# Vibration of Piezoelectric ZnO-SWCNT Nanowires

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

^{3}[22]) are the mass densities of SWCNT (per unit area of lateral surface) and the ZnO layer (per unit volume), respectively; $\mathsf{\lambda}$ is the radius coordinator, z the axial coordinator and $\mathsf{\theta}$ the angular coordinator of the cylindrical polar system. Here the subscripts ‘cnt ‘and ‘zno’ represent the parameters of SWCNT and the ZnO layer. In addition, $r=\sqrt{{\mathsf{\gamma}}^{2}+{R}^{2}-2R\mathsf{\gamma}\mathrm{cos}{\mathsf{\theta}}_{zno}+{\left({z}_{cnt}-{z}_{zno}\right)}^{2}}$ represents the distance between the differential element on the CNT and the one on the ZnO coating layer. The interphase vdW pressure p can then be calculated by:

_{11}(207 GPa) is the elastic modulus, e

_{31}(−0.51 $\mathrm{C}\cdot {\mathrm{m}}^{-2}$) is the piezoelectric constant and k

_{33}(7.88 $\times {10}^{-11}\mathrm{F}\cdot {\mathrm{m}}^{-1}$) is the dielectric constant of ZnO [25]. In addition, the effective axial force acting on the ZnO coating layer is [8]:

_{1}and A

_{2}are the vibration amplitudes of the SWCNT and the ZnO layer, respectively, m is the number of half wave number (or mode number) of the vibration, x is the axial coordinator, $\mathsf{\omega}$ is the angular frequency and i is the imaginary unit. Substituting solution (6) into Equation (4) leads to a system of algebraic equations:

**T**he condition for nonzero solutions of A

_{1}and A

_{2}reads $\mathrm{det}M=0$. Solving this characteristic equation yields the frequency $f=\mathsf{\omega}/2\mathsf{\pi}$ which is a function of the mode number m, the applied voltage U and the vdW interaction coefficient c.

## 3. Results and Discussions

#### 3.1. Influence of the Piezoelectric Effect

#### 3.2. Effect of Interphase VdW Interaction

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of the hybrid nanowire (HNW) structure where a single-wall carbon nanotube (SWCNT) of radius R is coated by a cylindrical layer of ZnO of thickness t. The SWCNT and ZnO layer are bonded via the van der Waals (vdW) interaction with equilibrium interspacing s.

**Figure 2.**Frequencies of piezoelectric HNWs with interphase vdW interaction where the voltage applied is in the range of (−0.2 V, 0.003 V), and the inner SWCNT has a radius (

**a**) R = 0.68 nm and (

**b**) R = 2.51 nm. The insets show the results for U = 0.001, 0.002 and 0.003 V.

**Figure 3.**Frequency ratio $f/{f}_{0}$ calculated at U = −0.2, −0.1, 0 V. The inset shows the corresponding results associated with U = 0.001, 0.002, 0.003 V.

**Figure 4.**Frequencies calculated without considering the interphase vdW interaction for the HNWs where the SWCNT radius R is (

**a**) 0.68 nm and (

**b**) 2.51 nm, respectively.

**Figure 5.**Frequency ratio ${f}_{vdw}/{f}_{n-vdw}$ calculated for the HWNs with the SWCNT radius R equal to (

**a**) 0.68 nm and (

**b**) 2.51 nm.

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**MDPI and ACS Style**

Xiao, Y.; Wang, C.; Feng, Y. Vibration of Piezoelectric ZnO-SWCNT Nanowires. *Nanomaterials* **2016**, *6*, 242.
https://doi.org/10.3390/nano6120242

**AMA Style**

Xiao Y, Wang C, Feng Y. Vibration of Piezoelectric ZnO-SWCNT Nanowires. *Nanomaterials*. 2016; 6(12):242.
https://doi.org/10.3390/nano6120242

**Chicago/Turabian Style**

Xiao, Yao, Chengyuan Wang, and Yuantian Feng. 2016. "Vibration of Piezoelectric ZnO-SWCNT Nanowires" *Nanomaterials* 6, no. 12: 242.
https://doi.org/10.3390/nano6120242