## Abstract

The paper presents a detailed study on the thermal vibration of a single-walled carbon nanotube by using different beam models of continuum mechanics, together with the law of energy equipartition, and the molecular dynamics simulations. The basic finding of the study is the relation, derived by using the Timoshenko beam model and the law of energy equipartition, between the temperature and the root-of-mean-squared (RMS) amplitude of thermal vibration at any cross section of the carbon nanotube. The molecular dynamics simulations show that both the Euler beam model and the Timoshenko beam model can roughly predict the thermal vibration of lower order modes for a relatively long carbon nanotube. However, the Timoshenko beam model, compared with the Euler beam model, offers a much better prediction of the RMS amplitude of the thermal vibration near the fixed end of the carbon nanotube. For the thermal vibration of a relatively short carbon nanotube or higher order models of a relatively long carbon nanotube, the difference between the Timoshenko beam and the Euler beam in dynamic prediction becomes obvious, and the Timoshenko beam model works much better than the Euler beam model.

## 1. Introduction

Carbon nanotubes have been attracting continuous interest for over a decade since their discovery (Iijima 1991). Carbon nanotubes exhibit superior mechanical and electronic properties over any known materials and hold substantial promise for new super-strong composite materials, among others (Qian *et al.* 2002). Treacy *et al.* (1996) estimated Young’s modulus of isolated carbon nanotubes by measuring, in the transmission electron microscope, the amplitude of their intrinsic thermal vibrations. Krishnan *et al.* (1998) presented the relationship between Young’s modulus, the size and the standard deviation of the vibration amplitude at the tip of a carbon nanotube at a given temperature. Hsieh *et al.* (2006) investigated the intrinsic thermal vibrations of a single-walled carbon nanotube via molecular dynamics simulations. The recently developed nanotechnology enables one to construct small devices working at a molecular level and activated by the thermal fluctuation. One example is the so-called Brownian ratchet, where a particle moves unidirectionally over a switchable anisotropic potential (Astumian 1997). Xu *et al.* (2006) studied thermally driven large-amplitude fluctuations in carbon nanotube-based devices by using molecular dynamics simulations.

Recent years have witnessed numerous applications of the Timoshenko beam model to analyse the vibration and wave propagation of carbon nanotubes. For example, Yoon *et al*. (2004, 2005) studied the flexural wave propagation and vibration of a multi-walled carbon nanotube based on the model of the Timoshenko beam with the rotary inertia and the shear deformation taken into account. Wang & Hu (2005) found that the non-local elastic model of the Timoshenko beam, compared with other beam models, can offer a better prediction of the dispersion of the flexural wave in carbon nanotubes. To the best of our knowledge, however, neither experimental studies nor numerical simulations have been available for the validation of the Timoshenko beam model in studying the thermal vibration of any carbon nanotubes.

The primary objective of this study is to check the validity of the Timoshenko beam model together with the law of energy equipartition in studying the thermal vibration, simulated by the molecular dynamics, of a single-walled carbon nanotube in argon atmosphere. For this purpose, §2 presents the root-of-mean-squared (RMS) amplitude of the stochastically driven vibration of a Timoshenko beam, which will be used to model single-walled carbon nanotubes. Then, §3 gives the molecular dynamics simulation for the thermal vibration of a carbon nanotube under an argon atmosphere. Section 4 outlines a comparison between the analytical results and the numerical results. Finally, the paper ends with concluding remarks in §5.

