## Abstract

Carbon nanotubes are nanostructures that promise much in the area of constructing nanoscale devices due to their enhanced mechanical, electrical and thermal properties. In this paper, we examine a gigahertz oscillator that comprises a carbon nanotube oscillating in a uniform concentric ring or bundle of carbon nanotubes. A number of existing results for nanotube oscillators are employed to analyse the design considerations of optimizing such a device, and significant new results are also derived. These include a new analytical expression for the interaction per unit length of two parallel carbon nanotubes involving the Appell hypergeometric functions. This expression is employed to precisely determine the relationship between the bundle radius and the radii of the nanotubes forming the bundle. Furthermore, several pragmatic approximations are also given, including the relationships between the bundle radius and the constituent nanotube radius and the oscillating tube radius and the bundle nanotube radius. We also present a simplified analysis of the force and energy for a nanotube oscillating in a nanotube bundle leading to an expression for the oscillating frequency and the maximum oscillating frequency, including constraints on configurations under which this maximum is possible.

## 1. Introduction

Carbon nanotubes are recently discovered nanostructures which have generated considerable research into their potential applications. They have outstanding mechanical properties, such as high strength, low weight, as well as high electrical and thermal conductivities, all of which promise many applications exploiting nanoscale mechanics and electronics. However, before nanoscale devices incorporating carbon nanotubes can be implemented, a more complete understanding of their interaction with themselves and their environment is needed.

One potential application of carbon nanotubes is the nanoscale gigahertz oscillator. This device was inspired by the experiments of Cumings & Zettl (2000), which show that the inner shell of a multi-walled carbon nanotube will slide with almost zero friction. These results are confirmed by Yu *et al*. (2000) and multi-walled carbon nanotube oscillators are described in detail by Zheng & Jiang (2002). These results are also confirmed in the molecular dynamics simulations of Legoas *et al*. (2003) and Rivera *et al*. (2003, 2005) who investigated double-walled carbon nanotube oscillators. There are also classical applied mathematical investigations into the mechanics of carbon nanotube oscillators. Baowan & Hill (2007) and Baowan *et al*. (in press) studied the double-walled carbon nanotube oscillators. The present authors (Cox *et al*. 2007*a*–*c*) investigated atoms, spherical and spheroidal fullerenes oscillating in a single-walled carbon nanotube, while Hilder & Hill (2007*a*,*b*) studied oscillators involving carbon nanotori.

Recently, a new configuration of carbon nanotube-based oscillators has been proposed, which is that of the carbon nanotube bundle oscillator. Kang *et al*. (2006) proposed a new oscillator based on a single-walled carbon nanotube oscillating within a bundle of similar carbon nanotubes, which they investigated using molecular dynamics simulation. Following the methodology of Cox *et al*. (2007*a*–*c*), we study here the mechanics of single-walled carbon nanotube bundles and derive expressions for the acceptance and the suction energies of carbon nanotubes oscillating within the carbon nanotube bundles. The definition of a carbon nanotube bundle employed here is wider than that adopted by Kang *et al*. (2006), and the present paper contains a number of results that are widely applicable and involve new analytical expressions for the interaction between parallel carbon nanotubes in terms of a series of Legendre functions or equivalently in terms of a single Appell hypergeometric function of two variables. The advantage of adopting this wider definition of a bundle in terms of a ring of *N* nanotubes is that the results for the usual *N*=6 emerge from the analysis, which can then be used for the nearest neighbour calculations for bundles in the usual triangular lattice. Moreover, future technology and fabrication techniques may well enable other nanotube bundle geometries to become available, which are not currently realizable. Therefore, it is useful to analyse the most general case so that interesting phenomena may be identified even if it is not considered feasible within the context of the present-day technology.

In §2, we define the Lennard-Jones potential and provide an overview of our approach to calculating the total van der Waals interaction based on the continuous approximation. In §3 we derive an expression for the energy of a bundle and use this to determine the bundle radius dependence on the radii of the constituent tubes. In §4 we investigate the interaction between a nanotube and a bundle which we use to prescribe the optimal bundle size for three specific oscillating nanotubes as well as the general approach to determine the optimal configuration for an oscillating nanotube of any radius. This is extended in §5 to a detailed analysis of the force and the energy for a nanotube being sucked into a nanotube bundle which shows characteristics not previously identified for the double-walled nanotube oscillators. Finally in §6, we discuss the results and present the conclusions of this work. The detailed derivations of the analytical expressions for the interaction energy of an atom and a nanotube and the interaction energy per unit length of two parallel nanotubes are contained in appendices A and B, respectively.

