J. Phys. Chem. 1994,98, 9297-9300
9297
Intermolecular Motion in Solid C70: A Molecular Dynamics Simulation Study Michiel Sprik' IBM Research Division, Zurich Research Laboratory, 8803 Riischlikon, Switzerland
Michael L. Klein Department of Chemistry and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 191 04-6323 Received: April 4, 1994; In Final Form: June 3, 1994"
The rotational and intermolecular translational dynamics in solid C ~ isOanalyzed using the trajectories of a molecular dynamics simulation of state points in the cubic high-temperature phase, the rhombohedral partially ordered intermediate phase, and the fully ordered monoclinic low-temperature phase. The rotational dynamics in the cubic phase is found to be fast with correlation times of the order of 10 ps and modest anisotropy. The monoclinic phase is characterized by a libron band centered around 2 meV. The orientational dynamics in the rhombohedral phase is uniaxial. However, the velocity autocorrelation spectra for the intermediate phase resemble more closely the results for the high-temperature than the fully ordered phase.
Introduction
Plastic crystals are molecular solids consisting of comparatively rigid and highly symmetric molecules close-packed on fcc or hcp lattices. At high temperatures the orientational degrees of freedom are disordered. Upon cooling, one or several phase transitions are observed in which the orientations are ordered. These structural transformations are accompanied by only minor distortions of the translational lattice. Solid c60 is almost a perfect example of such behavior. Above 260 K the crystal lattice is fcc with nearly complete orientational disorder. Below 260 K the structure becomes simple cubic with Pa3 symmetry, preserving the fcc structure for the molecular centers but with the four molecules in the conventional fcc unit cell having different average orientations.l-3 The rotational dynamics is closely coupled to the change in molecular organization. Above 260 K the motion is diffusive and approximately isotropic with surprisingly short correlation times (12 ps at 300 K4). Below 260 K the molecular frames oscillate about their equilibrium configurations (libration) with infrequent, activated jumps between the various symmetry-related orientations. The orientational correlation time can become several orders of magnitude larger (-60 ns at 200 K) compared to the disordered phasee4,5 With decreasing temperature the reorientational motion is eventually frozen in and only the librational dynamics remain^.^,^ This is the basic picture of solid c60 that has emerged from e~perimentl-~ and computer simulations-10 (for a review see ref 11). However, this picture is somewhat oversimplified. More recent experiments3 have revealed that there is a second, inequivalent, stable orientation in the Pa2 structure which is thermally accessible above 100 K. The drastic slowing down of the orientational relaxation at lower temperatures leads to nonequilibrium occupation of the alternative set of orientations resulting in a "glass" transition around 90 K." The thermal behavior of solid C70 follows a similar pattern. The details of the orientational structure and dynamics are richer as a consequence of the prolate spheroidal shape of the molecule. Ordering of the orientational degrees of freedom proceeds in stages with a partially ordered intermediate phase I1 inserted between the high-temperature rotator phase I and the fully ordered low-temperaturephase I I P 3 (seealso ref 11). Theexperimental characterization of these structures and the determination of the
* Abstract published in Advance ACS Abstracts, August 15, 1994. 0022-3654/94/2098-9297$04.50/0
transition temperatures have been hampered by the occurrence of metastable states and large thermal hysteresis. It has now been established that the stable, disordered phase I has a fcc lattice.'* The intermediate phase I1 is rhombohedral with the molecular fivefold symmetry axes aligned along a ( 11 1) direction and little or no ordering of the rotations about the body axis. The symmetry of phase I11 is monoclinic.12 These results are in accordance with predictions of computer ~imu1ations.l~ Not only do theoretical and experimental estimates of the transition temperatures show appreciable discrepancies, the various experiments are not in full agreement either, presumably due to differences in the thermal history of the samples. Here, we will quote the figures from,I2 which finds the 1-11 transition at 345 K and the 11-111 transition at 295 K. The elongated shape of the molecule and the corresponding reduction in the symmetry of the ordered phase is reflected in anisotropies in the orientational dynamics. Resolving and quantifying these anisotropies is a challenge to experimentalists and theorists alike. Evidence from pSR (muon spin relaxation/ rotation/resonance) measurements showed that uniaxial rotation persists down to low temperatures (100 K). With increasing temperature, orientational disorder sets in, ending in almost isotropic rotational motion in the high-temperature phase.13J6 In recent nuclear magnetic resonance (NMR)17and inelastic neutron scattering (INS) studied8 the distinction between the dynamics in the ordered and disordered phase was investigated in more detail. It was verified that the rotational motion in phase I is diffusive and near-isotropic with short correlation times ( 5 ps at 340 K),17 comparable to values for c60. In the ordered phase, uniaxial dynamics prevails. The N M R estimate of the correlation time for reorientation about the fivefold axis in the monoclinic phase I11a t 250 K is 2 ns and becomes shorter in the rhombohedral phase II.17 At 250 K and below, I N S can identify a libration of about 2 meV.7J8 The librations soften as the temperature increases, merging into a single quasielastic line at 300 K.l* Simulation and Model