## 2. Timoshenko beam model

This section starts with the dynamic equation of a Timoshenko beam with uniform cross section placed along direction *x* in the frame of coordinates (*x*,*y*,*z*), *w*(*x*,*t*) being the displacement of section *x* of the beam in direction *y* at the moment *t* (Thomson 1972)
2.1a
and
2.1b
where *E* is Young’s modulus, *I* is the moment of inertia for the cross section, *ρ* is the mass density, *A* is the cross-section area of the beam, *G* is the shear modulus, *φ* is the slope of the deflection curve when the shearing force is neglected, *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 boundary conditions of a cantilever beam clamped at *x*=0 are
2.2

To study the free vibration of a Timoshenko beam, let the dynamic deflection and slope be given by
2.3
where represents the deflection amplitude of the beam, and the slope amplitude of the beam due to bending deformation alone. Let
2.4
Substituting equation (2.3) into equation (2.1), one obtains
2.5a
and
2.5b
where
2.6
In the case of [(*r*^{2}−*s*^{2})^{2}+4/*b*^{2}]^{1/2}>(*r*^{2}+*s*^{2}), the solutions of equations (2.5a) and (2.5b) read (Huang 1961)
2.7a
and
2.7b
where
2.8
In the case of [(*r*^{2}−*s*^{2})^{2}+4/*b*^{2}]^{1/2}<(*r*^{2}+*s*^{2}), then equations (2.7a) and (2.7b) should be replaced by (Huang 1961)
2.9a
and
2.9b
where
2.10
In equations (2.7a), (2.7b) and (2.9a), (2.9b), only one-half of the constants are independent since they are related by equations (2.5a) and (2.5b) as follows:
2.11a
2.11b
2.11c
and
2.11d

For the case of [(*r*^{2}−*s*^{2})^{2}+4/*b*^{2}]^{1/2}>(*r*^{2}+*s*^{2}), the natural frequency of the cantilever beam yields the following equation (Huang 1961):
2.12
Solving equation (2.12) for *b* gives an infinite number of *b*_{n},*n*=1,2,…, which determines the *n*th natural frequency of the Timoshenko beam in this case. The *n*th normal mode for the cantilever beam is (Huang 1961)
2.13a
and
2.13b
where
2.14a
2.14b
2.14c
and
2.14d
For the case of [(*r*^{2}−*s*^{2})^{2}+4/*b*^{2}]^{1/2}<(*r*^{2}+*s*^{2}), the *n*th natural frequency *ω*_{n} of the Timoshenko beam can be determined from (Huang 1961)
2.15
The *n*th normal mode for the cantilever beam is
2.16a
and
2.16b
where
2.17a
2.17b
2.17c
and
2.17d
In equations (2.12)–(2.17*a*–*d*), one has
2.18a
and
2.18b

Now, the study turns to the stochastic vibration of the Timoshenko beam from the viewpoint of energy analysis. The total energy *E*_{n} contained in the *n*th vibration mode can be found by calculating the elastic energy at the instant of maximal deflection when the cantilever is momentarily stationary, i.e. e^{jωnt}=1,
2.19
From the law of energy equipartition, there is an average energy of *kT*/2 per degree of freedom for all of the relevant lateral vibration modes. There are both elastic and kinetic energy degrees of freedom in a vibration mode then, on average, 〈*E*_{n}〉=*kT* for each vibration mode, with *E*_{n} yielding the Boltzmann distribution (Krishnan *et al.* 1998). From equation (2.19), it is easy to get
2.20
Then, the RMS amplitude of the *n*th mode at *x* is
2.21

As the vibration modes are mutually independent, the vibration profile for the combined modes is also a Gaussian distribution, with standard deviation given by the sum of the variances. The RMS amplitude of the carbon nanotube at *x* is
2.22

## 3. Molecular dynamics model

In order to check the applicability of the above Timoshenko beam model to studying the thermal vibration of a single-walled carbon nanotube, this section presents the model of molecular dynamics simulations of a carbon nanotube under an argon atmosphere as shown in figure 1.

