Structure of armchair single-wall carbon nanotubes under hydrostatic pressure

Based on the helical and rotational symmetries and Tersoff potential, the structural parameters, i.e., bond lengths and bond angles, have been investigated for armchair single-wall carbon nanotubes. The bond lengths and bond angles are determined for several radii tubes of various lengths. Results for armchair tubes show that one bond length is greater than that of the graphite while the other is smaller. Furthermore, the tube length is found to have significant effects on these bond lengths and bond angles. We have also recalculated the variation of these bonds under hydrostatic pressure. With the application of pressure, the bond lengths compress and the larger bond length decreases faster with pressure in comparison to the shorter one. As a consequence, at some critical pressure the bond lengths become equal. An analysis regarding the cross-sectional shape of the nanotubes and its pressure dependence has also been done. At some particular pressure, the first transition from circular to elliptical cross section takes place. For (10,10) tube the first transition pressure is found to be equal to $2.2\mathrm{GPa}$.

Figure 1

(a) A schematic side view of armchair SWNT and (b) a part of armchair tube indicating two types of C–C bonds, these are labeled as $b_1$ and $b_2$, and three bond angles $\alpha$, $\beta$, and $\gamma$. (c) Carbon atoms $i$, $j$, and $k$, the corresponding bonds $i-j$ and $i-k$, and bond angles ${\theta }_{ijk}$.

Figure 2

(a) Normalized bond lengths with the tube radius for armchair $(n,n)$ single-wall nanotubes, where $n=3$, 5, 7, 10, 15, 20, 30, 40, and 50. The solid line shows the behavior of $b_1$ and the broken line that of $b_2$. The dotted line shows the results using one bond length. (b) Bond angles versus $n$. (c) The curvature energy $E_c$ as a function of the tube radius.

Figure 3

(a) Normalized bond lengths $b_1∕b_0$ and $b_2∕b_0$, (b) bond angles, and (c) curvature energy $E_c$ as a function of the length to radius ratio for armchair $(n,n)$ SWNTs.

Figure 4

Normalized bond length $b_1∕b_0$ and $b_2∕b_0$ with applied hydrostatic pressure for (a) (5,5), (b) (10,10), (c) (15,15), and (d) (30,30) SWNTs.

Figure 5

Bond angles with applied hydrostatic pressure for (a) (5,5), (b) (10,10), (c) (15,15), and (d) (30,30) SWNTs.

Figure 6

Critical pressure as a function of the (a) tube radius and (b) bond length $b_c$.

Figure 7

(a) Compression along the tube axis and circumferential direction under applied pressure for (10,10) SWNT. (b) Bulk moduls as a function of the tube radius at zero pressure for (5,5), (10,10), (15,15), and (30,30) tubes.

Figure 8

Energy as a function of $n$ for armchair $(n,n)$ SWNTs having circular cross section $(b_e∕a_e=1.0)$ and various elliptical cross sections.

Figure 9

(a) Energy with pressure for (10,10) tube. AB and CD curves are corresponding to circular and elliptical cross section, respectively. First transition pressure $P_T$ as a function of (b) tube radius with elliptical aspect ratio equal to 0.99 for armchair SWNTs and (c) elliptical aspect ratio for (10,10) tube. (d) Bulk modulus versus $n$ for armchair $(n,n)$ SWNTs. Similar curves are obtained for other radii tubes.