In the present communication we report some results for the dynamics of solid C70 obtained from molecular dynamics (MD) simulation data used previously to explore the structure and energetics of the low-pressure s01id.l~This simulation consisted of a long series of runs, starting with a 256-molecule sample of the high-temperature disordered phase (phase I) assuming a fcc 0 1994 American Chemical Society
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structure for the lattice. This system was cooled in small steps using constant temperature, constant pressure MD techniques. The 1-11 transition to the partially ordered rhombohedral phase was observed at 390 K, whereas the 11-111 transition to the monoclinic phase occurred at 200 K. The MD trajectories at three temperatures corresponding to state points in phase I (400 K), I1 (300 K), and I11 (10 K), respectively, were stored for subsequent numerical analysis of the dynamics. Technical details of the calculation and the intermolecular interaction model can be found in ref 14. Here, we will highlight a particular aspect of the model that is responsible for an interesting detail of the structure of the ordered phases and also has implications for the dynamics, particularly in phase 11. In previous simulation studies of solid c 6 0 it was demonstrated that a simple model consisting of a rigid atomic framework with shortrange (12-6) pair potentials centered on the C atoms yields a tetragonal ground state instead of the experimental simple cubic structure.*J9 By assigning fractional negative charges to the center of the short (6:6) bonds with the positive counter charges either on the C atomsg or long (5:6) bondsI9 it is possible to stabilize the simplecubicstructure. The model used in our C,josimulations9 had an additional feature. We also assigned a 12-6 interaction to the negative charge sites on the 6:6 bonds. The motivation for this extension is largely empirical. The additional short-range repulsion increases the order4isorder transition temperature and enhances the first-order character of the transition. In particular, our extended model is able to reproduce the discontinuity of the lattice parameters.”J The model used for C70 is an extrapolation of the c60 model. In the case of c 7 0 the contribution of the electrostatics is less important, particularly in the partially ordered phase I1 where the electrostatic energy is negligible. The “bulge” in the short bonds has a frustrating effect on the alignment of the long axis. Instead of an equilibrium orientation exactly along a (1 1 1 ) direction, the fivefold axis is a t a small angle (=15’) to the trigonal crystal axis, with a preference for residing in one of the three planes defined by the (1 11) axis and the intersecting ( 1 10) directions. In the monoclinic phase this apparent instability is resolved by aligning half of the molecules along one of these three ( 1 10) directions, while the small angle of the other half of the molecules to the ( 1 1 1 ) direction is relaxed to 2er0.l~ In ref 20 the C70 dynamics was studied using the simple interaction model with C-C 12-6 interactions only. Use of the extended model has a marked effect on the phase I1 dynamics as a result of the increased disorder (see below).
Results and Discussion In Figures 1-3 we give the results for the time correlation of the fivefold (“long”) axis nl and a vector n, (“short axis”) fixed in the molecular frame perpendicular to n1. In order to facilitate direct comparison to N M R results, second-order polynomials of the axis coordinates have been used; the curves in Figures 1-3 correspond to the time correlation functions of I = 2 tensors. Figure l a shows the long and short-axis correlations at 400 K in phase I obtained from a 200 ps trajectory. The decay of nl is slower than that of n,. The estimate of the correlation times is T I = 15 f 2 and T , = 6 f 1 ps. The modest anisotropy is compatible with experiment which also seems toindicate that theorientational motion in the disordered phase is not completely isotropic.~7J* Moreover, the time scale of the motion is comparable to the T~ = 5 ps value obtained from N M R at 340 Kin phase 1.17 Focusing on the short-time behavior, Figure l b shows the spectrum of the time derivative of the 1 = 2 tensors. A curious feature of these spectra is the (minor) deviation from the Lorentzian frequency dependence of simple exponential relaxation. Both the nl and n, motions exhibit a small harmonic component a t very low frequency (VI = 5 and V, = 1.5 cm-1). The shift away from zero frequency can be compared to the rotational equivalent of the oscillation
Sprik and Klein
T I ps I
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/
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”
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v/cm-1 Figure 1. Long- and short-time dynamics in the disordered phase I at 400 K. The solid curves indicate results of the autocorrelation of the fivefold axis of the c70 molecule, and the dashed curves show the autocorrelation of a vector perpendicular to the fivefold axis. (a) The real-time autocorrelation of a I = 2 tensor and (b) the spectrum of the corresponding velocity autocorrelation. 0.10
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Figure 2. Spectrum of the orientational velocity autocorrelation in the fully ordered phase 111 at 10 K. Solid and dashed curves distinguish again the dynamics of vectors parallel and perpendicular to the fivefold axis, respectively.
