Proc. R. Soc. A (2007) 463, 477–494
doi:10.1098/rspa.2006.1772
Published online 11 October 2006
Mechanics of atoms and fullerenes
in single-walled carbon nanotubes.
II. Oscillatory behaviour
B Y B ARRY J. C OX , N GAMTA T HAMWATTANA *
AND
J AMES M. H ILL
Nanomechanics Group, School of Mathematics and Applied Statistics,
University of Wollongong, Wollongong, New South Wales 2522, Australia
The discovery of carbon nanotubes and C60 fullerenes has created an enormous impact on
possible new nanomechanical devices. Owing to their unique mechanical and electronic
properties, such as low weight, high strength, flexibility and thermal stability, carbon
nanotubes and C60 fullerenes are of considerable interest to researchers from many scientific
areas. One aspect that has attracted much attention is the creation of high-frequency
nanoscale oscillators, or the so-called gigahertz oscillators, for applications such as ultrafast
optical filters and nano-antennae. While there are difficulties for micromechanical
oscillators, or resonators, to reach a frequency in the gigahertz range, it is possible for
nanomechanical systems to achieve this. This study focuses on C60–single-walled carbon
nanotube oscillators, which generate high frequencies owing to the oscillatory motion of the
C60 molecule inside the single-walled carbon nanotube. Using the Lennard-Jones potential,
the interaction energy of an offset C60 molecule inside a carbon nanotube is determined, so as
to predict its position with reference to the cross-section of the carbon nanotube. By
considering the interaction force between the C60 fullerene and the carbon nanotube, this
paper provides a simple mathematical model, involving two Dirac delta functions, which
can be used to capture the essential mechanisms underlying such gigahertz oscillators. As a
preliminary to the calculation, the oscillatory behaviour of an isolated atom is examined.
The new element of this study is the use of elementary mechanics and applied mathematical
modelling in a scientific context previously dominated by molecular dynamical simulation.
Keywords: carbon nanotubes; fullerenes C60; gigahertz oscillators;
Lennard-Jones potential; Dirac delta functions
1. Introduction
Carbon nanotubes and fullerene molecules have generated considerable impact
on nanotechnology and particularly for the creation of possible new nanodevices.
Their small size together with their special and unique mechanical properties
make both carbon nanotubes and fullerenes potential materials for many uses in
nanomechanical systems. One aspect that has attracted much attention is the
creation of nanoscale oscillators or the so-called gigahertz oscillators. We note
that while there are difficulties for micromechanical oscillators, or resonators, to
* Author for correspondence ([email protected]).
Received 29 March 2006
Accepted 17 August 2006
477
This journal is q 2006 The Royal Society
478
B. J. Cox et al.
reach frequencies in the gigahertz range, it is possible for nanomechanical
systems to achieve this. Cumings & Zettl (2000) experimented on multi-walled
carbon nanotubes, where they removed the cap from one end of the outer shell
and attached a moveable nanomanipulator to the core in a high-resolution
transmission electron microscope. By pulling the core out and pushing it back
into the outer shell, they report an ultra low-frictional force against the intershell
sliding, which is also confirmed by Yu et al. (2000). Cumings & Zettl (2000) also
observed that the extruded core, after release, quickly and fully retracts inside
the outer shell owing to the restoring force resulting from the van der Waals
interaction acting on the extruded core. These results led Zheng & Jiang (2002)
and Zheng et al. (2002) to study the molecular gigahertz oscillators, where the
sliding of the inner shell inside the outer shell of a multi-walled carbon nanotube
can generate oscillatory frequencies up to several gigahertz.
Based on the results of Zheng et al. (2002), the shorter the inner core
nanotube, the higher the frequency. As a result, Liu et al. (2005) investigated the
case where, instead of using multi-walled carbon nanotubes, the high frequency is
generated by a fullerene C60 oscillating inside a single-walled carbon nanotube.
A fullerene C60, commonly known as a buckyball, comprises 60 carbon atoms
bonded to approximately form a sphere. We refer the reader to Dresselhaus et al.
(1996) for details and properties of fullerenes. Further, in contrast to the multiwalled carbon nanotube oscillator, the C60–nanotube oscillator appears not to
suffer the rocking motion, which is associated with the high frictional effect.
While Qian et al. (2001) and Liu et al. (2005) use the molecular dynamical
simulation approach to study this problem, this paper employs elementary
mechanical principles utilizing the continuum approximation arising from the
assumption that the discrete atoms can be smeared across the surface, to provide
a model for the C60–single-walled carbon nanotube oscillator.
Further, we adopt Newton’s second law for the oscillatory motion of the C60
molecule of radius b and oscillating in a carbon nanotube of radius a and length
2L, namely
d2 Z
m f 2 Z FvdW ðZÞKFr ðZÞ;
ð1:1Þ
dt
where Z is the distance between the centres of the fullerene C60 and the carbon
nanotube; m f, the total mass of the fullerene C60; FvdW(Z ), the van der Waals
restoring force; and Fr(Z ), the total sliding resistance force experienced by the
C60 molecule. Since experiments have shown for multi-walled carbon nanotube
oscillators that Fr(Z ) is extremely small when compared with the restoring force,
Zheng & Jiang (2002) neglect Fr(Z ) in their calculation. However, Zheng et al.
(2002) incorporate the sliding resistance force in the model of Zheng & Jiang
(2002), where Fr(Z ) is assumed to depend on the chirality of both shells in the
intershell sliding.
