## Abstract

Based on an equivalent continuum cylindrical shell model, we have predicted the intermittent transformation between radial breathing and flexural vibration modes in a single-walled carbon nanotube. It is found that the radial breathing and flexural vibration modes may appear intermittently, in certain circumstances, when the dominant parameters of the problem are in the instable region of the Mathieu stability diagram. The coupled nonlinear differential equations of the radial breathing and the flexural vibration modes are presented to a fourth-order approximation, which is then solved using a finite series expansion method. Intermittent transformation between radial breathing and flexural vibration modes is predicted, which may influence the physical properties of the carbon nanotube and the Raman spectroscope measurements.

## 1. Introduction

Carbon nanotubes (CNTs) have received increased attention for their distinct mechanical, chemical, thermal, optical and electronic properties (Treacy *et al*. 1996; Tans *et al*. 1998; Rao *et al*. 2000; Plentz *et al*. 2005; Liu *et al*. 2007) since the discovery of CNTs in 1991. In various applications of CNTs, their mechanical and physical properties and their designated functions are closely related to the vibration characteristics of CNTs. For example, on one hand, it was found that strong interactions between the electronic and vibrational states of a single-walled carbon nanotube (SWCNT) are important for its optical properties, which are fundamental for the operation and modelling of nanophotonic devices based on SWCNTs (Plentz *et al*. 2005). Some dominant vibration modes (e.g. radial breathing mode, RBM) in a SWCNT can be excited by passing electrons through a CNT (i.e. electric tunnelling; LeRoy *et al*. 2004). On the other hand, the vibration characteristics of CNTs have been realized using Raman spectroscopy when the photons in a CNT resonate with an electronic state in a Raman process. Attention was paid to the four most distinctive modes observed in Raman spectra, i.e. the RBM, the D band, the G band and the G′ band, as shown in fig. 4 of Dresselhaus *et al*. (2002). Lower frequency modes, e.g. the flexural mode in the circumferential plane of a SWCNT as shown in fig. 2 of Rao *et al*. (1997), received less attention in spectrum analysis.

The RBM, associated with first-order Raman scattering (Dresselhaus *et al*. 2005), is one of the most likely vibration modes of carbon atoms in a SWCNT. It is a bond-stretching, out-of-plane phonon mode in which all the carbon atoms move simultaneously in the radial direction. The RBM frequency, *ω*_{RBM}, is in the range of 120–350 cm^{−1} for SWCNTs with diameters between 0.7 and 2.0 nm. The Raman spectrum of the RBM is independent of the laser excitation and is considered to be an important means of determining nanotube diameters. A frequently used empirical relationship between RBM frequency and radius of a SWCNT, *a* (nm), is given by *ω*_{RBM}=*A*/*a*+*B*, where the parameters *A* and *B* are constants determined by Raman spectrum measurements. When interactions among CNTs in a SWCNT bundle are accounted for, then *A*=234 cm^{−1} nm and *B*=10 cm^{−1} (Milnera *et al*. 2000; Jorio *et al*. 2001), and for SWCNTs on an oxidized Si substrate, *A*=248 cm^{−1} nm and *B*=0 cm^{−1} (Dresselhaus *et al*. 2002). The above empirical relationship between *ω*_{RBM} and *a* is used to identify the existence and diameter distributions of SWCNTs in a carbon material. It has been found that the RBM may contain small non-radial components and interact with other high energy modes (Mohr *et al*. 2007), and thus weak links exist between RBM frequency and other parameters (e.g. nanotube chirality, strain level and the surrounding environment such as pressure and interactions among CNTs).

In this paper, we employed a nonlinear continuum elastic shell model to study the intermittent transformation between the RBM and the flexural vibration mode. It is found that the initial RBM vibration of a SWCNT may be transferred to flexural modes with subsequent intermittent transformations between them under certain conditions. The implications of this intermittent vibration mode transformation in nanotube dynamics are discussed.

## 2. Continuum elastic cylindrical shell model of SWCNTs

A continuum elastic cylindrical shell model has been successfully used to study the vibration of CNTs (Mahan 2002; Wang *et al*. 2004, 2005; Chico *et al*. 2006), in which the movement of the material point in a continuum model represents the movement of the centre of the Brillouin zone of a CNT. Simulation efficiency is the main advantage of using the continuum elastic cylindrical shell model to study the intermittent transformation between the RBM and the flexural vibration mode in SWCNTs. It is estimated that the first vibration mode transformation in a SWCNT occurs at a time of the order of picoseconds (10^{−12} s) or even longer, as shown in §3. However, the time steps in the molecular dynamics (MD) simulations are limited to the order of femtoseconds (10^{−15} s) by the vibration modes of the atoms. Therefore, it seems that continuum methods are more realistic for the study of long-term vibration of CNTs. On the other hand, experimental methods and instruments are not readily available to record such vibration mode transformation in the time domain, although an ultra-short light pulse technique (sub-10 fs) has been reported to excite and detect the time-domain vibrations of SWCNTs in a time duration of the order of picoseconds (Gambetta *et al*. 2006).