in the translational time correlation function of simple liquids such as argon. The ratio of the frequencies vl/vs = 3.3 is similar to the ratio T ~ / T= ~2.5, suggesting that the anisotropies in the short and long-time dynamics are related. Skipping phase I1 for the moment, we proceed to the discussion of the dynamics of the fully ordered phase at 10 K. Here, the reorientational motion is quenched, and both relaxation times T I and T , are infinite on the MD time scale. Figure 2 shows the spectrum of the velocity autocorrelation evaluated from a 1 = 2 tensor as in Figure 1 b. The motion is characterized by a broad libration band centered around v = 15 cm-l = 1.9 meV, in good agreement with the INS libron peaks (2 meV) of ref 18. The nl and& spectra overlap, implying that it will be difficult to separate the two types of motion in experiment.
Intermolecular Motion in Solid C70
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9299
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Figure 4. Spectrum of the velocity autocorrelation function of the molecular center of mass coordinates in phase I at 400 K (dashed), phase I1 at 300 K (dotted), and phase I11 at 10 K (solid).
0
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v/ cm-1 Figure 3. Long-time dynamics (a) in the partially ordered phase I1 at 300 K. (b) Spectrumofcorrespondingvelocityautocorrelation. Definition of the curves and correlations functions as in Figure 1.
Figure 3, a and b, shows the results for the long- and shorttime dynamics a t the 300 K state point in phase I1 obtained from a 100 ps run. As a consequence of the alignment along the ( 1 11 ) direction, the reorientation of the fivefold axis is frozen and 71 is, in effect, infinite. In contrast, the rotational motion about this axis is still rapid. The correlation time rS = 10 f 2 ps is of the same order of magnitude as the value in phase I a t higher temperatures (Figure la). The velocity spectra of ns in phase I and I1 are also very similar, including the low-frequency harmonic component. Despite the ordering of the fivefold axis, the same holds true for the nl dynamics. The nl spectrum has a striking resemblance to thedisordered phase picture (Figure 1b), showing little evidence of remnants of the librons of phase I11 (Figure 2). In an attempt to rationalize this somewhat paradoxial finding, we return to our earlier observation that the ordering of the long axis in our model is not complete. The small residual angle between nl and the trigonal crystal direction leads to a precessional motion about the ( 111 ) axis with an increased probability of finding nl pointing in one of the three equivalent directions defined by the local trigonal symmetry. This large-amplitude disordered rotational motion is superimposed on the librational dynamics and will broaden the libron lines. In our simulation, the perturbation is evidently sufficiently serious to overdamp the librons and suppress the harmonic motion completely. The predicted absence of librational modes in phase I1 seems to be contradicted by the INS results in ref 18, where the librational splitting of the central line could be followed from 10 K up to 260 K. At this relatively low temperature the sample in ref 18 is believed to enter a state in which phase I coexists with the ordered phases.13J8 A clear 11-111 transition could not be detected, presumably due to similar nonequilibrium effects. Hence, it may be possible that the high-temperature librons observed in ref 18 in fact originate from metastable phase I11 domains. Recent N M R experiments21 seem to support our picture of the phase I1