## 2. Lennard-Jones potential for nanotube bundle

### (a) Lennard-Jones potential

The van der Waals interaction energy function is modelled here using the 6-12 Lennard-Jones potential functionwhere *ρ* is the distance between the interacting atoms and *A* and *B* are the Lennard-Jones attractive and repulsive constants, respectively. By modelling each individual molecule as a surface with uniform atomic distributions, *η*_{1} and *η*_{2}, the total interaction for two molecules is given bywhere *S*_{1} and *S*_{2} represent the two nanotube surfaces and *ρ* denotes the distance between the surface elements d*S*_{1} and d*S*_{2} on each surface.

The continuum approach to calculating the van der Waals interactions from the Lennard-Jones potential function has been employed for many configurations by Girifalco (1992), Henrard *et al*. (1999), Girifalco *et al*. (2000), Hilder & Hill (2007*a*,*b*), Baowan & Hill (2007) and Baowan *et al*. (in press), as well as in the previous cited work of the present authors (Cox *et al*. 2007*a*–*c*). In this paper, we provide new results that have not appeared previously in these works, including new analytical expressions for the interaction energy of parallel tubes, as well as the ideal configurations for various geometries of carbon nanotube bundle oscillators.

### (b) Nanotube bundle geometry

Kang *et al*. (2006) defined a nanotube bundle as a closely packed array of aligned tubes in a triangular lattice. By denoting one nanotube as the centre of the bundle, hexagonal rings of nanotubes can be defined in the network. Henrard *et al*. (1999) also performed their calculations assuming that each nanotube is surrounded by six neighbours. As will be seen shortly, we require a more general definition of a bundle, and therefore for this work we adopt the following definition. As shown in figure 1, we assume a nanotube bundle as an integral number *N* of carbon nanotubes each of the same length 2*L* and radius *r*, and all lying parallel to each other. The axis of each tube is assumed to be equidistant from a common bundle axis, which is taken to be the *z*-axis, and this distance *R* is termed the bundle radius. The nanotube axes are also assumed to be symmetrically spaced around the cylinder defined by the bundle axis and the radius *R*. Therefore, the *i*-th tube in the nanotube bundle has a surface in the Cartesian coordinates (*x*, *y*, *z*) given by(2.1)where *i*∈{1, …, *N*}, 0≤*θ*_{i}≤2*π* and −*L*≤*z*_{i}≤*L*. However, it is convenient for some of the analysis in this paper to assume a semi-infinite tube, with 0≤*z*_{i}<∞, or a completely infinite tube, with −∞<*z*_{i}<∞. The numerical values of the various constants used in this model are shown in table 1, where the values of *A* and *B* are the same as those used by Kang *et al*. (2006) and the values of the nanotube radii for various tube types are taken from Cox & Hill (2007) with a C–C bond length of 1.42 Å.

## 3. Nanotube bundle energy

### (a) Potential for two nanotubes

For the purpose of determining the equilibrium spacing in a nanotube bundle, we assume that the tubes are doubly infinite and begin by calculating the interaction between an infinite nanotube and an atom lying at a perpendicular distance *ρ* from the tube axis. The interaction energy is derived in appendix A and is given by(3.1)where and *P*_{v}(*z*) denotes the usual associated Legendre function of degree *v*.

Using (3.1) we may derive an analytical expression for the interaction potential per unit length, *E*_{tt}, of two parallel carbon nanotubes. We suppose that the distance between the axes is denoted by *δ* and the radii of the tubes are *r*_{1} and *r*_{2}. The interaction potential can be calculated by substituting into equation (3.1) and then integrating over the circumference of the second cylinder. In appendix B, two analytical expressions are derived for this potential: one in the form of a series of associated Legendre functions given by equation (B 4) and the other in terms of the Appell hypergeometric functions of two variables that can be expressed as(3.2)where and is an Appell hypergeometric function of two variables of the second kind as defined by Erdélyi *et al*. (1953). We comment that (3.2) is a new analytical expression that has not appeared previously in the literature.

