Keywords: carbon nanotubes, cellular automata, nanoscale materials.

Properties of the carbon nanotube structure have been extensively examined in several analytical and numerical studies. The single-wall carbon nanotube is described as a rolled-up sheet of carbon atoms that are arranged in a hexagonal ring. The tubes are capped by a hemisphere section drawn from a spherical (fullerene) arrangement of carbon atoms. The presence of carbon atoms in a planar 2-D arrangement permits large out-of-plane distortions while retaining extremely high in-plane stiffness against deformation. The multi-wall nanotube can be thought of as a number of concentric single-wall nanotubes, obtained by the roll-up of a stack of sheets consisting of hexagonal rings of carbon atoms. These have similar structural properties as the SWNT. Depending upon the orientation of carbon atoms with respect to the tube axis, SWNT can be either conductors or semi-conductors. Additionally, the hollow structure and the extremely lightweight large surface area prove to be highly beneficial in bio-chemical applications.

Exploratory efforts designed to develop nanoscale devices that take advantage of these diverse properties are being currently pursued. However, in order to take full advantage of the unique properties of nanomaterials, it is important to gain a more complete understanding of methods that can accurately characterize these materials. Simulation based tools have been pursued by many researchers in this context, and their successful implementation will play a pivotal role in the design and development of nanomaterials and nanoscale devices.

Among the most widely pursued methods to simulate the behavior of nanomaterials is the atomistic molecular dynamics (MD) approach. This approach represents the dynamics of atoms or molecules of the materials by a discrete solution of Newton's classical equations of motion. The interatomic forces required for the equations of motions are obtained on the basis of interaction energy functions. For carbon-carbon interactions, multibody force field functions such as the Tersoff-Brenner [1] have been widely adopted. Similar functions for C-H interactions have also been established [2]. The Tersoff-Brenner potential allows for bonded reactions to be reactive, accommodating formation and breakage of bonds. MD simulation codes have been parallelized to allow for a very large number of atoms (105-108) to be considered in the computations.

The present paper describes a cellular automata based numerical model [3] to study the behavior of a single-walled nanotube and to compute intrinsic properties such as an equivalent Young's modulus. This model of computation is ideally tailored for implementation on parallel machines. The approach assumes that a computational domain can be subdivided into a number of discrete cells, with state variables associated with each cell. Collectively, these cell states define the state of the entire domain, and which may be evolved through application of local rules of interaction that apply to a defined neighborhood around each cell. These local rules of interaction are based on first principles, and are repeatable at every site in the solution domain. The key issue here is, therefore, to obtain the appropriate rule of interaction for modeling the behavior of SWNT's.

In the present paper, each site in the cellular model was assumed to be occupied by a carbon atom, with its position representing the state variable for the site. The interaction rules were based on a minimization of the bonding energy between an atom and its three neighboring atoms, and from which a new position can be derived. The bonding energy was calculated using the empirical formulation suggested by Tersoff; a quasi-second order numerical approach was used to find a solution to the energy minimization problem. The Young's modulus obtained from the proposed approach compares well against published results and those obtained from a molecular dynamics simulation. The approach promises to be effective in computing other properties of such nanostructures, and can be extended to multi-wall configurations.

1

Brenner, D.W., Shenenderova, O.A., Areshkin, D.A., "Reviews in Computational Chemistry", K.B. Lipkowitz and D.B. Boyd (eds), 213, VCH Publishers, New York, 1998.

2

Garison, B.J., Srivastava, D., "Potential Energy Surfaces for Chemical Reactions at Solid Surfaces", Ann. Rev. Phys. Chem, 46, 373-394, 1995. doi:10.1146/annurev.pc.46.100195.002105

3

Wolfram, S., "Cellular Automata and Complexity", Addison-Wesley Publishing Company, 1994.

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