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A structural mechanics approach for the analysis of carbon nanotubes

Chunyu Li, Tsu-Wei Chou *

Department of Mechanical Engineering, Center for Composite Materials, University of Delaware, 126 Spencer Laboratory, Newark, DE 19716-3140, USA

Received 14 March 2002; received in revised form 3 January 2003

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. 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Carbon nanotube; Nanomechanics; Molecular mechanics; Force ﬁelds; Atomistic modeling; Structural mechanics

1. 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

International Journal of Solids and Structures 40 (2003) 2487–2499 w.elsevier.com/locate/ijsolstr

*Corresponding author. Tel.: +1-302-831-2423/2421; fax: +1-302-831-3619. E-mail address: chou@me.udel.edu (T.-W. Chou).

0020-7683/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0020-7683(03)00056-8 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), tightbinding 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,b) 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.

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2. 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 the stiﬀness matrix method for linearly elastic space frame problems, which is relevant to the present studies.

For an element in a space frame as shown in Fig. 1, the elemental equilibrium equation can be written as following (Weaver and Gere, 1990):

where are the nodal displacement vector and nodal force vector of the element, respectively and K is the elemental stiﬀness matrix. The matrix K consists of following submatrices:

K ¼ Kii Kij

KTij Kjj ; ð4Þ where

Fig. 1. Illustration of a beam element in a space frame. C. Li, T.-W. Chou / International Journal of Solids and Structures 40 (2003) 2487–2499 2489

It is observed from the above elemental stiﬀness matrices that when the length, L, of the element is known, there are still four stiﬀness parameters need to be determined. They are the tensile resistance EA, the ﬂexural rigidity EIx and EIy and the torsional stiﬀness GJ. In order to obtain the deformation of a space frame, the above elemental stiﬀness equations should be established for every element in the space frame and then all these equations should be transformed from the local coordinates to a common global reference system. Finally, a system of simultaneous linear equations can be assembled according to the requirements of nodal equilibrium. By solving the system of equations and taking into account the boundary restraint conditions, the nodal displacements can be obtained.

3. 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 case, each carbon atom has three nearest neighbors. The atomic structure of nanotubes can be described in terms of the tube chirality, or helicity, which is deﬁned by the chiral vector ~CCh and the chiral angle h. In Fig. 2, we can visualize cutting the graphite sheet along the dotted lines and rolling the tube so that the tip of the chiral vector touches its tail. The chiral vector, also known as the roll-up vector, can be described by the following equation:

where the integers ðn;mÞ are the number of steps along the zigzag carbon bonds of the hexagonal lattice and

~aa1 and ~aa2 are unit vectors. The chiral angle determines the amount of twist in the tube. The chiral angles are 0 and 30 for the two limiting cases which are referred to as zigzag and armchair, respectively (Fig. 3).

Fig. 2. Schematic diagram of a hexagonal graphene sheet (Thostenson et al., 2001).

2490 C. Li, T.-W. Chou / International Journal of Solids and Structures 40 (2003) 2487–2499

In terms of the roll-up vector, the zigzag nanotube is denoted by ðn;0Þ and the armchair nanotube ðn;nÞ. The roll-up vector of the nanotube also deﬁnes the nanotube diameter.

The physical properties of carbon nanotubes are sensitive to their diameter, length and chirality.

In particular, tube chirality is known to have a strong inﬂuence on the electronic properties of carbon nanotubes. Graphite is considered to be a semi-metal, but it has been shown that nanotubes can be either metallic or semi-conducting, depending on tube chirality (Dresselhaus et al., 1996). The inﬂuence of chirality on the mechanical properties of carbon nanotubes has also been reported (Popov et al., 2000; Hernandez et al., 1998).

4. 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 about the elastic and sectional properties of the carbon–carbon bonds and the material properties. Therefore, It is imperative to establish a linkage between the microscopic computational chemistry and the macroscopic structural mechanics.

4.1. Potential functions of molecular mechanics

From the viewpoint of molecular mechanics, a carbon nanotube can be regarded as a large molecule consisting of carbon atoms. The atomic nuclei can be regarded as material points. Their motions are regulated by a force ﬁeld, which is generated by electron–nucleus interactions and nucleus–nucleus interactions (Machida, 1999). Usually, the force ﬁeld is expressed in the form of steric potential energy. It depends solely on the relative positions of the nuclei constituting the molecule. The general expression of

Fig. 3. Schematic diagram of (a) an armchair and (b) a zigzag nanotube (Thostenson et al., 2001). C. Li, T.-W. Chou / International Journal of Solids and Structures 40 (2003) 2487–2499 2491

the total steric potential energy, omitting the electrostatic interaction, is a sum of energies due to valence or bonded interactions and nonbonded interactions (Rappe et al., 1992):

UrþX

UhþX

UxþX Uvdw; ð9Þ where Ur is for a bond stretch interaction, Uh for a bond angle bending, U/ for a dihedral angle torsion, Ux for an improper (out-of-plane) torsion, Uvdw for a nonbonded van der Waals interaction, as shown in Fig. 4. There has been a wealth of literature in molecular mechanics devoted to ﬁnding the reasonable functional forms of these potential energy terms (Rappe et al., 1992; Brenner, 1990; Mayo et al., 1990; Cornell et al., 1995). Therefore, various functional forms may be used for these energy terms, depending on the particular material and loading conditions considered. In general, for covalent systems, the main contributions to the total steric energy come from the ﬁrst four terms, which have included four-body potentials. Under the assumption of small deformation, the harmonic approximation is adequate for describing the energy (Gelin, 1994). For sake of simplicity and convenience, we adopt the simplest harmonic forms and merge the dihedral angle torsion and the improper torsion into a single equivalent term, i.e., where kr, kh and ks are the bond stretching force constant, bond angle bending force constant and torsional resistance respectively, and the symbols Dr, Dh and D/ represent the bond stretching increment, the bond angle change and the angle change of bond twisting, respectively.

4.2. Linkage between sectional stiﬀness parameters and constants of force ﬁelds

In a carbon nanotube, the carbon atoms are bonded to each other by covalent bonds and form hexagons on the wall of the tube. These covalent bonds have their characteristic bond lengths and bond angles in a

Fig. 4. Interatomic interactions in molecular mechanics. 2492 C. Li, T.-W. Chou / International Journal of Solids and Structures 40 (2003) 2487–2499

three-dimensional space. When a nanotube is subjected to external forces, the displacements of individual atoms are constrained by these bonds. The total deformation of the nanotube is the result of these bond interactions. By considering the covalent bonds as connecting elements between carbon atoms, a nanotube could be simulated as a space frame-like structure. The carbon atoms act as joints of the connecting elements.

In the following, we establish the relations between the sectional stiﬀness parameters in structural mechanics and the force constants in molecular mechanics. For convenience, we assume that the sections of carbon–carbon bonds are identical and uniformly round. Thus it can be assumed that Ix ¼ Iy ¼ I and only three stiﬀness parameters, EA, EI and GJ, need to be determined.

Because the deformation of a space frame results in the changes of strain energies, we determine the three stiﬀness parameters based on the energy equivalence. Notice that each of the energy terms in molecular mechanics (Eqs. (10)–(12)) represents an individual interaction and no cross-interactions are included, we also need to consider the strain energies of structural elements under individual forces. According to the theory of classical structural mechanics, the strain energy of a uniform beam of length L subjected to pure axial force N (Fig. 5a) is

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