### (b) Equilibrium position for a bundle of nanotubes

In §2 we defined a nanotube bundle, which we assume adopts a lowest energy level configuration, i.e. we determine the value of the interspacing distance *δ*, for which the total potential energy *E*_{B} is minimized. This means that for a particular bundle number *N* and tube radius *r*, the van der Waals interactions will prescribe the distance *δ* between tubes and therefore the bundle radius *R*. We note that the bundle might be held in place by nanotube–nanotube interactions in a larger lattice of nanotubes or, alternatively, may be held in place by structural components, such as intertube bridging (Kis *et al*. 2004). For the purposes of this analysis, we do not prescribe the precise method for fixing the bundle in place, but regardless of the method of implementation, it is assumed that the least energy configuration is adopted for any particular values of *N* and *r*.

The total interaction potential for the bundle, *E*_{B}, is defined as the sum of all the constituent interactions, i.e.(3.3)where *E*_{tt}(*δ*) is given by (3.2) which is the tube–tube energy for two nanotubes of radius *r* at a distance *δ* apart. The above sum is actually dominated by the nearest neighbour interactions and so for most purposes we can approximate (3.3) byand a graph of the bundle energy *E*_{B} versus radius *R* is shown in figure 2 for three different sized carbon nanotubes.

For the purposes of actually constructing nanotube bundles, we are interested in the minimum energy configuration. Accordingly, in figure 3, we plot the bundle radius *R* against the nanotube radius *r* for various numbers of nanotubes *N*. It can be seen from this figure that the relationship between the bundle radius and the nanotube radius is almost linear and it is scaled by a factor based on the number of nanotubes comprising the bundle *N*. If we term the equilibrium interstice *λ* as the distance between two tubes when the van der Waals energy is minimized, this distance varies slightly as a function of the nanotube radius but generally lies between 3.10 and 3.16 Å. With this simplification and the previous comment that the nearest neighbour interactions dominate the bundle energy *E*_{B}, and using the cosine rule for an isosceles triangle with two sides of length *R* at an angle of 2*π*/*N* and base of length 2*r*+*λ*, the bundle radius *R* can be approximated byWe comment here that the calculated value of the interstice distance *λ* agrees very well with the experimental measurement of Thess *et al*. (1996) who found *λ*=3.15 Å for (10,10) nanotubes, as well as with the theoretical calculations of Henrard *et al*. (1999; *λ*=3.2 Å for (10, 10) nanotubes), Girifalco *et al*. (2000; *λ*=3.12–3.17 Å for a range of nanotubes) and Šiber (2002; *λ*=3.2 Å for (10, 10) nanotubes).

## 4. Interaction of carbon nanotube with nanotube bundle

### (a) Interaction potential

In this section, we calculate the interaction potential between a single-walled carbon nanotube of radius *r*_{0} and a nanotube bundle formed from *N* nanotubes each of radius *r*. The single nanotube is located with its axis parallel to the *z*-axis but offset by a distance *ϵ* in the direction *ϕ* relative to the direction of the *x*-axis, i.e. the axis of the nanotube can be given parametrically as . Therefore, the distance between the axis of the single nanotube and that of the *k*-th nanotube in the nanotube bundle is given by

In this case, since the single nanotube radius *r*_{0} is not necessarily the same as all the nanotubes forming the bundle, we need to calculate the interaction potential between two parallel tubes of differing radii as shown in equation (3.2). Therefore, if we define *W*_{t} as the suction energy per unit length of the single carbon nanotube interacting with the bundle, then we may write(4.1)

Now assuming that the equilibrium position for the single nanotube is located along the bundle axis, then *W*_{t} becomes simplywhich means that *N*, the number of nanotubes in the bundle, determines the magnitude of the suction energy but not the location of any extrema. Therefore, we may determine a relationship between the radii of the nanotubes forming the bundle, *r*, and the bundle radius *R*, for which the suction energy *W*_{t} is maximized for a particular single nanotube of radius *r*_{0}. By plotting this against the data derived in §3 for optimal nanotube bundles, we can determine the optimum configurations for nanotube–nanobundle oscillators, in the sense that the suction energy per unit length is a maximum.

