3568
J. Phys. Chem. 1992,96, 3568-3570
kJ/mol). Pan et a1.2 report a somewhat lower (167.8 f 5.4 kJ/mol) value, but their measurements were also on a mixture of Ca and C7* The present measurements were carried out with pure Cso, and the agreement between different samples and runs suggests that our value is reliable. The trend toward a lower enthalpy of sublimation in mixtures of Ca and C70 and the small magnitude of the difference are consistent with the picture of a
solid solution held together by van der Waals forces.
Acknowledgment. We thank Mr. R. Viswanathan and Dr. D. Darwin Albert Raj for their assistance in the mass spectrometric measurements and useful discussions. We also thank Mr. I. Kaliappan and Mr. R. Dhamodaran for their help in the preparation and purification of c 6 0 samples.
Nonstatlstical Energy Flow Dynamics in Slllcon Clusters as a Result of Bond Directionality Thomas A. Holme* and William J. Leei Department of Chemistry, University of South Dakota, Vermillion, South Dakota 57069 (Received: January 21, 1992)
Energy flow dynamics for two model clusters containing 39 silicon atoms are reported. When nondirectionalpairwise interactions are assumed between silicon atom, energy placed in one region is rapidly shared among all atoms of the cluster, the statistical expectation. By contrast, when directionality is introduced such that silicon prefers to bond with tetrahedral bond angles, energy flow from a locally excited region to the remainder of the cluster is quite slow. This nonstatistical tendency could have implications for several observed characteristics of silicon microclusters.
Silicon microcluster chemistry has been the subject of signficant experimental'-I0 and investigation. While other elemental microclusters have also been studied, silicon attracts special attention, due at least in part to its covalent bonding nature. This very nature also imparts the constraint of directional bonding, preferably along tetrahedral bond angles. In this report, we will note that this constraint appears to have important dynamical consequences whose ramifications in the study of silicon microclusters may in turn prove to be crucial. Two distinct issues within silicon microcluster chemistry may be impacted by the findings presented here. Silicon clusters have been observed to have unusual fragmentation patterns,I-j with fragmentation favoring six atom pieces, as opposed to the sequential individual atom fragmentation observed for other microclusters." The silicon fragmentation pattern changes to favor 5-1 1 atom fragments after the cluster size exceeds 30 silicon atoms.3 This observation may be connected to evidence that clusters above this range are structurally distinct from the smaller cluster^.^ Considerable theoretical effort has been put forth to help elucidate the nature of magic numbers," including those implied by the fragmentation behavior. In addition to the structural probes of silicon microclusters, their chemical reactivity patterns have also been in~estigated.~-~ Reaction rates were observed to vary dramatically between clusters differing by only one atom,5 though the extent of the variation has been found to depend on the experimental condition^.^^^ Annealing long after cluster formation has also been shown to effect the outcome of silicon microcluster chemistry.s Among the reagents used to attack silicon microclusters, oxygen has provided some of the most enticing observations. For clusters below some threshold value, roughly SiM,oxygen repeatedly etches the cluster leading to only dimers of Si0 or Si? At largers sizes, greater than Si3s, adduct formation becomes the dominant product channel. JarroldlO has suggested that dynamical factors contribute to this observation; small clusters are unable to accommodate the exothermicity of the reaction and break apart within the proposed scenario. The calculations reported in this work were undertaken to help determine the nature of the energy flow dynamics suggested by Jarrold's explanation. 'Present address: School of Medicine, University of Iowa, Iowa City, IA 52245.
