Thermally induced nonlinear vibration of single-walled carbon nanotubes

The thermally induced nonlinear vibration of single-walled carbon nanotubes (SWNTs) was investigated, focusing on the peak broadening and/or multiple peaks of the frequency spectra. From Poincaré maps of SWNT tip trajectories, we observed two distinguishable patterns of vibration: planar and nonplanar (whirling) motion which appeared repeatedly. The alternation of the pattern is sometimes accompanied by a change of rotational direction. Proposed approximate solutions of nonlinear beam equations can reproduce well the alternation of the patterns for both cantilevered and suspended SWNTs. By using this analytical approach, we have found that multiple peaks of experimentally observed frequency spectra are due to the repeated and sudden alternations of two modes, induced by nonlinear effects.

Figure 1

Visualization of motion from MD simulation. The averaged coordinate of each carbon ring perpendicular to tube axis is amplified five times to clearly visualize the motion: (a) schematic figure of the system; (b) nonplanar motion; (c) planar motion.

Figure 2

Fixed boundary using LJ potential spring. Each atom has three springs along Cartesian axis and the force is calculated using LJ potential function with the coefficients as in Table S2 in Supplemental Material B [43].

Figure 3

Schematic figure to measure the rotation exchange.

Figure 4

Poincaré map of $(5,5)$ SWNT with $7.6\mathrm{nm}$ length at $300\mathrm{K}$ from MD (1)–(8), low-pass filtered results (a)–(h), and analytical solution (i)–(viii); (1) $2.5–3\mathrm{ns}$, (2) $6.5–7\mathrm{ns}$, (3) $10–10.5\mathrm{ns}$, (4) $15–15.5\mathrm{ns}$, (5) $19–19.5\mathrm{ns}$, and (6) $22.5–23\mathrm{ns}$ in MD; (a) $2.5–3\mathrm{ns}$, (b) $6.5–7\mathrm{ns}$, (c) $10–10.5\mathrm{ns}$, (d) $15–15.5\mathrm{ns}$, and (f) $22.5–23\mathrm{ns}$ in low-pass filtered; (i) $2.5–3\mathrm{ns}$, (ii) $6.5–7\mathrm{ns}$, (iii) $10–10.5\mathrm{ns}$, (iv) $15–15.5\mathrm{ns}$, (v) $19–19.5\mathrm{ns}$, and (vi) $22.5–23\mathrm{ns}$ in alaytical solution.

Figure 5

Angular velocity plots by MD for (5,5) SWNT 7.6 nm; (a) $50\mathrm{K}$, (b) $100\mathrm{K}$, and (c) $300\mathrm{K}$.

Figure 6

Comparison of motion type exchange in Poincaré map; N.P. and P. indicate nonplanar and planar motion period. Tr. is transient period to inverse the rotational direction; (a) MD simulation and (b) analytical solution. The motion at (i)–(vi) is compared in Fig. 4.

Figure 7

Frequency response of first transverse mode of MD and analytical solution results along the $x$ and $y$ axes; black line (hollow square) is $50\mathrm{K}$, green line (cross) is $100\mathrm{K}$, and red line (hollow circle) is $300\mathrm{K}$ of (5,5) $7.6\mathrm{nm}$ case: (a) MD simulation along the $x$ axis. (b) Analytical solution along the $x$ axis. (c) MD simulation along the $y$ axis. (c) Analytical solution along the $y$ axis.

Figure 8

$C_0$ plot of the results from $50\mathrm{K}$ (square), $100\mathrm{K}$ (cross), and $300\mathrm{K}$ (hollow circle) for cantilevered and suspended case with fitted result.

Figure 9

$C_0$ fitting for comparison with experimental results for Eichler et al. [6] (cross) and Sazonova et al. [18] (circle) and triangle is MD result with suspended SWNT.

Figure 10

Resonance frequency change for each temperature in MD simulation: (a) cantilevered SWNT (5,5) $7.6\mathrm{nm}$; (b) cantilevered SWNT (5,5) $23.5\mathrm{nm}$.

Figure 11

Comparison of frequency response from the result of Eichler et al. [6] (black line, upper) and from analytical solution (red line, lower): The suspended CNT with $L=2\mu \mathrm{m}$ and $\mathrm{diameter}=1.5\mathrm{nm}$ with thickness $0.0706\mathrm{nm}$. Fitting parameters are in Supplemental Material B [43]. * mark represents the peak from noise: (a) with the amplitude from free thermal motion at $50\mathrm{mK}$; (b) with the amplitude at $1\mathrm{K}$.

Figure 12

Comparison of frequency response from the experiment done by Sazanova et al. [18] (black) and that from analytical solution (red): The suspended CNT with $L=1.75\mu \mathrm{m}$ and $\mathrm{diameter}=1\mathrm{nm}$. Parameter for analytical solution is in Supplemental Material B [43].