Anomalous Young’s modulus of a nanotube

An analytic theory has been developed to predict and explain the mechanical responses of nanotubes (NTs) as their structure changes from the perspective of thermodynamics and continuum medium mechanics. It was found that the surface energy of the inner surface increases while the surface energy of outer surface decreases with decreasing diameter of NTs, which leads to the anomalous behavior of Young’s modulus of NT that increases with decreasing diameter. For the same material, the Young’s modulus of NT with thin shell is higher than that of one with thick shell, the Young’s modulus of NT is higher than that of nanowire (NW) with the same size, and both of the Young’s moduli of NT and NW are larger than that of bulk. Theoretical predictions are consistent with experiments.

Figure 1

(a) Schematic illustration of nanotube (NT) and (b) the core-shell structural model. The surface unit cell of NT is shown in (c). Here, $h$ is the diameter of an atom and $d$ and $D$ are the inner and outer diameters of NT.

Figure 2

Dependence of surface energy of NT on the diameter. Curve A is in the inner surface, while curve B is in the outer surface. The necessary parameters of copper used in the case are from Refs. 22, 29, 30: $h=0.2556\mathrm{nm}$, $H_m=13.1\mathrm{kJ}{\mathrm{mol}}^{-1}$, $S_{mb}=9.65\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$, ${\gamma }_0=1.592\mathrm{J}{\mathrm{m}}^{-2}$, $C_{11}=167.38\mathrm{GPa}$, $C_{12}=124.11\mathrm{GPa}$, $V_s=7.11{\mathrm{cm}}^3{\mathrm{mol}}^{-1}$, and $E=130\mathrm{GPa}$.

Figure 3

The relationship between the ratio of $E∕E_0$ and the diameter of NT. The thickness of NT is assumed at $1\mathrm{nm}$ in A and $2\mathrm{nm}$ in B.

Figure 4

Dependence of Young’s modulus of NT on $D$. (a) $D=5\mathrm{nm}$ in A, $D=10\mathrm{nm}$ in B, and $D=20\mathrm{nm}$ in C. The gray dots with E, F, and G denote the corresponding value of NWs.