## Abstract

The group velocities of longitudinal and flexural wave propagations in single- and multi-walled carbon nanotubes are studied in the frame of continuum mechanics. The dispersion relations between the group velocity and the wavenumber for flexural and longitudinal waves, described by a beam model and a cylindrical shell model, are established for both single- and multi-walled carbon nanotubes. The effect of microstructures in carbon nanotubes on the wave dispersion is revealed through the non-local elastic models of a beam and a cylindrical shell, including the second-order gradient of strain and a parameter of microstructure. It is shown that the microstructures in the carbon nanotubes play an important role in the dispersion of both longitudinal and flexural waves. In addition, the non-local elastic models predict that the cut-off wavenumber of the dispersion relation between the group velocity and the wavenumber is approximately 2×10^{10} m^{−1} for the longitudinal and flexural wave propagations in both single- and multi-walled carbon nanotubes. This may explain why the direct molecular dynamics simulation cannot give a proper dispersion relation between the phase velocity and the wavenumber when the wavenumber approaches approximately 2×10^{10} m^{−1}, much lower than the cut-off wavenumber for the dispersion relation between the phase velocity and the wavenumber predicted by continuum mechanics.

## 1. Introduction

Interest in carbon nanotubes has grown rapidly since their discovery (Iijima 1991). Recent studies have indicated that carbon nanotubes exhibit superior mechanical and electronic properties over any known materials, and hold substantial promise for new super-strong composite materials, among others. For instance, carbon nanotubes have an exceptionally high elastic modulus (Treacy *et al*. 1996), and sustain large elastic and failure strains (Yakobson *et al*. 1996; Wong *et al*. 1997). Apart from an extensive experimental study to characterize the mechanical behaviour of carbon nanotubes, theoretical or computational modelling of carbon nanotubes has received considerable attention. The current computational modelling approaches include both atomistic and continuum modelling.

Among the research of continuum modelling of carbon nanotubes, numerous studies have concentrated on the static mechanical behaviour, such as buckling, of carbon nanotubes using the elastic models of a beam and a cylindrical shell (Qian *et al*. 2002). Several research teams implemented the elastic models of beam to study the dynamic problems, such as vibration and wave propagation (Poncharal *et al*. 1999; Popov *et al*. 2000; Yoon *et al*. 2002, 2003*a*,*b*, 2004), of carbon nanotubes. Some other teams used the elastic models of a cylindrical shell to study the vibration and wave dispersion relations of carbon nanotubes (Natsuki *et al*. 2005, 2006; Wang *et al*. 2005; Dong & Wang 2006). Furthermore, a multi-walled carbon nanotube was modelled as an assemblage of cylindrical shell elements connected throughout their lengths by distributed springs to investigate the elastic waves of very high frequency in carbon nanotubes (Chakraborty 2006), and to give the dispersion relation between the group velocity and the wavenumber. Their studies showed that both elastic models of a beam and a cylindrical shell are valid to describe the vibration or wave propagation of carbon nanotubes in a relatively low-frequency range.

Recognizing such a limit mainly coming from the microstructures in carbon nanotubes, several researchers (Sudak 2003; Zhang *et al*. 2004, 2005; Wang 2005; Wang *et al*. 2006*a*; Xie *et al*. 2006) established non-local elastic models with the second-order gradient of stress taken into account so as to describe the vibration and wave propagation of both single- and multi-walled carbon nanotubes, in a higher frequency range. The molecular dynamics simulations showed that these models worked much better than the elastic models in a relatively high-frequency range. However, the molecular dynamics simulations also showed that the microstructures of a carbon nanotube might have a very significant influence on the high frequency waves such that non-elastic models could not predict the wave dispersion well (Wang & Hu 2005; Wang *et al*. 2006*b*).

The concept of group velocity may be useful in understanding the dynamics of carbon nanotubes, since it is related to the energy transportation of wave propagation. To the best knowledge of the authors, however, there is little work on the group velocity of wave propagation in carbon nanotubes with the effect of microstructures taken into consideration. The primary objective of this work is to study the group velocity of the longitudinal and flexural wave propagations in carbon nanotubes, so as to examine the effect of microstructures of a carbon nanotube on the wave dispersion from the viewpoint of group velocity or energy transportation.

This work deals with the dispersion relations between the group velocity and the wavenumber for both longitudinal and flexural waves in single- and multi-walled carbon nanotubes via the non-local elastic models of both Timoshenko beams and cylindrical shells. The study focuses on the comparison between the non-local elastic and the elastic models in predicting the dispersion relation of the group velocity with respect to the wavenumber. For this purpose, §2 presents the dispersion relation between the group velocity and the wavenumber of longitudinal waves in a single-walled carbon nanotube from a non-local elastic model of cylindrical shell, which includes the second-order gradient of strain in order to characterize the microstructures of the carbon nanotube. Section 3 gives the dispersion relation between the group velocity and the wavenumber of flexural waves in a single-walled carbon nanotube using a non-local elastic model of a Timoshenko beam. Section 4 turns to the dispersion relation between the group velocity and the wavenumber of longitudinal waves in a multi-walled carbon nanotube on the basis of a non-local elastic model of multi-cylindrical shells, which also takes the second-order gradient of strain into account. Similarly, §5 gives the dispersion relation between the group velocity and the wavenumber of flexural waves in a multi-walled carbon nanotube from a non-local elastic model of multi-Timoshenko beams. Finally, in §6, the paper ends with some concluding remarks.

## 2. Group velocity of a longitudinal wave in a single-walled carbon nanotube

To describe the effect of microstructures of a carbon nanotube on its mechanical properties, it is assumed that the model of the carbon nanotube is made of a kind of non-local elastic material, where the stress state at a given reference location depends not only on the strain of this location but also on the higher order gradient of strain, so as to take the influence of the microstructures into account. The simplest constitutive law to characterize the non-local elastic material in the one-dimensional case reads (Askes *et al*. 2002)(2.1)where *E* is Young's modulus; *σ*_{x} is the axial stress; and *ϵ*_{x} is the axial strain. The material parameter characterizes the influence of microstructures on the constitutive law of the non-local elastic materials, and *d* describes the inter-particle distance (Askes *et al*. 2002) and is the axial distance between two neighbouring rings of carbon atoms when the single-walled carbon nanotube is modelled as a non-local elastic cylindrical shell.

The single-walled carbon nanotube can be considered as a thin cylindrical shell, where the bending moments can be neglected for simplicity. The non-local elastic model of a cylindrical shell with the second-order gradient of strain taken into consideration is as follows (Wang *et al*. 2006*b*):(2.2a)(2.2b)(2.2c)where *x* is the coordinate in the longitudinal direction; *u*, *v* and *w* are the displacement components in the longitudinal, tangential and radial directions, respectively; *R* is the radius of the cylindrical shell; *ρ* is the mass density; and *υ* is Poisson's ratio. It is interesting that equation (2.2*b*) is not coupled with equations (2.2*a*) and (2.2*c*) such that the torsional waves in the cylindrical shell are independent of the longitudinal and radial waves.

Now consider the motions governed by the coupled dynamic equations in *u* and *w*, and let(2.3)where ; is the amplitude of longitudinal vibration; and is the amplitude of radial vibration. Furthermore, *ω* is the angular frequency and is the wavenumber related to the wavelength *λ* via . Substituting equation (2.3) into equations (2.2*a*) and (2.2*c*) yields(2.4)The determinant of the coefficient matrix of equation (2.4) gives(2.5)where . Solving equation (2.5) gives the two branches of the wave dispersion relation as follows:(2.6)Then, the group velocity reads(2.7)

