Simple model of van der Waals interactions between two radially deformed single-wall carbon nanotubes

The van de Waals potential energy between two parallel infinitely long radially deformed single-walled carbon nanotubes is calculated within the Lennard-Jones approximation. The radial deformations are described with analytical shapes in order to facilitate the calculations. The most preferred mutual orientations are determined in all considered cases in terms of their potential well depths, equilibrium distances, and geometrical parameters. It is found that the interaction evolves in such a way as to keep the distance between the interacting surfaces comparable to the graphene-graphene distance in graphite. In addition, the universal graphitic potential concept is extended to radially deformed carbon nanotubes. These results may be used as a guide for future experiments to investigate interactions between deformed carbon nanotubes.

Figure 1

Schematic drawing of two parallel tubes with radii $R_1$ and $R_2$ separated by a distance $g$ between their surfaces. The intercenter distance is $d$, and the distance between two surface elements is $r$. The axis of each tube is along $z$, perpendicular to the $x\text{−}y$ plane.

Figure 2

Interaction potential $V$ (meV) per unit length as a function of the separation distance $d$ (Å) for several pairs of SWNTs with different radii. The equilibrium distance is found to be at $g_0\sim 3.2\mathrm{\AA }$.

Figure 3

Schematic drawing of parallel elliptically deformed tubes with major semiaxis $a_1$ and $a_2$ and minor semiaxis $b_1$ and $b_2$, separated by an intercenter distance $d$. The distance between two surface elements is $r$, and the angles $\theta$ and $\phi$ describe their relative orientation. Each CNT axis is along $z$ perpendicular to the $x\text{−}y$ plane.

Figure 4

vdW potential per unit length as a function of the distance separation $d$ between the centers of the ellipses for different values of $\phi$ for $\theta =0°$.

Figure 5

(a) Minimum interaction potential $\mid V_0\mid$ per unit length and (b) the equilibrium distance $d_0$ as a function of the angle $\phi$ for different values of $\theta$

Figure 6

(a) Model shape of the cross section of a flattened nanotube, consisting of two flat sections (similar to graphene planes in graphite) connected by two elliptical caps. (b) Schematic drawing of the two parallel flattened deformed tubes, separated by an intercenter distance $d$. The distance between two surface elements is $r$, and the angles $\theta$ and $\phi$ describe their relative orientation. Each tube axis is along $z$ perpendicular to the $x\text{−}y$ plane.

Figure 7

vdW potential per unit length as a function of the distance separation $d$ between the centers of the flattened tubes for different values of $\phi$ when $\theta =0°$.

Figure 8

(a) Minimum interaction potential $\mid V_0\mid$ per unit length and (b) the equilibrium distance $d_0$ as a function of the angle $\phi$ for different values of $\theta$.

Figure 9

(a) Model shape of the cross section of a peanutlike deformed nanotube, consisting of four smoothly connected elliptical sections. (b) Schematic drawing of two parallel peanutlike deformed tubes, separated by an intercenter distance $d$. The distance between two surface elements is $r$, and the angles $\theta$ and $\phi$ describe their relative orientation. Each tube’s axis is along $z$ which is perpendicular to the $x\text{−}y$ plane.

Figure 10

vdW potential per unit length as a function of the distance separation $d$ between the centers of the peanutlike tubes for different values of $\phi$ when $\theta =0°$.

Figure 11

(a) Minimum interaction potential $\mid V_0\mid$ per unit length and (b) equilibrium distance $d_0$ as a function of the angle $\phi$ for different values of $\theta$.

Figure 12

(a) Equilibrium intercenter distances for all the deformed shapes as a function of their relative orientation. The angle $\theta$ is kept constant $(\theta =0°)$ and the angle $\phi$ undergoes a full rotation range from 0° to 360° degrees. The intercenter distances are also shown. (b) The universal potential curves per unit length for perfect and radially deformed nanotubes; $\rho$ is the distance between the centers of the two interacting tubes when $g=0$.