A structural mechanics approach for the analysis of carbon nanotubes

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Abstract

This paper presents a structural mechanics approach to modeling the deformation of carbon nanotubes. Fundamental to the proposed concept is the notion that a carbon nanotube is a geometrical frame-like structure and the primary bonds between two nearest-neighboring atoms act like load-bearing beam members, whereas an individual atom acts as the joint of the related load-bearing beam members. By establishing a linkage between structural mechanics and molecular mechanics, the sectional property parameters of these beam members are obtained. The accuracy and stability of the present method is verified by its application to graphite. Computations of the elastic deformation of single-walled carbon nanotubes reveal that the Young’s moduli of carbon nanotubes vary with the tube diameter and are affected by their helicity. With increasing tube diameter, the Young’s moduli of both armchair and zigzag carbon nanotubes increase monotonically and approach the Young’s modulus of graphite. These findings are in good agreement with the existing theoretical and experimental results.

Introduction

The advancement of science and technology has evolved into the era of nanotechnology. The most distinct characteristic of nanotechnology is that the properties of nanomaterials are size-dependent. Due to the extremely small size of nanomaterials, the evaluation of their mechanical properties, such as elastic modulus, tensile/compressive strength and buckling resistance, presents significant challenges to researchers in nanomechanics. While the experimental works has brought about striking progress in the research of nanomaterials, many researchers have also resorted to the computational nanomechanics. Because computer simulations based on reasonable physical models cannot only highlight the molecular features of nanomaterials for theoreticians but also provide guidance and interpretations for experimentalists. It is still an ongoing and challenging process to identify effective and efficient computational methods with respect to specific nanomaterials.

Among the many nanostructured materials, carbon nanotubes have attracted considerable attention. This kind of long and slender fullerene was first discovered by Iijima (1991). They can be produced by an array of techniques, such as arc discharge, laser ablation and chemical vapor deposition. A recent review of the processing and properties of carbon nanotubes and their composites is given by Thostenson et al. (2001). From the viewpoint of atomic arrangement, carbon nanotubes can be visualized as cylinders that rolled from sheets of graphite. They assume either single-walled or multi-walled structures and their helicity may also be different (Iijima and Ichlhashi, 1993; Bethune et al., 1993). Since the discovery of carbon nanotubes, much attention has been given to the investigation of their exceptional physical properties (Thostenson et al., 2001; Harris, 1999). It has been revealed that the conducting properties of carbon nanotubes depend dramatically on their helicity and diameter (Terrones et al., 1999), and the stiffness, flexibility and strength of carbon nanotubes are much higher than those of conventional carbon fibers (Treacy et al., 1996; Salvetat et al., 1999; Iijima et al., 1996). The extraordinary properties of carbon nanotubes have motivated researchers worldwide to study the fundamentals of this novel material as well as to explore their applications in different fields (Ajayan and Zhou, 2001).

Besides the great deal of experimental works on carbon nanotubes, many researchers have pursued the analysis of carbon nanotubes by theoretical modeling (Harris, 1999; Saito et al., 1998). These modeling approaches can be generally classified into two categories. One is the atomistic modeling and the major techniques include classical molecular dynamics (MD) (Iijima et al., 1996; Yakobson et al., 1997), tight-binding molecular dynamics (TBMD) (Hernandez et al., 1998) and density functional theory (DFT) (Sanchez-Portal et al., 1999). In principle, any problem associated with molecular or atomic motions can be simulated by these modeling techniques. However, due to their huge computational tasks, practical applications of these atomistic modeling techniques are limited to systems containing a small number of molecules or atoms and are usually confined to studies of relatively short-lived phenomena, from picoseconds to nanoseconds.

