J. Phys. Chem. 1991, 95, 6750-6751
6750
approach alp0 predicts that with no added salt 4 increases linearly with increasing [surfactant] provided that reaction occurs wholly in the micellar pseudophase. Experimental Section The rate constants are from ref 2. The simulationswere carried out as described in ref 3, and values of k2"' are from plots of k, against calculated values of [H+],.
Acknowledgment. We are graW to a number of organizations for financial support: both C.A.B. and L.S.R.to the NSF U. S.-Latin American Cooperative Program-Brazil; C.A.B. to the Organic and Molecular Chemistry Program of NSF;and L.S.R. to the Busch and Biological Sciences Research Fund of Rutgers University, the donors of the Petroleum Research Fund, administered by the American Chemical Society, Research Corporation, and National Institutes of Health (GM32972).
Molecular Dynamics Simulations of SdM Buckminsterfulleretnes Ailan Cbeng and Michael L.Klein* Department of Chemistry and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323 (Received: June 5, 1991; In Final Form: July 1 1 , 1991)
The results of constant-pressure molecular dynamics (MD) calculations are presented for facecentered-cubic solid buckminsterfullerene based on an atom-atom intermolecular potential acting between rigid C, molecules. At room temperature, potential parameters fitted to the oaxis compressibility of graphite yield a lattice constant in agreement with the measured value. The molecular dynamics trajectories are analyzed to obtain estimates of the reorientational relaxation time and the lattice vibrational frequency distribution. Below 200 K,orientational freezing is observed on the MD time scale.
The carbon cluster Cm (Buckminsterfullerene) can now be prepared more or less routinely in macroscopic quantitie~l-~This possibility has lead to a flurry of research activity directed at characterizing the properties of the pure solid and studying its chemistry.c6 For example, it is now established that under ambient conditions solid Cbois a facaomtered-cubic crystal7$ with lattice parameters a = 14.17 A and that the individual 'Bucky balls" are rotating.'-" However, on cooling to only 249 K, there appears to be a phase transition, as evidenced by the appearance of additional X-ray diffraction peaks.* Calorimetric measurements* and more recent NMR data" suggest a phase transition at 260 K. The effect of applying pressure has also been investigated, and rudimentary information is now available conceming the compressibility of the ~ o l i d . ~ . ' ~ The purpose of the present Letter is to see to what extent classical molecular dynamics calculations based on a simple ( I ) Kroto, H. Science 1988, 242, 1139-1145. (2) KrHtschmer, W.; Lamb, L. D.; Foatiropoulos, K.; Huffman, D. R. Nature. 1990. 347. 354-358. . --(3) Krlltnchmer, W.; Fastiropoulos, K.: Huffman, D. R. Chem. Phys. Lett. 1990,170,167-170. (4) Hawkim. J. M.:M e w .. A,:. Lewis. T. A.: Loren.. S.:. Hollander. F. J. Sciitice 1991,2S2.31i-313. (5) Holczer, K.; Klein, 0.;Gruner, 0 . ; Thompson, J. D.; Diedcrich, F.; Whetten, L. R. Submitted for publication in Phys. Rm. b i t . (6) H?+, K.;Klein, 0.;Hung, S. M.:Kaner, R. E.;Fu, K. J.; Whetten, R. L.: Dledench. F. Science 1991. 252. IlS&l157. (7) Fischer, J: E.; Heiney, P. A.;M k h i e , A. R.; Romanow, W. J.; Denenstein, A. M.; McCauley, J. P., Jr.; Smith, A. B., 111; Cox, D. E. Science 1991.252. -.---.1288-1 - -~290. (8) Heiney, P. A.; Fischer, J. E.; McGhie, A. R.; Romanow, W. J.; Denenstein, A. M.:McCaulcy, J. P., Jr.: Smith, A. B., 111; Cox, D. E. Phys. Rev. Lett. 1991,66, 291 1-2914. (9) Yannoni, C. S.;Johnson, R. D.; Meijer, G.; Bethune, D. S.;Salem, J. R.J. Phys. Chcm. 1991, 95,9-10. (IO) Tycko, R.: Haddon, R. C.; Dabbagh, G.; Glarum, S.H.; Douglas, D. C.; Mujm, A. M. J . Phys. Chem. 1991, 95, 518-520. ( I I ) Tycko, R.; Dabbagh, G.; Fleming, R. M.; Haddon, R. C.; Makhija, A. V.; Zahurak, S. M. Submitted for publication in Phys. Reu. Lett. (12) Duclos, S.;Brister, K,; Haddon, R. C.; Kortan, A. R.; Theil, F. A. Nature 1991, 351, 380-382.