dynamics consisting of precessional jumps of the fivefold axis about the (1 11) direction. Finally, in Figure 4 we compare the spectra of the velocity autocorrelation of the centers of mass of the molecules. The translational (phonon) dynamics in phase I1 is almost identical to the result for phase I and shows no particular characteristic features. The phase I11 spectrum has more structure with a sharp low-frequency peak at 20 cm-I which is absent in phases I and 11,suggesting that thismode is related to the monoclinicsymmetry breaking. The translational spectra confirm our earlier conclusion that the dynamics in phase I1 is dominated by the disordered degrees of freedom and is more similar to the high-temperature phase I than to the fully ordered phase 111. Conclusion
The relatively simple intermolecular potential derived in our later work9J4 is able to rationalize some of the dynamical information already available for solid C70. In particular, we have almost quantitative accordance with the measured libron frequencies in phase I11 and the reorientation dynamics in phases I and 11. Acknowledgment. We thank K. Prassides, R. Blinc, A. Cheng, D. Neumann, J. Copley, R. Tycko, P. Heiney, A. Smith, and J. Fischer for giving us details of their work before publication. This work was supported, in part, by NSF/DMR 91-20668. References and Notes (1) Heiney, P. A.; Fischer, J. E.; McGhie, A. R.; Romanow, W. J.; Denenstein, A. M.; McCauley Jr., J. P.; Smith 111, A. B.; Cox, D. E. Phys. Rev. Lett. 1991, 66, 2911. (2) David, W. I. F.; Ibberson, R. M.; Matthewman, J. C.; Prassides, K.; Dennis, T. J. S.; Hare, J. P.; Kroto, H. W.; Taylor, R.; Walton, D.R. M. Nature 1991, 353, 147. (3) David, W. I. F.;Ibberson,R. M.;Dennis,T. J.S.;Hare, J.P.;Prassides, K. Europhys. Lett. 1992,18, 219. ( 4 ) Tycko, R.; Haddon, R. C.; Dabbagh, G.; Glarum, S. H.; Douglass, D. C.; Mujsce, A. M. J. Phys. Chem. 1991, 95, 518. Tycko, R.; Dabbagh, G.; Fleming, R. M.; Haddon, R. C.; Makhija, A. V.; Zahurak, S . M. Phys. Rev. Lerr. 1991, 67, 1886. (5) Yannoni, C. S.;Johnson, R. D.; Meijer, G.; Bethune, D. S.; Salem, J. R. J. Phys. Chem. 1991.95.9. Johnson, R. D.; Yannoni, C. S.; Dorn, H. C.; Salem, J. R.; Bethune, D. S. Science 1992, 255, 1235. (6) Neumann, D. A.; Copley, J. R. D.; Kamitakahara, W. A.; Rush, J. J.; Cappelletti, R. L.; Coustel, N.; McCauley Jr., J. P.; Fischer, J. E.; Smith 111, A. B.; Creegan, K. M.; Dox, D. M. J . Chem. Phys. 1992, 96, 8631. (7) Renker, B.; Gompf, F.; Heid, R.; Adelmann, P.; Heiming, A.; Reichardt, W.; Roth,G.;Schrober, H.;Reitschel, H. Z . Phys. 1993,890,325. (8) Cheng, A.; Klein, M. L. Phys. Rev. 1992,845, 1889. (9) Sprik, M.; Cheng, A.; Klein, M. L. J . Phys. Chem. 1992, 96, 2027. (IO) Cheng, A.; Klein, M. L.; Parrinello, M.; Sprik, M. In New Methods for Modelling Processes within Solids and their Surfaces; Catlow, C. R. A., et al., Eds.; Oxford University Press: London, 1993; p 133.
9300 The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 (11) Fischer, J. E.; Heiney, P. A. Phys. Chem. Solids 1992,63, 1333. , E.;Ricketts(12) Vaughan,G. B.M.; Heiney,P. A.; Fischer,J. E.; L ~ z dD. Foot, D. A.; McGhie, A. R.; Hui, Y.W.; Smith, A. L.; Cox, D. E.; Romanow, W. J.; Allen, B. H.; Coustel, N.; McCauley Jr, J. P.; Smith 111, A. B. Science 1992,254,1350. Vaughan, G. B. M.; Heiney, P. A,; Cox, D. E.; Fischer, J. E.; McGhie, A. R.; Smith, A. L.;Strongin, R. M.; Cichy, M. A.; Smith 111, A. B. Chem. Phys. 1993,178, 599. (13) Christides,C.;Thomas,I.M.;Dennis,T. J.S.;Prassides,K.Europhys. Lett. 1993,22, 611. (14) Sprik, M.; Cheng, A.; Klein, M. L. Phys. Rev.Lett. 1992,69,1660.
Sprik and Klein (1 5) Prassides, K.;Dennis, T. J. S.; Christides, C.; Roduner, E.; Kroto, H. w.; Taylor, R.; Walton, D.R. M. J . Phys. Chem. 1992,96,10600. (16) Dennis,T. J. S.; Prassides, K.;Roduner, E.; Cristofolini, L.;DeRenzi, R. J . phYs. Chem. 19939 979 8553. (17) Tycko,R.;Dabbagh,G.;Vaughan,G.B. M.;Heiney,P.A.;Strongin, R. M.; Cichy, M. A.; Smith 111, A. B. J. Chem. Phys. 1993, 99,7554. (18) Christides, C.; Dennis, T. J. S.; Prassides, K.;Cappelletti, R. L.; Neumann, D. A.; Copley, J. R. D. Phys. Rev. 1994,B49, 2897. Martin, R. M.Phys. Rev. Lett. 1992.68, 1551. (19) Lu,J. P.; Li, X.-P.; (201 Chena, A.; Klein. M. L.Phys. Reu. 1992,846,5948. (21) BlincrR. Private communication.
