## Abstract

Owing to their unusual properties, carbon nanostructures such as nanotubes and fullerenes have caused many new nanomechanical devices to be proposed. One such application is that of nanoscale oscillators which operate in the gigahertz range, the so-called gigahertz oscillators. Such devices have potential applications as ultrafast optical filters and nano-antennae. While there are difficulties in producing micromechanical oscillators which operate in the gigahertz range, molecular dynamical simulations indicate that nanoscale devices constructed of multi-walled carbon nanotubes or single-walled carbon nanotubes and C_{60} fullerenes could feasibly operate at these high frequencies. This paper investigates the suction force experienced by either an atom or a C_{60} fullerene molecule located in the vicinity of an open end of a single-walled carbon nanotube. The atom is modelled as a point mass, the fullerene as an averaged atomic mass distributed over the surface of a sphere. In both cases, the carbon nanotube is modelled as an averaged atomic mass distributed over the surface of an open semi-infinite cylinder. In both cases, the particle being introduced is assumed to be located on the axis of the cylinder. Using the Lennard-Jones potential, the van der Waals interaction force between the atom or C_{60} fullerene and the carbon nanotube can be obtained analytically. Furthermore, by integrating the force, an explicit analytic expression for the work done by van der Waals forces is determined and used to derive an acceptance condition, that is whether the particle will be completely sucked into the carbon nanotube by virtue of van der Waals interactions alone, and a suction energy which is imparted to the introduced particle in the form of an increased velocity. The results of the acceptance condition and the calculated suction energy are shown to be in good agreement with the published molecular dynamical simulations. In part II of the paper, a new model is proposed to describe the oscillatory motion adopted by atoms and fullerenes that are sucked into carbon nanotubes.

## 1. Introduction

The discovery of carbon nanotubes by Iijima (1991) has given rise to speculation on many new potential nanodevices. Owing to the unique mechanical properties of carbon nanotubes, such as high strength, low weight and flexibility, both multi- and single-walled carbon nanotubes promise many new applications in nanomechanical systems. However, owing to a lack of theoretical understanding of their precise behaviour and also their behaviour when they interact with their environment, there remain many fundamental challenges incorporating carbon nanotubes into a system.

The interaction of carbon nanotubes and C_{60} fullerene molecules is of particular interest as it has been proposed as a possible configuration for nanoscale oscillators which operates in the gigahertz range. Cumings & Zettl (2000), Yu *et al*. (2000) and Zheng & Jiang (2002) show that the sliding of the inner shell inside the outer shell of a multi-walled carbon nanotube can generate oscillatory frequencies in the gigahertz range. While there are difficulties for micromechanical oscillators to reach a frequency in the gigahertz range, it is possible for nanomechanical systems to achieve this. Building on this work, Qian *et al*. (2001) and Liu *et al*. (2005) use molecular dynamical simulations to examine the consequences of decreasing the length of the inner core to the practical limit of a C_{60} fullerene molecule. These studies show that a C_{60} molecule located on the axis of a nanotube and a short distance away will be sucked into the nanotube and spontaneously begin oscillatory motion.

In this paper, we investigate the nature of the suction force and develop an acceptance condition which can be used to determine if the suction force for a particular configuration will result in the particle being sucked completely into the nanotube. In addition, we also provide an expression for the total suction energy imparted to the particle in the form of an increased velocity as a result of the suction force. We define the suction energy (*W*) as the total work performed by van der Waals interactions on a molecule entering a carbon nanotube. In certain cases, as detailed in §§3 and 4, the van der Waals force becomes repulsive as the entering particle crosses the tube opening. In these cases, we define the acceptance energy (*W*_{a}) as the total work performed by van der Waals interactions on the particle entering the nanotube, up until the point that the van der Waals force once again becomes attractive. In part II of this paper, a new model is proposed which describes the subsequent oscillatory motion that the particle adopts after it has been sucked into the nanotube. This model takes the initial velocity determined from the suction energy and factors into this the restoring suction force experienced at each end of the nanotube and a frictional term to provide a reasonably complete description of the oscillatory motion. The major new contribution of these papers is the use of elementary mechanical principles and classical applied mathematical modelling techniques to formulate explicit analytical criteria and ideal model behaviour in a scientific context previously only elucidated through molecular dynamical simulation. While van der Waals interactions have been calculated previously using classical approaches, these papers extend the analysis and provide explicit expressions for the acceptance condition and suction force, which have not previously appeared in the literature. The model of oscillatory motion which appears in part II of the paper is completely novel and shown to be in agreement with molecular dynamical simulations.