Figure 1 shows the dispersion relation between the group velocity and the wavenumber of longitudinal waves in an armchair (5, 5) and (10, 10) single-walled carbon nanotubes. Now, the product of Young's modulus *E* and the wall thickness *h* is *Eh*=346.8 Pa m, and Poisson's ratio is *υ*=0.20 for a (5, 5) and a (10, 10) single-walled carbon nanotube. In addition, one has *r*=0.0355 nm when the axial distance between two neighbouring rings of atoms is *d*=0.123 nm. The product of the mass density *ρ* and the wall thickness *h* yields *ρh*≈760.5 kg m^{−3} nm. The radii of (5, 5) and (10, 10) single-walled carbon nanotubes are 0.34 and 0.68 nm, respectively. There is a slight difference between the non-local theory and the classical theory of elasticity for the lower branches. The group velocity decreases rapidly with an increase in the wavenumber. The group velocity goes to zero for the (5, 5) and (10, 10) single-walled carbon nanotubes when the wavenumber approaches to and or so, respectively. This may explain the difficulty that the direct molecular dynamics simulation can only offer the dispersion relation between the phase velocity and the wavenumber up to and for the (5, 5) and (10, 10) single-walled carbon nanotubes, respectively. For the upper branches of the dispersion relation, the difference can hardly be identified when the wavenumber is lower. However, the results of the elastic cylindrical shells remarkably deviate from those of the non-local elastic cylindrical shells with an increase in the wavenumber. Figure 1*a*,*b* shows the intrinsic limit , instead of for the maximum wavenumber owing to the microstructures. This can explain the contradiction that the cut-off longitudinal wave predicted by the non-local elastic cylindrical shell is , but the molecular dynamics simulation offers the dispersion relation up to the wavenumber only (Wang *et al*. 2006*b*).

## 3. Group velocity of a flexural wave in a single-walled carbon nanotube

This section starts with the dynamic equation of a non-local elastic model of a Timoshenko beam of infinite length and uniform cross section placed along the *x* direction in the frame of coordinates , with being the displacement of a section *x* of the beam in the *y* direction at the moment *t* (Wang & Hu 2005)(3.1)where *G* is the shear modulus; *φ* is the slope of the deflection curve when the shearing force is neglected; *A* is the cross-sectional area of the beam; is the moment of inertia for the cross section; *β* is the form factor of shear depending on the shape of the cross section; and *β*=0.5 holds for the circular tube of the thin wall (Timoshenko & Gere 1972). The other parameters are the same as those defined in §2.

To study the flexural wave propagation in an infinitely long beam, let the dynamic deflection and slope be given by(3.2)where represents the amplitude of deflection of the beam and is the amplitude of the slope of the beam due to the bending deformation alone. As defined in §2, *ω* is the angular frequency of wave and is the wavenumber related to the wavelength *λ* via . Substituting equation (3.2) into equation (3.1) yields(3.3)If there exists at least one non-zero solution of equation (3.3), one arrives at(3.4)

Solving equation (3.4) for the angular frequency *ω* gives two branches of the wave dispersion relation(3.5)where , and . The group velocity reads(3.6)where(3.7)and(3.8)

Figure 2 shows the dispersion relations between the group velocity and the wavenumber of flexural waves in an armchair (5, 5) and (10, 10) single-walled carbon nanotubes. The product of Young's modulus and the wall thickness is and Poisson's ratio is , and it follows that . In addition, the material parameter *r*=0.0355 nm. The product of the mass density and the wall thickness yields *ρh*≈760.5 kg m^{−3} nm. For the (5, 5) single-walled carbon nanotube, the product of the mass density and the section area yields , the product of the mass density and the moment of inertia for the cross section yields , and it follows that . For the (10, 10) single-walled carbon nanotube, the product of the mass density and the section area yields , the product of the mass density and the moment of inertia for the cross section yields , and it follows that . For both lower and upper branches of the dispersion relation, the results of the elastic Timoshenko beam remarkably deviate from those of the non-local elastic Timoshenko beam with an increase in the wavenumber. Figure 2*a*,*b* shows again the intrinsic limit of the wavenumber , instead of . This fact explains why the cut-off flexural wave predicted by the non-local elastic cylindrical shell is , but the direct molecular dynamics simulation only gives the dispersion relation up to the wavenumber (Wang & Hu 2005).