In this research, we focus on the radial and flexural movements of the Brillouin centre in a circumferential plane (i.e. a plane perpendicular to the longitudinal direction) of a SWCNT. When the ratio of length to radius of the SWCNT is sufficiently large (i.e. for a relatively long SWCNT), the radial and flexural movements in the circumferential plane of the cylindrical shell are reduced to a plane-strain problem. In future studies, the model is presented for plane-strain rings, which, however, can be applied to plane-stress rings when *E*/(1−*ν*^{2}) is replaced by *E*, where *E* and *ν* are Young's modulus and Poisson's ratio, respectively.

Goodier & McIvor (1964) and McIvor (1966) predicted the transformation between an initial RBM and a flexural bending mode in an elastic cylindrical shell. It was shown that the coupled dynamic equations between the radial and tangential displacements can be simplified into a typical Mathieu differential equation. The RBM couples with the flexural bending modes intermittently when the system parameters are in the instable region of the Mathieu stability diagram. Based on the method proposed in Goodier & McIvor (1964) and McIvor (1966), we add more nonlinear terms in the dynamic equations to obtain coupled equations that contain the fourth-order terms to avoid the divergence of the solution.

In order to capture the radial and in-plane tangential elastic responses of a thin plane-strain ring, a Lagrange description of the in-plane deflection is employed in a polar coordinate system, as shown in figure 1. The original point P on the mid-plane of the ring cross-section is represented by coordinates (*a*, *θ*) at time *t*=0. Point P is deformed to current position P^{*} at time *t* represented by coordinates (*r*, *ϕ*). The coordinates of the current position P^{*} can be expressed as(2.1a)and(2.1b)

Non-dimensional radial and tangential displacement variables *ς* and *ψ* are introduced, where(2.2)

The membrane strain *ϵ* and curvature *κ* are(2.3)and(2.4)in which the primes indicate derivatives with respect to angle *θ*. Under the assumptions of small deformation for a thin shell and the negligence of the fourth- and higher order terms in equation (2.3) and the third- and higher order terms in equation (2.4),1 membrane strain and curvature can be simplified to(2.5)and(2.6)respectively. The underlined third-order term in equation (2.5), which was not considered by Goodier & McIvor (1964) and McIvor (1966), is required to obtain governing equations for *ς* and *ψ* to the third order. Following the same procedure as Goodier & McIvor (1964), we obtain the following differential equations of *ς* and *ψ*, which govern the dynamic motion of the plane-strain ring:(2.7)and(2.8)in which dots above symbols mean derivatives with respect to the reduced time *τ*, where , , *ρ* is the mass density and for a plane-strain ring, where *E* and *ν* are Young's modulus and Poisson's ratio, respectively. (Note. *E*_{1} is replaced by *E* for the dynamic equations of a plane-stress ring.) The term is a non-dimensional thickness and is normally very small for thin cylindrical shells. All the underlined terms in equations (2.7) and (2.8) come from the third-order term in the membrane strain in equation (2.5). The term *ς*^{(4)} is the fourth-order partial derivative with respect to *θ* and *P* is the applied internal pressure pulse that will be substituted by a given initial radial velocity (or an initial radial deflection) in future studies.

The extra term *Λ* in equation (2.7) is(2.9)

When initial conditions are given, equations (2.7) and (2.8) can be solved numerically. Detailed numerical investigations of these nonlinear differential equations will be presented in a separate paper. The series expansion method (Goodier & McIvor 1964; McIvor 1966) is employed in this study.

According to Goodier & McIvor (1964) and McIvor (1966), pure extensional and pure inextensional (i.e. flexural) modes can be taken into account by the following series solutions of *ς* and *ψ*:(2.10)and(2.11)where terms *a*_{n}(*τ*)cos *nθ* and *c*_{n}(*τ*)cos *nθ* represent the vibrations of pure extensional and pure flexural modes, respectively. It should be noted that only cosine terms are included in equations (2.10) and (2.11) for simplicity. Furthermore, *c*_{1} cos *θ* in equations (2.10) and (2.11) is discarded since it corresponds to rigid translation.

