Axisymmetric and beamlike vibrations of multiwall carbon nanotubes

Several vibration problems of multiwall carbon nanotubes (MWNTs) are studied in detail based on a multiple-elastic shell model. According to recent data available in the literature, an updated value of bending stiffness for single-wall carbon nanotubes (SWNTs) is suggested, which is in a much better agreement with atomistic model for phonon-dispersion relation of SWNTs. For axisymmetric vibrations (with circumferential wave number $n=0$), it is found that longitudinal $\left(L\right)$ modes of individual tubes of a MWNT have almost identical frequencies and are usually coupled with each other through Poisson-ratio effect-induced radial $\left(R\right)$ vibrations and interlayer van der Waals interaction. Especially in the transition zone of $R$- and $L$ modes, the significant Poisson-ratio effect leads to mixed $R–L$ modes with comparable longitudinal and radial displacements. On the other hand, for beamlike vibrations (with $n=1$), the present multiple-shell model is found to be in good agreement with the multiple-beam model for almost coaxial bending $\left(B\right)$ modes of large- and small-radius MWNTs and noncoaxial $B$ modes of small-radius MWNTs (e.g., of the outermost radius less than $2\mathrm{nm}$), with relative errors less than 10%. However, for high-order noncoaxial modes of large-radius MWNTs, the relative errors between the two models increase up to 50% in extreme cases due to larger non-beamlike deformation of the cross section while both models give similar overall vibration modes through the entire length of MWNTs. In particular, for lower circumferential wave numbers $(n=0–10)$, the lowest frequency always corresponds to the minimum half-axial wave number $m=1$ for simply supported end conditions. When the wave vector decreases, the lowest frequency decreases and the associated mode shifts from an $R$ mode with larger $n$ to a coaxial $B$ mode with $n=1$.

Figure 1

The comparison between the continuum model (Ref. 13) and the multiple-shell model with $D=2\mathrm{eV}$ for phonon-dispersion relations of SWNT (10, 10). Here, $K(=m\pi ∕L)$ is wave vector, $r$ is the radius of SWNT (10, 10), and $f_{\mathit{RBM}}$ is RBM frequency of the SWNT.

Figure 2

The comparison between the atomistic model (Ref. 4) and the multiple-shell model with (a) $D=2\mathrm{eV}$ and (b) $D=0.85\mathrm{eV}$ for phonon-dispersion relations of SWNT (10, 10). Here, $K(=m\pi ∕L)$ is wave vector and $T$ represents the shortest repeat distance between two atomic cells along axis of the SWNT.

Figure 3

Axisymmetric mode frequencies $(n=0)$ of (a) example 1; (b) example 3; (c) example 2; and (d) example 4.

Figure 4

Amplitude ratios associated with frequencies (a) $L_1$ & $R_1$ (inset) and (b) $L_2$ & $R_2$ (inset) shown in Fig. 3a for example 1.

Figure 5

Comparison between the multiple-beam model and the multiple-shell model when $n=1$ for (a) coaxial $B$ mode and noncoaxial $R$ mode of example 1 ($\alpha =2$, Ref. 8); (b) coaxial and noncoaxial $B$ modes of example 3 ($\alpha =2$, Ref. 8); (c) coaxial $B$ mode and four noncoaxial $R$ modes of example 2 $(\alpha =3)$; and (d) coaxial and noncoaxial $B$ modes, and noncoaxial $R–T$ combined modes of example 4 $(\alpha =3)$.

Figure 6

Amplitude ratios associated with (a) frequency $B$ and (b) frequency $R$ shown in Fig. 5a for example 1, and (c) frequency $B_1$ and (d) frequency $B_2$ shown in Fig. 5b for example 3.

Figure 7

Dependence of six frequencies on $K{r}_2$ for example 1. For a given $n$ $(n⩾1)$, the lowest frequency, the second lowest frequency, …, and the highest frequency are shown in (a), (b), …, and (f), respectively. ($R$: radial mode, $L$: longitudinal mode, $T$: torsional mode, $B$: bending mode, and $R–T$: radial and torsional combined mode).

Figure 8

Dependence of six frequencies on $K{r}_2$ for example 3. For a given $n$ $(n⩾1)$, the lowest frequency, the second lowest frequency, …, and the highest frequency are shown in (a), (b), …, and (f), respectively. ($R$: radial mode, $L$: longitudinal mode, $T$: torsional mode, $B$: bending mode, and $R–T$: radial and torsional combined mode).

Figure 9

The lowest frequency and associated modes for examples 1 to 4.