The Journal of
Physical Chemistry
0 Copyright, 1992, by the Americon Chemical Society
VOLUME 96, NUMBER 9 APRIL 30,1992
LETTERS Disintegration and Formation of Cs0 C. Z. Wang,* C. H. Xu, C. T. Cban, and K. M. Ho Ames Laboratory, USDOE, and Department of Physics and Microelectronics Research Center, Iowa State University, Ames, Iowa 5001 I (Received: January 8, 1992; In Final Form: March 6, 1992)
The disintegration and formation of the Cm fullerene are studied using molecular dynamics simulations. The interactions between carbon atoms are described with a tight-binding potential model that yields structural and vibrational properties of the molecule in good agreement with experimental data. The simulations show that Cbo is stable against spontaneous disintegration up to 5000 K. The cage formation process is also observed by cooling and compressing 60 carbon atoms from the gas phase.
The Cbomolecule has received considerable attention recently due to the breakthrough in synthesisl that makes it possible to produce quite pure crystals of the material to permit measurement of its physical properties and to test theoretical calculations. Recent experimental and theoretical studies2-' have verified that the ground-state geometry of the Cm molecular is a truncated icosahedron consisting of 20 six-membered rings and 12 fivemembered rings, named "buckminsterfullerene" or "buckyball", as proposed several years ago by Kroto et a1.* The molecules (1) Kriltschmer, W.; Lamb, L. D.; Fostiropoulos, K.; Huffman, D. R. Nature 1990,347, 354. ( 2 ) Yannoni, C. S.;Johnson, R. D.; Meijer, G.; Bethune, D. S.;Salem, J. R. J. Phys. Chem. 1991, 95, 9. (3) Yannoni, C. S.;Bernier, P. P.;Bethune, D. S.;Meijer, G.; Salem, J. R.J. Am. Chem.Soc. 1991, 113, 3190. (4) Tycko, R.; Haddon, R. C.; Dabbagh, G.; Glarum, S.H.; Douglas, D. C.; Mujsce, A. M. J. Phys. Chem. 1991, 95, 518. (5) Weaver, J. H.; Martins, J. L.; Komeda, T.; Chen, Y.; Ohno, T. R.; Kroll, G. H.; Troullier, N.; Haufler, R. E.; Smalley, R. E. Phys. Reo. Lett. 1991, 66, 1741. (6) Zhang, Q.;Yi, J.-Y.; Bernholc, J. Phys. Reu. Lett. 1991, 66, 2633. (7) Feuston, B. P.;Andreoni, W.; Parrinello, M.: Clementi, E . Phys. Reu. B 1991, 44, 4056. ( 8 ) Kroto, H. W.; Heath, J. R.; OBrien, S.C.; Curl, R. F.; Smalley, R. E. Nature 1985, 318, 162.
are also found to crystallize into close-packed structures bound by van der Waals interacti0ns.I Although the structural, vibrational, and electronic properties of this novel material have been well characterized,'-I0 the microscopic mechanisms of formation and fragmentation of the Cbomolecules are still not well understood. In this paper, we report on a study of the disintegration and formation of the Cbomolecule using tight-binding molecular dynamics (TBMD) simulations. Newton's equations of motion for atoms involved in the tight-binding molecular dynamics are derived from a Hamiltonian with the form pi2
H(Pil)
occupied
?I;;; + C n
(+nIHTe(Fil)I+n)
+ Ercp((7iI)
(1)
where {Ti} denotes the positions of the atoms (i = 1, 2, ...,N), and Pi stands for the momentum of atom i. The first term in (1) is the kinetic energy of the ions, the second term is electronic band-structure energy calculated by a parametrized tight-binding (9) Bethune, D. S.;Meijer, G.; Tang, W. C.; Rosen, H. J.; Golden, W. G.; Seki, H.; Brown, C. A.; de Vries, M. S.Chem. Phys. Leu. 1991, 179, 181. (10) Krltschmer, W.; Fcstiropoulos, K.; Huffman, D. R. Chem. Phys. Lcrr. 1990, 170, 167.