In figure 4, the relationship between the nanotube radius *r* and the bundle radius *R* for three nanotubes of type (5, 5), (8, 8) and (10, 10) is shown. The points of intersection with the optimal bundle configurations from §3 prescribe the optimum oscillators in terms of a maximized suction energy per unit length of the single nanotube.

As described in §3, we can approximate the bundle radius *R* in terms of the radius of the oscillator *r*_{0} and the radius of the constituent tubes *r* in the bundle. In this case, we may express the approximate relationship by(4.2)where, as before, *λ* is the intertube equilibrium distance, which generally varies from 3.1 to 3.16 Å. Combining this with the expression for the minimized bundle energy allows us to derive the following approximate formula for the radius of constituent tubes in a bundle of number *N* to maximize the suction energy for an oscillating tube of radius *r*_{0}:We note that for *N*=6 it follows immediately that *r*≈*r*_{0}, as expected.

### (b) Suction of nanotubes into bundles

Here, we determine a condition that a carbon nanotube, which is initially outside a nanotube bundle, will be accepted into the vacancy located at the centre of the bundle, according to the definition given in the beginning of §2*b*. For bundles of sixfold symmetry, *N*=6, the configuration of the problem is as shown in figure 5. We assume that the carbon nanotubes forming the bundle are of semi-infinite length and the central nanotube is of length 2*L*_{0} and is centred on the *z*-axis at a position *Z*, which can be inside or outside the bundle. The Cartesian coordinates of a typical point on the middle tube is , where *r*_{0} is the tube radius, 0≤*θ*_{0}<2*π* and −*L*_{0}≤*z*_{0}≤*L*_{0}. The Cartesian coordinates (*x*, *y*, *z*) of the nanotubes in the bundle are shown in equation (2.1), where in this case 0≤*z*_{i}<∞, as shown in figure 5. Owing to the assumed symmetry of the problem, we need to consider only the interaction between the middle tube and one of the carbon nanotubes in the bundle to obtain the potential energy. As shown in figure 10, we look at the tube with coordinates , which is in the middle of the bundle, and the tube *i*=1, which has coordinates . If *E* denotes the energy of the interaction between the centre tube and the first tube, then the total interaction energy of the system can be obtained by *E*_{tot}=*NE*, where *N* is the number of tubes in the bundle, which are symmetrically located around the centre tube.

Using the Lennard-Jones potential and the continuum approach, we obtain the interaction energy *E* as(4.3)where *η* is the mean atomic density of a nanotube and *ρ* denotes the distance between two typical surface elements on each nanotube, which is given by(4.4)Here, we write (4.3) as(4.5)where the integrals *M*_{n} (*n*=3, 6) are defined by(4.6)where and . Details for evaluating the integrals *M*_{n} can be found in Baowan *et al*. (in press). We note that the case of *R*=0 gives rise to the scenario of the suction of a nanotube to form a concentric tube, and that the above integrals can be evaluated analytically in terms of hypergeometric functions as shown in detail by Baowan *et al*. (in press). In this paper, with *R*>0, these integrals are determined numerically. We also note that on looking at an offset nanotube inside a carbon nanotube of infinite length, an integral of the form arises, for which Baowan *et al*. (in press) attempted analytical evaluation. We also comment that when the nanotube of finite length is well inside the semi-infinite nanotube bundle, the energy *E* approaches simply the potential energy *E*_{n} per unit length, which is given by equation (3.2), multiplied by the nanotube length, i.e. when then .

