Atomistic simulations of formation and stability of carbon nanorings

Atomistic simulations of the formation and stability of nanorings through the energy relaxation of geometrically folded single-walled carbon closed rings are performed using the second-generation reactive bond-order potential. It is found that the critical diameter for forming a stable nanoring can be made significantly smaller than that observed in experiments. The critical diameter for an armchair nanoring is smaller than that for a zigzag nanoring with the same nanotube diameter. The effect of torsion on a nanoring reduces its critical diameter. A large flattening of the nanotube cross section is found to be effective for the reduction in stress and stiffness of the nanoring. In addition, the instability of a nanoring always starts with the formation of short wavelength ripples on the compressed side of the nanotube. Subsequently, some ripples will develop into buckles, resulting in buckling failures.

Figure 1

Critical nanoring diameter $D_c$ vs nanotube diameter $d$ for armchair configurations without torsion, armchair configurations with torsion and zigzag configurations with torsion. For the same nanotube diameter, the addition of torsion is of a positive effect for nanoring stability and the critical diameter with the armchair type is smaller than that with the zigzag type.

Figure 2

The deformation sequence of an armchair (5,5) nanoring with an initial diameter $D=7.86\mathrm{nm}$ without torsion during relaxation. (a) Before relaxation, (b) the shrinkage of nanoring diameter and the flattening of the cross section, (c) the appearance of ripples, and (d) the development of buckles.

Figure 3

Number of ripples $N$ vs the nanoring diameter $D$ of the (6,6) type without torsion. The diameters of the nanorings in the inset are 5.89, 7.86, 9.82, and 11.79 nm, respectively. A linear relationship is observed between the number of ripples $N$ and the nanoring diameter $D$.

Figure 4

Final stable configuration of the (6,6) ring with a ring diameter of 7.86 nm with torsion. The stable ring adopts a regular polygonal shape. Large curvature changes are concentrated at the apexes to accommodate most of the bending deformation, while the sides of the polygon remain nearly straight and thus deform only slightly.

Figure 5

Deformation sequence of a zigzag (7,0) nanoring with an initial diameter $D=7.86\mathrm{nm}$ with torsion during relaxation. (a) Before relaxation, (b) the shrinkage of nanoring diameter and the flattening of the nanotube cross section, (c) the appearance of apexes, and (d) the development of buckles. The apex spacing with torsion is larger than the ripple spacing without torsion.

Figure 6

Unstable configuration at ripple formation for the same initial ring as Fig. 4 but without torsion. The spacing of apexes in Fig. 4 is much larger than the spacing of ripples without torsion shown here.

Figure 7

Flattened cross sections for different nanotube configurations and nanoring diameters $D$. (a) (4,4) and $D=7.86\mathrm{nm}$, (b) (4,4) and $D=15.72\mathrm{nm}$, (c) (5,5) and $D=11.79\mathrm{nm}$, (d) (5,5) and $D=15.72\mathrm{nm}$, (e) (6,6) and $D=7.86\mathrm{nm}$, and (f) (6,6) and $D=11.79\mathrm{nm}$. The cross sections for (a), (b), (c), and (d) are the final stable configurations while the cross sections for (e) and (f) are the configurations just prior to the ripple formation. It can be seen that the distance between the two flattening nanotube walls is approximately constant and is in fact 0.34 nm.

Figure 8

Critical cross-sectional shape of an elastic tube under bending for instability predicted by elastic shell theory. The dotted circle stands for the initial shape of an elastic tube cross section with diameter $d$. The solid ellipse stands for the critical cross section for bending instability; the major axis is $9d∕7$ and the minor axis is $7d∕9$ Ref. 24.