In the corresponding molecular dynamics models, the interatomic interactions are described by the Tersoff–Brenner potential (Brenner 1990), which has been proved applicable to the description of mechanical properties of single-walled carbon nanotubes. The structure of the Tersoff–Brenner potential is as follows:
3.1
where *r*_{ij} is the distance from atom *i* to atom *j*; *V*_{R}(*r*_{ij}) and *V*_{A}(*r*_{ij}) are the repulsive and attractive terms given by
3.2a
and
3.2b
Here, *S*_{ij}=1.29, *D*_{ij}=6.325 eV, *β*_{ij}=15 nm^{−1}, *r*_{0}=0.1315 nm, *f*_{ij}, *D*_{ij}, *S*_{ij}, *β*_{ij} are scalars; *f*_{ij}(*r*_{ij}) is a switch function used to confine the pair potential in a neighbourhood with radius of *r*_{2} as follows:
3.2c
where *r*_{1}=1.7 Å and *r*_{2}=2.0 Å. In equation (3.1), reads
3.2d
3.2e
and
3.2f
where *θ*_{ijk} is the angle between bonds *i*–*j* and *i*–*k*, *δ*=0.80469, *a*_{0}=0.011304, *c*_{0}=19 and *d*_{0}=2.5. In addition, the C–C bond length in the model is 0.142 nm.

The van der Waals interaction either between two argon atoms or between a carbon atom and an argon atom is described by the Lennard–Jones pair potential
3.3
where *ε*_{C−C}=3.19×10^{−3} eV, *σ*_{C−C}=3.345 Å (Rafizadeh 1974), *ε*_{Ar−Ar}=1.032×10^{−2} eV, *σ*_{Ar−Ar}=3.822 Å (Cleary & Mayne 2006), and *σ*_{C−Ar}=(*σ*_{C−C}+*σ*_{Ar−Ar})/2.

The Verlet algorithm in the velocity form with time step 1 fs is used to simulate the thermal vibration of carbon nanotubes
3.4
where *R* represents the position of atoms, *V* the velocity of atoms, *a* the acceleration of atoms and *δt* the time step. In the simulation, the periodic boundary conditions are applied to argon atmospheres. The displacements of nanotube of each section are observed every 1 ps, the total simulation time is *T*_{tat} ps and the RMS amplitude of the cross section of carbon nanotube at *x* is
3.5
where *w*_{i}(*x*) is the amplitude of the cross section of carbon nanotube at *x*.

## 4. Comparison between analytical and numerical results

To predict the thermal vibration of a single-walled carbon nanotube from the analytical results in §2, it is necessary to know Young’s modulus *E* and the shear modulus *G* or Poisson’s ratio *υ* in advance. The previous studies based on the Tersoff–Brenner potential gave a great variety of Young’s moduli of single-walled carbon nanotubes from the simulated tests of axial tension and compression (Guo *et al*. 2006). When the thickness of the wall was chosen as 0.34 nm, for example, 1.07 TPa was reported by Yakobson *et al*. (1996), 0.8 TPa by Cornwell & Wille (1997) and 0.44–0.50 TPa by Halicioglu (1998). Meanwhile, Young’s modulus determined by Zhang *et al*. (2002) on the basis of the nanoscale continuum mechanics was only 0.475 TPa when the first set of parameters in the Tersoff–Brenner potential was used. Wang *et al.* (2006) reported that Young’s modulus was about 0.46 TPa, based on the higher order Cauchy–Born rule. Hence, it becomes necessary to compute Young’s modulus and Poisson’s ratio again from the above molecular dynamics model for the single-walled carbon nanotubes under the static loading.

For the same thickness of wall, Young’s modulus computed by using the first set of parameters in the Tersoff–Brenner potential was 0.46 TPa for the armchair (5,5) carbon nanotube and 0.47 TPa for the armchair (10,10) carbon nanotube for the axial tension. Furthermore, the simulated test of pure bending gave the product of effective Young’s modulus *E*=0.39TPa and Poisson’s ratio *υ*=0.22 for the armchair (5,5) carbon nanotube and *E*=0.45TPa and *υ*=0.20 for the armchair (10,10) carbon nanotube (Wang & Hu 2005). In the numerical simulations of this section, Young’s moduli and Poisson’s ratios obtained from the simulated test of pure bending for those two carbon nanotubes were used. For the single-walled carbon nanotubes, the wall thickness is *h*=0.34nm and the mass density of the carbon nanotubes is *ρ*=2237kg m^{−3}.

