95, 8939. (22) Ohtani, S.; Hasebe, T. Chem. Lett. 1986, 1283. (23) Bartell, L. S.; Valente, E. J.; Dibble, T. S. J. Phys. Chem. 1990,94, 1452. Note that the phases identified in this reference as I and 11 are in fact phases I11 and IV, the same phases seen in the present work. (24) Goyal, P. S.; et al. Acta Phys. Pol. A 1974, 46, 399. (25) Aksnes, D. W.; Ramstad, K.; Bjorlykke, 0. P. Magn. Reson. Chem. 1988, 26, 1086. (26) Bartell, L.S.; Heenan, R. K.; Nagashima, M. J. Chem. Phys. 1983, 78, 236. (27) Valente, E. J.; Bartell, L.S. J. Chem. Phys. 1983, 79, 2683. (28) Cluster diameters reported herein were corrected for broadening due to the finite size of the electron beam (-0.03 mm) by the method of ref 10. (29) Turnbull, D. In Solid Stare Physics; Academic Press: New York, 1956; Vol. 3. (30) Fisher, J. C.; Hollomon, J. H.; Turnbull, D. J . Appl. Phys. 1948, 19, 775. (31) Raghavan, V.; Cohen, M. In Treatise on Solid Stare Chemistry; Hannay. N. B., Ed.; Plenum Press: New York, 1975; Vol. 5 . (32) Buerger, M. J. In Phase Transformations in Solids; Smoluchowski, R., Mayer, J. E.,Weyl, W. A,, Eds.; Wiley: New York, 1951.
J. Phys. Chem. 1992,96, 8610-8613
8610
(33)El Adib, M.; Descamps, M.; Chanh, N . B. Phase Transitions 1989, 14,85. (34)Timmermans, J. J . Chem. Phys. Solids 1961,18, 1. (35)Kashchiev, D. Surf. Sci. 1969,14,209. (36) Kelton, K. F.; Greer, A. L.; Thompson, C. V. J . Chem. Phys. 1983, 79,6261. (37) Dove, M. T.; Pawley, G. S.J . Phys. C 1983,17,6581. (38)Angell, C. A.; Dworkin, A.; Figuiere, P.; Fuchs, A.; Szwarc, H. J . Chim. Phys. 1985,82,773. (39)Kashchiev, D.; Verdoes, D.; van Rosmalen, G.M. J . Cryst. Growfh 1991, 110, 373. (40) Bartell, L. S.; Dibble, T. S.; Hovick, J. W.; Xu, S.In The Physics and Chemistry of Finite Systems: From Clusters to Crystals; Jena, P., Rao, B. K.; Khanna, S.N., Eds.; Klewer Academic Publishers: Dordrecht, in press. (41)Xu,S.; Bartell, L. S. Unpublished research. (42)Note that the coefficient of thermal expansion used to calculate the temperature of clusters of bcc SeF6 in ref 17 is derived from the uncertain results of ref 13. (43)Gspann, J. In Physics of Electronic and Atomic Collisions; Datz, S., Ed.; Hemisphere: Washington, DC, 1976.
(44) Klots, C. J . Phys. Chem. 1988,92,5864. (45) Bartell, L. S.J. Phys. Chem. 1990,94,5102. (46) Bartell, L. S.;Machonkin, R. A. J . Phys. Chem. 1990,94,6468. (47)Shi, X.Ph.D. Thesis, University of Michigan, Ann Arbor, Michigan, 1988. (48)Eucken, A.;Schroder, E. Z . Phys. Chem. (Munich) 1938,41B,307. (49) Hirth, J. P.; Lothe, J. Theory of Dislocations, 2nd ed.; Wiley-Interscience: New York, 1982. (50) Turnbull, D. J . Appl. Phys. 1950,21, 1022. (5 1) Partington, J. R. An Advanced Treatise on Physical Chemistry; Longmans, Green: London, 1951;Vol. 2. (52) Bartell, L. S. J . Phys. Chem. 1992,96, 108. (53) Spaepen, F. Acta Metall. 1975, 23, 729. (54) Fuchs, A. H.; Pawley, G. S. J . Phys. (Paris) 1988,49,41. ( 5 5 ) Urban, S.;et al. Phys. Status Solidi A 1972,10, 271. (56) Rudman, R.; Post, B. Mol. Cryst. 1968.5, 95. ( 5 7 ) O'Reilly, D. E.; Peterson, E. M.; Scheie, C. E.; Seyfarth, E. J . Chem. Phys. 1972,59,3576. (58) Schwartz, M. H.; Andres, R. P. J . Aerosol. Sci. 1976,7,281. (59)French, R. J.; Bartell, L. S. Unpublished research.
