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Yu, P. The Parallel Single-Walled Carbon Nanotubes. Encyclopedia. Available online: https://encyclopedia.pub/entry/17019 (accessed on 08 August 2024).

Yu P. The Parallel Single-Walled Carbon Nanotubes. Encyclopedia. Available at: https://encyclopedia.pub/entry/17019. Accessed August 08, 2024.

Yu, Peishi. "The Parallel Single-Walled Carbon Nanotubes" *Encyclopedia*, https://encyclopedia.pub/entry/17019 (accessed August 08, 2024).

Yu, P. (2021, December 13). The Parallel Single-Walled Carbon Nanotubes. In *Encyclopedia*. https://encyclopedia.pub/entry/17019

Yu, Peishi. "The Parallel Single-Walled Carbon Nanotubes." *Encyclopedia*. Web. 13 December, 2021.

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For small structures on the scale of nanometers, the intermolecular van der Waals (vdW) interaction can play a leading role in some cases. Since their discovery, carbon nanotubes (CNTs) have shown great application prospects in various fields with their excellent physical and mechanical properties.

vdW interaction
cohesive energy
vibration

The molecular dynamics (MD) simulation software LAMMPS (Large-scale Atomic Molecular Massively Parallel Simulator) ^{[1]} has been widely used to predict the remarkable thermomechanical properties of CNTs, such as the influence of grain boundaries on the mechanical properties of polycrystalline carbon nanotubes ^{[2]} and the transversely isotropic thermal properties of carbon nanotubes containing vacancies ^{[3]}^{[4]}. The existing research results show that in a micro system composed of carbon nanotubes (CNTs) bundles, the van der Waals (vdWs) interaction between adjacent carbon tubes has a great impact on the mechanical behavior of the system ^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}.

With the influence of finite size and boundary effects, the internal vdW bonding energy has an important impact on the mechanical behavior of CNT bundle micro systems ^{[11]}^{[12]}^{[13]}. Therefore, clarifying the bond energy between tube bundles of finite size has important guiding significance for the design and application of microsystems ^{[14]}^{[15]}^{[16]}^{[17]}^{[18]}^{[19]}^{[20]}.

In previous work, Zhao et al. obtained the analytical solution of the bonding energy per unit length of infinite parallel single-walled CNTs based on the continuous model, and analyzed the influence of CNT diameter on the bonding energy and the equilibrium distance of the interface ^{[8]}. However, due to the limited length of CNTs in actual microsystems, the scale effect of length on the cohesive energy between tubes needs to be further explored.

Zhang et al. ^{[12]} prepared suspended single-walled CNT array devices and found that this structure has unique mechanical behavior. Based on the molecular mechanics model, Chang ^{[13]} proposed an anisotropic shell model to reveal the mechanical properties of CNTs, and studied the young’s modulus, Poisson’s ratio and radial breathing mode of single-walled CNTs, which laid a foundation for further study of the influence of the bonding energy between parallel carbon tubes on microsystems. Rueckes et al. ^{[21]} studied a suspended single-walled CNT array and found that the array has good characteristics of switchable and bi-stable device elements, and the mechanical stability of this structure is determined by the vdW interaction between single-walled CNTs. Wang and Hu further studied the thermodynamic vibration between monolayer graphene sheets based on the nonlocal elastic plate model ^{[22]}. Parallel CNT systems have prospective application as micromechanical systems.

For nano-mechanical systems, the shear deformation effect has been studied recently in the literature. Al-Furjan et al. presented the vibrational characteristics of a rotating orthotropic piezoelectric nanodisk ^{[23]}. Li et al. analyzed and tested the quasi-static compression and hygrothermal stability of BMI/CE co-cured composite lattice cylindrical shell ^{[24]}. Zhang et al. designed and fabricated an ultra-lightweight beam string structure made of carbon fibre-reinforced polymer (CFRP) composites ^{[25]}. Dai et al. investigated the vibrations of non-polynomial viscoelastic composite open-type shells under residual stresses ^{[26]}. Zhang et al. dealt with the vibration and low-velocity impact responses of functionally graded graphene nanoplatelet-reinforced composite panels on a viscoelastic foundation ^{[27]}.

The vdW energy variation with the distance between two carbon atoms is shown in **Figure 1**. For *r* < *r _{0}*, the repulsive force is dominant, defining the repulsive domain, whereas the attractive force is dominant for

The continuum model and coordinate system of two parallel single-walled CNTs are established as shown in **Figure 2**a. The radii of the CNTs are *r*_{1} and *r*_{2} respectively, and the shortest distance between parallel CNTs is *h*.

Molecular dynamics (MD) simulations were conducted using the LAMMPS software package ^{[1]} to verify the analytical model, as shown in **Figure 2**b. The adaptive intermolecular reactive empirical bond order (AIREBO) potential was used ^{[28]}, which evaluates the covalent carbon–carbon bonding interaction by the well-established REBO potential, and the non-bonded van de Waals interaction was described by the 12-6 Lennard-Jones (LJ) potential (σ = 0.34 nm and ε = 0.0028 eV) ^{[29]}^{[30]}. ReaxFF was mainly used for the chemical reaction atomic simulations, as the Tersoff potential cannot describe the non-bonded van de Waals interactions between CNTs. The AIREBO potential function with an LJ cut-off radius of 10.2 Å was adopted in all the simulations ^{[31]}^{[32]}. The Polak–Ribiere version of the conjugated gradient algorithm ^{[33]} was used to optimize the initial positions of atoms. The temperature of system was controlled using Nose–Hoover thermal bath coupling ^{[34]}^{[35]} (coupling constant 0.1 ps, time step 0.5 fs) for 0.5 ns. The time step used in the simulations was set at 0.5 fs.

For two parallel CNTs with different tube diameters, the vibration frequencies are predicted by the analytical model as shown in **Figure 3**. The resonance frequency in **Figure 3**a for low-frequency vibration decreases with the increase in tube diameter. For high-frequency vibration in **Figure 3**c the resonance frequency reaches the maximum value when r_{2} = (20,20), which is among the several prediction results including r_{2} = (5,5)~(50,50). The amplitude ratio in **Figure 6**b,d show remarkable dependence and significant nonlinearity with the change in the diameters of the two CNTs.

The influence of vibration modes (takes the first three orders into account) on the resonant frequency for the bi-CNT system with different tube diameters was obtained and is shown in **Figure 4**. It can be seen from **Figure 4**a,c that the order has a significant influence on the resonant frequency of carbon nanotubes, which show an increasing tendency in the resonant frequency with the increase in order. The amplitude ratio presents a decreasing dependence on the order, as shown in **Figure 4**b,d. This confirms that the system is nonlinear because the vdW interaction is nonlinear.

The proposed model focuses on the vibration of CNTs with large slenderness ratios. Thus, the deformation of the cross section is neglected in the continuum beam model. However, if the prerequisite of the large slenderness is not satisfied (for short CNT with large radius), the geometric nonlinearity would be influential on the vibration properties and cannot be neglected. The scope of applications of the continuum mechanics model at the nano scale is always an important topic and has been the subject of many studies in the field ^{[6]}^{[8]}^{[36]}.

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