The dynamics of fullerene multifragmentation

Supr0molecular Science 2 (1995) 135sl40 1 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 0968-5677/95/%10.00

The dynamics of fullerene multifragmentation Jiirgen Schulte* Michigan State University, National Superconducting MI 48823, USA (Revised 20 November 1995)

Cyclotron Lab., East Lansing,

The molecular dynamics statistical trajectory ensemble (MDSTE) method has been employed to study the fragmentation of fullerenes in a cross-beam collision experiment. Differential fragmentation crosssections and double differential cross-sections of the final time step of the large scale molecular dynamics statistical trajectory ensemble simulation with improved statistics is presented. The envelope of the crosssections of the small collision fragments (Fullerene multifragmentation: J. Schulte (N < 28)) follow a scaling law with a(C,) M N-“, with I. = 1.54. An odd-even alteration with enhanced even number size fragments (N < 12) is found with a change of the alteration sequence starting at N = 15. The small and large fragments of the fullerene collisions are carbon chains and intertwined carbon chains with vibrational excitations close to that of an ideal cluster gas where about 50% of the collision momentum is transferred to internal vibrational energy, and a strong correlation of fragment mean coordination number and their respective excitation has been found. PACS: 36.40+d, 35.10b, 82.40d, 32.80n

1

INTRODUCTION

The molecular dynamics statistical trajectory ensemble (MDSTE) simulation’ of colliding fullerenes resembles a cluster-cluster cross-beam experiment similar to that performed by Campbell et ~1.~. In their experimental set up a beam of *Cl0 (the asterisk denotes that the clusters are thermally excited) clusters, with an estimated 30eV average thermal excitation energy, was crossed with a gas target of cold C6o clusters. The experiment was performed at some 102eV center of mass (corn) collision energy. The fragment mass distribution was measured where good resolution had been obtained for fragment sizes N = 50 and larger. The simulation of *Cl0 + Cbo collisions showed an acceptable resolution of the small fragments (N d 15) and a limited resolution of the larger fragments. The nature of this behavior was very obvious since our simulation provided a relatively small *Cl0 beam current (number of collisions trajectories) only, however, the number of collision trajectories at our final simulation time step was sufficiently large to provide a good resolution of the small collision fragments. The thermal excitation, kinetic energy, internal structure (mean coordination number), and angular distribution as a function of fragment size was monitored over the entire course of the collision. We show the cross-sections at our last simulation time step *Current Research

address: Electron Devices Department, Hitachi Laboratory, Kokobunji-shi, Tokyo 185, Japan

Central

(t = 2.1 ps) where the deep inelastic *Cl0 + C,, collisions are believed to have finished3. The improved statistics of the cross-sections were calculated from 128 collision trajectories, where for the interaction of carbon atoms a hybrid of an empirical potential and the density functional linear combination of atomic orbitals in the local density approximation was employed4. This approach has been successfully applied to the simulation of ion-trap experiments on C+ + Dz5, and to the deuterium capture in C6s + O2 reactions6, respectively. Various investigations have been performed to study possible reaction paths of coagulating and colliding C@, molecules7’8. Only recently, however, the possibility of integrating over the entire reaction cross-section of ensembles of colliding clusters and molecules, interacting via accurate many-body forces, has become available’ and may serve as a tool and guide to understand molecular collision experiments, as well as to provide information useful for selected cluster fabrication, and possible information on the reverse reaction, i.e. carbon nucleation. The presented *Cl0 + C60 collision simulation has been carried out at 500eV center of mass collision energy. Thus, the total kinetic energy per atom in this simulation is low enough (4.41 eV) such that electronic excitation is not expected to occur3. In Section 2 we give a brief summary on the preparation of the clusters, and their respective collision trajectories. In Section 3 we present results of the MDSTE

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simulation of *C& + CeO collisions at 500eV center of mass collision energy. The fragmentation cross-sections, the mass resolved fragment vibrational excitation and mean coordination number, and the translational kinetic energy distribution, respectively, are presented and discussed. Results of this study are summarized in the conclusion in Section 4.