The Lagrange equation,(2.12)where *q*_{i}, a generalized coordinate, is used to derive the governing equations of the unknown coefficient functions *a*_{0}(*τ*), *a*_{n}(*τ*) and *c*_{n}(*τ*). When fifth- and higher order terms are neglected, the kinetic energy *T* and strain energy *U* for a unit length plane-strain ring are obtained based on equations (2.5) and (2.6), i.e.(2.13)and(2.14)where(2.15)

The term *Δ* in equation (2.14) represents all the fourth-order terms consisting of both *ψ* and *ς* and their spatial derivatives, as well as *α*^{2} (*α*^{2}≪1). This term was usually completely or partially discarded in previous studies (Goodier & McIvor 1964; McIvor 1966). However, our numerical results have shown that it is essential to include a complete fourth-order term in strain energy to ensure that solutions of the dynamic equations are stable.

Substituting series expressions from equations (2.10) and (2.11) into equations (2.13) and (2.14), then applying equation (2.12) with *a*_{0}(*τ*), *a*_{n}(*τ*) and *c*_{n}(*τ*), taken as the generalized coordinate *q*_{i}, the following governing differential equations for *a*_{0}(*τ*), *a*_{n}(*τ*) and *c*_{n}(*τ*) are obtained:(2.16)(2.17)and(2.18)

We will focus on the initial transformation from RBM to flexural vibration mode and the subsequent intermittent transformations between them. Without losing generality, the following initial conditions for *a*_{0}(*τ*), *a*_{n}(*τ*) and *b*_{n}(*τ*) (*n*≥2) are used:(2.19)where *v*_{0} is the initial radial velocity of the ring, which initiates the RBM as the dominant response mode of the ring in the early stage. The term *v*_{n} represents imperfections of the initial uniform radial velocity, which is normally at least two orders smaller than *v*_{0}.

The fourth-order Runge–Kutta method is employed to solve equations (2.16)–(2.18) under the initial conditions of equation (2.19).

## 3. Numerical results and discussion

When the initial imperfections are discarded, i.e. *v*_{n}=0, *n*≥1, in equation (2.19), the RBM solution is given by *a*_{0}=*v*_{0}*c*^{−1} sin *τ*, or(3.1)and thus the RBM frequency of the plane-strain ring is or(3.2)where , in which *c*_{l}=2.998×10^{8} m s^{−1} is the speed of light. The values *A*=116.8 cm^{−1} nm when *Eh*=360 J m^{−2}, *ρh*=7.72×10^{−7} kg m^{−2} and *ν*=0.19 are taken, in which *h* is the representative wall thickness of a nanotube.2 Therefore, the RBM frequency for a (10, 10) armchair SWCNT (*a*=0.678 nm) is 171.5 cm^{−1} (or *f*_{0}=5.16 THz with a vibration period of *T*=194 fs), which agrees with other predictions (e.g. *ω*_{RBM}=169.0 cm^{−1} in Lawler *et al*. (2005) using first principle calculation and *ω*_{RBM}=165.0 cm^{−1} in Kuzmany *et al*. (1998)). It is shown that the isolated nonlinear thin-shell model cannot account for the environmental effects represented by the non-zero parameter *B* in the frequently used empirical equation *ω*_{RBM}=(*A*/*a*)+*B*.

When the initial imperfections in equation (2.19) exist, RBM is usually the dominant vibration mode.3 However, it has been shown by Goodier & McIvor (1964) that there exists an instable region in the Mathieu stability diagram defined by two non-dimensional parameters and , in which the initial RBM will be subsequently transformed to flexural modes with wavenumber *n* in equations (2.10) and (2.11). Goodier & McIvor's (1964) conclusion is based on the simplified forms of equations (2.16)–(2.18), in which the fourth-order term *Δ* and the third-order term with coefficient *α*^{2} (i.e. the underlined terms) in the strain energy integral in equation (2.14) were omitted and only the pure flexural imperfections were considered, i.e. *a*_{n}≡0 in equations (2.10) and (2.11). Our numerical studies have shown that the mode transformation criterion based on the Mathieu stability diagram is valid at least for the first transformation from RBM to flexural vibration mode when the underlined terms in equation (2.14) are included. However, the influence of the underlined terms on the subsequent intermittent transformations may not be ignored, especially when numerical simulations based on Goodier & McIvor's (1964) equations become divergent. A selected numerical example for a plane-strain ring with *a*/*h*=20.4 at initial velocity of *v*_{0}/*c*=0.0075 is presented to illustrate the significant influence of the underlined terms in equation (2.14) on the intermittent transformations between RBM and flexural mode, as shown in figures 2 and 3. A similar divergence problem is observed when McIvor's (1966) model is applied, but an inclusion of the fourth-order terms ensures the convergence of the solution. This, together with a detailed study of the coupled nonlinear equations (equations (2.16)–(2.18)), which will be presented in a separate paper, convinces us that the fourth-order terms should be included in the study of the intermittent transformation between RBM and flexural mode.

