J. Phys. Chem. 1993,97, 9969-9972
9969
Structure and Dynamics of Molecular Clusters. 3. Monte Carlo Studies of Small Clusters of TeFs hrjing cbuko and Lawrence S . Bartell’ Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 Received: April 27, 1993; In Final Form: July 12, 1993’
Melting and freezing transitions of clusters of tellurium hexafluoride containing 12, 13, and 14 molecules have been investigated using a seven-site intermolecular interaction potential. Various diagnostic methods were applied to simulations of heating and cooling stages including caloric curves, the Lindemann 6 index, PawleyFuchs projections, and indices of translational and rotational diffusion. Results differed markedly from those for clusters of similar size obtained in simulations of argon and benzene systems and from experiments on charged clusters of SFa. All of the latter systems exhibited special stability at the magic number 13, but TeFa clusters did not. The hexafluoride molecules rotated freely down to the lowest temperatures studied where the cluster’s translations were all but frozen. A similar rotational freedom was not exhibited by benzene. When clusters were warmed they became progressively more fluxional, but the transition to liquidlike behavior was more gradual than for the other systems. These differences in behavior of small clusters of the different systems are correlated with the preferred molecular packing arrangements and linked to the phase behavior of large clusters where the hexafluorides exhibit a much more diverse structural chemistry.
Introductioo Electron diffraction studies of large molecular clusters generated by the condensation of vapor in supersonic flow have provided illuminating insights about condensed matter.’“ One remarkable observation is that many clusters so formed can be produced in any of several structural types, or phases, depending upon the conditions of the supersonic flow.’ An understanding of the underlying molecular behavior requires more detailed information than can be provided by the diffraction intensities, alone. The critical condensation nuclei are believed to be no larger than a dozen molecules and liquidlike under typical conditions of condensation.8 Somewhere in their growth clusters become large enough to adopt a bulklike structure, which may be liquid or crystalline. For clusters of polyatomic molecules the size at which bulklike crystalline phases become stable may be much smaller than is the case for van der Waals clusters of atom^,^,^ but few details about this behavior are known. In our experiments the critical stage of growth occurs inside the supersonic nozzle governing the flow, and electron diffraction is blind to this crucial stage of events. In order to begin to acquire a grasp of processes in condensing aggregates that we are unable to follow experimentally, we decided to carry out Monte Carlo simulations of molecular clusters of the size of critical nuclei and larger. Such an approach would allow us to investigate systematically how clusters of different substances behave as they pass through their formative stages. Onesystemof spacial interest is that of chalcogen hexafluorides. Clusters of these simple, highly symmetric molecules have been seen in a variety of different crystalline packings, including bodycentered cubic, trigonal, monoclinic, and orthorhombic?Jo and there are indications that even more phases will ultimately be observed. Not all of the phases seen in clusters have yet been found in the bulk for the same compounds. Therefore, the hexafluorides provide a rich opportunity for testing ideas about molecular packing and about the model intermolecular interactions invoked. Among the hexafluorides,TeFs attracted notice because of some anomalous properties discussed elsewhere.” In the following we investigate the propertiesof clusters of tellurium hexafluoride containing 12, 13, and 14 molecules as a first step in a study of the formativestagesofgrowing molecular aggregates. .Abstract published in Aduunce ACS Absrrucrs. September 1, 1993.