2 PREPARATION

OF TRAJECTORIES

orientations, thermal excitations, and impact parameters. From the statistical point of view, in order to achieve meaningful fragmentation statistics, and some convergence, one might expect to calculate a large number of trajectories for a system of 120 atoms. Fortunately, the studied system is not just composed of simple structureless 2-body interacting point particles, but of particles with electronic structure that favor strong directional chemical bonds. The simple combinatory assumption of fragmentation statistics obtains strong constraints by the geometry and the bond properties’of the studied systems. Thus, the basic fragmentation characteristics can be shown with using a rather limited number (128) of ensemble trajectories.

The fragmentation cross-section of the *C& + C60 cross-beam experiment is modeled by employing the molecular dynamics statistical trajectory ensemble (MDSTE) method developed for parallel computing’. In the MDSTE approach reaction cross-sections are calculated in general by integrating over a statistical ensemble of collision trajectories where independent distribution functions are used for the internal degrees of freedom (vibrational energy) of individual clusters, and external degrees of freedom such as impact parameters, collision energy and relative cluster collision orientation. In the presented case, individual trajectories are integrated by simultaneously solving Newton’s equation of motion for the propagation of atoms and the Schrodinger equation for electron interaction. For the interaction of carbon atoms we take a hybrid of an empirical potential and the density functional linear combination of atomic orbitals in the local density approximation regime by Seifert and Jones4. In this regime the bond energies are overestimated by a factor of two. Thus, one has to be careful with comparing absolute energies. There is no problem, energy differences as however, with comparing represented in the bond breaking process and individual relative cluster stabilities. Since in a cross-beam experiment the properties of beam and clusters are well defined, a reduced version of the original MDSTE simulation approach can be applied. The cross-beam collision of the carbon clusters is modeled by an ensemble of collision trajectories where the initial conditions of every trajectory are taken from the proper statistical distribution. The initial geometric cluster configuration, for all C,, clusters, is the fullerene LDA equilibrium structure. The carbon atoms of every thermally excited cluster are Maxwell-Boltzmann velocity given an internal distribution such that the ensemble of thermally excited clusters has an initial mean internal cluster kinetic energy of 30eV. The thermally excited clusters, and the cold clusters, are placed at a center of mass distance of 20 atomic units (au), and were given 10 fixed impact parameters between 1 and 14au. After an initial random rotation about the internal axes of the clusters, the clusters were given a relative collision velocity according to the corn collision energy of 500eV. The equation of motion is calculated using the Verlet algorithm with a step size of At = 50atu (1 atu = 2.4 x lo-l7 s). Cross-sections are calculated by integrating the trajectories over all collision

From the MDSTE simulation data at our final time step (t = 2.1 ps) of the *C& + C6,, collisions we calculated the differential fragmentation cross-sections, the double differential cross-sections of the fragment mass and kinetic energy angular distribution, and the fragment mass resolved mean coordination number and vibrational excitation. In the following, with respect to kinetic energies (translational and vibrational), we refer to the respective mean kinetic energy per atom. In this context, with quasi-elastic, inelastic and deep inelastic collision, we mean ‘Cl, + C6, collisions that have suffered energy loss to vibrational excitation of fully intact Cc0 clusters, to collisions where one or both of the colliding C6e clusters have lost a minor amount (Q 4) of atoms only, and to collisions where one or both of the colliding clusters have suffered severe fragmentation or loss of the original structure. For better visualization of the more interesting cross-sectional features of our simulation, the cross-sections of the large cluster compounds have been omitted. During the collisions the C,, clusters suffer quasielastic, inelastic and deep inelastic collisions. The quasi-elastic collisions leave behind fully intact C6e clusters that are vibrationally excited with energy E* = 2.1 eV (Figure I, r.h. scale). About 20% of the E* = 2.1 eV quasi-elastic scattered clusters experienced a delayed multifragmentation process after being detached again from their respective collision partner. The delayed fragmentation of the quasi-elastic scattered clusters contributes mostly to the population of small fragments N < 15 with a preferable fragmentation channel for N = 10 fragments. The inelastic scattered C6e clusters have maintained their structural integrity as well, and they show very similar excitation and scattering behavior as the quasi-elastic scattered clusters do. The deep inelastic scattered Cc0 clusters suffered immediate fragmentation into small and medium size fragments, and in some cases have formed few large cluster compounds.