Solubility and Micelle Formation of Bolaform-Type Surfactants: Hydrophobic Effect of Counterlon Yoshikiyo Moroi,*st Yoshio Murata, Yuji Fukuda, Yoshifumi Kido, Wataru &to, and Mitsuru Tanaka Department of Chemistry, Faculty of Science, Kyushu University 33, Higashi-ku, Fukuoka 81 2, Japan, and Department of Chemistry, Faculty of Science, Fukuoka University, Johnan- ku, Fukuoka 81 4-01, Japan (Received: April 27, 1992; In Final Form: June 22, 1992)
Aqueous solubility and critical micelle concentration (cmc) of 1,l'-( 1,w-tetradecanediyl)bis(pyridinium)alkane-1-sulfonates
fl=
4 , 6,8. 10, 12, 1 4
were measured at various temperatures, and the effect of hydrophobicity of counterion on solubility, the critical micelle concentration (cmc), the micelle temperature (MTR or Krafft point), and aggregation number of micelle was examined. The MTR was determined as 36, 39, and 36 "C for decane, dodecane, and tetradecanesulfonates, respectively, while the one for the rest was below 5 "C. Plots of log cmc against the carbon number of the counterions gave a straight line from 8 to 14, from whose slope the alkane chain of these counterions was found to locate in hydrophobic micellar core. The micelle aggregation number at 45 "C monotonously increases, with increasing hydrophobicity of the alkane- 1-sulfonate counterion, from 27 (octane) to 46 (tetradecane). The molecular weights of micelles with the counterions of less than eight methylene groups were too small to determine.
Introduction Properties of aggregates, or micelles, of ionic surfactants are strongly influenced by the kind of surfactant ion and counterion. Most counterions of conventional anionic surfactants so far investigated have been alkali or alkaline earth metal ions, having their electrical charge concentrated within a very small volume. On the other hand, anionic surfactants with nonmetallic cationic counterions of nonconcentrated, diffuse1q2or separate,3s4charges were found to have physicochemical properties much different from those of conventional surfactants. The divalent cationic counterions, whose charge separation is long enough, for example, apart by 14 methylene groups, can be regarded as bolaform-type cationic surfactant ion from another point of view, since they are able to assemble due to their own hydrophobic interaction among the long methylene chains. From this standpoint the former anionic surfactant ion then becomes the counterion. The effect of alkane chain length of the counterion on solubility and micelle formation is another interest in selfassembly of bolaform-type surfactants. There have appeared From their results the several papers on bolaform following have been found: (1) more than 12 methylene groups in hydrophobic chain are necessary for bolaform surfactants to aggregate,5*6(2) two ionic groups of bolaform surfactant locate Kyushu University.
at micellar surface via loop-type structure,4~'Jo(3) counterion dissociation from micelles is relatively large,'*" (4) their cmc's are smaller than those of ionic surfactants with a half number of carbon atoms of hydrophobic chain of bolaform surfactants,'oJ1 and ( 5 ) the aggregation number of micelles is also rather small compared with usual micelles? The above findings seemingly result from a loose packing of bolaform surfactant molecules in micellar state due to their chemical structure. In spite of these findings, any systematic information has not been reported yet on the effect of hydrophobicity of counterions of bolaform surfactants on their aqueous solubility and micelle formation. The aim of this work is to make clear this hydrophobicity effect using alkanesulfonate ions (CnH2n+lSO),n = 4, 6, 8, 10, 12, 14) as the anionic counterion of cationic bolaform surfactants. All experimental results as for n = 14 are those from our previous paper^.^,^ Experimental Section Preparation of Surfactants. The starting materials of 1,l'( 1,w-tetradecanediyl)bis(pyridinium) alkane- 1-sulfonates were 1,l'-( 1,w-tetradecanediyl)bis(pyridinium) dichloride
and silver alkane-1-sulfonates. The former is a kind gift from 0022-365419212096-8610$03.00/0 0 1992 American Chemical Society