## 4. Group velocity of a longitudinal wave in a multi-walled carbon nanotube

This section deals with the propagation of longitudinal waves in a multi-walled carbon nanotube via a non-local elastic model of multi-cylindrical shells, including the second-order gradient of strain. The dynamic equations of the *k*th layer for an *N*-walled carbon nanotube are (L. F. Wang *et al*. 2007, unpublished data)(4.1a)(4.1b)(4.1c)where *x* is the coordinate in longitudinal direction; *u*_{k}, *v*_{k} and *w*_{k} are the displacement components in the longitudinal, tangential and radial directions of the *k*th tube, respectively; and *R*_{k} is the radius of the *k*th tube. Other parameters are those defined in §§2 and 3.

As only infinitesimal vibration is considered, the net pressure due to the van der Waals interaction is assumed to be linearly proportional to the deflection between two layers, i.e.(4.2)where *N* is the total number of layers of the multi-walled carbon nanotube and *c*_{kj} is the van der Waals interaction coefficient, which can be determined through the Lennard-Jones pair potential (Jones 1924)(4.3)where , =3.407 Å and is the distance between two interacting atoms. In this study, the coefficient of the van der Waals interaction comes from the Lennard-Jones pair potential suggested by He *et al*. (2005)(4.4)where(4.5)and(4.6)

Consider the motions governed by the coupled dynamic equations in *u*_{k} and *w*_{k}, and let(4.7)where is the amplitude of longitudinal vibration and is the amplitude of radial vibration. As defined in §2, *ω* is the angular frequency of the wave and is the wavenumber related to the wavelength *λ* via . Substituting equation (4.7) into equations (4.1*a*) and (4.1*c*) yields(4.8a)(4.8b)The dispersion relation can be obtained by solving the following eigenvalue equation:(4.9)where and is an identity matrix and the entries in the matrix are(4.10a)(4.10b)(4.10c)(4.10d)(4.10e)From equation (4.9), the dispersion relation between the group velocity and the wavenumber of longitudinal waves in the multi-walled carbon nanotube can be numerically determined by(4.11)

Figure 3 presents the dispersion relation between the group velocity and the wavenumber of longitudinal waves in a double-walled carbon nanotube with radii and , and in a four-walled carbon nanotube with radii , , and . The product of Young's modulus and the wall thickness is , and Poisson's ratio is for both the (5, 5) and (10, 10) single carbon nanotubes. In addition, one has when the axial distance between two neighbouring rings of atoms is . The product of the mass density and the wall thickness yields *ρh*≈760.5 kg m^{−3} nm. Now, there is a slight difference between the theory of non-local elasticity and the classical theory of elasticity for the lower branches. The group velocity decreases rapidly with an increase in wavenumber. When the wavenumber approaches approximately , the group velocity of all these lower branches tends to zero. For the upper branches of the dispersion relation, the difference is almost invisible when the wavenumber is lower. However, the results of the elastic cylindrical shells remarkably deviate from those of non-local elastic cylindrical shells with an increase in the wavenumber. Figure 3*a*,*b* shows again the intrinsic limit of wavenumber yielding , instead of .