Figure 2 shows the RMS amplitude of the thermal vibration of the first four modes of a (10, 10) armchair carbon nanotube of 24.6 nm long at 300 K. Here, the symbol *E* represents the results predicted by using the Euler beam model in appendix B, and the symbol *T* represents results predicted by using the Timoshenko beam model. In figure 2, no difference between the results of the Euler beam model and the Timoshenko beam model can be found for the first mode of vibration, and little difference can be observed for the second mode of vibration. The differences between the results of the two beam models become more and more obvious with an increase in the mode order. The RMS amplitude predicted by the Timoshenko beam model is larger than that predicted by the Euler beam model.

Figure 3 presents the RMS amplitudes of the thermal vibration of the first four modes of a (10, 10) armchair carbon nanotube of 4.9 nm long at 300 K. Here, the symbols *E* and *T* represent the same models as in figure 2. Figure 3*a* indicates that the difference between the results of the Euler beam model and the Timoshenko beam model looks obvious even for the first mode of vibration. Figure 3*b*–*d* shows again that the difference for the beam models becomes more and more obvious with an increase in the mode order. The RMS amplitude predicted by the Timoshenko beam model is also larger than that predicted by the Euler beam model.

Figure 4 illustrates the RMS amplitudes of thermal vibration of armchair carbon nanotubes of different sizes at 300 K, where figure 4*a* is for a (10,10) armchair carbon nanotube of 14.5 nm long, figure 4*c* is for a (10,10) armchair carbon nanotube of 9.6 nm long, figure 4*e* is for a (5,5) armchair carbon nanotube of 9.6 nm long and figure 4*b*,*d*,*f* are the zoom views of figure 4*a*,*c*,*e*, respectively. Here, the symbols *E* and *T* have the same meaning as before, and MD1 and MD2 are the molecular dynamics simulations in *x* and *y* directions, respectively. The parameters of the Euler beam model and the Timoshenko beam model are the same as those given previously. In the molecular dynamics simulation, different densities of argon were used, but the difference among those results was not obvious. It can be seen that both the Euler beam model and the Timoshenko beam model can roughly predict the thermal vibration of carbon nanotubes in figure 4. In the zoomed view of the fixed end of figure 4*b*,*d*,*f*, however, the results show that the Timoshenko beam model, compared with the Euler beam model, offers a much better prediction of the RMS amplitude of thermal vibration simulated by using molecular dynamics. For shorter carbon nanotubes, the difference between the results of the Timoshenko beam model and the Euler beam model is obvious. And the Timoshenko beam model can give a better prediction of the thermal vibration than the Euler beam model for shorter carbon nanotubes.

## 5. Concluding remarks

The Timoshenko beam model, together with the law of energy equipartition, enables one to establish the analytical relation between the temperature and the RMS amplitude of thermal vibration at any cross section of the carbon nanotube. The molecular dynamics simulations of both (5, 5) and (10, 10) armchair carbon nanotubes show that the Timoshenko beam model, compared with the Euler beam model, is able to offer a much better prediction for the thermal vibration of those carbon nanotubes when the vibration modes are higher and the carbon nanotubes are shorter, especially at the fixed end of a carbon nanotube.

## Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under grant nos. 10702026, 60910007.

## Appendix A. Vibration profile of the tip of a carbon nanotube (Krishnan *et al.* 1998)

This appendix presents the calculation of the vibration profile of the tip of a carbon nanotube. For a classical simple oscillator of amplitude *D*_{n}, the oscillator position *y* at the moment *t* is given by
A1
where, as before, *D*_{n} is the amplitude that depends on the energy of the oscillator, and *ω*=2*π*/*f*, with *f* being the frequency. In the interval between *y* and *y*+d*y*, the oscillator spends a time d*t*, which is found by taking the derivative of equation (A1)
A2
or
A3

The probability *P*(*u*_{n},*y*) d*y* of finding the oscillator between *y* and *y*+d*y* when the amplitude is *D*_{n} is proportional to the time spent in this interval d*t*. After normalization, it is found that
A4
*P*(*D*_{n},*y*) reaches a peak at the extrema *y*=±*D*_{n} and has a minimum at *y*=0. However, the energy of the system, and hence the amplitude *D*_{n}, is changing in a stochastic manner with an increase in time. Therefore, one needs also to average over all the possible values of *D*_{n} that the system can be adopted. Furthermore, one must also average over all of the activated modes.

