J. Phys. Chem. 1992,96,6133-6135 (13) Reuter, W.; Peyerimhoff, S. D. Chem. Phys. 1992, 160, 11-24. (14) Herzberg attributed the nonresolved rotational structure in the B (3d) Rydberg state of I2CH2to predissociation. See: Herzberg, G. Proc. R. SOC. (London) 1961, A262,291-317. (15) Carter, S.;Handy, N. C. Mol. Phys. 1984,52, 1367-1391. (16) Bunker, P. R.; Jensen, P.; Kraemer, W. P.; Beardsworth, R. J. Chem. Phys. 1986,85, 3724-3731.
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(17) Smyth, K. C.; Taylor, P. H. Chem. Phys. Lett. 1985, 122, 518-522. (18) Celii, F. G.; Butler, J. E. Annu. Reu. Phys. Chem. 1991,42,643-684. (19) (a) Johnson, R. D., 111; Long, G. R.; Hudgens, J. W . J. Chem. Phys. 1987,87, 1977-1981. (b) Johnson, R. D., 111; Long, G. R.; Hudgens, J. W. J. Chem. Phys. 1988,89, 3930. (20) Forster, R.; Hippler, H.; Hoyermann, K.;Rohde, G.; Harding, L. B. Chem. Phys. Lett. 1991, 183,465-470.
On the Way to Fullerenes: Molecular Dynamics Study of the Curling and Closure of Graphitic Ribbons D.H.Robertson,* D.W. Brenner, and C.T.White Theoretical Chemistry Section, Code 6179, Naval Research Laboratory, Washington,DC 20375-5000 (Received: May 4, 1992)
The short-time behavior of isolated graphitic ribbons is simulated at high temperature using a model hydrocarbon potential. These ribbons show large instantaneous deviations from planarity that often result in the formation of open-ended hollow carbon structures representing good fullerene precursors. While confirming the importance of pentagon formation in the production of these precursors, these results also point to the central role of relatively high temperatures in these processes.
Sparked by the synthetic breakthrough of Kratschmer, Huffman, and co-workers yielding macroscopic amounts of research on the fullerenes (discovered 5 years earlier3) has continued to accelerate. Despite the pace of this research, the mechanisms leading to the facile formation of highly symmetric fullerene cages in the chaotic conditions of the carbon arc remain to be ~larified."~In this Letter, following ideas advanced by Smalley, Curl, Kroto, and co-workers,eE we use molecular dynamics (MD) simulations to study the behavior of graphitic fragments thought to condense in the cooling carbon vapor prior to fullerene formation. These fragments are treated as ribbons as an idealization of their probable low-symmetry, far-from-circular shapes. At the conditions present in the carbon arc, we find that such ribbons can exhibit large thermal fluctuations from planarity without fragmentation. These fluctuations, when s u p plemented by the curvature induced by pentagons, often cause these isolated strips spontaneously to form open-ended hollow carbon structures. Once formed, these hollow structures represent good fullerene precursors. Graphitic ribbons are modeled using a bond-order type empirical hydrocarbon potentiallo which accurately predicts the bonding and energetics of solid diamond lattices and graphite sheets, as well as hydrocarbon molecules, while still allowing reactions to occur. The reactive aspect of the potential is important both to model correctly the reactive edges of these ribbons and to allow for any possible fragmentation at high temperature. Although this potential has already been shown to provide a good model for a wide variety of properties of fullerenes and related struct u r e ~ , to ~ ~assess - ~ ~further its reliability for the present study we examined the energetics of the inversion of the corranulene molecule ( C a l 0 ) . This molecule is similar in shape to one-third of a buckminsterfullerenecluster. Recent studies have shown that corranulene can rapidly fluctuate between two curved minima at room temperature14-a behavior that is related to the ribbon motions discussed below. First principles calculations yield an energy difference between the minimum-energy curved structures and a planar intermediate of 8.8-1 1.0 kcal/mol.l5J6 Our empirical potential predicts a difference of 10.3 kcal/mol, in excellent agreement with these calculations. The MD simulations are performed by integrating Newton's equation of motion with an accurate high-order Nordsieck predictor-corrector method.l 7 The starting conditions are generated by assuming the velocities of the individual atoms in the relaxed