Our approach in this paper is to further investigate the mechanical behaviour of the van der Waals interaction between single-walled carbon nanotubes and separately both unbonded atoms and C_{60} fullerene molecules. The calculation of the atom–nanotube interaction is provided as an exposition of the method in the simplest form, which is then expanded to the case of the spherical fullerene molecule. However, the atom–nanotube model may still be applied to cases where a nanotube is interacting with a small particle which can be considered as a point mass. In the following section, we introduce the Lennard-Jones potential and the usual approach of assuming an average surface density of carbon atoms. In §3, we first determine the Lennard-Jones potential for a single atom being introduced along the axis of the carbon nanotube, and this is used to derive an acceptance energy to determine whether the atom will be sucked into the nanotube or not, and the suction energy which is a measure of the total increase in the kinetic energy experienced by the introduced atom. In §4, the same approach is applied to the case of a C_{60} fullerene. Again, both the acceptance and suction energies are determined and the results are compared with previous molecular dynamical simulation studies. Finally, in §5, conclusions are given which show good agreement with some studies but there exist some discrepancies with others.

## 2. Potential function

The non-bonded interaction energy is obtained by summing the interaction energy for each atom pair,(2.1)where *Φ*(*r*_{ij}) is a potential function for atoms *i* and *j* at distance *r*_{ij} apart. In the continuum approximation, carbon atoms are assumed to be uniformly distributed over the surface of the molecules. As a result, the double summation in equation (2.1) can be replaced by a double integral, which averages over the surfaces of each entity(2.2)where *n*_{1} and *n*_{2} represent the mean surface density of atoms on each molecule and *r* denotes the distance between two typical surface elements d*Σ*_{1} and d*Σ*_{2} on each molecule. Two empirical potentials commonly used are the Lennard-Jones potential and the Morse potential. While this paper adopts the Lennard-Jones potential to determine the van der Waals interaction force, we refer the reader to Wang *et al*. (1991) and Qian *et al*. (2002) for details of the Morse potential and its applications.

The classical Lennard-Jones potential for two atoms at a distance *r* apart is given by(2.3)where *A* and *B* are the attractive and the repulsive constants, respectively. Equation (2.3) can be written in the form(2.4)where *σ* is the van der Waals diameter. The equilibrium distance *r*_{0} is given by(2.5)and the well depth, *ϵ*=*A*^{2}/(4*B*). The Lennard-Jones potential has been used in different configurations, including the interactions between two identical parallel carbon nanotubes (Girifalco *et al*. 2000), between carbon nanotube bundles (Henrard *et al*. 1999), between a carbon nanotube and a C_{60} molecule (both inside and outside the tube) (Girifalco *et al*. 2000) and between two C_{60} molecules (Girifalco 1992). The values of interaction Lennard-Jones constants for atoms in graphene–graphene, C_{60}–C_{60} and C_{60}–graphene are shown in table 1 (Girifalco *et al*. 2000).

## 3. Interaction of an atom located on the axis of a single-walled carbon nanotube

In order to study the suction of particles into carbon nanotubes more generally, we start from the ideal situation of a single atom. This situation may not be physically meaningful but serves to demonstrate the underlying ideas. The two questions that must be addressed are: first, is the suction force sufficient to have the atom accepted into the tube?; and second, what is the magnitude of the energy imparted to the atom by this interaction? In this and §4, we demonstrate that the analysis employed in the case of the atom is applicable to the more complicated geometry of an approximately spherical molecule, and therefore the reduced complexities make it a useful and instructive exercise.

In an axially symmetric cylindrical polar coordinate system (*r*, *z*), an atom is assumed located at (0, *Z*) which might be inside or outside the carbon nanotube assumed to be of semi-infinite length, centred around the positive *z*-axis and of radius *a*. The parametric form of the equation for the surface of the carbon nanotube is (*a*, *z*), where *z*≥0. As shown in figure 1, the distance *ρ* between the atom and a typical surface element of the tube is given by *ρ*^{2}=*a*^{2}+(*Z*−*z*)^{2}. Owing to the symmetry of the problem, we are only concerned with the force in the axial direction, , where *F*_{vdW} is the van der Waals interaction force defined by(3.1)

Consequently, the interaction force between an atom located on the *z*-axis and all the atoms of the carbon nanotube is given by(3.2)where *n*_{g} is the uniform surface density of carbon atoms in a graphene structure such as a carbon nanotube. Since and , equation (3.2) becomes(3.3)

We note that is a continuous function with zeros at *Z*=±*Z*_{0}, where(3.4)and *Z*_{0} is real only when *a*≤*a*_{0}, where . In figure 2, we plot for carbon nanotubes of various radii, which illustrates that as the radius of the nanotube increases beyond *a*_{0} (in this case *a*_{0}≈3.443), the value of remains positive for all values of *Z*.

