Journal of Undergraduate Research
Volume 5, Issue 5 - February 2004

Fragmentation of Fullerenes

Ryan Chancey

ABSTRACT

Fragmentation of various fullerenes by collision with a model surface was studied using classical dynamics and interatomic potentials that included three-body interactions. Trajectories were run for a large ensemble of impact angles and fullerene orientations relative to the surface, and for a number of incident molecular kinetic energies. The intent was to study the fragmentation patterns and to compare with studies on the scaling of fracture processes in macroscopic solid samples. The detailed results show a strong dependence on the fullerene orientation, indicating that a significant ensemble of trajectories should be run, and then analyzed statistically. We report the results obtained and the conclusions drawn therefrom.

INTRODUCTION

The symmetric structures of the fullerenes have received considerable recent attention due to possible practical implications in the field of molecular electronics [2]. The resistance of a single C60 has been successfully measured [3], and a single fullerene has been used as a transistor within a gold nanojunction [4]. The related structures of carbon nanotubes also have promising implications in the field of molecular electronics [5]. The stability of the fullerenes is crucial to their use in molecular electronic devices, and it is likely that information obtained regarding the stability of fullerenes may have implications for the stability of carbon nanotubes.

In the present work, we compliment previous experimental studies of the structural stability and fragmentation of macroscopic and mesoscopic objects by simulating the surface-impact fragmentation of four fullerenes, C60, C24, C100, and C240. We simulate fragmentation over a wide range of impact energies, a task which proves experimentally impractical. The simulated fragment distributions of the fullerenes have both a quantitative and qualitative dependence on impact energy, which helps to explain previous experimental results.

Ref. [1] links to a website showing snapshots from a simulated surface-impact fragmentation of a C60 fullerene at 300eV and 100eV, exhibiting significant fragmentation and no fragmentation, respectively. The latter energy is far larger than the bond energies of the C60, but we see from the snapshots that inelasticity is manifested by a conversion into vibrational energy without destruction to the fullerene structure. This is demonstrated further by the MPEG trajectory animations featured in Ref [1].

SIMULATION METHOD

The investigated fullerenes were modeled using the respective number of point atoms subject to the well-known Tersoff classical interaction potential [6]. The Tersoff potential is an empirical form which contains two-body and angle-dependant three body contributions and has parameters that were chosen to optimize the description of solid carbon, silicon, and germanium. Initial static configurations of the fullerenes were obtained from experimentally-reported structures [7] and relaxed to their equilibrium positions consistent with the Tersoff potential. The structureless wall with which the fullerenes collided was modeled by a potential of the form:

V = Vo[1-tanh(γz)],    (1)

where z is a Cartesian coordinate normal to the wall. The other parameters of this potential were chosen such that all atoms bounce back from the wall and the wall is comparable in steepness to a diamond surface. Earlier studies [8] support our use of such wall, indicating that qualitative results are not sensitive to the details of the wall description or lack of structure.

Each simulation was initiated with the fullerene position outside of the wall potential, in its static equilibrium geometry, and with all atoms assigned a uniform velocity normal to the wall and consistent with the desired impact energy. Each simulation was started with a different random orientation of the fullerene, leading to a distribution of fragmentation outcomes.

The dynamics were simulated by a computer program written by Frank E. Harris. The program conducts stepwise integration of the classical equations of motion, using 0.1 femtosecond timesteps. Most simulations were run for 2 picoseconds, or 20,000 timesteps, and fragmentation was determined by the positions of the carbon atoms at the end of the simulation. We have defined a fragment as a group of atoms each of which has a nonzero interaction with some other member of the group. A few simulations were extended to 4 picoseconds to examine the completeness of fragmentation after 2 picoseconds, and we found that very little fragmentation and essentially no recombination would occur after the first picosecond.

For C60, fragmentation data at each energy were accumulated by running batches of 1000 trajectories, with the orientations in each batch random with respect to those in other batches. Eight batches were run, resulting in overall statistics on 8000-trajectory samples. After analysis, it was determined that the statistics did not vary with sample size for batches of 1000 trajectories or more. Consequently, sets of 1000 trajectories were run for each of the other fullerenes. Due to our use of a structureless wall, an efficient computer program, and an IBM SP supercomputer, we were able to conduct a large number of simulations.

Figure 2 shows the percentage of each of the fullerenes that remains unfragmented as a function of impact energy per carbon atom.

STABILITY OF FULLERENES

We characterize the stability of the fullerenes by determining the impact energy at which 50% of the fullerenes in a batch have fragmented, and we denote this energy Ecrit. Figure 2 shows the percentage of each of the fullerenes that remains unfragmented as a function of impact energy per carbon atom. Furthermore, Figure 1 shows Ecrit/n (where n is the number of carbon atoms in each fullerene) plotted against n. As shown, the C60 fullerene appears more stable with respect to fragmentation than other investigated fullerenes. 		Figure 1 shows Ecrit/n (where n is the number of carbon atoms in each fullerene) plotted against n. As shown, the C60 fullerene appears more stable with respect to fragmentation than other investigated fullerenes.

FRAGMENT DISTRIBUTION

The fragment distributions from the collisions of the fullerenes against a hard wall were plotted at various impact energies and the data was reduced to yield representative functional forms. Figure 2 shows one such plot, that for C60. Upon examination of the plots, we found a distinct dependence of fragment distribution upon impact energy. Therefore, we have divided the fragment distribution into three energy regimes and defined the three regimes as follows:

Regime I: This regime comprises the low impact energies, where the fragment distribution appears mirror symmetric between the large and small fragments. The data indicate a preference for the break-off of small fragments, and we see that large and small fragments are well fitted by an exponential function with the numerical value of the exponent decreasing with increasing impact energy, consistent with the fact that greater energy leads to more extensive fragmentation.