0022-3654f 93 f 2097-9969$04.00f 0
Procedure Simulations were carried out with a Monte Carlo (MC) program differing only in minor details from one described in detail by Jorgensen and co-workers.12J3 Model functions for intermolecular interactions have been reported elsewhere.]] Incorporatingpairwise-additiveLennard-Jones functions between all interatomic sites, they have been found to give an excellent representation of the crystal structure of the bulk orthorhombic phaseloas well as a realistic account of the dynamics of the phase change from bcc to monoc1inic.l’ Initial configurations were approximately spherical clusters given the high-temperature bcc structure of the bulk.10 From initial temperatures of 30 K maintained over 2 million equilibration cycles, clusterswere heated by 5 K every million trial moves until they began to fragment. Move sizes were increased as the temperature was increased to maintain an acceptance rate of approximately 48%. Thermodynamic averages were accumulated each step, and molecular coordinates were saved every 2000 trial moves. In another series of runs the relaxed clusters at 30 K were cooled 5 K every 2 million trial moves to 15 K. From the initial configurations at 30 K, clusters quickly relaxed to more stable conformations but did not achieve full equilibrium for several hundred thousand cycles. For the second million cycles they had settled down enough to warrant the computation of average properties. Analyses of the runs included the conventional indicators of melting, namely, the configurational energy and the Lindemann index 6 as functions of the temperature, and Pawley-Fuchs projections14to illustrate distributions of molecular orientations. The variant of the Lindemann index applied was*
with rljthe distance between centers of mass of adjacent molecules (those separated by no more than 7.5 A). New indices RD(r) and RD(e),based on a random walkmodel, were introduced to diagnose respectively translational and rotational diffusion. These indices were constructed to be independent of the arbitrary move size selected. R&), as has been discussed in detail elsewhere,8 is defined as the ratio
between mean-square displacements ( A+)N of molecules un0 1993 American Chemical Society
Chuko and Bartell
9970 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993
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Figure 1. Potential energies of TeF6 clusters plotted every 2000 trial moves (every dump) over the course of isothermal Monte Carlo runs of 2 million trial move8 at 15 K. From top to bottom, results for 12-, 13-, and 1Cmolecule clusters. No special stability of the 13-molecule, icosahedral cluster is evident.
0
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Figure 2. Temperature dependence of mean potential energies of TeF6 clusters during heating runs. From top to bottom, results for 12-, 13-, and 1Cmolecule clusters.
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dergoing N , attempted trial moves to the product of Nw and the mean-square displacement ( @)1 of a singlestep. &(e) is defined by the relation (3) in which A0 is the angular displacement of a given bond in TeF6 from its initial direction, in NaGc steps accepted for that molecule, and a.4 is the (maximum) angular move in a given step. As discussed elsewhere? the mean-square vibrational and librational displacements associated with the translational and oscillatory motions of the molecules in cages of their neighbors must be subtracted from the diffusional mean-squaredisplacementsbefore eqs 2 and 3 are invoked. Also, because ( (AO)z) can quickly attain its saturation value of [(rz- 4/21 when diffusion is quite free, and because diffusional displacements are by their very nature subject to large fluctuations, the average mean-square displacements in eqs 2 and 3 are those averaged over many different origins of walks, for displacements that are not too large. In eq 3 the factor 2/3 arises because the MC program randomly selects only one of the three axes for a molecular rotation each move, and the bond direction moves over a two-dimensional domain. The factor (aA2/3)represents the mean-square angular displacement per step about a selected axis that would result from taking a series of unbiased random diplacements between 4 . 4 and +UA. If molecular rotations in the simulations were completely free, that is, if the only molecular moves that were rejected were rejected solely because of translational, not rotational displacements,then the index RD(0)would be unity. Accordingly, when the index approaches unity, it is safe to say that rotational diffusion is essentially free. When the index is very small, so is the diffusion.
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TIK Figure 3. Temperature dependence of relative root-mean-square intermolecular distance fluctuations, 6, of N-membered TeF6 clusters during heating runs: circles, N = 12; filled squares, N = 13; triangles, N = 14.
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When the clusters relaxed from their initial configurations (fragments of bcc crystals), they quickly arranged themselves into more stable polytetrahedral structures, making the 13molecule cluster more or less icosahedral. The 14-molecule cluster, however, remained trapped in a low-lying metastable structure for the first million cycles and then abruptly underwent a transition toa conformationwith a potential energy 1.6% lower. Configurational energies during the course of an isothermal run at 15 K are plotted in Figure 1 for what are believed to be the equilibrium structures. Figure 1 and the caloric curves of Figure 2 show scarcely a hint of special stability for the 13-molecule pseudoicosahedron, for the energies of the three sizes of cluster are almost equally spaced. In the examples of argon and benzene, the energies of the 13-molecule aggregates lay much closer to those for the 14-molecule than to the 12-molecule clusters.
Figure 4. Temperature dependence of index of translational diffusion, R&), for TeF6 clusters during heating runs: circles, N = 12; filled squares, N = 13; triangles, N = 14.