The integral of represents the work done by the van der Waals forces which are imparted onto the atom in the form of kinetic energy. For the atom to be accepted into the nanotube, the sum of its initial kinetic energy and that received by moving from −∞ to −*Z*_{0} needs to be greater than that which is lost when the van der Waals force is negative (i.e. in the region −*Z*_{0}<*Z*<*Z*_{0}). We term this the acceptance energy (*W*_{a}), which allows us to write the acceptance condition as(3.5)where *m*_{0} is the mass of the atom and *v*_{0} is its initial velocity, and(3.6)

Employing the substitution *Z*=*a* tan *ψ*, this integral is changed into the following form:(3.7)where . Evaluation of this integral gives the acceptance energy in explicit form as(3.8)

Assuming that the atom is initially at rest, the acceptance condition becomes simply *W*_{a}>0. Using the values from table 2, we can calculate the acceptance energy for various radii of nanotube which is graphed in figure 3 using the values of *Z*_{0} as graphed in figure 4. We comment that the acceptance energy is positive for tubes of radius *a*>3.276 Å. This radius value is smaller than that of a (5, 5) carbon nanotube, where we use the usual notation (*n*, *m*) and *n*, *m* are positive integers representing the helicity of a carbon nanotube. Since (5, 5) is the smallest carbon nanotube expected to be physical (Dresselhaus *et al*. 1996, pp. 769–776), we therefore conclude that all physical carbon nanotubes will accept a single atom from rest. However, for a nanotube with radius less than this size (e.g. a (7, 2) nanotube with a radius of *a*=3.206 Å), our model predicts that it would not accept an atom by suction force alone and the atom would need to possess an initial velocity for it to overcome the negative acceptance energy. We note that owing to the symmetrical nature of the restoring force that the atom would experience at the other end of a physical carbon nanotube, any initial velocity would remain intact and therefore oscillatory motion would not occur and the atom would pass straight through the carbon nanotube. We also note that when *a*>*a*_{0}, the force graph does not cross the axis and therefore *Z*_{0} is not real, in which case the atom will always be accepted by the nanotube.

Once the issue of the nanotube accepting the atom has been determined, we next consider the change in kinetic energy (i.e. velocity) owing to the van der Waals force experienced by the atom passing through the tube opening. As can be seen in figure 2, the force is only appreciable within a few tube radii either side of the tube end (|*Z*|≲10 Å) and outside of this region the van der Waals force is negligible. If we term the total work done by van der Waals interaction, the suction energy (*W*), it can be readily calculated as the total integral of from −∞ to ∞ which is a good approximation where the atom starts more than 10 Å outside of the tube end and moves to a point more than 10 Å within the nanotube. It can be seen that this is just equation (3.7) with the upper limit of the integral (*ψ*_{0}) replaced with *π*/2 and therefore by evaluation of this integral we have(3.9)

Assuming that the atom is initially at rest, the increase in its velocity (*v*) can be calculated directly from the kinetic energy formula and is explicitly given by(3.10)

We note that care must be taken when calculating *v*, as the value inside of the parentheses may be negative. In this case, the atom loses energy when entering the tube and in this situation will decelerate upon entering the tube. By differentiating equation (3.9), it is possible to calculate the tube radius *a*_{max} which will give the maximum suction energy and therefore the maximum velocity on entering the tube. This occurs for a value of radius *a*_{max} which is given by(3.11)and for our values of *A* and *B*, we have *a*_{max}≈3.739 Å. We also comment that both (6, 5) and (9, 1) nanotubes have a radius of *a*=3.737 Å which is very close to *a*_{max}.

In figure 5, we graph the suction energy for various carbon nanotubes illustrating a maximum value occurring at *a*=*a*_{max}. We also note that *W* is positive for any value of radius *a*>3.210 Å, which means that there is a range of nanotube radii 3.210<*a*<3.276 Å for which *W* is positive but *W*_{a} is negative. In other words, an atom accepted into a nanotube with a radius in this range would experience an increase in velocity. However, the atom would not be sucked in from rest owing to the magnitude of the repulsive component of the van der Waals force experienced as it crosses the tube opening. We comment that we do not expect physical nanotubes with radii falling within this range as we assume that (5, 5) is the smallest physical carbon nanotube (Dresselhaus *et al*. 1996, pp. 769–776).

