Oscillation of nested fullerenes (carbon onions) in carbon nanotubes

Abstract

Nested spherical fullerenes, which are sometimes referred to as carbon onions, of I _h symmetries which have N(n) carbon atoms in the nth shell given by N(n) = 60n ² are studied in this paper. The continuum approximation together with the Lennard-Jones potential is utilized to determine the resultant potential energy. High frequency nanoscale oscillators or gigahertz oscillators created from fullerenes and both single- and multi-walled carbon nanotubes have attracted much attention for a number of proposed applications, such as ultra-fast optical filters and ultra-sensitive nano-antennae that might impact on the development of computing and signalling nano-devices. Further, it is only at the nanoscale where such gigahertz frequencies can be achieved. This paper focuses on the interaction of nested fullerenes and the mechanics of such molecules oscillating in carbon nanotubes. Here we investigate such issues as the acceptance condition for nested fullerenes into carbon nanotubes, the total force and energy of the nested fullerenes, and the velocity and gigahertz frequency of the oscillating molecule. In particular, optimum nanotube radii are determined for which nested fullerenes oscillate at maximum velocity and frequency, which will be of considerable benefit for the design of future nano-oscillating devices.

Appendices

Appendix A

Interaction of carbon atom and spherical fullerene

To obtain the interaction energy for nested fullerenes, we first consider the interaction between a spherical fullerene and a single carbon atom at a point P inside the fullerene, as shown in Fig. 11. We assume that the carbon atoms on the fullerene are uniformly distributed over the surface of the molecule. Assuming that the atom is located at the point P as shown in Fig. 11, we find that the potential for a carbon atom interacting with a spherical fullerene is given by

$$ E^\ast = -Q_6 + Q_{12}, $$

(A.1)

where Q _n (n = 6, 12) are defined by

$$ Q_n = C_n \eta_{f1} \int_\Sigma \frac{1}{p^n}d\Sigma, $$

(A.2)

where p denotes the distance from a carbon atom to a typical surface element dΣ of the spherical molecules. The constants C ₆ and C ₁₂ are the Lennard-Jones potential constants A and B respectively, and η _f1 represents the surface density of carbon atoms on the spherical fullerene. This approach has been used in Ruoff and Hickman (1993), for which their calculation is based on Mahanty and Ninham (1976), to determine the interaction of a spherical fullerene and a graphite plane. For an atom located inside the spherical molecule of radius b as shown in Fig. 11, the distance p is given by p ² =  b ² +  r ² +  2brcos ϕ, for which we have p dp =  −brsin ϕ dϕ. Thus (A.2) becomes

$$ \begin{aligned} Q_n =& C_n\eta_{f1} \int^{\pi}_0 \frac{2\pi b\sin\phi}{p^n}b d\phi =-\frac{2\pi bC_n\eta_{f1}}{r}\int^{b-r}_{b+r}\frac{1}{p^{n-1}}dp\\ =& \frac{2\pi bC_n\eta_{f1}}{r(2-n)}\left\{ \frac{1}{(b+r)^{n-2}} - \frac{1}{(b-r)^{n-2}}\right\}. \\ \end{aligned} $$

(A.3)

Fig. 11

Atom inside a spherical fullerene

As a result, we find from (A.1) that the potential energy of a spherical fullerene interacting with a carbon atom is given by

$$ E^\ast = \frac{2\pi\eta_{f1}b}{r} \left\{\frac{A}{4}\left(\frac{1}{(b+r)^4} - \frac{1}{(b-r)^4} \right) - \frac{B}{10}\left(\frac{1}{(b+r)^{10}} - \frac{1}{(b-r)^{10}} \right) \right\}. $$

(A.4)

For an atom outside a spherical fullerene as shown in Fig. 12 with p given by p ² =  b ² +  r ² − 2brcos ϕ, we find that following the calculation given for the case of an interior atom, the interaction energy is also found to be governed by (A.4). Upon evaluating E ^* (A.4) numerically for various size of fullerenes, we find that the atom has a preferred position, which is where its energy is minimum which is at an inter-atomic distance from the surface of the fullerene. For both cases of atom either inside or outside a spherical fullerene, the atom is most likely to be at approximately 3.4 Å away from the inner or outer surface of the fullerene respectively.

Fig. 12

Spherical fullerene interacting with a carbon atom