carbon ribbon are initially distributed with arbitrary directions in 3-D space according to a Boltzmann distribution. The dynamics of these isolated ribbons are then followed at constant energy to simulate their motion in the carbon vapor between collisions. Assuming the conditions in the carbon arc reported by Haufler et a1.8 (a temperature of 1000-2000 OC and a pressure between 100 and 200 Torr), the average time baween collisions of He, C, or Cm will be on the order of nanosmnds.I* The 250-ps length of our simulations is an order of magnitude less than this collisional time. The initially planar ribbons used in this study are depicted in Figure 1. These ribbons, with sizes ranging from 32 to 108 atoms, have varying lengths and widths. In addition, 1-fold coordinated atoms were included at the edges of most of these ribbons to allow for the dynamic formation of edge pentagon^.^ The effective temperature of an isolated ribbon (herein referred to as the temperature) is defined as the average kinetic energy per degree of freedom in units of temperature K, and u is the standard deviation in the average distance of the atoms to the fragment center of mass. u is large for planar ribbons, varies with changes in curvature, and approaches zero as the atoms become equidistant from the center of mass as in very symmetric fullerenes. At low temperatures all these ribbons remain close to planar, showing only small out-of-plane fluctuations. This is illustrated in Figure 2 for the 108-atom ribbon where the solid line is u for this ribbon plotted as a function of time for a trajectory started at 300 K. The observable fluctuations in the solid line correspond to only minor atomic displacements from planarity-less than 0.5 nm. No pentagonal rings form over the course of this trajectory because the temperature is too low to cause sufficient movement of any of the 1-fold coordinated edge atoms to allow the formation of another bond at the ribbon's edges. However, at higher temperatures pentagonal rings do form as illustrated in Figures 3 and 4. In Figure 3a, u is plotted for a 108-atom ribbon trajectory started at an initial temperature of 1150 K. During the first 15-20 ps, the internal temperature, plotted in Figure 3b, increases to near 2500 K because of the exothermic process of bond formation along the edges of the ribbon producing five-membered rings. During this time, the amplitude of the fluctuations from planarity (Figure 3a) increases dramatically from both this internal heating and the curvature induced by the edge pentagons. These large fluctuations are illustrated in Figure 4 where several snapshots of this trajectory are shown
0022-365419212096-6133$03.00/0 0 1992 American Chemical Society
6134 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992
c32
c41
c56
G59
c66
(373
Letters
,-.
G84
c94
b) 28.5 ps
c ) 39.0 ps
d ) 77.5 ps
Figure 4. Snapshots of the curling and closure of the 108-atom ribbon started at an initial temperature of 1150 K at times of (a) 0.5, (b) 28.5, (c) 39.0, and (d) 77.5 ps.
TABLE I: Summary of 250-ps M D Trajectories size ratio closed time of closure (ps)
c108
Figure 1. Initial relaxed configurations for ribbons simulated in this study. 5.0
c32 c4l c56
cs9 c66
Cl3
C82
4.5
2
a) 0.5 ps
C84 c94 GO8
4.0
014 014 115 0/5 114 418 115 415 015 316
91.5 212.5 65.5, 67.25, 86.75, 230.5 76.5 37.5, 156.0, 157.0, 211.0 70.5, 182.0, 186.5
a Tabulated are the ribbon size, number of trajectories exhibiting closure out of the total number of trajectories for that ribbon size, and the times at which this closure occurred.
v
b
3.5
t
3.0
0
"
20
40
60
60
I 100
Time (ps) Figure 2. u for the 108-atom ribbon at 300 K (solid line) and 94-atom ribbon at 2500 K (dashed line).