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atomatom potential can account for the physical properties of Buckminsterfullerene. Anticipating our results, we will see that this atomatom potential, with parameters fitted to the c-axis compressibility of graphite, gives a remarkably good account of the existing experimental data.7 Indeed, the agreement is sufficiently good that we have been motivated to present some information on the dynamical properties of the crystal. The simulations were carried out using the constant-pressure Parrinello-Rahman equations of The translational degrees of freedom were solved by using a third-order Gear predictor-corrector algorithm, and the rotational degrees of freedom, expressed by quaternions, were generated by using a fourth-order Gear a1g0rithm.I~ Each Cm molecule is treated as a rigid molecule with bond lengths lI = 1.37 A and 1, = 1.448 A, obtained from high-level quantum chemistry calculation^.^^*^^ This choice yields a 7. I-A diameter for the nuclear framework of the C,~molecule. Pairs of carbon atoms on different molecules are considered to interact via a Lennard-Jones (12-6) potential, with parameters e = 28.0 K and u = 3.4 A taken from early work on graphite.'* These parameters agree quite well with a recent fit19to the sublimation energym and lattice c o n ~ t a n t .Although ~ the molecule is nearly spherical, the interaction between two molecules depends on their relative orientation. The dimer energy minimum occurs between 9.7 and 10.3 A, with energy ranging from -2200 to -2900 K. The potential energy barrier for rotational motion can be estimated (13) Parrinello, M.;Rahman, A. Phys. Rev. Lett. 19so,45,1196; J. Appl. Phys. 1981,52,7182. (14)No&, S.: Klein, M.L. Mol. Phys. 1983,50, 1055. Impey, R. W.; Sprik, M.;Klein, M.L. J. Chem. Phys. 1905.83, 3638. (1 5) Gear, C. W. Numerical Initial Value Problems In OrdlnOrY D U f b ential Equations; PrenticuHall: Englewood Cliffa, NJ, 1971. (I 6) Scuseria, G. E. Chem. Phys. Lett. 1991, 176, 423-427. ( 17) Fowler, P.W.; Lazmeth, P.: Zanasi, R. Chem. Phys. Lett. 1998,165, 19. (1 8) Steclc, W. A. The Interaction of Gases wlth Solid Surfaces; Pergamon: New York, 1974. (19) Giriflco, L. A. Submitted for publication in J. Phys. Chem. (20) Pan, C.; Sampson. M.P.; Chai, Y.; Hauge, R. H.; Margrave, J. L. J . Phys. Chem. 1991, 95, 2944-2946.