The carbon nanotube vibration is essentially relaxed phonons that are in an equilibrium at an ambient temperature *T*. The probability that the system is in the state *m* of energy is given by the Boltzmann factor
A5
where J s is the Planck constant. The frequency *ω*_{n} of a vibrating nanotube of density *ρ* is given by
A6
The energy in the *n*th mode is therefore quantized in units of . For typical nanotubes in this paper , for the lower modes, which is important for the RMS amplitude of thermal induced vibration of a carbon nanotube. Thus, to a very good approximation, one has
A7
If one sets and , then *W*(*E*_{n}) d*E*_{n} is the continuum limit of *W*(*m*) in equation (A7), and *W*(*E*_{n}) d*E*_{n} is the probability between *E*_{n} and *E*_{n}+d*E*_{n} of energy in the *n*th mode,
A8
This result is expected when the thermal average number of phonons, , is high. The average energy is *E*_{n}=*kT*, the half of which comes from the kinetic energy degree of freedom, and the other half from the elastic energy degree of freedom.

The stochastically averaged probability amplitude is therefore A9 which, from equations (A4) and (A9), reads A10 From equation (B4), one has A11 where . Therefore, one reaches A12 The substitution A13 ensures that the condition holds. Thus, one has A14 The integral above can be easily worked out to give the Gaussian form A15 Using equation (A11), one has the standard deviation A16

## Appendix B. Stochastic vibration of a cantilever Euler beam (Krishnan *et al.* 1998)

This appendix starts with the dynamic equation of an Euler beam of uniform cross section placed along direction *x* in the frame of coordinates (*x*,*y*,*z*), with *w*(*x*,*t*) being the displacement of section *x* of the beam in direction *y* at the moment *t*. Upon the assumption of small amplitudes, the motion of a vibrating Euler beam is governed by the following partial equation of the fourth order (Thomson 1972)
B1
where *E* is Young’s modulus, *I* is the moment of inertia for the cross section, *ρ* is the mass density and *A* is the cross-section area of the beam. The boundary conditions for a cantilever Euler beam are
B2

The solution for the *n*th mode reads (Thomson 1972)
B3
where , , *α*_{n}*L*=1.8751,4.6941,7.8548,10.9955,14.1372,…, and *D*_{n} is a constant decided from initial conditions.

The total energy *E*_{n} corresponding to the *n*th mode can be found by calculating the elastic energy at the instant of maximal deflection when the cantilever is momentarily stationary for e^{jωnt}=1
B4

From the law of energy equipartition, there is an averaged energy of *kT*/2 per degree of freedom for all of the relevant lateral vibration modes. Because there are both elastic and kinetic energy degrees of freedom for all of the relevant lateral vibration modes, then, on average, 〈*E*_{n}〉=*kT* holds for each vibration mode, with *E*_{n} yielding the Boltzmann distribution. It is straightforward to show that each mode of a stochastically driven oscillator has a Gaussian probability profile and the standard deviation of the vibration amplitude of the free tip of the Euler beam is given by (Krishnan *et al*. 1998)
B5
Then, the RMS amplitude of the *n*th mode at *x* reads
B6

As all vibration modes are independent, their contributions can be added incoherently. To average in coherently over all the vibration modes, one simply adds the variances to get another Gaussian distribution with the standard deviation given by
B7
So, the RMS amplitude of the stochastic vibration of the Euler beam at *x* can be obtained.

## Footnotes

- Received November 16, 2009.
- Accepted February 4, 2010.

- © 2010 The Royal Society