## 4. Interaction of a fullerene sphere located on the axis of a single-walled carbon nanotube

In this section, we model the interaction between an approximately spherical fullerene molecule and a carbon nanotube in the continuum approximation obtained by averaging over the surface of each entity. By performing the average of the Lennard-Jones potential over the sphere, we find that the potential energy for an atom on the tube interacting with all atoms of the sphere radius *b* is given by(4.1)where the derivation of *Q*_{n} is given in appendix A with coefficients *C*_{6}=*A* and *C*_{12}=*B*, and *ρ* is the distance between a typical tube surface element and the centre of the fullerene, as shown in figure 6. Substituting from equation (A 1) and simplifying gives(4.2)where *n*_{f} is the mean surface density of carbon atoms for the fullerene molecule. From figure 6, the van der Waals interaction force between the fullerene molecule and an atom on the tube is of the form *F*_{vdW}=−∇*P*, and therefore, we have the axial force(4.3)

As a result, the total axial force between the entire carbon nanotube and the fullerene sphere is given by(4.4)and since , we have . Thus, equation (4.4) can be simplified to give(4.5)

Now by placing the fractions over common denominators, expanding and reducing to fractions in terms of powers of (*ρ*^{2}−*b*^{2}), it can be shown that(4.6)

(4.7)

Substituting these identities in equation (4.5) gives a precise expression for the *z* component of the van der Waals force experienced by a fullerene located at a position *Z* on the *z*-axis as(4.8)where . This is the corresponding expression for the sphere as equation (3.3) is for the atom. However, in this case, determining the roots of analytically is not a simple task owing to the complexity of the expression and the order of the polynomial involved. However, in general, the function for the sphere behaves very much like that for the atom as figure 7 demonstrates and there will be at most two real roots of the form *Z*=±*Z*_{0} and these roots will only exist when the value of *a* is less than some critical value *a*_{0} for some particular value of the parameter *b*. In the case of a C_{60} fullerene, if *b*=3.55 Å, then *a*_{0}≈6.509 Å. As in the previous section, the integral of represents the work imparted to the fullerene and equates directly to the kinetic energy. Therefore (as before), the integral of equation (4.8) from −∞ to *Z*_{0} represents the acceptance energy (*W*_{a}) for the system and would need to be positive for a nanotube to accept a fullerene by suction force alone. If the acceptance energy is negative, then this represents the magnitude of initial kinetic energy needed by the fullerene in the form of the inbound initial velocity for it to be accepted into the nanotube. To calculate this acceptance energy, we make the change of variable . Then, and , and the limits of the integration change to −*π*/2 and which yields(4.9)where . However, in the case of the sphere, a value of *Z*_{0} cannot be specified explicitly and must be determined numerically. Once determined, it can be substituted in the expression for *W*_{a} for any value of parameters where *a*<*a*_{0}. In figure 8, we graph the acceptance energy for a C_{60} fullerene and a nanotube of radii in the range 6.1<*a*<6.5 Å, using the values of *Z*_{0} as graphed in figure 9. We comment that *W*_{a}=0 when *a*≈6.338 Å and nanotubes which are smaller than this will not accept C_{60} fullerenes by suction force alone. Therefore, this model predicts that a (10, 10) nanotube (*a*=6.784 Å) will accept a C_{60} fullerene from rest; however, a (9, 9) nanotube (*a*=6.106 Å) will not. This shows reasonable agreement with Hodak & Girifalco (2001) who determined that a nanotube with a radius less than 6.27 Å cannot be filled with C_{60} molecules, and Okada *et al*. (2001) who interpolated a value of approximately 6.4 Å as the minimum radius for a nanotube to encapsulate C_{60} molecules. We note that although our model predicts a minimum radius of 6.338 Å, all three models agree that (9, 9) nanotubes will not accept C_{60} molecules from rest but (10, 10) nanotubes will. Qian *et al*. (2001) report that firing C_{60} molecules at speeds of up to 1600 m s^{−1} is insufficient to have them penetrate carbon nanotubes with (6, 6), (7, 7) or (8, 8) configurations. For the largest of these, (8, 8), with a radius *a*=5.428 Å, the acceptance energy predicted by our model is *W*_{a}=−252 eV. This equates to firing the C_{60} fullerene at the unlikely speed of more than 8200 m s^{−1}. Therefore, we conclude that the acceptance energy requirements predicted here are in agreement with the calculations of Hodak & Girifalco (2001) and those derived from the molecular dynamical simulation of Qian *et al*. (2001). We also note that when *a*>*a*_{0}, the force graph does not cross the axis and, therefore, *Z*_{0} is not real and in this case the fullerene will always be accepted by the nanotube.

