One of the most puzzling aspects of fullerenes is how such complicated symmetric molecules are formed from a gas of atomic carbons, namely, the atomistic or chemical mechanisms. Are the atoms added one by one or as molecules (C_{2}, C_{3})? Is there a critical nucleus beyond which formation proceeds at gas kinetic rates? What determines the balance between forming buckyballs, buckytubes, graphite and soot? The answer to these questions is extremely important in manipulating the systems to achieve particular products. A difficulty in current experiments is that the products can only be detected on time scales of microseconds, long after many of the important formation steps have been completed. Consequently, it is necessary to use simulations, quantum mechanics and molecular dynamics, to determine these initial states. Experiments serve to provide the boundary conditions that severely limit the possibilities. Using quantum mechanical methods (DFT) we derived a force field (MSXX FF) to describe one-dimensional (rings) and two-dimensional (fullerene) carbon molecules. Combining DFT with the MSXX FF, we calculated the energetics for the ring fusion spiral zipper (RFSZ) mechanism for formation of C_{60} fullerenes. Our results shows that the RFSZ mechanism is consistent with the quantum mechanics (with a slight modification for some of the intermediates).
Introduction
One of the most puzzling aspects of fullerenes (C_{60}, C_{70}, etc.) is how such complicated symmetric molecules are formed from a gas of atomic carbons [atomcarbon], namely, the atomistic or chemical mechanism. Are the atoms added one by one or as molecules (C_{2}, C_{3})? Is the C_{60} fullerene formed by adding C_{1}, C_{2}, or C_{3} to some smaller fullerene or is C_{60} formed by isomerization of some type of precursor molecule C_{60}? Is there a critical nucleus beyond which formation proceeds at gas kinetic reates? What determines the balance bwtween forming buckyballs, buckytubes, graphite, and soot? The answer to these questions might lead to means of manipulating the systems to achieve particular products.
A difficulty in current experiments [Jarrold 1993, Jarrold 1994, Bowers 1993a, 1993b] is that the products can only be detected on times scales of µs, long after many of the important formation steps have been completed. Consequently, it is necessary to use first principles quantum mechanical theory to determine these initial states; however, the experiments serve to provide boundary condition that severely limint the possibilities, making the use of first principles theory practicable.
Calculations
We selected density functinal theory (DFT) as the best compromise between accuracy and speed for studying these systems. We use the Becke gradient corrected exchange and the gradient corrected correlation functional of Lee, Yang, and Parr. [Johnson 1993] The calculations were carried out using PS-GVB with the 6-31G* basis set [Rignalda 1995].
Because the carbon rings play a central role, we studied how the structures and energetics of such rings changed with size and extracted a force field (denoted as the MSX FF) that would reproduce the DFT energetics and structures. This MSX FF would be used later in conjunction with the DFT calculations on various multiring systems to estimate the energetics of the full 60 atom systems without the necessity of DFT on the complete system.
The calculations on ring systems up to C_{60} are shown in Figure 1. The energies quoted here are cohesive energy par carbon atom. In calculating these energies we used as our reference the triplet C atom, calculated by LSDA.
Figure 1. Cohesive energy per carbon atom.
We found that
For n=4m, the minimum energy structure has a polyacytelene geometry of alternating single and triple bonds. The bond length difference is from 0.5 Å to 0.9 Å. We find that inclusion of correlation reduces the dimerization amplitude, similar to the case in polyacetylene [Konig 1990]. Comparing to the DFT geometry, Hartree-Fock (HF) gives too large of a bond alternation, along with too large of angle alternations. Our HF calculation gives a bond difference of 0.16 Å, in agreement with that of Feyereisen et al. Feyereisen 1992]. As for angle alternation, for C_{20}, HF gives 160°-164° [Raghavachari 1993] while DFT gives 161.5°-162.5°.
For n=4m+2, the minimum energy structure has the polyallene geometry with equal bond lengths. This is due to the resonance between the two structures, involving the pi-bond perpendicular to the plane and pi-bond parallel to the plane.
C_{4m+2} is more stable than C_{4m}. But as n --> infinity, the difference in E_{coh} decreases to zero, leading to E_{infinity}^{sp1}=6.56eV.
Both the polyacetylene and polyallene structures involve sigma-bonds that are sp^{1} hybrids, which prefer linear geometries. Thus we expect a strain energy proportional to
Indeed we found strain energy increase linearly with 1/n^{2} with slopes of 63.3 eV/n^{2} for 4m and 40.1 eV/n^{2} for the 4m+2, respectively. Both converge to E_{infinity}^{sp1}=6.56eV
The force field took the following form:
Here q_{r1}(l)=R_{i}(l)-R_{i0}(l) is the bond strain term, where for n=4m, i=1 is the triple bond and i=2 is the single bonds; for n=4m+2 their bonds are equivalent. The angle strain term is q_{theta}(l)=theta(l)-theta_{0}(l) We use the periodic boundary condition so that, with theta_{0} = pi, n/2+1=1, where n is the total number of atoms in the system and n/2 is the number of unit cells. E_{0} is a reference energy corresponding to zero strain energy structure. (infinite linear chain) Comparing E_{MSX} with E_{DFT} for several structures, we can derive the force field parameters.