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3 DIFFERENTIAL FRAGMENTATION CROSSSECTIONS OF FULLERENE COLLISIONS

Fullerene multifragmentation:

Figure 1 Double differential fragmentation cross-sections o (in (a~)~) of the simulated *C& + C,, collisions. Cross-sections are shown of the impact parameter integrated, ensemble averaged, vibrational excitation energy per atom E’ (in units of eV, r.h. scale) and coordination number n, (1.h. scale), respectively, for every fragment size

In order to monitor development of cluster structures over the entire collision event, we introduced the mean coordination number n,. The perfect fullerene C6e cluster has a coordination number n, = 3, the carbon ring and the infinite carbon chain have the coordination number n, = 2, and the carbon dimer has the coordination number IZ,= 1. Coordination numbers between these values represent non-ideal structures, and they are to be looked at individually in order to be certain about specific structures. At the end of our simulation we found elongated carbon chains (n, M 2) for fragments 2 < N d 56, and intertwined (n, M 2.5) C,, clusters (fullerenes that lost their structural integrity in favour of intertwined carbon chain structures similar to those found by Gon and Tomanek’ in a thermal bath simulation of evaporating isolated CeO clusters near T,), and fullerene structure (n, = 3) maintaining clusters of size 56 6 N < 60. Due to the large cross-section that C6e contributes to n, = 3, and better visibility of the more interesting crosssection features, the cross-section for n, = 3 is not shown in Figure 1. In this carbon cluster ensemble calculation where the total number of fragments (nr = 577) does not allow us to look at every individual fragment, the mean coordination number cross-section H, turned out to be very helpful in monitoring formation and loss of structures. As indicated in Figure 1 there was a very strong correlation between the structure of clusters and fragments, and their respective vibrational excitation. We found the fragments vibrationally excited with E* z 1 eV, and the quasi-elastic and inelastic scattered clusters with vibrational excitation 1 eV < E’ d 2.1 eV with a considerable cross-section of quasi-elastic scattered ChO clusters at E* = 2.1 eV. Notice that the collision has started with cold (E* = OeV) and

J. Schulte

relatively cold (E* = 0.5 eV) CeOclusters sharing a total of 4.4eV translational kinetic energy. Considering this we noticed that in the quasi-elastic and inelastic scattered clusters, 50% of the initially available kinetic energy was transferred into vibrational modes. Figure 2 shows the integrated translational kinetic energy per atom as a function of the fragment size for the small and medium size fragments. If we assume a center of mass thermal source of fragmentation with energy 2.1 eV, we find a kinetic energy distribution as represented in the solid curve in Figure 2. From the solid curve in Figure 2 we see that for the small fragments N < 12, the kinetic energy distribution of the fragments may be described as that of an ideal cluster gas with a thermal source of 2.1 eV. Notice that in our simulation we follow the *Cl0 + C,, collision trajectories only, i.e. there are no subsequent fragmentC6e collisions in our simulation, and no collisions of fragments of one set of *Cl, + C,, collision with fragments of another set of *C& + C’,, collision trajectories. Also, from the historical course of the simulation we found only a negligible amount of fragment-fragment within collisions individual *C& + CbOcollision trajectories. This may explain the deviation from the ideal cluster gas case of some prominent small fragment sizes and the medium size fragments. In a real world experiment where frequent subsequent collisions occur, however, we suspect it is likely to find an ideal fragment cluster gas distribution in this collision. fragmentation cross-section (mass From the distribution) in Figure 3 we see that the fragmentation process has produced sets of enhanced fragmentation cross-sections. For the small fragment sizes 2 < N d 12 we find an odd-even alteration with distinct enhanced cross-section for even numbered fragment sizes. The odd-even alteration sequence seems to change with larger fragment sizes with the N = 15 fragment crosssection as the first enhanced odd numbered fragment.