As theclusters were warmed, they became increasinglydistorted and fluxional. Their caloric curves are presented in Figure 2. Whatever transformation is occurring is gradual and protracted. An indicator of transformation that is considered to be much more sensitive to melting than the configurational energy is the relative root-mean-square distance fluctuation, the Lindemann 6, which is reproduced in Figure 3. For reasons discussed elsewhere,8its value at the threshold of melting of clusters of rigid polyatomic molecules is appreciably lower than the value of 0.1 that is often assumed to apply to atomic clusters. A more fundamental, but intinsically noisier, gauge of melting is molecular diffusion. Figures 4 and 5 depict the diffusional indices RDfor translational and rotational diffusionof the three clusters. Again,
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 9971
Structure and Dynamics of Molecular Clusters
- -9 m 5
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T/ K Figure 5. Temperature dependenceof index of rotational diffusion, RD(e), for TeF6 clustersduring heating runs: circules,N = 12; filled squares, N = 13; triangles, N = 14.
Figure 6. Pauley-Fuchs projections of directions of Te-F bonds of all molecules in a 13-moleculeclusterof TeF6 at 30 K,including points from 200con~utivesavedconfigurationsover 400 000 trial moves. Preferred molecular orientations slowly evolve and correspond to no known solidstate structure.
the meltinglike transition for translational motions is very indistinct and spread over a wide temperature range. Rotations, however, appear to be rather free down to temperatures well belowthoseat which translationaldiffusionsubsides. Even though the rotational diffusion is large at all temperatures examined, Pawley-Fuchs projections14(Figure 6) imply that some molecular orientations are weakly preferred over others. These do not correspond closely to any known crystalline phase, however. When clusters were cooled back down, their indicators of melting and diffusion clasely retraced the paths exhibited in the heating stages, demonstratingno hysteresis. This reciprocal response is illustrated for the potential energy of the 13-molecule cluster in Figure 7.
At the lowest temperatures examined the clusters were solidlike in that their molecules showed no tendency to interdiffuse. Nevertheless, rotational motions persisted. There would have been little point in continuing calculations to lower temperatures to see whether rotations could be arrested because the corresponding molecular librational energies would have become much lower than the librational zero-point energies executed in real clusters of TeF6. Latticevibrational energies for bulk, crystalline hexafluorides do not become fully classical until approximately 100 K.15 In small, more loosely bound clusters the classical limit would be reached at a lower temperature.
-17' 0
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T/K Figure 7. Comparison between mean potential energies of 13-molecule cluster of TeF6 in heating and in cooling runs: filled squares, heating (except for the coldest four points; see text); empty squarcs, cooling.
As the clusters were heated, they softened and showed unmistakable signs of transforming into liquidlike aggregates. The transitions were more gradual than they have been reported to be for clusters of the same size made up of Lennard-Jones spheres'"19 or of benzene.20 This difference may in part be a consequence of the unceasing rotations of the quasispherical hexafluorides. It is certainly related to theshapeof themolecules, for it is this molecular property which is responsible for the fact that the bulk compound freezes to a bcc structure1*rather than to the cubic closest packed (fcc) structure adopted by the rare gases. This intrinsicdifference between octahedral and spherical molecules is at the root of their different spectrum of stability as clusters grow. Clusters of the size ( T ~ F ~ )fail N to exhibit the special stability at N = 13 enjoyed by the rare gases and benzene. A plausible explanation is that, at N = 13, spherical and quasispherical molecules naturally arrange into a compact icosahedral aggregate to maximize the number of close contacts. Adding or removing a molecule spoils the packing density. An icosahedron can be produced by a very modest deformation of a cuboctahedron, and a cuboctahedron is a 13-moleculefragment of the fcc structure and of benzene's crystal structure. Therefore, the icosahedral configuration is particularly favorable for rare gases and substances with related structures. Large systems of TeF6, by contrast, spontaneouslyorganizeinto bcc or orthorhombic crystals where the coordination is entirely different.lO Accordingly, fragments of TeF6 with 13 molecules are not especially stable relative to those with 12 or 14 if the natural packing geometry is that of the crystal. The fact that the quasi-icosahedral form adopted by the molecules at low temperatures is not closely related to a characteristic crystal structure of TeF6 probably accounts for the rotational freedom in even the coldest clusters. The absence of a natural enmeshing of the molecules corresponds to a low barrier to rotation. It reasonable to apply the same arguments to clusters of sF6, a substance also freezing into bcc crystals. Nevertheless, for this compound N = 13 has been seen as a magic number in a mass spectrometric study.21 The clusters investigated were positively charged ions, however, consisting of 12 molecules Of sF6 and one ion, SF$+. The ion, in all probability, was located in the cluster's center. Being a charged trigonal bipyramid, it presented quite a different template for the other molecules to pack around than a neutral octahedron would have. Therefore, there is no contradiction between the mass spectrometric and the present results. Other aspects of the lack of special stability of the 13-molecule cluster can be seen in Figures 3-5. If 13 had been a magicnumber, the corresponding cluster would have been expected to exhibit a smaller molecular mobility and a greater reluctance to melt than the 12- and 14-molecule clusters. The Lindemann 6 index of Figure 3, if anything, suggests a weak tendency in the opposite direction and the index of translational diffusion of Figure 4 is