Now, we apply the present model to predict the intermittent transformation between RBM and flexural vibration mode in SWCNTs. We used values of *Eh*=360 J m^{−2}, *ρh*=7.72×10^{−7} kg m^{−2} and *ν*=0.19 that were employed in the MD simulation in Yakobson *et al*. (1996) for SWCNTs. The corresponding phonon velocity of a SWCNT is . Because the flexural vibration mode belongs to a bending mode, it is necessary to use an effective thickness to calculate the aspect ratio, , in which *h*_{eff}=0.066 nm (Yakobson *et al*. 1996; Ru 2000).

A (10, 10) armchair SWCNT is chosen as an example to show the intermittent transformation history of vibration modes. As shown in figure 4, a (10, 10) SWCNT has undergone transformation of vibration modes three times within a duration of approximately 18 ps, where *v*_{0}/*c*=0.015 and *v*_{n} is two orders smaller than *v*_{0} in the initial condition equation (2.19). The time for the occurrence of the first transformation is nearly 3 ps. Many simulations for different SWCNTs with different initial conditions have been conducted and the results show that the initiation time of the first transformation depends on two factors, *α* and *v*_{0}/*c*. Figure 4*c* gives the radial deflection of the (10, 10) armchair nanotube in both time and frequency domains. Two distinct frequencies are observed, which correspond to a RBM and a flexural vibration mode with *n*=4, as shown in figure 4*a*,*b*, respectively. The frequency of the first excited flexural vibration mode is half the RBM frequency, which supports Goodier & McIvor's (1964) prediction.

We applied available theoretical models (i.e. Goodier & McIvor (1964) and McIvor (1966) and the present model) to simulate the vibration mode transformations of armchair SWCNTs. It is found that the initial radial velocity *v*_{0} needs to be large enough in order to initiate such transformations. In these simulations, the initial imperfection velocities are at least two orders smaller than *v*_{0}. The critical values of *v*_{0} for the initiation of vibration mode transformation for a range of armchair SWCNTs are calculated and given in table 1. The first excited flexural mode and its frequency, as well as the frequency of the RBM, are also given in table 1.

The legitimacy for the application of the thin-shell equation to CNTs has been shown in many publications (e.g. Mahan 2002; Wang *et al*. 2004, 2005; Chico *et al*. 2006). The largest error comes from the local treatment of the deformation that cannot take the influences of the long-range forces of nanotube atoms into account. Wang & Hu (2005) applied non-local elastic constitutive equations and Euler and Timoshenko beam theories to study the flexural wave propagation in the longitudinal direction of a SWCNT. They compared local and non-local beam models with MD results for a (10, 10) SWCNT and found that a local Euler model gives good predictions of the phase velocity when the wavelength is greater than 20 nm, while a local Timoshenko model gives good predictions when the wavelength is greater than 3 nm. For a (5, 5) SWCNT, the minimum wavelengths for good predictions from the Euler and Timoshenko models are reduced to roughly 10 and 1.5 nm, respectively (Wang & Hu 2005); this implies that the non-dimensional number, *λ*/*h*, where *λ* is the wavelength and *h* is the thickness of the beam, may be more representative to define the valid range of non-local Euler and Timoshenko models. We employ this ratio to give an indirect verification for the valid use of the present thin-shell model to study the dynamic flexural response in the circumferential plane of a SWCNT. The minimum value of *λ*/*h* for the valid use of Euler beam theory in the longitudinal flexural wave propagation in a (10, 10) nanotube is *λ*/*h*=*λ*/2*a*=20 nm/(2×0.678 nm)=14.75, according to Wang & Hu (2005).4 Taking the (10, 10) armchair SWCNT as an example, the most likely excited flexural mode in the circumferential plane is *n*=4, which corresponds to a wavelength of 2*πa*/4=1.065 nm. We estimate that the corresponding *λ*/*h* ratio is *λ*/*h*_{eff}=1.065 nm/0.066 nm=16.14, which is greater than the 14.75 in Wang & Hu (2005).