Figure 2 Impact parameter integrated, ensemble averaged, kinetic energy per atom Ek (in units of eV) as a function of the fragment size. The solid curve represents the kinetic energy distribution of an ideal cluster gas assuming the source of the fragmentation has a thermal energy of 2.1 eV

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N

Figure 3 Differential fragmentation simulated *C& + Cm collisions

cross-sections

(in (au)2) of the

Because of the very finite size of the colliding clusters (N = 60) and the limited number of calculated trajectories, we cannot guarantee the accuracy of the cross-section for the medium and large size fragments (15 < N < 60). Thus, our fragmentation cross-sections for these fragment sizes may be viewed as trends only. Because of the very finite size of the colliding clusters, we speculate that it is unlikely to find long sequences of oddeven alteration of enhanced fragmentation cross-section for the medium size clusters. A current investigation with improved statistics for the medium size fragments will shed some more light into this subject. The center of mass (corn) angular scattering crosssections of the clusters and the fragments, respectively, is shown in Figure 4. We find the quasi-elastic and inelastic scattered clusters preferably scattered at corn 0 = 20 (and correspondingly at 8 = 160) which is mostly due to the fact that we have integrated the impact cross-section over a limited maximum parameter only. Particular enhanced scattering crosssection at 0 = 80 (and somewhat less pronounced at 8 = 20, 40, 140, 160) can be found for the scattering of the carbon dimers. In our simulation we observed that the dimers are released at the very early stage of the collision shortly after the CGO clusters encountered contact. The delayed produced fragments of sizes N = 10 may be collected at their enhanced corn scattering angle 8 M 40 (0 M 140). At a somewhat larger scattering region (0 = 45) the largely populated N = 15 fragments can be observed as well. With more improved statistics and a closer analysis of the scattering cross-sections we speculate that with the shown trend of the fragment scattering (Figure 4) proceeding, each small (N < 15) cluster fragment size may be found with enhanced scattering cross-section at a respective characteristic corn scattering angle. Since we have found that the prominent small fragment sizes follow an almost ideal cluster gas kinetic distribution, the formation of the small fragment sizes were given further investigation. A power-law function

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has been fitted to the envelope of the fragmentation cross-sections of fragments with N < 28 shown in Figure 3. We found that the cross-sections of this simulation follow the power-law a(C,) 0: N-” with i M 1.54. In a recent experiment by LeBrun et al.” on the multifragmentation of Ceo clusters, a power-law fall-off in the production cross-section of small carbon fragments, a(&) 0: N-‘, has been discussed for fragment sizes N < 20. In this experiment a vapor target of Cbo clusters has been bombarded with highly ionized Xe ions with center of mass energies of some hundred MeV. In both the experiment and the accompanying fragmentation model based on a twodimensional bond-percolation calculation a scaling coefficient 1 M 1.3 has been found. LeBrun et al. compared the scaling coefficient found in the COO fragmentation process to the critical exponent r = 2 known from percolation and phase transition theory, and to the critical exponent i z 2.6 found in nuclear fragmentation experiments”. In our ab initio based simulation with a fit along the envelope of the most prominent fragmentation cross-sections we find 1 M 1.54 for fragment sizes N 6 28. It is interesting to see that the value of A is significantly lower than r for both the simple bond-percolation model calculation and the presented elaborated simulation. In the bondpercolation model calculation the authors attributed the low l-value to the finiteness and the periodic boundary conditions of the model fullerene. The C,, cluster is a cluster having atoms on the surface only. Thus we may consider the fragmentation process as a ‘surface ablation’ process. In fact in our model simulation we may speak of a one-step ablation process considering the CGoclusters do not have a filled core, and very few secondary fragment collisions have been observed in our simulation. Notice that our fragmentation scaling coefficient is close to the scaling coefficient A = 1.68 found in high

