As shown in

Figure 1, a HNW is comprised of an inner single CNT (SWCNT) of radius

R and a ZnO coating layer of thickness

t. The length

L of the HNW is assumed to be the same as those of the SWCNT and ZnO layer (

Figure 1). The SWCNT interacts with the ZnO layer with the interphase vdW interaction which is characterized by the Lennard-Jones (L-J) 6–12 potential.

where

$\sqrt[6]{2}\mathsf{\sigma}$ is the equilibrium distance between the atoms,

$\mathsf{\epsilon}$ is the depth of potential and

r is the distance between two reference points. For the ZnO-CNT interaction the two constants can be estimated by

${\mathsf{\sigma}}_{\mathrm{ZnO}-\mathrm{CNT}}=\frac{1}{2}\left({\mathsf{\sigma}}_{\mathrm{ZnO}-\mathrm{ZnO}}+{\mathsf{\sigma}}_{\mathrm{C}-\mathrm{C}}\right)$ and

${\mathsf{\epsilon}}_{\mathrm{ZnO}-\mathrm{CNT}}=\sqrt{{\mathsf{\epsilon}}_{\mathrm{ZnO}-\mathrm{ZnO}}{\text{\xd7}\mathsf{\epsilon}}_{\mathrm{C}-\mathrm{C}}}$ where the constants of the L-J potential between C atoms are

${\mathsf{\sigma}}_{\mathrm{C}-\mathrm{C}}=4\mathrm{A}$ and

${\mathsf{\epsilon}}_{\mathrm{C}-\mathrm{C}}=0.0065\mathrm{eV}$ [

19]. The constants of ZnO are not available but

$\mathsf{\sigma}$ and

$\mathsf{\epsilon}$ of Au–O, Mg–O, Al–O, Ca–O, Fe–O, Si–O, K–O, Na–O are found to fall in the range of (3.67A, 2.267A) and (0.0024eV, 0.0180eV) [

19]. It is thus reasonable to assume that

${\mathsf{\sigma}}_{\mathrm{ZnO}-\mathrm{ZnO}}$ and

${\mathsf{\epsilon}}_{\mathrm{ZnO}-\mathrm{ZnO}}$ are of the order of magnitude 3A and 0.01eV, respectively. Then based on the above mentioned equations we obtained

${\mathsf{\sigma}}_{\mathrm{ZnO}-\mathrm{C}}\approx 3.5\mathrm{A}$ and

${\mathsf{\epsilon}}_{\mathrm{ZnO}-\mathrm{C}}\approx 0.008\mathrm{eV}$. Following a similar procedure in [

13] the cohesive potential at the CNT-ZnO interface due to the vdW interaction can be derived as:

where

${\rho}_{cnt}$ (

$\left(2{.27\mathrm{g}/\mathrm{cm}}^{3}\right)\times 0.34\mathrm{nm}$ [

20,

21]) and

${\rho}_{zno}$ (5.61g/cm

^{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:

where

${s}_{eq}$ denotes the equilibrium inter-phase spacing associated with the lowest value of

$\mathrm{\Phi}$ (and zero vdW interaction), and

$2\mathsf{\pi}RL$ is approximately equal to the area of the SWCNT-ZnO interface provided that the HNW is long and thin, i.e.,

$L>20R$. For a very small deviation from the equilibrium interphase spacing

${s}_{eq}$ the vdW pressure

$p\approx c\left(s-{s}_{eq}\right)=c\left({w}_{cnt}-{w}_{zno}\right)$ where

${w}_{cnt}$ and

${w}_{zno}$ represent the transverse deflection of the inner SWCNT and the outer ZnO layer, respectively. The vdW interaction coefficient

$c=1.136\times {10}^{11}$ N/m and

$c=0.943\times {10}^{11}$ N/m can be evaluated based on Equations (2) and (3) for the SWCNTs of radius 0.68 nm and 2.51 nm, respectively. As will be shown below the coating thickness

t > 10 nm is considered. In this case the influence of thickness

t on

c is small and thus was not considered in the present work.

In this work we considered long and thin HNWs where the constituent SWCNT and coating layer can be treated as two Euler beams whose dynamic equations are coupled via the vdW interaction terms. In addition, as shown in

Figure 1 an electrical voltage

U was applied in the transverse direction to the HWN.

here the equivalent bending stiffness of the SWCNT is

${\left(EI\right)}_{cnt}=\mathsf{\pi}\cdot {E}_{cnt}\left({R}^{2}+{\left(\frac{{t}_{cnt}}{2}\right)}^{2}\right)R\cdot {t}_{cnt}$ where

${E}_{cnt}=3.5\mathrm{TPa}$ is the equivalent Young’s modulus and

${t}_{cnt}=0.1\mathrm{nm}$ is the effective thickness of SWCNTs [

23,

24]. The bending stiffness of the ZnO beam is

${\left(EI\right)}_{zno}=\frac{\pi {d}^{4}}{64}\left(1-{(1-\frac{2t}{d})}^{4}\right)\left({c}_{11}+\frac{{e}_{31}^{2}}{{k}_{33}}\right)$ [

8], where

d $=2\left(R+s+t\right)$ is the diameter of the HNW; c

_{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]:

where

${\mathsf{\sigma}}_{x}^{0}$ is the residual axial stress in the ZnO layer, which is assumed to be zero in the present study. In Equation (4),

$\mathsf{\tau}$ is time,

${\left(\rho S\right)}_{cnt}=2\mathsf{\pi}R{t}_{cnt}{\rho}_{cnt}$ and

${\left(\rho S\right)}_{zno}=\mathsf{\pi}\left(d-t\right)t\cdot {\rho}_{zno}$ are the mass densities per unit length of the SWCNT and the ZnO coating layer respectively.

For simply supported HNWs the solution to Equation (4) takes the form:

where

A_{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: