Report on Atomistic Modeling of Bonding in Carbon-Based Nanostructures

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1 Report on Atomistic Modeling of Bonding in Carbon-Based Nanostructures Timothy Stillings Department of Physics, Astronomy and Materials Science Missouri State University Advisor: Ridwan Sakidja Abstract In this study, modern DFT calculations are compared with classical molecular dynamics calculations using Tersoff and AIREBO potentials to assess their accuracy as a means to model carbon structures, specifically C70. Further assessment models interactions between two C70 molecules. It was found that the DFT calculations provided the most accurate results, but calculations done with AIREBO potentials correlate well with the DFT calculations. Tersoff potentials, on the other hand did not perform well and overestimated the bond lengths. An observation of the charge redistribution as a result of inter-molecule interaction between C70 s was made revealing an interaction which may be originated from the formation of pi bond. Electronic Structure Calculations To calculate the bonding interaction, the team made use of VASP 6 (Vienna Ab-Initio Simulation Package) where the electron density of the Carbon atoms is calculated in accordance to the Kohn-Sham equation of total energy (Hamiltonian) of the electrons of the Carbon atoms: where ε i is the orbital energy of the corresponding Kohn Sham orbital, φ i, and the density for an N-particle system is At the ground state (OK), the electronic structure calculations make it possible to find the most relaxed atomic positions which would corresponds to the lowest level of total energy and therefore the most stable configuration. Once these atomic positions are found, comparisons can then be made in the bonding characteristics between the individual C atoms that make up the nanostructure of the C70 molecule. Density Functional Theory (DFT) calculations, such as those done by VASP, have been known to not sufficiently account for long-range dispersion interactions. In VASP code, however, there is an additional van der Waals correction 1 that can be included to remedy this problem. Such corrections were included as part of the calculation in an effort to further verify the result. The results of VASP calculations should then be compared to the results from classical MD simulations. The comparison will examine the accuracy of the interatomic potentials being used in the classical simulations, especially as they pertain to the interatomic distances. In particular,

2 the well-known and published interatomic-potentials of Tersoff 2 and AIREBO implemented in LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) 3. As a means to quantify the variation in the bond lengths within or between nanostructures, radial distribution function (RDF) was used. RDF is the ratio between the discretized densities of the atoms versus a uniform density radially, and describes how charge density varies as a function of distance from a reference C atom. This was used primarily to evaluate the shortest interatomic distance (C-C bond length) which is the range of 1.4 to 1.5 Angstroms. The atomic trajectory data collected after the energy minimization process is then used to construct the RDF plots. All calculations were performed in the STAMPEDE Supercomputer Cluster at Texas Advanced Computer Center (TACC) in Austin, TX. LAMMPS Potentials Detailed implementations of the potentials are explained in LAMMPS manual, the following is a summary. The Tersoff style computes a 3-body Tersoff potential with a close-separation pairwise modification based on a Coulomb potential and the Ziegler-Biersack-Littmark universal screening function (ZBL), giving the energy E of a system of atoms as:

3 The f_f term is a fermi-like function used to help smoothly connect the ZBL repulsive potential with the Tersoff potential. There are 2 parameters used to adjust it: A_F and r_c. A_F controls how sharp the transition is between the two, and r_c is the cutoff for the ZBL potential. The AIREBO pair style computes the Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO) Potential of Stuart 4 for a system of carbon and/or hydrogen atoms The AIREBO potential consists of three terms: The minimization procedure in LAMMPS By iteratively adjusting atom coordinates, LAAMPS can perform an energy minimization of the system, thereby finding the configuration with the lowest potential energy: where the first term is the sum of all non-bonded pairwise interactions including long-range Coulombic interactions, the 2nd thru 5th terms are bond, angle, dihedral, and improper interactions respectively, and the last term is energy due to fixes which can act as constraints or apply force to atoms, such as thru interaction with a wall. Iterations are terminated when the energy stopping criteria is satisfied. Results For C70 there are multitude of bond lengths that are experimentally found, ranging from Angstrom to Angstroms [R7]. Fig 1b shows that there is quite of variation with respect to distances obtained from different types of simulations. The C-C bonds obtained by Tersoff clearly overestimated the lengths. AIREBO on the other hand seems to produce a much better match, although it does only gives two large peaks. DFT TERSOFF AIREBO

4 (a) (b) Figure 1: a) Atomic model of the C70 molecule. b) Comparison of RDF s from relaxed atomic positions of isolated C70 molecule using classical MD code (LAMMPS) with 1) Tersoff Potential and 2) AIREBO Potential versus 3) generated from the electronic structure calculations (VASP). Note that AIREBO results more closely match that from electronic structure calculations. The DFT calculations appear to produce the best results, with three major peaks and one minor peak. More accurate calculations typically done by quantum chemistry simulations identified more minor peaks 7, but overall the DFT and to some extent, a classical MD approach using AIREBO potential provide a good estimate. It should be noted that these distances may influence the reactivity of these molecules and thus it is particularly significant. Especially when modeling the physical properties of C-based composite where the filler materials may be made of the C- based nanostructures. From the results, it is quite obvious that using Tersoff provides unsatisfactory results, and would not be recommended in the modeling of assemblies of these type of materials. DTF calculations were ran to compare the RDF of a C70 molecule with and without accounting for the effect of the long-range van der Waals interaction. From Figure 2 it should be noted that no significant differences were apparent. This is potentially a result of the fact that the interactions taking place were primarily due to the sigma bonding of C-C within a C70 molecule, which is less influenced by a relatively weaker van der Waals bond. vdw No vdw Figure 2: Comparison of RDF for C70 molecule with or without van-der Waals correction in the DFT calculations. As a further calculation, the bonding mechanism between two C70 molecules was assessed. For this, the RDF was used as a means to detect deviation with respect to the bond lengths. In the absence of inter-molecular bonding, there we be no variation to detect. Figure 3a provides an example in which two C70 molecules are separated past the reach of the C-C interaction. Figure 3b shows no change in the RDF from one C70 molecule to that of two C70 molecules, aside from the intensity.

5 (a) Figure 3 a) An example of relaxed positions of a pair of non-interacting C70 s B) The RDF from the two C70 s showing peaks essentially matched that of one C70 s (with 2x area). Figure 4 shows the input for the LAMMPS calculation modeling an example of a pair of sidebonded C70 molecules. The model shown in Figure 5a, the output of the energy minimization from LAMMPS, confirms that a number of C atoms engaged inter-molecular bonding between the C70 molecules. This results in a stretching of the affected bond lengths. This can also be seen in the RDF plot in Figure 5b where distinct extra small peaks, which are noted with arrows, from bonding between the two C70 molecules are present and correspond to the deformed C-C bonds near the bonded region of bond lengths of ~1.5 Angstroms. (b) Figure 4 Two C70 side-bonded from MD simulations (with the AIREBO potential) which is also accompanied by deformed C-C bonds within each C70 s at the vicinity of the bonded side regions.

6 Figure 5: a) An illustration showing the formation of a new tetrahedral C cluster connecting the sidebonded two C70 from one pentagon from one C70 molecule (circled) to the hexagon on the other C70 molecule, b) Comparison of RDF from relaxed C70 & 2 side-bonded C70 s. Figure 6 shows the results from LAMMPS calculations for the edge-bonded C70 pairs showing the formation of inter-molecular bonds that forms a number of tetrahedral C clusters. It is possible that the edge part of C70 is more reactive than the side part of C70 because the side part has a flatter side due to the hexagonal belt formation. A further examination is clearly warranted. Such a preference may affect the physical properties of assemblies of these nanostructures. (a) (b) Figure 6: A formation of two tetrahedral C clusters connecting near-edge-bonded C70 pairs b) a closer look at the RDF (boxed in a) corresponding to the deformed area. To further analyze the bonding mechanism, the charge density of the relaxed C70, as shown in Figure 7a, was plotted. Here, based on the contour plots of the charge density, a strong directional correlation between C atoms is present, presumably representing the sigma bonds. Note these calculations result from the charge density and do not account for the molecular orbitals. In addition, further examination charge density of the edge-bonded C70 pair as shown in Figure 7b. The map, using the same contour levels, clearly shows the presence of electron

7 density accumulation between the two molecules. It is clear, however, the density is much weaker, which can be understood from the fact that the bond length now is stretched. It is possible that the more diffuse charge density accumulation can be attributed to the pi bond. Figure 7: a) A contour plot of the charge density for the relaxed Carbon atoms in C70 molecules indicating the presence of both sigma and pi C-C bonds b) the same type of plot on the near edge-bonded C70 pairs showing a weaker and diffuse bonding, presumably pi-type connecting the two molecules. (Iso surface levels: 0.1, 0.15 and 0.25 e - /Å 3 ) Summary 1. DFT calculations of the relaxed C70 molecule provided the most accurate representation of the bonding and bond lengths within the molecule. 2. The classical MD relaxation using the AIREBO potential seems to correlate well with the results from DFT calculations. On the other hands, less accurate Tersoff potential calculations overestimate the bond lengths and may be questionable to model chemical reaction or reactivity for C70 molecules. 3. An observation of the charge redistribution as a result of inter-molecule interaction between C70 s was made revealing an interaction which may be originated from the formation of pi bond. Biography Timothy Stillings is a senior from Missouri State University where he is seeking a degree in Physics from the Department of Physics, Astronomy and Materials Science. Tim is originally from Ava Missouri but currently resides in Springfield. After graduation, Tim plans on seeking a Master s degree from Missouri State University Acknowledgments Tim would like to thank the Department of Physics, Astronomy, and Materials Science at Missouri State University, Dr. Ridwan Sakidja, and the Missouri Space Grant Consortium for funding this project.

8 References [R1] M. Dion, H. Rydberg, E. Schröder, D. C. Langreth, and B. I. Lundqvist, Phys. Rev. Lett. 92, (2004). [R2] J. Tersoff, Phys Rev B, 37, 6991 (1988). [R3] [R4] Stuart, Tutein, Harrison, J Chem Phys, 112, (2000) [R5] Brenner, Shenderova, Harrison, Stuart, Ni, Sinnott, J Physics: Condensed Matter, 14, (2002). [R6] [R7] Alexander V. Nikolaev, T.John S. Dennis, Kosmas Prassides, Alan K. Soper, Chemical Physics Letters, Volume 223, Issue 3, 17 June 1994, Pages