The theoretical study of cluster dynamics is not new. There have been numerous investigations of Ar clusters,I* helping to elucidate such fundamental issues as the origins of melting. There have also been some investigations using molecular dynamics calculations with silicon clusters.12-16 Early calculations were concerned primarily with describing stable structures using simulated annealing,l* and subsequent investigations have assessed the effects of finite temperatures on structure^.'^ Vibrational spectra have been calculated,14 and a b initio molecular dynamics methods have been used to investigate cluster melting a t tem(1) Bloomfield, L. A,; Freeman, R. R.; Brown, W. L. Phys. Rev. Lerr. 1985, 54, 2246. (2) Zhana. O.-L.: Liu. Y.; Curl. R. F.; Tittel, F. K.; Smalley, R. E. J . Chem. Physr1988,88, 1670. (3) Jarrold, M. F.; Honea, E. C. J. Phys. Chem. 1991, 95, 9181. (4) Jarrold, M. F.; Constant, V. A. Phys. Rev. Lerr. 1991, 67, 2994. (5) Elkind, J. L.; Alford, J. M.; Weiss, F. D.; Laaksonen, R. T.; Smalley, R. E. J . Chem. Phys. 1981,87, 2397. ( 6 ) Jarrold, M. F.; Power, J. E.; Creegan, K. M. J. Chem. Phys. 1989,90, 3615. (7) Creegan, K. M.; Jarrold, J. F. J . Am. Chem. SOC.1986, 112, 3678.
Alford, J . M.; Laaksonen, R. T.; Smalley, R. E. J . Chem. Phys. 1991, 94, 2618. (8) Maruyama, S.;Anderson,&. R.; Smalley, R. E. J . Chem. Phys. 1990, 93, 5349. Anderson, L. R.; Mamyama, S.;Smalley, R. E. Chem. Phys. L e r r . 1991, 176, 348. (9) Jarrold, M. F.; Ray, U.; Creegan, K. M. J . Chem. Phys. 1990,93,224. (10) Jarrold, M. F. Science 1991, 252, 1085. (11) Phillips, J. C. J . Chem. Phys. 1988,88,2090. Jelski, D. A,; Wu, 2. C.; George, T. F. Chem. Phys. Lett. 1988,150,447. Kaxiras, E. Phys. Rev. Drr. 1990,64, 551. Patterson, C. H.; Messmer, R. P. Phys. Reu. B 1990,42, 7530. Jelski, D. A.; Swift, B. L.; Rantala, T. T.; Xia, X.; George, T. F. J . Chem. Phys. 1991, 95, 8552. (12) Blaisten-Barojas, E.; Levesque, D. Phys. Rev. B 1986, 34, 3910. (13) Feuston, B. P.; Kalia, R. K.; Vashishta, P. Phys. Rev. B 1987, 35, 6222. (14) Sankey, 0. F.; Miklewski, D. J.; Drabold, D. A.; Dow, J. D. Phys. Reo. B 1990, 41, 12750. (15) Ballone, P.; Andremi, W.; Car, R.; Parrinello, M. Phys. Reu. LeU. 1988, 60, 271. (16) Mistirotis, A. D.; Flytzanis, N.; Farantos, S.C. Phys. Reu. B 1989, 39, 1212. Chelikowsky, J. R.; Glassford, K. M. Phys. Reu. B 1990, 41, 12750. Phillips, J. C. Phys. Reu. B 1991, 44, 1538. (17) See: Barnett, R. N.; Landman, U.; Rajagopal, G. Phys. Rev. L e f f . 1991, 67, 3058 and references therein. (18) Wales, D. J.; Berry, R. S. J . Chem. Phys. 1990, 92, 4283. Adams, J. E.; Stratt, R. M. J . Chem. Phys. 1990,93, 1332. Stillinger, F. H.; Stillinger, D. K. J . Chem. Phys. 1990, 93, 6013. Rick, S . W.; Leitner, D. C.; Doll, J. D.; Freeman, D. L.; Frantz, D. D. J . Chem. Phys. 1991, 95, 6658.