In a similar fasion we can derive the force field for the sp^{2} bonded carbons. The optimum structure for bulk carbon is graphite, which has each carbon bonded to three others (sp^{2} bonding) to form hexagonal sheets stacked on each other. The fullerenes structures can be considered as finite two dimensional anologs, in which each carbon is distorted (strained) from its preferred planar configuration. Since the strain should be proportinal to the square of the planar distortion angle, delta psi, we expect that the strain energy should scale as 1/n.
We have performed the DFT(Becke/LYP) calculations on C_{n} fullerenes with n=20, 32 and 60. Figure 1 shows the cohesive energies per carbon atom.
Extrapolating the calculated cohesive energy to n ---> infinity leads to a cohesive energy per sp^{2} carbon of E_{coh}^{sp2} = 7.71 eV. This can be compared to the experimental cohesive energy of a single graphitic sheet of E^{sheet}_{coh}=7.74 eV. This is derived from the experimental cohesive energy [CRC Handbook] of graphite of E^{graphite}=7.8 eV plus total Van der Waals attraction of E^{vdw} = 0.056 eV between sheets calculated using the graphite force field. [Guo 1992]
Now that we have the energy and force field of both sp^{1} and sp^{2} hybridized carbon we can get the energetics of any carbon clusters. Adding the entropic contribution within harmonic approximation using FF, we get the free energy of various species at different temperature, which dictate the thermal equilibrium distribution of these species. Our population analysis is shown in Figure 2.
Figure 2. Population analysis of species.
For studying formation reaction sequence we adopt two level of models, a fine one and a coarse one, as explained below.
(A) Fine model:
The energies of a structure were computed via the following procedures
that combine the DFT with MD as illustrated in Figure 3.
Figure 3. Paths.
The reaction from two C_{30} ring (I) to C_{60} bicyclic ring molecules (II) is achieved via an intermediate III. For each of I, II and III, the system is partitioned into two parts: part A involving bond lengths changes, and part B involving continium deformation.
The energy of I is determined directly from Figure 1.
The energy difference between I and III is a strain energy which can be calculated using the MSX FF.
The energy difference between III and II is calculated in two parts. (a) part A: DFT calculations are carried out on the reaction of two C_{6}H_{2} molecules to form to the 4-membered ring, C_{12}H_{2}. (b) part B: we calculate the corresponding change in going from III to II using MSX FF. Then we combine A and B to get the energy difference between III and II.
Thus the energy of C_{60} bicyclic ring (II) can be calculated by II--III--I.
(B) Coarse model:
We extend the MSX FF to include terms capable of describing the different bonding schemes. The key components are the additive energy terms for the dangling bond and the energy cost for bending a triple bond to form a 1,2-benzyne. Our FF are defined as follows:
We have chosen E_{0} = 60 epsilon_{1}, as zero point. Here, n_{2} is the sp^{2} bonded carbons, n_{2} (epsilon_{1} - epsilon_{2}) gives the energy gained by converting sp^{1} bonded carbon into spn^{2} bonded carbon, with epsilon_{1} = -6.56 eV and epsilon_{2} = -7.71 eV. d_{1} is the energy of a dangling bond relative to the sigma-bonded state, n_{R} number of of such dangling bond(radicals); d_{2} is the energy of an atom participating bended planar pi-bond relative to the sigma-bonded state and n_{sigma pi} is the number of such atoms. We use the Benson-like scheme to evaluate d_{1} and d_{2} [Guo 1992] and found d_{1} = 2.32 eV and d_{2} = 1.64 eV. E^{str} (n_{2}) is the strain energy and it is evaluated at the minimum energy structure.
We would use the fine model for the initial steps in the C_{60} formation. As the reaction take off and begin to release more and more energy, we switch to the coase one.
The spiral model of fullerene formation
At the beginning atomic carbons combine themselves to form dimers and trimer, C_{2}, C_{3}. These would then grow into linear chain of carbons C_{n}, etc., for n<10. [Hutter 1994] When n>10 the carbon cluster prefer ring structure [Hutter 1994] because beyond n>10 the energy gain in killing the dangling bonds at the two ends over compensate for the strain energy incured by folding up the chain. At around n>30 the ring structures give way to fullerene structures,[Bowers 1993a, 1993b] because replacing more pi-bond by sigma-bond over compensate for the strain of folding the 2-D net.
One process of C_{60} formation, as suggested by Jarrolds experiments, [Bowers 1993a, 1993b, Jarrold 1994] is to combine two C_{30} rings to form a bycyclic C_{60} ring, which in turn isomerized into a C_{60} fullerene. This unimolecular reaction will be the focus of our study.
As a mnemonic for referring to the various structures, we will simply denote the ring sizes of a structure. Thus the simple C_{60} ring is denoted as 60, while the double ring system, 1, is 30+4+30. This notation does not uniquely describe a structure, but it is for the species we will consider. We will take the reference energy to be E_{o}=60 epsilon_{1}, where epsilon_{1}=-6.56 eV.