In figures 6–9, we plot the total potential energy *E*_{tot} and the van der Waals force *F*_{vdW} for a (5, 5) carbon nanotube of length 2*L*_{0}=30 Å interacting with a sixfold symmetry (5, 5) carbon nanotube bundle, for which the tubes in the bundle are assumed to be of semi-infinite length. It can be seen from figures 6–9 that the acceptance of a single carbon nanotube into a bundle strongly depends on the bundle radius *R*. As shown in figure 6, for *R*=9.47 Å, the single nanotube will not be accepted into the bundle due to the high energy barrier both outside and inside the bundle. By increasing *R* to 9.48 Å, we see that there is an energy well that will suck a single nanotube inside. However, as the other end of the nanotube approaches the bundle edge, the nanotube encounters an energy barrier that prevents the nanotube from being completely sucked by the nanotube bundle. For *R*=9.49 Å, the single nanotube is accepted into the bundle and just experiences a small repulsion as the far end of the nanotube crosses over into the bundle. As shown in figure 4, the optimal bundle radius *R* for a single (5, 5) carbon nanotube interacting with a bundle of sixfold symmetry is *R*=9.941 Å. Using this value, we plot the energy and force for a single carbon nanotube entering the centre of the bundle, as shown in figures 8 and 9. We observe that the tube is accepted into the bundle without any repulsion effects at the tube extremities. We comment that the area under the graphs in figures 7 and 9 represent the work (energy) done by the van der Waals forces. For the single nanotube to be completely accepted into the bundle, it is required for this area to be greater than zero.

## 5. Oscillating nanotubes in bundles

### (a) Force and energy of oscillating nanotube

In this section, we consider a single carbon nanotube oscillating in the middle of a bundle of finite length carbon nanotubes of *N*-fold symmetry. We assume that the centre of the oscillating tube remains on the *z*-axis during its motion. In a cylindrical polar coordinate system, a typical point on the oscillating tube has coordinates , where *r*_{0} is the tube radius, −*L*_{0}≤*z*_{0}≤*L*_{0} and *Z* is the distance between the centre of the oscillating tube and the origin. The coordinates of the nanotubes in the bundle are given by (2.1), where in this case −*L*≤*z*_{i}≤*L*. In this analysis, we follow the work by Cox *et al*. (2007*b*), which analysed the frictions of such systems and found its effect to be negligible in the high-velocity regime. Other energy dissipation through radial breathing modes and secondary modes of vibration will have a dampening effect, but these are ignored in the present model. The numerical results from the earlier work using this approach were found to be in excellent agreement with molecular dynamics studies, although it does not predict the decaying envelope of vibration which comes about from these dampening effects.

As discussed in §4, due to the assumed symmetry of the problem, we need to consider only the interaction between the oscillating tube, which is located at the centre of the bundle, and one of the carbon nanotubes in the bundle surrounding the oscillating tube. As shown in figure 10, we examine the tube with coordinates , which is in the middle of the bundle, and the tube *i*=1, which has coordinates . Again, the total interaction energy of the oscillating nanotube inside the bundle is given by *E*_{tot}=*NE*, where *E* is the energy of the interaction between the two nanotubes and *N* is the number of tubes in the bundle, which are located symmetrically around the inner oscillating nanotube.

Using the Lennard-Jones potential and the continuum approach, we obtain the interaction energy *E* as(5.1)where *η* is the mean atomic density of a nanotube and *ρ* denotes the distance between two typical surface elements on each nanotube, which is given by (4.4). Again, we can rewrite (5.1) as(5.2)where the integrals *J*_{n} (*n*=6, 12) are defined by(5.3)where and . The integral *J*_{n} can be evaluated analytically as shown in detail by Baowan *et al*. (in press).

In the case when *R*=0, we have the scenario of an oscillating nanotube inside another nanotube and the integrals *J*_{n} can be evaluated analytically in terms of hypergeometric functions, as shown by Baowan & Hill (2007). Since *R*>0 in this paper, we evaluate these integrals numerically.

In figures 11 and 12, we plot the total energy *E*_{tot} and the van der Waals force *F*_{vdW} for a (5, 5) carbon nanotube oscillating in a sixfold symmetry (5, 5) carbon nanotube bundle. Note that we use *R*=9.941 Å, which is the optimal bundle radius for the interaction of a (5, 5) nanotube and a bundle of sixfold symmetry (figure 4). A similar behaviour to that of the double-walled carbon nanotube oscillators studied by Baowan & Hill (2007) is obtained here. The single carbon nanotube has the minimum energy at *Z*=0 inside the bundle. By pulling the tube away from the minimum energy configuration in either direction, the van der Waals force tends to propel the nanotube back towards the centre of the bundle. Accordingly, we obtain an oscillatory motion of the nanotube inside the bundle. We also observe that when , we obtain the peak-like forces that are similar to those obtained by Cox *et al*. (2007*a*,*b*) for a C_{60} oscillating inside a single-walled nanotube.