Q 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 18, 1991 6751
Letters
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Figure 1. Pressure as a function of fcc lattice constant for solid Cm Circles are from static calculations, asterisks are from constant-pmure simulation, and squares are the experimental data.' by studying the dimer. Rotation of one molecule for fixed centersf-mass separation (10.0 A) yields barriers ranging from 100 to 300 K. Due to the large number of atoms in each molecule, the simulation cell contained only 32 Ca molecules, which were arranged on a 2 X 2 X 2 fcc lattice. To reduce computer time, the C-C pair potentials were truncated at 9.5 A. Center-of-mass and angular velocities were scaled to the desired temperature during a run of 6000 time steps, 5.0 X lo-" s each. After this, 4000 configurations were saved for the analysis of various properties. Constant-pressure simulations have been performed at room temperature but at two different pressures. The average (fcc) lattice parameter obtained, 14.14 A at zero pressure and 13.70 A at P = 17.5 kbar, are in excellent agreement with X-ray scattering data, 14.17 A at zero pressure and 13.70 A at P = 12.0 f 1.O kbar? The configuration energy obtained at P = 0.0,142.3 kJ/mol, is 15% less than the reported heat of sublimation.m The latter result suggests the need to increase the c parameter of the potential. Representative configurations from the MD runs were used to calculate the pressure as a function of lattice constant. The results are shown in Figure 1, together with MD results reported above and experimental dataa7This rough calculation provides an additional test of potential model. It seems that the simple potential model produces a surprisingly good account of the bulk compressibility. Visualization of the simulation results on a graphics terminal clearly indicates that the C, molecules rotate rapidly at room temperature. (Recall the rotational barrier is less than 300 K.) This finding is consistent with X-ray and NMR data.'-'' The calculated room-temperaturerotational diffusion coefficient is 1.5 X IOi2 s-'. However, at high pressure (P = 17.5 kbar), the rotational diffusion coefficient falls dramatically to 5 X loios-l. Linear and angular velocity autocorrelation functions have been examined in order to probe the single particle and collective dynamics of the "Bucky balls". The room-temperature twofold (C2) axis reorientation relaxation time, estimated by computing the appropriate time correlation function, is 5 ps, which can be compared with 12 ps derived from NMR data." Fourier transformation of the velocity autocorrelation function (see Figure 2) fies the density of phonon states. Due to the system size, there are only a few allowed lattice modes and the calculated spectrum is sparse. Since Cbomolecules are rotating, the room-temperature density of states for librational modes is peaked at zero frequency (see Figure 2). We have also carried out several calculations at zero pressure for the range of temperatures down to 80 K in order to study the freezing of the reorientational motion. The rotational diffusion coefficient is a convenient monitor of any orientational freezing. The variation of calculated values with temperature is shown in Figure 3. Rotational motion slows down dramatically as the temperature drops. The low-temperature results show that C, molecules freeze below about 200 K. Orientational freezing is accompanied by the appearance of a band of librational modes (see Figure 2). By comparing with the rotational motion calcu-
F i p e 2. Calculated translational (bold) and librational (dotted) densities of states at 340 and 100 K for fcc CWusing a 32-moleculesystem
(see text).
0
100
200
300
400
T (K)
Figure 3. Calculated rotational diffusion coefficients as a function of temperature. On the M D time scale the Bucky balls freeze below 200 K.
lated at high pressure, it is clear that the effect of applying pressure bears some similarity to cooling the system: both dramatically slow down the rotational motion. A more detailed study is currently underway to clarify the nature of the orientational ordering at low temperature by employing the sixth-order spherical harmonic functions, Yh The most recent NMR data suggest partial ordering sets in at 260 K. Constant-pressure molecular dynamics simulations based on an atom-atom pair potential give lattice parameters (equation of state) in good agreement with experimental data.' Information on the reorientation relaxation time and the phonon density of states is presented. Orientational freezing of this highly spherical molecule is observed, which is in qualitative accord with NMR and X-ray e x p e r i m e n t ~ . ~ .The ~ J ~ low-temperature librational modes are predicted to form a band centered around 10 cm-l. Acknowledgment. We are grateful to our colleagues in the University of Pennsylvania Laboratory for Research on the Structure of Matter for sharing their new results on fullerites at our Thursday morning meetings. Special thanks go to J. E. Fischer, P. A. Heiney, and E. Mele for their insights and to P. W. Fowler and R. Tycko for keeping us fully informed about their own studies. This research was supported by the National Science Foundation under Grants CHE 87-22481, CHE 88-15130, and DMR 88-19885.