Figure 4 Double differential sections in units of (au)2

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center of mass angular

scattering

cross-

Fullerene

energy collision of basalt on basalt’*, and as well close to the scaling coefficient ,? = 1.70 found in data of the mass distribution of asteroids in the planetary systemi3. Considering the fragmentation of asteroids and basalt as a combination of a weathering and ablation process Htifner and Mukhopadhyayi4 have shown that for these fragmentation processes the value of the scaling coefficient 1 is always smaller than two. Htifner and Mukhopadhyay assumed that the colliding basalt stones crash into a fragmentation distribution D(m, A) with m being the fragment mass, A the step-number of subsequent collisions, and D(m, 0) = 6(m - M). During the crashing process a fraction q of the distribution, presumably the surface, is ablated and the remainder (1 - q) D(m, A) is again crushed into (1 - q) D(m, 2A), and the fraction q( 1 - q) D(m, A) is in turn ablated and so forth. The resulting fragmentation distribution is the sum of all ablated matter14, D(m) = qF(l

- q)“-‘D(m,nA).

(1)

tl=l

Assuming that the fragmentation distribution D(m, A) in equation (1) follows the Kolmogorov scaling theory and weathering distribution15,

multifragmentation:

J. Schulte

process according to equation (3) a hypothetical ablation ratio of q = 0.65 would result in same powerlaw exponent A = 1.7 as has been found for basalt and asteroids, respectively. In such an assumed collision of carbon matter, an average 35% of each collision remainder would serve as new large core of subsequent collisions. If we assume a pure weathering fragmentation process according to equation (2), due to the very finite size of the molecular clusters considered, only a limited number nA of subsequent collision-fragmentation processes are available, and correspondingly few overlapping distributions will contribute to the total fragmentation spectrum. Thus, the power law receives some logarithmic corrections, however, it would still be the leading term. It would be interesting to see at what size carbon clusters show a characteristic fragmentation (bulk) coefficient and whether it converges to ,J M 1.7. Also interesting would be to see from what size, at sufficient high collision energy, carbon clusters follow the percolation and phase transition theory, i.e. i 3 2.

4 CONCLUSIONS D(m, nA) =

3-u

A4 a-’

M(n - l)! C-) m

(2)

x [(Z - a)ln(X)]+‘@MHiifner and results in

Mukhopadhya

found

that

m), equation

(1)

(3) which is a pure power-law distribution, i.e. no logarithmic terms are involved. Notice that only for a subsequent one-step weathering process, i.e. no weathering and crushing, equation (2) will result in a power-law m -’ as well, with A = a - 1. In fact, if we assume that in the Htifner and Mukhopadhya weathering-ablation process (equation (3)) almost all matter follows the ablation fragmentation path, i.e., q + 1, equation (3) approaches the one-step process power law of equation (2). The parameter a depends on the fragmentation physics, and mass conservation requires it to be less than three which means 2 < 2. If in our simulation we would have found subsequent collision induced fragmentation, or if there were an additional source of collision available (such as molecules inside the CGO cage, or CN molecules from CeO fragmentation processes on collision other trajectories), i.e. q < 1, we would have found a EL-value we somewhat larger than /z M 1.54. If in our simulation assume the one-step fragmentation parameter a found in our simulation (a = 2.54) being characteristic for the fragmentation of small carbon clusters, and if we further assume that (compact as well as fullerene-like) carbon clusters follow in general a fragmentation

