Molecular Simulations of Carbon Nanotubes

Simulation of Single-walled Carbon Nanotube

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Simulation of Single-walled Carbon Nanotube
 
 
 
 
 

Graph of energy versus time for simulation with varying initial momentum

Abstract

The objective of this project was to simulate carbon nanotubes using cluster potentials involving a spring-like potential for two-body interactions and a harmonic oscillator term for three-body interactions. The cylindrical hexagonal structure was initialized with the initial conditions.During initialization, particle interactions were tracked using both nearest neighbor and bond matrices.These matrices facilitated force calculations. The code was written such that the length of the nanotube, the number of carbons per ring, the magnitude of the initial momentum per particle, and the simulation time were entered by the used.The simulation was visualized by outputting the position of the carbon particles to the Chime software; the potential, kinetic, and total energy of the system were also monitored.A variety of initial conditions were tested, and the nanotube configuration was stable for all test cases. This code will serve as the starting point for future simulations involving carbon nanotubes.

Introduction

Carbon nanotubes are a relatively new material with a wide range of potential applications, varying from one-dimensional quantum wires to micro-cylinders for gas storage.Due to their unusual configuration, the properties of the nanotubes have been theoretically calculated but not widely studied.In addition, the effects of the conformational strain introduced by the curvature of the nanotube are not widely understood.Interactions between carbon nanotubes to form bundles and ropes have been observed, but the factors influencing inter-nanotube interactions have not been studied.Preliminary work in molecular simulations of carbon nanotubes have included both molecular dynamic studies and grand canonical Monte Carlo simulations.The Monte Carlo simulations have been used to a large extent to study gas adsorption onto the carbon surface.Molecular dynamic studies have been used to investigate the mechanism for formation of carbon nanotubes in laser ablation using a density functional approach (Charlier, et al., Science, 275, 1997).Terrones, et al., used tight-binding molecular dynamics and Monte Carlo simulations to study the coalescence of single-walled carbon nanoubes by introducing defects along the nanotube that led to coalescence of adjacent nanotubes (Science 288, 2000).Mao and Sinnot studied the molecular diffusion of hydrocarbons through the nanotubes using MD simulations(J. Phys. Chem. B. 104, 2000).

Molecular simulations of nanotubes remain largely unexplored and could be used to optimize formation conditions and properties.Simulations that would be applicable to my research project include:(1) surface diffusion studies (along both the interior and exterior surfaces); (2) the interactions and dynamics in multi-walled carbon nanotubes; (3) expanding the work of Charlier et al., to study the mechanism of nanotube formation in a catalytic system; and (4) studying inter-nanotube interactions that lead to bundle formation and introducing intercalating metals to vary the spacing.

Simulation

The code currently is a constant energy simulation; however, future versions may include the addition of a thermostat and/or barostat to simulation constant temperature and pressure conditions. 

The initial conditions of the system are designed to be a cylindrical structure in which the particles are bonded in a hexagonal matrix along the surface of the cylinder.This was intended to simulate a single-wall carbon nanotube in which the sp2-hybridized carbon atoms are in a hexagonal matrix.The user inputs both the length of the nanotube and the number of atoms per ring; the initial carbon-carbon spacing was held fixed for the simulations.This information was then used to calculate the diameter of the ring and the position of the particles.Along a ring, the carbon spacing was varied from one to two arclengths to result in the hexagonal structure, up to the specified amount per ring.For the next ring, the carbon atoms were offset to again account for the hexagonal structure.As the positions of the carbon atoms were determined, both a nearest neighbor matrix and bond matrix were set up to facilitate force calculations.At this time, periodic boundary conditions have not been used, but the code could be easily modified by including these in the nearest neighbor and bond matrices. 

The molecular bonds in the carbon nanotubes were treated as semi-rigid with the ability to fluctuation around their equilibrium position.Due to the geometry of the graphitic carbon present in the nanotube, the carbons have an equilibrium bond distance and an equilibrium bond angle.Therefore, the potential was separated into two- and three- body terms.In this way, the carbon atoms will be allowed to fluctuate around their equilibrium distance and bond angle.Both the two- and three- body interactions were treated as harmonic oscillators: 

The two-body forces were calculated using the bond matrix; whereas the three-body forces were calculated using the nearest neighbor matrix. The hexagonal structure of the graphitic carbon, which results in three nearest neighbors for each carbon atom, simplified the calculation of the three-body term; once two atoms were specified there was only one possible third particle.Thus, the three-body terms were calculated by cycling through the nearest neighbor matrix.The values of the spring constants k₂ and k₃, were set equal to 1 and 0.3, respectively.The equilibrium positions, r_o and q₀, were set equal to 1 and 2.094, respectively.

Data Analysis

The main objective of this project was to setup the nanotube lattice and ensure that it remains intact during the simulation.Therefore, the particle positions were written to a .xyz file in order to observe the position of the carbon atoms.This file is shown to the left.Secondly, the potential, kinetic, and total energies of the particles were monitored with simulation time.This graph is given to the left for several cases of the initial conditions.

References

J.-C. Charlier, A. DeVita, X.Blasé, R. Car, “Microscopic Growth Mechanisms for Carbon Nanotubes,” Science, 275, 1997.

Z. Mao and S.B. Sinnot,“A Computational Study of Molecular Diffusion and Dynamic flow through Carbon Nanotubes,” J. Phys. Chem. B. 104, 2000. 

M. Terrones, H. Terrones, F. banhart, J.-C. Charlier, P.M. Ajayan, “Coalescence of Single-Walled Carbon Nanotubes,” Science 288, 2000.