Chuko and Bartell
9972 The Journal of Physical Chemistry,Vol. 97, No. 39, 1993
neutral in this respect. The index of rotational diffusion of Figure 5 indicates virtually free molecular rotation at all temperatures except the very lowest where the rotation in the 13-molecule cluster is marginally lower. Even though it has become commonplace to speak of the ‘melting” and ‘phase transitions” of clusters of fewer than a dozen molecules, it it not obvious that such terminology is really appropriate for the present clusters. Phase transitions in bulk matter are sharp, and those attributed to small atomic clusters, so far, have at least manifested their presence by a somewhat localized change in the slope of the caloric curve or other indicator of change. For the present clusters there is assuredly a change from something like a soft solid to a highly fluxional liquidlike mass, but the change is too gradual to resemble a true transition. If the “melting point” were taken to be the temperature at which the Lindemann index is in the range 0.084.09 suggested as a threshold for small rigid polyatomic molecules,8all three clusters would be considered to melt in the vicinity of 50-60 K. Indeed, these melting points lie close to an extension of the linear trend of TmvsN-l/3established elsewhereUin MD simulations of much larger TeF6 clusters containing 350,250,150, and 100molecules. On the other hand, R&), the index of translational diffusion, does not drop rapidly as the clusters are cooled below this temperature as it does in large clusterss and in the bulk. The index does, however, slowly increase to values characteristic of bulk liquids as the temperature is increased. Clusters of TeFs, then, are qualitatively different from the other clusters studied previously, in the slow but continuousway in which their properties are altered in response to heating. As suggested above, this gradualness may in part correspond to the fact that the polytetrahedral solidlike structures adopted by default by the very small clusters are already more disrupted from the natural packing Of TeFs molecules than is the case for argon and benzene. Another point of view is that TeF6 differs in that its large liquid clusters, upon continued cooling, first freeze to a bcc structure and then undergo another distinct transition to a monoclinic structure of similar coordination. As clusters decrease in size, both transition temperatures drop and become more diffuse22 until they finally overlap, becoming doubly blurred. Another aspect of cluster response different from that seen in normal melting and freezing is the lack of hysteresis encountered in heating and cooling cycles. At the high heating and cooling rates of the runs, a hysteresis indicative of a nucleation barrier to crystallization would have been expected for the case of true freezing. The solidification of the present small clusters is probably more akin to that of glass formation than to crystallization. Of course, nucleation in a bit of matter containing only a dozen moleculesis not entirely credible. Yet, the thermal history of the 1Cmolecule cluster makes the concept of critical configuration seem not completely preposterous. This cluster initially relaxedquicklyfromitsbcc-likestructure toa metastablestructure but transformed into its stable solidlike structure only after a million trial steps. Although systematicsimulationsof small clusters of polyatomic molecules with more or less realistic potential functions have only just begun, several points worth noting have emerged. It is clear that small clusters, in the size range of the critical nuclei formed in thecondensationofvapor, have properties quitedifferent from those of the bulk. It would be remarkable if it were otherwise. But the differences between the clusters are not simply washed
out by the smallness of size. Clusters of different molecules, molecules as similar as spherical and quasi-spherical,display very different structural and thermodynamic properties. Furthermore, significant clues about why these differences arise were provided by the bulk properties. Although the present results for clusters of TeFa are so undramatic as to appear prosaic, the very lack of conspicuous features in their diagnostic curves makes them different from the other cluster systems that have been studied. Their indifference to adopting the simple close-packed polytetrahedral configurations seen in benzene and argonlike clusters (manifested in their failure to show special stability at N = 13) underlies this behavior and is an augury of the striking solid-state transitions found in clusters of TeFd of just twice the diameter.22 What has been learned so far is insufficient to elucidate in detail the basis of the control that the experimentalist is able to exercise over the phases of clusters generated in supersonic flow.’ Nevertheless, insights are being acquired about the molecular behavior in small clusters, and a systematic study of molecular clusters as a function of size, temperature, and chemical composition should lead to a better understanding of the nonquilibrium growth of condensing matter.