### (b) Oscillatory frequency

By adopting the simplified model of motion for oscillating nanotubes proposed by Baowan & Hill (2007), we can show that when *L*≥*L*_{0} the van der Waals force *F*_{vdW} experienced by the oscillating nanotube can be approximated bywhere *H*(*z*) is the Heaviside unit step function and *W*_{t} is the suction energy per unit length, which we call the suction force, as shown in (4.1). From this model and following Baowan & Hill (2007), we may show that assuming the nanotube is initially at rest and extruded by a distance *d* out of the nanotube bundle, the resulting oscillatory frequency *f* is given bywhere *M* is the mass of the oscillating nanotube, which is given by , and *m*_{0} is the mass of a single carbon atom.

As found in Baowan & Hill (2007), the maximum frequency occurs when the extrusion distance satisfies the relationship *d*=(*L*−*L*_{0})/2. In this case, the maximum frequency *f*_{max} is given by(5.4)However, there are certain limitations on those oscillators that can attain this frequency. First, the extrusion distance *d* must be less than the length of the oscillating nanotube 2*L*_{0}. This leads to an upper limit on the ratio of the bundle length *L* to the oscillator length *L*_{0}, i.e. *L*<5*L*_{0}, and the bundle oscillators longer than this will not be able to attain the theoretical maximum frequency *f*_{max} shown in (5.4). Second, a sensible lower limit is required on the extrusion distance. The reason for this is that when *L*=*L*_{0} extremely high frequencies are theoretically achievable by choosing very small extrusion distances *d*. However, for an oscillator to be practical, the total displacement of the oscillating nanotube needs to be measurable and this represents a practical limit on the design of such oscillators. In this paper, we denote this practical limitation on the minimum extrusion distance as *d*_{min}, and therefore we conclude that equation (5.4) only applies when the bundle length lies within the limits(5.5)We also comment that the constraints (5.5) have not appeared previously in the literature and they apply equally well to the double-walled oscillators, such as those studied by Baowan & Hill (2007); Baowan *et al*. (in press), as well as the bundle oscillators presented here.

A comparison of the results of the present model with the molecular dynamics study of Kang *et al*. (2006) shows reasonable overall agreement, considering the assumptions of the model presented here. First, our model is one of mechanics only and takes no account of thermal effects. Therefore, to minimize thermal effects, we compare our results with the simulations performed with an initial temperature of *T*=1 K, the results of which are presented in Kang *et al*. (2006). Although Kang *et al*. (2006) do not report specific values for the frequency of this experiment, they can be determined from the measurement of the aforementioned figure in the article. That simulation involved an oscillating (5, 5) nanotube in a sixfold nanotube bundle also comprising (5, 5) nanotubes, and both the bundle and the oscillating nanotube have the same half-length *L*=*L*_{0}=15 Å. In the molecular dynamics simulation of Kang *et al*. (2006), an initial displacement (which we termed as the extrusion distance) of 15 Å quickly reduces to a reasonably stable 6.3 Å maximum displacement within the first 200 ps. By the end of the 200 ps simulation, the oscillation period has settled to approximately 13.8 ps which equates to an oscillatory frequency of 72 GHz. Using our model with an extrusion distance of 15 Å and the same nanotube and bundle configuration as in the simulation, we obtain *W*_{t}=0.3912 eV Å^{−1} which, for an extrusion distance of 15 Å, produces a frequency of *f*=51.6 GHz. However, comparing our model with the actual extrusion distance after the oscillation has stabilized, we have an extrusion distance of approximately 6.3 Å, which corresponds to an oscillatory frequency of approximately *f*=79.6 GHz. This result is reasonable considering that the present model ignores any thermal effects and energy losses due to other modes of vibration.