0022-365419212096-3568%03.00/0 0 1992 American Chemical Society
Letters peratures above 3000 K.Is Ultimately, each of these investigations has been directed toward determining the structural nature of clusters as opposed to their energy flow dynamics. In addition, a number of potential energy functions for interactions of silicon atoms within a cluster have been developed in tandem with these efforts.I6 We wish to view the nature of the cluster dynamics from a different perspective. Since the chemical behaviorlo and perhaps the fragmentation pattern~l-~ have implied dynamic effects, we wish to determine whether more than structural information is needed to understand the nature of silicon microclusters. In an intuitive sense, the directionality of bonding in silicon clusters represents a most distinct character relative to other elemental clusters. Does this directional tendency have any effect on the dynamical behavior of silicon clusters? We have attempted to gauge this question in qualitative terms by a comparison of energy flow dynamics for two models of a Si39cluster. O u r computations indicate that there is an impact on energy flow due to the constraint toward tetrahedral angles, which at low temperatures (Le. below melting, which occurs above 2000 K15) may play a crucial role in the nature of both fragmentation and chemistry of silicon microclusters. Our choice of model potentials is directed by the desire to have distinct qualitative features, and neither interaction potential chosen is expected to provide quantitative information. The first interaction potential is a pairwise, Lennard-Jones (LJ) potential. While this choice of potential will not reproduce any structural features of silicon clusters, it represents to this study an important control model. It contains no directional bonding disposition and has parameters chosen for silicon from the two-body interactions within Tiller and co-workers' models.Ig O u r second model is built upon the interaction potential of Stillinger and Weber (SW).20 This potential includes three-body terms which impart directionality. The qualitative trends associated with bond directionality are well evidenced using this function, which has the advantage of being characterized by previous calculation^.'^-^^ Since this functional form has been given in detail in many previous studies, we will not report it in this Letter. Using both models, a minimum-energy structure was determined using a steepest descent algorithm. Repeated heating and annealing was used to verify the structure was a robust minimum. On the SW surface, a number of local minima were found and the calculations reported were carried out on the lowest energy, most robust minimum. For both models the cluster was chosen to be very cold, with no zero-point vibrational energy. For these very cold clusters, energy was introduced by pulling and releasing an atom on the surface of the microcluster. Trajectories were propagated using a Verlet algorithm,*' and energy conservation to better than 104% was observed for all trajectories. Using this method, roughly 80 trajectories for the SW potential were carried out, using different atoms to provide the initial energy. In each trajectory, energy placed in a local region remained in thot local region for a number of vibrational periods. We have chosen, as an example, a trajectory for which energy movement was greatest and depicted its dynamics in Figure 1. A total of 3.0 eV of energy was added by pulling out a surface atom with relatively large numbers of nearest neighbors, six. Figure la shows the kinetic energy of the entire cluster over roughly 11 vibrational periods. Figure 1b shows the kinetic energy of only the 13 atoms "local" to the initial excitation (withii roughly the first two nearest neighbors). Kinetic energy is used so that there is no ambiguity associated with the partitioning of the potential energy arising from the three-body interactions. This choice is adequate since we were careful to begin with very cold clusters prior to the initial excitation. There is a remarkable similarity in these two plots. The numeric value for the kinetic energy differs only by a factor of 3 associated with averaging over 3 times as many atoms for (19) Pearson. E.;Takai, T.; Halicioglu, T.; Tiller, W. A. J . Cryst. Growth 1984, 70, 33. (20) Stillinger, F. H.; Weber, T. A. Phys. Reu. B 1985, 31, 5262. (21) See: Allen, M.P.; Tildaley, D. J. Computer Simulations ofuququids; Oxford University Press: Oxford, 1989; Chapter 3.
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3569
T I M E STEPS(X 1 . 0 ~ 4 )
-1
I
I
TIME STEPS(x .1.OE4) Figure 1. Energy flow dynamics for a Si,9 cluster model which uses the Stillinger-Weber potential." Panel a shows total kinetic energy of the cluster averaged over all 39 atoms. Panel b shows the sum of the kinetic energies for only the closest 13 atoms. Through the first 10 vibrational periods, the structure and extent of energy flow are limited to the region local to the initial excitation.
the whole cluster in Figure la. (Thus, by choosing 13 atoms for the local region we can have the most straightforwardcomparison.) Since a silicon cluster will have a mean vibrational frequency in the vicinity of 400 cm-l,14 10 periods represent roughly 0.8 ps. Thus, for times scales long as compared to a gas-phase chemical energy remains localized within this model of a silicon cluster. Each trajectory calculated using the SW potential showed this behavior. In some cases,when the surface atom used to impart the initial energy had relatively few nearest neighbors, the same localization was observed with even fewer atoms participating (our minimum structure is slightly ~ylindrical~~). Thus, when constrained to uphold tetrahedral bond angles and at relatively low temperatures, nonstatistical energy flow dynamics are observed. By contrast, the LJ model, when provided energy in the same manner, rapidly disperses it to the entire cluster. That an LJ model would behave statistically is not surprising in light of numerous studies of Ar clusters using the same functional form.18 As a better guide for comparison over longer time scales, we show results obtained using both models in Figure 2. In these plots, we define the local region in each model to be the 13 atoms nearest the initial excitation and calculate for a representative trajectory the percentage of the kinetic energy that is accounted for by atoms within this local region. Figure 2a shows this percentage for the LJ cluster. As noted, energy rapidly departs from the local region and within just a few vibrational periods the local region possesses 33% of the kinetic energy on average. Thus,the LJ model behaves in a statistical manner. By contrast, Figure 2b shows this per(22) Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics, 2nd Oxford, 1988. (23) Lubinga. S.; Holme, T. A. Work in progress.