## 5. Group velocity of flexural waves in a multi-walled carbon nanotube

This section starts with the dynamic equation of a non-local elastic model of multi-Timoshenko beams of infinite length and uniform cross section placed along the *x* direction in the frame of coordinates . The dynamics equations of the *k*th tube for an *N*-walled carbon nanotube are(5.1a)(5.1b)where is the displacement of section *x* of the *k*th tube in the *y* direction at the moment *t*; *φ*_{k} is the slope of the deflection curve of the *k*th tube when the shearing force is neglected; *A*_{k} is the cross-sectional area of the *k*th tube; is the moment of inertia for the cross section of the *k*th tube; *β* is the form factor of shear depending on the shape of the cross section; and *β*=0.5 holds for the circular tube of the thin wall (Timoshenko & Gere 1972). is the van der Waals interaction coefficient for the interaction pressure per unit axial length and is estimated based on an effective interaction width (Ru 2000)(5.2)where can be obtained by equation (4.4).

To study the flexural wave propagation, consider the deflection and the slope given by(5.3)where represents the amplitude of deflection of the *k*th tube and is the amplitude of the slope of the *k*th tube due to bending deformation alone. As in §2, *ω* is the angular frequency of wave and is the wavenumber related to the wavelength *λ* via . Substituting equation (5.3) into equation (5.1*a*) and (5.1*b*) yields(5.4)Under the assumption that there exists at least one non-zero solution of equation (5.4), one arrives at(5.5)where is an identity matrix and the entries in the matrix are(5.6a)(5.6b)(5.6c)(5.6d)(5.6e)From equation (5.5), the dispersion relation between the group velocity and the wavenumber of flexural waves in a multi-walled carbon nanotube can be numerically obtained using .

Figure 4 presents the dispersion relation between the group velocity and the wavenumber of flexural waves in a double-walled carbon nanotube with radii and , and in a four-walled carbon nanotube with radii , , and . The product of Young's modulus and the wall thickness is , and Poisson's ratio is and it follows that . The product of the mass density and wall thickness yields *ρh*≈760.5 kg m^{−3} nm, and the form factor of shear . Table 1 shows the parameters *ρA*_{k}, *ρI*_{k} and *EI*_{k} of the double-walled carbon nanotube. Table 2 gives the parameters *ρA*_{k}, *ρI*_{k} and *EI*_{k} of the four-walled carbon nanotube. For the lower and upper branches of the dispersion relation, the difference is almost invisible when the wavenumber is lower. However, the results of the elastic model of Timoshenko beams remarkably deviate from those of the non-local elastic model of Timoshenko beams with an increase in the wavenumber. Figure 4*a*,*b* shows again the intrinsic limit , rather than for the maximum wavenumber owing to the microstructures.

## 6. Concluding remarks

The paper presents a detailed study on the dispersion relation between the group velocity and the wavenumber for the propagation of longitudinal and flexural waves in single- and multi-walled carbon nanotubes, on the basis of elastic and non-local elastic cylindrical shells and Timoshenko beams. The study indicates that both elastic and non-local elastic models can offer the correct prediction when the wavenumber is lower. However, the results of the elastic model remarkably deviate from those given by the non-local elastic model with an increase in the wavenumber. As a result, the microstructures play an important role in the dispersion of both longitudinal and flexural waves in both single- and multi-walled carbon nanotubes.

The cut-off wavenumber of the dispersion relation between the group velocity and the wavenumber is approximately 2×10^{10} m^{−1} for both longitudinal and flexural waves in both single- and multi-walled carbon nanotubes. This contradiction is interpreted to show that the direct molecular dynamics simulation cannot give the dispersion relation between the phase velocity and the wavenumber when the wavenumber approaches approximately 2×10^{10} m^{−1}, which is much lower than the cut-off wavenumber of the dispersion relation between the phase velocity and the wavenumber predicted by the continuum mechanics.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under grants 10702026 and 10732040, the National Postdoctoral Foundation of China, the Postdoctoral Foundation of Jiangsu Provincial Government and the Ministry of Education of China under grants 705021 and IRT0534.

## Footnotes

- Received November 28, 2007.
- Accepted January 28, 2008.

- © 2008 The Royal Society