## 6. Conclusions

In this paper, we consider a new class of gigahertz oscillators comprising a carbon nanotube oscillating in a nanotube bundle. The definition of a nanotube bundle in this paper is wider than that appearing previously in the literature Kang *et al*. (2006), since we do not necessarily prescribe the ring of nanotubes to contain only six members. The Lennard-Jones potential is used to calculate the interaction energy between the nanotubes forming the bundle and an expression for the bundle radius is derived. In the process of determining the bundle energy, an expression (3.2) is produced for the interaction energy per unit length of two parallel nanotubes, which is both completely analytical and has not appeared previously in the literature. Following the exact analysis, approximate simplified formulae are given based on the assumption of ignoring all interactions other than those of the nearest neighbours. Such analysis is extended to determine the optimum bundle size for various oscillators comprising oscillating nanotubes of various radii, which is used to predict oscillator bundle configurations that optimize the suction energy and thus lead to a maximum oscillation frequency. We comment that despite the idealization in the present model in terms of ignoring friction and secondary vibrational modes, we justify our approach on the basis that the authors' earlier work (Cox *et al*. 2007*b*) agrees well with molecular dynamics studies in the high-velocity regime. While such effects will impose a decaying envelope on the primary mode of oscillation, which will affect the longer-term oscillatory behaviour, this will be a feature of all configurations and therefore the qualitative results of this analysis are still valid. We also present a simplified analysis of the force and energy for a nanotube oscillating in a nanotube bundle, which displays certain features not apparent in the double-walled nanotube oscillators. Using the new expression (3.2) for the potential energy per unit length, we derive expressions for the frequency of an oscillator, which depends on the extrusion distance, and we also determine a theoretical maximum frequency (5.4) for a particular oscillator. In addition, we identify limits on the bundle length (5.5), which apply to configurations that are able to attain this theoretical maximum frequency.

## Acknowledgments

The authors are grateful to the Australian Research Council for their support through the Discovery Project scheme and the provision of an Australian Postdoctoral Fellowship to N.T. and an Australian Professorial Fellowship to J.M.H. This work was motivated by a seminar from Prof. Quanshui Zheng of the Tsinghua University, and the authors wish to acknowledge Prof. Zheng for his many helpful comments and discussions.

## Interaction energy of an atom and nanotube

In this appendix, we determine an analytical expression for the interaction potential, *E*_{ta}, between a carbon nanotube and a single atom. The interaction energy for a nanotube and a single atom is given by the integral(A1)First, evaluating the *z* integration by substituting *z*=*λ* tan *ψ*, where we may show thatThen, by substituting *t*=sin^{2}(*ϕ*/2) we obtainwhich is the fundamental integral form for the hypergeometric function and thereforewhere *F*(*a*, *b*; *c*; *z*) is the usual hypergeometric function. Using the transform of Erdélyi *et al*. (1953, §2.9, eqn (2)), we may show thatFinally, using Erdélyi *et al*. (1953, §3.2, eqn (28)) we may transform this expression from a hypergeometric function to a Legendre function and show that(A2)where *P*_{v}(*z*) is the Legendre function of the first kind of degree *v* and by substituting (A 2) into (A 1) gives (3.1).

## Interaction energy for two nanotubes

In this appendix, we derive an analytical expression for the interaction energy per unit length of two parallel carbon nanotubes, *E*_{tt}, of radii *r*_{1} and *r*_{2} at a perpendicular distance *δ* between their axes. From (3.1) the potential energy is given by(B1)where the integrals *J*_{n} are given bywhere . Now substituting for *P*_{v}(*z*) using Erdélyi *et al*. (1953, §3.2, eqn (7)) giveswhich can be expanded in the form of a series to give(B2)where (*x*)_{k} is the Pochhammer symbol. Next, we consider the integral which we denote byand substituting *t*=sin^{2}(*θ*/2) we obtainwhere and . The integral is now in the fundamental integral form for a hypergeometric function and we have(B3)where *F*(*a*, *b*; *c*; *z*) is the usual hypergeometric function. As before, we apply Erdélyi *et al*. (1953, §2.9, eqn (2)) to giveand Erdélyi *et al*. (1953, §3.2, eqn (28)) which producesFinally, substituting this back into (B 2) produces(B4)

Alternatively, we may take (B 3) and substitute into (B 2) and expand the hypergeometric function as the serieswhich can then be written aswhere is an Appell hypergeometric function of two variables of the second kind (e.g. Erdélyi *et al*. 1953, pp. 224–245). Thus, finally the potential energy per unit length of two tubes is given by (3.2).

## Footnotes

- Received July 17, 2007.
- Accepted November 23, 2007.

- © 2008 The Royal Society