ed.; Oxford University Press:
3570 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992
I
75 -
50 -
TIME S T E P S 0 1.OE4)
Figure 2. Percentage of kinetic energy accounted for by the 13 atoms nearest the initial excitation for (a) a Lennard-Jones model and (b) the Stillinger-Weber model. When there is no directional bonding, the statistical limit of 33% is quickly reached (a), whereas when bond directionality is imparted, energy flow away from the local region is slow and not complete over many vibrational periods (b). Note that the time step for the more complicated SW potential is shorter than the LJ potential. The plots are constructed to roughly match vibrational periods for the clusters.
centage for the SW model. As evidenced in Figure 1, over the initial approximately 10 vibrational periods very nearly all of the energy remainsin the local region of the initial excitation. Leakage into the remainder of the cluster is slow and not complete even over more than 50 vibrational periods later. Nowtatistical behavior in energy flow dynamics is introduced as a result of directional bonding. Prior to discussing the possible implications of this result, we wish to note that these qualitative models do not reproduce the observed details of silicon microclusters. Nonetheless, the extent
Letters of nonstatistical behavior for the SW model is quite noteworthy. We are presently extending our calculations to include better potentials and more physical techniques for imparting initial energy by collisions with a rare gas atom.23 It is, however, important to realize that, for most interaction potentials proposed to date, interaction with atoms more than three bond lengths away is either very small or explicitly outside of the cutoff distance for the interaction potential.l6Vm One might argue that even if our model proves to be unable to describe silicon clusters per se, such nonstatistical behavior for any interaction potential is worthy of note. There are several possible implications of these results. First, they tend to confirm the hypothesis of Jarrold and co-workers that dynamics play a role in the observed chemistry of silicon clusters with oxygen.l0 The possibility of relatively long time localization of energy within a cluster certainly confirms the suggestion that energy accommodation, or lack thereof, may affect chemical observations. Second, the fragmentation behavior which favors groups of 6-1 1 atoms may be connected to these nonstatistical energy flow dynamics. While our method for imparting initial energy is decidedly different from those used e~perimentally,~" the definition of "local regions of excitation" resultant from constrained energy flow dynamics may be true irrespective of the source of energy. In our calculations, the local groups that participate in the dynamics typically consist of 5-13 atoms. This observation may be coincidental, but further work is under way to determine whether our results speak to the fragmentation behavior problem. Finally, we have noted that different sized local regions result when different atoms are used for the initial excitation in our calculations. Jarrold and Constant4have recently suggested that larger clusters grow by a different mechanism than smaller ones, leading to different overall shapes of the cluster. Since the size of the local region in our calculations is primarily dependent upon local geometry, such a transition is consistent with our earlier conjecture about fragmentation patterns. The nature of fragmentation changes precisely where the structural crossover occurs." If larger clusters are shaped so that local regions contain on the average more atoms, the energy flow dynamics might dispose such a cluster to fragment using numbers of atoms dictated by the local regions. Clearly, these possible implications for fragmentation behavior will require significantly more detailed investigations. We are currently extending our investigations to address some of these key questions posed by the nonstatistical energy flow dynamics observed for this model.
Note Added in Proof. Recently published results24may be related to our observations. Clusters retain memory of their origin after collisions at elevated temperatures. Acknowledgment. We are grateful to M. T. Berry for a critical reading of this manuscript. This material is based upon work supported by the National Science Foundation under Grant
STI-8902066. I
(24) Blaisten-Barojas, E.; Zachariah, M. R. Phys. Reu. B 1992.45, 4403.