Regime II: As impact energies are increased, the fragment distribution ceases to be symmetric, and the fraction of large fragments decreases because multiple fragmentation of fullerenes now occur. The distribution of large fragments is not well fit by an exponential function. However, the small-fragment distribution can still be fit by an exponential function, the exponent only weakly dependent on impact energy.

Regime III: At high impact energies, the entire fragment distribution is fitted well by an exponential distribution.

Based on the above definitions, we have determined the energies at which the transitions between the regimes occur for the four investigated fullerenes. The transition between regimes I and II occurs at impact energies near 3.4eV per carbon atom and the transition between regimes II and III occurs at impact energies near 5.3eV per carbon atom.

DISCUSSION

The qualitative fragment distribution of the investigated fullerenes is similar to the distribution resulting from the breakup of atomic nuclei. Ref. [9] describes the nuclear breakup mechanism, and we note that Regime I in our work corresponds to the “compound nucleus” regime in Ref. [9], Regime II corresponds to “fission-like processes” and Regime III corresponds to the “multifragmentation” and “nuclear gas” regimes. The similarities between our results and those predicted for a nuclear system in Ref [9] are striking, and it is likely that a similar description would be valid for fullerenes.

Fullerene fragmentation had been experimentally examined both by collisions with atoms, ions, or small molecules [10], and by surface-impact studies [11]. These experiments are carried out by aiming beams of fullerene ions aimed at target gases or surfaces, with the collision products identified by mass spectrometry. Physically, our simulated results are closer to the experimental surface impact studies than to the target gas impact studies. However, for studies of fragment distribution on larger scales, it has been determined that the distributions of macroscopic objects such as gypsum rods, plates, and balls, are not very dependent upon the fragmentation method [12].

Beck et al. [13] experimentally examined fullerene cations at impact energies ranging from 150eV to 1050eV and concluded that at energies near the fragmentation threshold, the fullerenes tended to eject small C2 units, while at higher impact energies, the fullerene “shatters” into a large number of smaller fragments. The experimentally determined fragment distribution is in qualitative agreement with our findings and shows similar changes in fragmentation as a function of impact energy. A quantitative comparison of the experimental and theoretical data proves impractical due to the limited precision with which the plots may be read.

CONCLUSIONS

We have conducted a classical molecular-dynamics simulation study of the fragmentation of the C24, C60, C100, and C240 fullerenes upon impact with a hard wall by using a Tersoff potential to model atomic interactions. The resulting fragment distributions depend both qualitatively and quantitatively on the impact energy. Our results reproduce observed fragment distribution obtained from experiments conducted within a limited range of impact energies. Furthermore, we have produced data for impact energies that lie outside of the experimentally acceptable range, and we have investigated fullerenes which are not available for experimentation. We have divided the fragmentation distribution into three distinct regimes and identified traits of each regime. Similar regimes and traits have been theoretically predicted and experimentally observed in the fragmentation of atomic nuclei. We conclude that that the fracture process of a fullerene more closely resembles that of an atomic nucleus than that of a larger object. Additionally, our results show that the C60 fullerene is more stable towards fragmentation than the other investigated fullerenes.

REFERENCES

  1. URL  http://www.qtp.ufl.edu/~chancey/WWW1/C60/

  2. C. Joachim, J.K. Gimzewski, and A. Aviram, Nature (London) 408, 541 (2000).

  3. C. Joachim, J.K. Gimzewski, R.R. Schlittler, and C. Chavy, Phys. Rev. Lett. 74, 2102 (1995).

  4. H. Park, J. Park, A.K.L. Kim, E.H. Anderson, A.P. Alivisatos and P.L. McEuen, Nature (London) 407, 57 (2000).

  5. M.S. Fuhrer, J. Nygard, L. Shih, M. Forero, Y.G. Yoon, M.S.C. Mazzoni, H.J. Choi, J. Ihm, S.G. Louie, A. Zettl, and P.L. McEuen, Science 288, 494 (2000).

  6. J. Tersoff, Phys. Rev. B 39, 5566 (1989).

  7. URL  http://www.ccl.net/cca/data/fullerenes

  8. Q.H. Tang, K. Runge. H.-P. Cheng, and F.E. Harris, J. Phys. Chem. A 106, 893 (2002).

  9. J.P. Bondorf, A.S. Botvina, A.S. Iljinov, I.N. Mishustin, and K. Sneppen, Phys. Rep. 257, 133 (1995).

  10. P. Hvelplund, L.H. Andersen, H.K. Haugen, J. Lindhard, D.C. Lorents, R. Malhotra, and R. Ruoff, Phys. Rev. Lett. 69, 1915, (1992).

  11. H.-G. Busmann, T. Lill, B. Reif, I.V. Hertel, and H.G. Maguire, J. Chem. Phys. 98, 7574 (1993).

  12. L. Oddershede, P. Dimon, anf J. Bohr, Phys. Rev. Lett. 71, 3107 (1993).

  13. R.D. Beck, J Rockenberger, P. Weis, and M.M. Kappes, J. Chem. Phys. 104, 3638 (1996).

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