Acknowledgment. This research was supported by a grant from the National Science Foundation. Numerical calculations were made possible by a generous allocation of computing time from the Michigan Computing Center. We are indebted to Mr. F. Dulles for considerable help in computations and to Prof. W. Jorgensen for the computer program and helpful comments about simulations. Registry No. TeFs, 7783-80-4 (supplied by author). References and Notes (1) Torchet, G.; Bouchier, H.; Farges, J.; de Feraudy, M. F.; Raoult, B. J. Chem. Phvs. 1984. 81. 2137. (2) Farges, J.; deFeraudy, M. F.; Raoult, B.;Torchet, G. J.Chem. Phys.
1986. 84. 349 1. (3) Torchet, G.; Farges, J.; de Feraudy, M. F.; Raoult, B. Ann. Phys. Fr. 1989, 14, 245. (4) Bartell, L. S.; Dibble, T. S.J. Phys. Chem. 1991, 95, 1159. (5) Bartell, L. S.; Dibble, T. S.; Hovick, J. W.; Xu, S. In The Physics and Chemistry of Finite Systems: From Clusters ?oCrystals;Jena, P., Rao, B. K., and Khanna, S.N., Eds.; Klewer Academic Publishers: Dordrecht, 1992; Vol. 12, p 71. (6) Dibble, T. S.;Bartell, L. S.J. Phys. Chem. 1992, 96, 8603. (7) Bartell, L. S.;Harsanyi, L.; Valente, E. J. J. Phys. Chem. 1989, 93, 6201. (8) Bartell, L. S.;Dulles, F. J.; Chuko, B. J.Phys. Chem. 1991,95,6481. (9) Bartell, L. S.; French, R. J. Rea Sci. Ins?rum. 1989, 60, 1223. (10) Bartell, L. S.; Powell, B. M. Mol. Phys. 1992, 75, 689. (11) Bartell, L. S.; Xu, S. J. Phys. Chem. 1991, 95, 8939. (12) Jorgensen, W. L.; Binning, R. C., Jr.; Bigot, B. J. Am. Chem. SOC. 1981, 103.4393. (1 3) Jorgensen, W. L.; Madura, J. D.; Seversen, C. J. J. Am. Chem. Soc. 1984, 106,6638. (14) Fuchs, A. H.; Pawley, G. S. J. Phys. (Paris) 1988, 49, 41. (15) Osbourne, D. W.; Schreiner, F.; Malm, J. G.; Selig, H.; Rochester, L. J . Chem. Phys. 1966,44, 2802. (16) Etters, R. D.; Kaelbcrer, J. B. J. Chem. Phys. 1977,66,3233; Phys. Rev. A. 1975, 11, 1068. (17) Davis, H. L.; Jellinek,J.; Berry, R. S. J. Chem. Phys. 1987,86,6456. (18) Beck, T. L.; Jellinek, J.; Berry, R. S. J. Chem. Phys. 1987,87,545. ( 1 9) Quirkc, N. Mol. Sfmular. 1988, I , 249. (20) Dulles, F. J.; Chuko, B.; Bartell, L. S. In The Physics and Chemistry of Finite Systems: From Clusters to Crystals, Jena, P., Roa, B. K., Khanna, S. N., Eds.; Klewer Academic Publishers: Dordrecht 1992; Vol. 1, p 393. (21) Echt, 0.;Flotte, A. R.; Knapp, M.; Sattler, K.; Recknagel, E. Ber. Bunsen-Ges. Phys. Chem. 1982.86, 860. (22) Xu, S.Ph.D. Thesis, University of Michigan, Ann Arbor, MI, 1993. ~