The other approach is the continuum mechanics modeling. Some researchers have resorted to classical continuum mechanics for modeling carbon nanotubes. For examples, Tersoff (1992) conducted simple calculations of the energies of fullerenes based on the deformation of a planar graphite sheet, treated as an elastic continuum, and concluded that the elastic properties of the graphite sheet can be used to predict the elastic strain energy of fullerenes and nanotubes. Yakobson et al. (1996) noticed the unique features of fullerenes and developed a continuum shell model. Ru, 2000a, Ru, 2000b followed this continuum shell model to investigate buckling of carbon nanotubes subjected to axial compression. This kind of continuum shell models can be used to analyze the static or dynamic mechanical properties of nanotubes. However, these models neglect the detailed characteristics of nanotube chirality, and are unable to account for forces acting on the individual atoms.

Therefore, there is a demand of developing a modeling technique that analyzes the mechanical response of nanotubes at the atomistic scale but is not perplexed in time scales. Such a modeling approach would benefit us in novel nanodevices design and multi-scale simulations of nanosystems (Nakano et al., 2001). In this paper, we extend the theory of classical structural mechanics into the modeling of carbon nanotubes. Our idea stems from that carbon nanotubes are elongated fullerenes, which were named after the architect known for designing geodesic domes, R. Buckmister Fuller. In fact, it is obvious that there are some similarities between the molecular model of a nanotube and the structure of a frame building. In a carbon nanotube, carbon atoms are bonded together by covalent bonds. These bonds have their characteristic bond lengths and bond angles in a three-dimensional space. Thus, it is logical to simulate the deformation of a nanotube based on the method of classical structural mechanics. In following sections, we first establish the bases of this concept and then demonstrate the approach by a few computational examples.

Section snippets

Brief review of structural mechanics for space frames

Structural mechanics analysis enables the determination of the displacements, strains and stresses of a structure under given loading conditions. Of the various modern structural analysis techniques, the stiffness matrix method has been by far the most generally used. The method can be readily applied to analyze structures of any geometry and can be used to solve linear elastic static problems as well as problems involving buckling, plasticity and dynamics. In the following, we briefly review

Structural characteristics of carbon nanotubes

A single-walled carbon nanotube (SWNT) can be viewed as a graphene sheet that has been rolled into a tube. A multi-walled carbon nanotube (MWNT) is composed of concentric graphitic cylinders with closed caps at both ends and the graphitic layer spacing is about 0.34 nm. Unlike diamond, which assumes a 3-D crystal structure with each carbon atom having four nearest neighbors arranged in a tetrahedron, graphite assumes the form of a 2-D sheet of carbon atoms arranged in a hexagonal array. In this

Structural mechanics approach to carbon nanotubes

From the structural characteristics of carbon nanotubes, it is logical to anticipate that there are potential relations between the deformations of carbon nanotubes and frame-like structures. For macroscopic space frame structures made of practical engineering materials, the material properties and element sectional parameters can be easily obtained from material data handbooks and calculations based on the element sectional dimensions. For nanoscopic carbon nanotubes, there is no information

Results and discussions

To verify the reliability and efficiency of the structural mechanics approach to the modeling of carbon nanotubes and to demonstrate its capability, we choose graphite sheets and single-walled carbon nanotubes as examples and calculate some of their basic elastic properties, such as Young’s modulus and shear modulus. In these computations, the initial carbon–carbon bond length is taken as 1.421 Å (Dresselhaus et al., 1995). The computational results are compared with the limited existing

Conclusions

A structural mechanics approach has been developed for modeling carbon nanotubes. A simple linkage between structural mechanics and molecular mechanics is established. In this approach, the computational strategy is essentially classical structural mechanics, but the theoretical concept stems from modern computational chemistry and the modeling is kept at the atomistic scale. Our computational results for elastic properties of carbon nanotubes are comparable to those obtained from other

Acknowledgements

This paper is dedicated to Professor William D. Nix. This work is partially supported by the National Science Foundation (ECS-0103012, Dr. Usha Varshney, Program Director) and the Army Research Office (DAAD 19-02-1-0264, Dr. Bruce LaMattina, Program Director).

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