0022-3654/92/2096-3563$03.00/00 1992 American Chemical Society
Letters
3564 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992
Hamiltonian &B([j;j), and the third term is a short-ranged repulsive energy representing the ion-ion repulsion and the correction to the double counting of the electron-electron interaction in the second term. Electronic degrees of freedom are explicitly involved in the force calculation but not explicitly involved in the dynamics. The latter feature allows a larger time step to be used in the s has been simulation. In this study, a time step of 0.7 X used. The tight-binding Hamiltonian HTB(lii)) is constructed using an orthogonal sp3 basis ffTB((ii))
=
i
a%,p
eaabia
+
i j a$=s.p
ua,j3(rij)alaaj@
(2)
where ai,ol represents the a orbital on ith atom, e, and t are the on-site energies of s and p orbitals, us&$, uSv(rij),up (rid),and u,(rij) are overlap parameters between two orbitals wg"ose center are separated by distance rij. The repulsive energy is expressed as a function of pair interaction potential t$(riJ),i.e., E,, = C A ~ , 4 ( r i jin) ]the , same spirit as the "embedded atom method". The above parameters and functions are determined by fitting to first-principles calculation results of electronic band structure and volume-dependent binding energies of various crystalline carbon phases with heavy weights on the low-lying energy structures, i.e., graphite, diamond, and the linear chain. The optimized parameters and scaling functions obtained from this fitting procedure not only reproduce well the binding energies and bond lengths of crystalline carbon with different coordination numbers, but also describe well the properties of several complex carbon systems far away from the ground state such as microclusters and liquid and amorphous carbon." The accuracy of the above tight-binding Hamiltonian is further tested by performing calculations of the structural and the vibrational properties of the Cm molecule. The structural optimization starts from a distorted buckyball with the atoms in the correct topological framework but with distorted bond angles and bond lengths. This initial configuration has a binding energy of 7.40 eV/atom. We heat the cluster up to 500 K and gradually cool it down to 0 K without any constraint. This optimization process led to an ideal fullerene structure. Two carbon-carbon bond distances are found: 1.40 8, for the "double bonds" (the shared hexagon edges) and 1.46 8, for the "single bonds" (the pentagon edges), which are in good agreement with the experimental (NMR) data3 and first-principles density functional calculation results6 of 1.40 and 1.45 A. The bond-angle distribution has a &peak at 108O and another at 120O. The cohesive energy of this optimized molecule is 8.00 eV/atom. We believe that the buckyball is the ground-state structure for C60since it is more stable than other conceivable arrangements such as the linear chain, ring, and graphitic fragments. For comparison, the cohesive energies of a 60-atom linear chain, ring or graphite fragment are 7.16, 7.19, and 7.58 eV/atom, respectively. Nevertheless, the cohesive energy of the optimized buckyball is still far from reaching the value of 8.41 eV/atom, the cohesive energy of the infinite graphite crystal according to the present Hamiltonian. The vibrational properties of the optimized buckyball a t T = 0 K have been calculated by solving the eigenvalues of a 180 X 180 force constant matrix derived from the Hamiltonian in eq 1. The results of the vibrational frequencies are classified according to the symmetry of the I, point groupi2and listed in Table I. We found that the 46 distinct vibrational frequencies are equally divided into even and odd parity modes. The modes with Ti, symmetry are infrared-active and those with A, and H, symmetries are observable by Raman spectroscopy. As one can see from Table I, the present theoretical results are in good agreement with the available infrared and Raman data.g*10 Using the above well-tested tight-binding molecular-dynamics scheme, we have studied the disintegration behavior of the C60 (1 1 ) For details of the tight-binding model,see: Xu, C. H.; Wang, C. Z.; Chan, C. T.; Ho, K. M. To be published. (12) Weeks, D. E.; Harter, W. G. J . Chem. Phys. 1989, 90, 4744.
TABLE I: Vibration Frequencies of the CcoMoleculea odd parity even parity I h label freq, cm-' 1, label freq, cm-' A, (1) 1610 (1470) A, (1) 968 509 (496) TI, (3) 1293 TI, (3) 1573 (1428) 823 1201 (1183) 5 20 602 (577) 485 (527) T,, (3) 1369 T,, (3) 1619 1197 880 1027 800 551 694 327 G , (4) 1524 G, (4) 1585 1338 955
1321 1100 807
H,(5)
a75
549 454 1653 (1575) 1538 (1428) 1273 (1250)
752 318
H, (5)
1135 (1099) 786 (774) 690 (710) 392 (437) 237 (273)
1642 1357 1230 768 633
516 360
"The numbers inside the parentheses next to the 1, group labels indicate the degeneracy of the corresponding group representations. Experimental Raman and infrared data from ref 9 are also listed (values inside the parentheses next to calculated frequencies) for the purpose of comparison. c
h
-6.5 v
2.
-7.0 C Q)
-7.5 -8.0
0
1000
2000
3000
4Ooo
5000
6Ooo
7000
temperature (K) Figure 1. Energy (per atom) of the C , molecule as a function of temperature. molecule as a function of temperature. The initial configuration is the optimized buckyball. When the temperature is increased, the valleys between the two bond lengths and between the two bond angles wash out rapidly and their distributions merge into a broad single peak already at a temperature of lo00 K. However, the buckyball is found to be very stable against disintegration. In Figure 1, we plot the total energy of the buckyball as a function of temperature. The energies have been averaged over 2000 MD steps for those T I4500 K and over 6000 M D steps for those T 2 5000 K, after 2000 MD steps for the thermal equilibrium at each temperature. Below 5000 K, the energy increased linearly with temperature, indicating that there is no phase change taking place below this temperature within the time interval considered. However, when the temperature is higher than 5000 K, the energy starts to deviate from the linear behavior and grows rapidly as the temperature is increased further. Consistently, we found that carbon-carbon bonds start to break when the temperature reaches 5500 K. At 6000 K,the disintegration process is found to occur rapidly. In Figure 2, we show three snapshots of the atomic configurations a t T = 6000 K and at a time interval of 1.4 ps. We observed that more and more of the rings are broken as a
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3565
Letters
(a)
(b)
(c)
Figure 2. Snapshot pictures (perspective view) of the Cmmolecule at T = 6000 K. The carbon atoms are connected by straight lines when the
interatomic distances are less than 1.8 A. The buckyball is gradually heated up from T = 0 K as shown in Figure 1. The snapshots are taken at (a) 1.4 ps, (b) 2.8 ps, and (c) 4.2 ps when the temperature is increased to T = 6OOO K.