We have investigated the fragmentation of C6e clusters in a *C& + CbO cross-beam experiment at 500eV corn collision energy. The presented simulation shows that in quasi-elastic scattered clusters almost 50% of the collision momentum is transferred to internal vibrational energy. During the collision some fullerenes lose their structural integrity causing the formation of fullerene-like objects and intertwined carbon chains process has taken internal (n, E 2.5). This restructuring kinetic energy from the large compounds, and is part of beginning an equilibration process. During this process, small size fragments (N d 10) are released. The monitoring of the mean coordination number cross-section II, has been proved to be very useful in this simulation. The fragments are found with almost fixed internal energies of E’ = 1 eV over a wide range of fragment size (10 d N d 56). The internal energies of some small fragments (N < 10) differ in the sense that they show an increase of thermal energy with increasing cluster size converging to a thermal energy of E* = 1 eV. Following the historical course of the collision we found delayed multifragmentation of the thermally highly excited E’ = 2.1 eV (‘26eVC60) quasi-elastic and inelastic scattered C,, clusters. In our simulation we find some of the small fragments preferably scattered at specific scattering angles, and it is speculated that with improved simulation statistics it may be expected that almost all of the small fragments will show preferred scattering angles. A lit along the envelope of the prominent small fragment sizes N < 28 shows the scaling behavior G(C,) 0: z@ with 1 = 1.54. The scaling behavior has been discussed in terms of weathering and ablation

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fragmentation processes, and we have speculated that with more compact and larger carbon clusters the scaling coefficient may approach the one (1 M 1.7) found for bulk objects like basalt stones or asteroids. Further investigation is required to determine how large the collision energy and the colliding clusters have to be in order to show a liquid-gas phase transition scaling with 1 M r > 2. Finally, this simulation has shown that the MDSTE approach along with appropriate interaction potentials is a useful tool to shed more insight into general questions of collision and fragmentation processes.

REFERENCES 1 2 3 4 5 6 I

8 9 10

ACKNOWLEDGEMENTS

11

The author is grateful to the Computer Science Departments at Texas A & M University and Michigan State University, the Parallel Computing Divisions of the San Diego Supercomputer Center, and the Computer Science Division at the Sandia National Laboratory for providing computational grants on their local parallel computers.

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Schulte, J. J. Appl. Physics 1994, 75(1 I), 7195 Campbell, E. E. B., Schyja, V., Ehlich, R. and Hertel, I. V. Phys. Rev. Lett. 1993, 70, 263 Schulte, J. Phys. Rev. 1994, B50(17), 12312 Seifert, G. and Jones, R. 0. Z. Phys. 1991, DZO, 77 and references therein Schulte, J. and Seifert, G. Chem. Phys. Lett. 1994, 221, 230 Seifert, G. and Schulte, J. Phys. Lett. 1994, Al& 365 Strout, D. L., Murry, R. L., Wu, C., Eckhoff, W. C., Odom, G. K. and Scuseria, G. E. Chem. Phys. Lett. 1993, 214(6), 576; Ballone, P. and Milani, P. Z. Phys. 1991, D19,439 Seifert, G. and Jones, R. 0. Z. Phys. 1991, D20,77 Gon, S. G. and Tomanek, D. Phys. Rev. Lett. 1994, 72(15), 2418 LeBrun, T., Berry, H. G., Dunford, R. W., Esbensen, H., Gernmell, D. S., Kanter, E. P. and Bauer, W. Phys. Rev. Lett. 1994,72,3965 Hirsch, A. et al. Phys. Rev. 1984, C29, 508; Phair, L., Bauer, W. and Gelbke, C. K. Phys. Rev. Lett. 1993, B314,271 Moore, H. J. and Gault, D. E. ‘Ann. Prog. Rep’, (Astrogeological Studies), US Geolog. Survey, Part B, 1965, p. 127 Dohnanyi, J. S. J. Geophys. Res. 1969, 74, 2531; Dohananyi, J. S. J. Geophys. Res. 1970, 75, 3468 Hiifner, J. and Mukhopadhyay, D. Phys. Lett. 1986, B173(4), 373 Kolmogorov, A. N. and Dokl, C. R. Akad. Nuuk. 1949, SSSR XXXI, 99

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