Following Jarrold, the first few steps in the reaction are as follows: (see Figure 4)
Figure 4. First few steps of reactions.
1'={30+4+30} ---> 2={30+4+6+30}. This is a Bergman diyne cyclization which forms a 6-membered 1,4 benzyne-like ring from two triple bonds. This leads to two isolated radical sites (sp^{2}-like orbtials in the plane, that cannot form a bond), and we find that this increases the energy by about 0.7 eV.
2={30+4+6+30} ---> 3={30+8+30}. This process kills two dangling bond by breaking one sigma bond and forming two pi bonds. This process is down hill by about 1.3 eV.
3={30+8+30} ---> 4={30+8+6+22}. This involves breaking an in-plane pi bond and forming a sigma bond. In the process there is bending of one triple bond to from a 1,2-benzyne-like ring which includes a new radical site. This process is uphill by 1.66eV. Jarrold postulated 4'which is 2.1 eV above the bicyclic rings from our calculation.
4={30+8+6+22} ---> 5={6+6+55}. This involves twisting open the original 4-membered ring. Then it is followed by relaxing the 50 carbon chain to reduce the strain energy. This {6+6+55} contains two dangling bonds. This process is downhill by about 0.67 eV.
Spiral growth around the {6+6}. As a first step 5={6+6+55} ---> 6={6+6+53+5}. This uses one of the sp^{2} orbitals of the 1,2-benzyne-like ring to attack a triple bond and form a new 5-membered ring. This process is down hill by 0.13 eV.
Continue the spiral growth to form C_{60} fullerene. The energies calculated using the extended MSX FF on these systems are shown in Figure 5 where we see that they are monotonically downhill. The overall gain of energy from {6+6+53+5} to C_{60} is about 30 eV, so that no barriers are expected to impede these steps. Figure 6 illustrates some of the intermediates between 5 and the fullerene.
Figure 5. Energies calculated by MSXX FF.
The driving force for the growth is the gain in forming sp^{2} sigma bond. The opposing forces are the energy lost by the radicals created along the way and the increasing strain energies.
The Jarrold mechanism represents an innovative major step forward in understanding the formation of C_{60} fullerene. Our energetic analysis shows that some of the reactions pathways have large energy barriers, however they never exceed the energy available to the unimolecular reaction.
The similar approach could be used to study the formation of other fullerenes, like C_{40}, C_{50}, C_{70}, etc..
Figure 6. Intermediates between the 5 and the fullerene.
Summary
Why C_{60} is so stable and how C_{60} fullerenes are formed, these are the two most interesting problems in basic fullerene research. We have studied the formation mechanism of C_{60} fullerenes using first principles calculation and molecular dynamics simulations. We have derived a force field (MSX FF) that is suitable to describe both the sp^{1} hybrid and sp^{2} hybrid carbons. Combining DFT and MD with MSX FF we found the relative thermal stability of various neutral isomers at each cluster size n and predicted the relative abundancy of these neutral species for thermal equilibrium. We identified a complete path to form a C_{60} fullerene from atomic carbons and calculated its energetics. Our approach is fully applicable to other possible reaction paths and other fullerenes.
References
Joanna M Hunter, James L Fye, Eric j. Roskamp, and Martin F.Jarrold J. Phys. Chem.98, 1810-1818 (1994)
Joanna M Hunter, James L Fye, and Martin F.Jarrold J. Chem. Phys.99, 1785-1795 (1993)
Gert Von Helden, Nigel G. Gotts and Michael T. Bowers, Nature363, 60 (1993); Gert von Helden, Ming-Teh Hsu, Nigel Gotts, and Michael T. Bowers, J. Phys. Chem97 8182-8192, (1993)
Benny G. Johnson, Peter M. BW. Gill, and John A. Pople, J. Chem. Phys.98 5612 (1993).
Murco N. Ringnalda, Jean-Marc Langlois, Burnham H. Greeley, Robert B Murphy, Thomas V. Russo, Christian Cortis, Richard P. Muller, Bryan Marten, Robert E. Donnelly, Jr., Daniel T. Mainz, Julie R. Wright, W. Thomas Pollard, Yixiang Cao, Youngdo Won, Gregory H. Miller, William A. Goddard III, and Richard A. Friesner, PS-GVB v2.2, Schrodinger, Inc., (1995)
G. Konig, G. Stollhoff Phys. Rev. Lett.65, 1239 (1990)
Martin Feyereisen, Maciej Gutowski, and Jack Simons, and Jan Alm, J. Chem. Phys.96, 2926 (1992)
K. Raghavachari el al., Chem. Phys. Lett.214, 357 (1993).
CRC Handbook
Yuejing Guo, Ph.D. Thesis, California Institute of Technology. (1992)
Jurg Hutter, Hans Peter Luthi, and Francois Diederich J. Am. Chem. Soc.116 750-756 (1994)
S. C. O'Brien, J. R. Heath, H. W. Kroto, R. F. Curl, and R. E. Smalley, Chem. Phys. Lett.132 99-102 (1986)