3008 2500 2000
1500
b 0
20
c 40
d
/ , , , , I , , , , 60 60 100
Time (ps) Figure 3. (a) u for the 108-atom ribbon at initial temperature of 1150 K. (b) Ribbon temperature as a function of time for this same trajectory. The vertical dotted lines a, b, c, and d correspond to the snapshots shown in Figure 4.
at times corresponding to the dotted vertical lines in Figure 3. Although the ribbon started in a planar configuration, by -25 ps it can be far from planar as shown by the 28.5- or 39.0-ps
snapshots. Finally, by -70 ps the low-frequency large-amplitude oscillatory behavior exhibited by u is quenched and u remains small, indicating a new structure that remains more closely equidistant from the center of mass. Although similar in shape to Figure 4b, this new structure has formed bonds across the ribbon's ends preventing it from uncurling as shown in Figure 4d. This fusion of the ribbon's ends also results in the observed temperature increase in Figure 3b near 70 ps. Summarized in Table I are results for all the ribbons shown in Figure 1. Each of these simulations was started with Boltzmann-distributedatom velocities to generate a temperature of lo00 K, yielding actual initial temperatures in the range 800-1200 K. With the exception of the two smaller strips, these results show that all those ribbons with 1-fold coordinated edge atoms have high closure probabilities (20-80%) within times from 37.5 to 212.5 ps. In contrast, the two ribbons without any 1-fold coordinated edge atoms, CS9and Cg4,did not form any open-ended hollow structures. These ribbons also did not form such structures even when the starting temperatures were increased to 2500 K to compensate for their lack of self-heating. This is illustrated in Figure 2 where the dashed line is u for the 94-atom ribbon for a trajectory started at a temperature of approximately 2500 K. Although this trajectory shows large fluctuations, they are insufficient to allow the ribbon's ends to join. Therefore, the presence of pentagonal rings formed along the edges of these ribbons significantly reduces the bamer to closure and consequently increases the probability that they will form good fullerene precursors. However, these and additional results also indicate that the existence of pentagons is not the only factor important in the formation of such precursors. First, as Table I shows, if the ribbon is too short, then it is unlikely to close even if a large number of pentagons form at its edges. In addition, even when these ribbons grow to a size for which the initial closure is exothermic and the barrier to this process is easily overcome at higher temperatures, this barrier may still effectively prevent closure at lower tem-
Letters peratures. For example, we have found that while the 108-atom ribbon readily closes at 2500 K this same ribbon with all the edge pentagons formed is never found to close if its temperature is reduced below 900 K. Thus, graphitic ribbons growing at these lower temperatures are less likely to yield good 40-100-atom precursors for fullerenes. Rather, at these lower temperatures, such fragments may continue to spiral, eventually forming larger spheroidal particles with several interior layers such as envisioned by Krotoa6 It may seem counterintuitivethat so many of the ribbons simulated spontaneously generate open-ended hollow structum. After all, these structures form while the ribbon is isolated, and hence statistically this process must be driven by an increase in entropy. But for the larger fragments the entropy of the higher temperature open-ended hollow structure is arguably higher than the lower temperature ribbon. This can be seen from the classical expres~ion’~ which provides an estimate of the change of entropy of the fragment, AS, accompanying the formation of the hollow structure. In eq 1, Eb represents the potential energy transformed to heat as the ribbon’s ends join, kB is Boltzmann’s constant, T i s the temperature of the ribbon isomer, and n is the number of lower frequency modes that significantly shift upward from a typical initial value, wR, in the ribbon to a new typical value, wH, in the hollow structure. Equation 1 assumes that each isomer can be approximately described in terms of its normal modes. Eb/(NkBT) is also taken as small, because Eb = 6 eV, T = 2000 K, and the number of normal modes, N, is =300 for the larger fragments. Equation 1 shows that the sign of M is determined by the competition between two terms. The first is positive because of the potential energy transformed to heat in the fragment when the ribbon’s ends join, while the second is negative because of the loss of freedom caused by closing the ribbon. Hence, if Eb is large, then AS should be positive because n and In ( w H / w R ) should be only weakly dependent on Eb Although the strain of closure makes Eb smaller for the shorter ribbons, for those that form hollow structures Eb/(kBT)= 30. In addition, for those ribbons the major fluctuations in Figure 3 show that wR = 5 cm-l so that wH/% should be less than 100 as q,should not exceed 500 an-’.% These estimates suggest that if n < 10, then M > 0. But n < 10 is plausible, because only the lower few ribbon modes should significantly displace upward as the hollow structure forms. This analysis implies that the hollow structure can be closer to quasi-equilibrium for the longer fragments. Hence, to prepare such fragments as ribbons is to prepare them further from equilibrium, in which case they can spontaneously close accompanied by an increase in entropy. If the preparation is viewed as a growth step, then the MD simulations show that high-temperature ribbons with edge pentagons exceeding a critical length are likely to relax to locally hot hollow structures that do not dissociate. The higher
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6135
temperatures of these structures are then returned to the temperature of the vapor by radiative and collisional processes occurring at longer times. The MD simulations also show that the presence of edge pentagons together with higher temperatures is a key ingredient in readily surmounting the barrier to closure prior to additional growth. Although the simulations reported in this paper do not show why Cs0 is so favored, they do suggest that the high temperatures-maintained locally by the confining buffer gas in the Krfftsch”an t e c h n i q u h r e essential for the copious production of good fullerene precursors from high-aspect-ratio graphitic fragments containing 40-1 00 atoms. In addition, the mass distribution of these precursors, and hence the mass distribution of the resulting fidlerenes and fulleremrelated structures that may grow from them, is predicted to shift upward if the maximum growth temperature is decreased.