The suction energy (*W*) for a fullerene can be determined from the same integral as the acceptance energy (4.9) except with the upper limit changed from *ψ*_{0} to *π*/2. In this case, the value for *J*_{n} becomes(4.10)where !! represents the double factorial notation such that and . Substitution and simplification gives(4.11)where . In figure 10, we plot the suction energy *W* for a C_{60} fullerene entering a nanotube with radii in the range 6<*a*<10 Å. We note that *W* is positive whenever *a*>6.27 Å and has a maximum value of *W*=3.242 eV when *a*=*a*_{max}=6.783 Å. We also comment that a (10, 10) carbon nanotube with *a*≈6.784 Å is almost exactly the optimal size to maximise *W* and therefore have a C_{60} fullerene accelerate to a maximum velocity upon entering the nanotube.

There are molecular dynamical simulations of fullerene oscillators and it is interesting to compare our findings. For a C_{60}–(10, 10) oscillator, we calculate the suction energy to be 3.243 eV. This corresponds to the C_{60} molecule being accelerated to a velocity of 932 m s^{−1}. This agrees reasonably well with Qian *et al*. (2001) whose molecular dynamical simulation demonstrates an initial velocity of around 840 m s^{−1}. For a C_{60}–(11, 11) oscillator, our model predicts an initial velocity of 798 m s^{−1}. This is in contrast to the molecular dynamical simulation of Liu *et al*. (2005) who report speeds in excess of 1200 m s^{−1}. While some of this discrepancy is attributable to the differences in the models, the disparity looks so great as to suspect the underlying validity of one of the models. In table 3, we give the suction energy (*W*) and velocity (*v*) predicted by our model for various oscillator configurations and comment that in all cases the velocities are well below those predicted by Liu *et al*. (2005).

## 5. Conclusions

This paper considers the two related problems of a single atom and a fullerene C_{60} molecule being introduced into an open single-walled carbon nanotube. The Lennard-Jones potential is used to calculate the van der Waals force. Owing to the short distance over which van der Waals forces operate, the semi-infinite tube can be used to model the open end of any carbon nanotube whose length is an order of magnitude greater than its diameter. This assumption is confirmed by the force graphs shown in figures 2 and 7. In both cases, we demonstrate that the shape of the force graph is similar with at most two real roots, the values of which are needed to determine the acceptance condition. In the case of the fullerene acceptance condition, the value of the roots must be determined numerically and standard root-finding techniques can achieve this in a straightforward manner. The total integral of the force gives the suction energy which will be imparted to the introduced atom or molecule in the form of kinetic energy and therefore the method provides an analytical solution to the problem of determining the initial velocity for an atom or fullerene molecule being introduced into a carbon nanotube.

The results are in excellent agreement with Hodak & Girifalco (2001), Okada *et al*. (2001) and Qian *et al*. (2001), and in particular the initial velocity in the molecular dynamics model of Qian *et al*. (2001) show good agreement with the value of the suction energy calculated here for a C_{60}–(10, 10) oscillator. However, there is some disagreement with the results reported by Liu *et al*. (2005), particularly in terms of the suction energy. The disagreement seems even more marked for larger nanotubes. Some of the disagreement is perhaps due to the differences in the adopted models, since here we have made the assumption that the nanotube and the fullerene are rigid bodies. However, it would seem that this only accounts for part of the discrepancy and further work is required to resolve these differences.

## Acknowledgments

The authors are grateful to the Australian Research Council for support through the Discovery Project Scheme and the provision of an Australian Professorial Fellowship for J.M.H. The authors also wish to acknowledge the help of Professor Julian Gale of Curtin University of Technology, for many helpful comments and discussions on this and other related work.

## Footnotes

- Received March 29, 2006.
- Accepted August 17, 2006.

- © 2006 The Royal Society

## References

## Notice of correction

Equation (4.10) is now presented in its correct form.

A detailed erratum will appear at the end of volume. 19 July 2007