function of time. Fragmentation of a Cz dimer from the cluster is clearly seen from Figure 2c. In order to understand the formation mechanism and the growth conditions of the Cso, we have also performed molecular dynamics simulations to study the formation of Cm from gaseous carbon atoms. We enclosed 60 carbon atoms in a hollow sphere of radius R. Perfect specular reflection occurs when the atoms hit the inner surface of the sphere. We start the simulation by heating the carbon atoms to very high temperatures (10 000 K) in a larger sphere (R = 9.22 A). The carbon atoms under this condition are found to be gas like. Then we gradually reduce the temperature and the radius of the sphere. When the temperature is reduced to 6OOO K within a sphere of radius 5.3 A, we found that polygonal rings nucleate rather rapidly from the originally loose linearchained cluster (Figure 3a-c). After that, still keeping the temperature at 6000 K, we found that the closing of the cage proceeds rather slowly and has to be accelerated in our simulation by reducing the sphere radius gradually from 5.3 to 3.832 A (Figure 3d-f). We note that the structure of the cage is very similar to that of the buckyball apart from some defects which are probably due to the rapidity of the compression and cooling in the simulation. Further cooling to T = 0 K results in a metastable Cm whose cohesive energy is higher than that of the optimized buckyball by 0.12 eV/atom. It is interesting to note that, while the unconfined Cm molecule disintegrates rapidly at 6000 K, reducing the volume available to the atoms leads to condensation into a carbon cage at the same temperature. Line filaments are favored by their large entropy over the cage structure and compression reduces the entropy difference, allowing the lower-energy cage structure to dominate. However, the large yield of Cm relative to C70in the experiment in spite of the fact that C,,,has a larger binding energy per carbon atom seems to indicate that, in addition to thermodynamics factor, kinetics factors are also important in the fullerene formation process. Another interesting point is that, although the hollow sphere is artifically introduced to confine the carbon atoms, preliminary calculations suggest that the inner curved surface of the sphere plays a role in facilitating the nucleation of the cage.I3 During (13) Preliminary nucleation studies with a cubic box yielded mostly graphitic fragments. We also note that a similar study by J. Chelikowsky (Phys. Rev. Leu. 1991, 67, 2970) nucleating Cm inside a cubic box also produced more imperfect clusters with dangling bond defects. Dangling bond defects are absent in our sphere-generated cages.
Figure 3. Perspective view of Cm cage formation proccss at T = 6000 K. The carbon atoms are connected by straight lines when the interatomic distances are less than 1.8 A. (a), (b), and (c) are snapshots at 2.8, 4.2, and 5.6 ps, respectively, and with the sphere radius R = 5.32 A. (d), (e), and (f) are typical snapshots when the sphere radius is reduced to 4.61, 3.90, and 3.832 A, respectively. Note a closed cage forms when R = 3.832 A. The total simulation time is around 30 ps.
collisions of hot loose filament-like clusters with cool condensed graphitic or caplike fragments, the momentum of the hot cluster deforms the cool fragments and produces a curved surface with decreasing radius of curvature which resembles the shrinking sphere in our simulations. Our results might suggest that the cooling and compression of the hot clusters in such collisions provides the nucleation mechanism for the fullerene cages. The role of the helium gas is to cool down the fragments after the collisions. In summary, we demonstrated that tight-binding molecular dynamics is both accurate and efficient enough to perform realistic simulations of Cm We show that the optimized Cm molecule is an ideal buckyball. The vibrational properties of the buckyball calculated from the present scheme compare well with experimental measurements. Through the simulation, we found that the buckyball is very stable against disintegration. We also studied the nucleation of buckyball-like clusters near a curved surface at T =Z 6000 K.
Acknowledgment. We thank Dr. B. N. Harmon for comments on the manuscript and Dr. J. R. Chelikowsky for communication of results prior to publication. This work is supported by the Director of Energy Research, Office of Basic Energy Sciences, including a grant of computer time on the Cray computers at Lawrence Livermore Laboratory and by the National Science Foundation under Grant No. DMR-8819379. Ames Laboratory is operated for US. Department of Energy by Iowa State University under Contract No. W-7405-ENG-82.