Unfortunately, interests in the Raman active modes are focused on the frequencies between the RBM frequency (equation (3.2)) and the G′ band (approx. 2700 cm^{−1}) due to their correlations to the geometrical structures and physical features of SWCNTs. Owing to technical difficulties, only limited publications have covered the lower frequency range that may be related to the flexural vibration mode (Rao *et al*. 1997). The E_{2g} phonon mode is one of the seven most intense Raman-active modes predicted for a (10, 10) nanotube, and which was not observed in Rao *et al*. (1997). This mode belongs to a flexural bending mode with *n*=2 and has a frequency of 22 cm^{−1}. Using first principle calculations of the hydrostatic pressure effects on the vibration modes of SWCNT bundle, Wu *et al*. (2006) showed the rich non-resonant Raman spectra in the frequency range lower than the RBM frequency that may include low frequency flexural vibration modes. In addition to the lack of information on the lower frequency spectra of SWCNTs, there is very little information on the vibration characteristics of CNTs in the time domain, which is mostly useful for studying the interactions between various vibration modes. Gambetta *et al*. (2006) used resonant sub-10 fs visible pulses to generate and detect the coherent phonons in SWCNT ensembles in the time domain and produced a power spectrum. Both RBMs and G bands were observed in the power spectrum, which agree, in principle, with the Raman spectrum with He–Ne excitation. It seems that the excitation method used in Gambetta *et al*. (2006) can successfully actuate RBM-dominated vibrations in SWCNTs, which is, in principle, the same as the initial conditions used in our analyses. Thus, it is worth investigating if the excitation energy can be adjusted to meet the instability criterion given below for the occurrence of intermittent transformation between RBM and flexural vibration mode.

In order to relate the critical initial velocity for the occurrence of the intermittent transformation between RBM and flexural mode to the excitation energy of each atom in a SWCNT, we calculate the corresponding energy for each atom in a (*m*, *m*) armchair CNT. The total mass of a (*m*, *m*) armchair CNT is 2*πalhρ*, where is the selected length of the tube and *a*_{CC}=0.142 nm is the C–C bond length. The total initial kinetic energy is . A (*m*, *m*) armchair CNT with length *l* contains 4 m carbon atoms. Thus, the energy per atom is given by , which can be re-expressed by(3.3)Using *Eh*=360 J m^{−2}, *ν*=0.19 (Yakobson *et al*. 1996) and 1 eV=1.602177×10^{−19} J m^{−2}, we obtain the relationship between the initial velocity and the minimum energy per atom for the occurrence of the intermittent transformation between RBM and flexural vibration mode, i.e. (eV). Values of the excitation energy needed for each atom are given in table 1 and figure 5. An exponential fit, , is suggested, which can be used for further numerical or experimental investigations of this problem.

## 4. Conclusions

We present the possible occurrence of intermittent transformations between RBM and flexural vibration mode in a SWCNT, based on a nonlinear elastic shell mode. When the initial excitation energy is beyond a critical curve, the initial RBM is transferred to a flexural vibration mode with lower frequency. The subsequent transformation between RBM and flexural mode is intermittent. Depending on the initial excitation energy, multiple flexural vibration modes may be involved. The predicted intermittent vibration mode transformation may influence the physical properties of the CNT and the Raman spectroscope measurements, which need to be confirmed by further numerical and experimental studies.

## Acknowledgments

M.X.S. acknowledges financial support from the Dorothy Hodgkin Postgraduate Awards (Shell–EPSRC Scholarship).

## Footnotes

↵It can be shown that the third-order terms in equation (2.4) will not contribute to dynamic governing equations even when they are substituted in equation (2.6).

↵Due to the ambiguity for the definition of the wall thickness of a nanotube, we used the in-plane stiffness (Ru 2000) and the line density of the nanotube wall.

↵If the system is conservative (e.g. an elastic response), the RBM response will last infinitely. Otherwise, the amplitude of the RBM response will be reduced gradually until the available energy in the ring is completely consumed through dissipation mechanisms (e.g. damping). We ignore any energy dissipation mechanisms in this study.

↵It should be noted that the diameter of the (10, 10) nanotube is taken as the thickness because Wang & Hu (2005) considered the flexural wave along the longitudinal direction of a SWCNT.

- Received October 5, 2007.
- Accepted February 29, 2008.

- © 2008 The Royal Society