Acknowledgment. We thank J. W. Mintmire, B. I. Dunlap, J. Milliken, and B. L. Holian for helpful discussions. D.H.R. acknowledges a NRC-NRL Postdoctoral Research Associateship. This work was supported by the ONR.
References and Notes (1) Kritschmer, W.; Fostiropoulos, K.; Huffman, D. R. Chem. Phys. Lett. 1990. 170. 167. (2) Kritschmer, W.; Lamb, L. D.; Fostiropoulos, K.; Huffman, D. R. Nature 1990, 347, 354. (3) Kroto, H. W.; Heath, J. R.: OBrien, S.C.: Curl. R. F.: Smallev. R. E. Nature 1985,318, 162. (4) Kroto, H. W.; McKay, K. Nature 1988, 331, 328. (5) Curl, R. F.; Smalley, R. E. Science 1988, 242, 1017. (6) Kroto, H. Science 1988, 242, 1 1 39. (7) Curl, R. F.; Smalley, R. E. Sci. Am. 1991, 265, 54. (8) Haufler, R. E.; Chai, Y.; Chibante, L. P. F.; Conceicao, J.; Jin, C.; Wang, L.; Maruyama, S.;Smalley, R. E. Mater. Res. SOC.Symp. Proc. 1991, 206, 627. (9) Chelikowsky, J. R. Phys. Rev. Lot?. 1991, 67, 2970. (10) Brenner, D. W. Phys. Rev. B 1990,42,9458. (11) DunlaD. B. I.: Brenner. D. W.: Mintmire.’ J. W.: Mowrev. ~ , R. , C.: Wh’ite; C. T. j.’Phys.‘Chem. 1991, 95,’5763. (12) Robertson, D. H.; Brenner, D. W.; Mintmire, J. W. Phys. Rev. B 1992,45, 12592. (13) Mowrey, R. C.; Brenner, D. W.; Dunlap, B. I.; Mintmire, J. W.; White. C. T. J. Phvs. Chem. 1991. 95. 7138. (14) Scott, L. f;Hashemi, M.’M.;’Bratcher, M. S.J. Am. Chem. Soc. 1992, 114, 1920. (15) Schulman, J. S.;Peck, R. S.;Disch, R. L. J. Am. Chem. Soc. 1989, 11 1, 5675. (16) Borchardt, A.; Fuchicello, A.; Kilway, K. V.;Baldridge, K. K.; Siegel, J. S. J. Am. Chem.Soc. 1992, 114, 1921. (17) Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equotions; Prentice-Hall: Englewood Cliffs, NJ, 1971; p 148. (18) See for example: McQuarrie, D. A. Sratistical Mechanics; Harper & Row: New York, 1976; pp 358-373. (19) Barrow, G. M. Physical Chemistry; McGraw-Hill: New York, 1966; p 259. (20) An estimate for wH is given by the 273-cm-I mode of Cm Bethune, D. S.;Meijer, G.; Tang, W. C.; Rosen, H. J.; Golden, W. G.; s k i , H.; Brown, C. A.; de Vries, M. S . Chem. Phys. Lett. 1991, 197, 181. ~~
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