In this paper, we study both cases with and without the frictional effect.
Owing to the symmetry of the problem, we are only concerned with the force in
the axial direction. In §3b, we show that for b!a/2L, the axial van der Waals
force Fztot ðZ Þ, equation (3.5), behaves like two equal and opposite Dirac delta
functions at the tube extremities ZZKL and ZZL. Accordingly, FvdW(Z ) used
in Newton’s second law can be approximated by
Fztot ðZÞ Z W ½dðZ C LÞKdðZ KLÞ;
ð1:2Þ
Proc. R. Soc. A (2007)
Mechanics of fullerenes and nanotubes II
479
where W is the constant pulse strength. In the case where the frictional force
is neglected, the C60 molecule oscillates inside the carbon nanotube with a
constant velocity, agreeing with the result of Qian et al. (2001), and the
instantaneous forces operating at the extremities of the tube serve only to
change the direction of the fullerene. Further, this model also predicts that the
oscillatory frequencies of the fullerene C60 should be in the gigahertz range. In
§3b, it is shown that the approximation (1.2) is able to capture the essential
mechanisms of a C60–nanotube oscillator and provide reasonable agreement to
the molecular dynamical studies of Qian et al. (2001) and Liu et al. (2005).
We further emphasize that the major contribution of this paper is the use of
elementary mechanics and mathematical modelling techniques in this context,
where prior studies predominantly involve molecular dynamical simulation. In
addition, we note that the modelling presented here differs from that given for
the C60–carbon nanotube oscillators (Qian et al. 2001; Liu et al. 2005) and other
proposed models for the multi-walled carbon nanotube oscillators (Zheng & Jiang
2002; Zheng et al. 2002; Rivera et al. 2003) in two salient aspects. Firstly, we
identify the resulting van der Waals force as approximately two delta functions
and from which we obtain the constant velocity of the C60 travelling inside the
carbon nanotube. Secondly, we model the frictional term for the spherical C60 as
that arising from a ring of contact of a certain prescribed length rather than the
full surface contact area as is done for the oscillating cylinder.
In the following section, we introduce the Lennard-Jones potential and the
continuum approach of assuming an average surface density of carbon atoms. In
§2, we also determine the location of the minimum potential energy of an offset
atom and an offset C60 molecule inside a carbon nanotube. In §3, we first study
the idealized situation of an atom moving along the carbon nanotube axis.
A similar approach is applied in the second part of §3 since the C60 fullerene
also assumed moving along the axis. The force generated from the LennardJones potential is used as the restoring force for both the atom and the C60
molecule, causing the oscillation of the particle between both ends of the carbon
nanotube. Although we neglect the frictional effect in §3, following Zheng et al.
(2002) a periodic frictional force is introduced in §4 for the case of the
oscillating fullerene. Finally, some conclusions are presented in §5.
2. Energy potentials for offset atoms and C60 fullerenes
Generally speaking and depending on the relative dimensions, a single atom and,
to a certain extent, a buckyball will tend to have a preferred position located
approximately one inter-atomic length from the surface of the carbon nanotube.
In this section, we formally confirm this by examining the Lennard-Jones
potential energy for an offset atom and an offset buckyball.
In continuum approximation, where carbon atoms are assumed to be
uniformly distributed over the surface of molecules, the non-bonded interaction
energy can be obtained from
ð ð
E Z n gn f
FðrÞ dSf dSg ;
ð2:1Þ
Sg Sf
Proc. R. Soc. A (2007)
480
B. J. Cox et al.
x
r
e
e
a
z
a
Figure 1. An offset atom inside a single-walled carbon nanotube.
where ng and nf represent the mean surface density of carbon atoms on a carbon
nanotube and a buckyball, respectively, and r denotes the distance between two
typical surface elements dSg and dSf on the two different molecules. The
potential function adopted here is the classical Lennard-Jones potential given by
A
B
FðrÞ ZK 6 C 12 ;
ð2:2Þ
r
r
where A and B are the attractive and the repulsive constants, respectively. The
Lennard-Jones potential has been used in different configurations, including the
interactions between two identical parallel carbon nanotubes (Girifalco et al.
2000), carbon nanotube bundles (Henrard et al. 1999), a carbon nanotube and a
C60 molecule (both inside and outside the tube; Girifalco et al. 2000) and two C60
molecules (Girifalco 1992).
(a ) An offset atom inside a single-walled carbon nanotube
Here, we determine the preferred position of an offset atom with reference to
the cross-section of a carbon nanotube. This position is where the atom has its
minimum potential energy. In an axially symmetric cylindrical polar coordinates,
without the loss of generality, we may assume that the atom is located at
(3, 0, 0), as shown in figure 1, and that the carbon nanotube of infinite extent
with a parametric equation of (a cos q, a sin q, z). We note that 3 is the assumed
distance of the offset atom from the central axis of the tube, a is the tube radius,
Kp%q%p and KN!z!N. In this case, the distance r from the atom to the
wall of carbon nanotube is given by
r Z ða2 C 32 K2a3 cos q C z 2 Þ1=2 :
ð2:3Þ
Thus, from equation (2.1), the potential energy E for the offset atom, which
interacts with the entire carbon nanotube, is of the form
ð p ðN A
B
E Z ang
K 6 C 12 dz dq:
ð2:4Þ
r
r
Kp KN
Now if we let lZ ða2 C 32 K2a3 cos qÞ1=2 , then rZ(l2Cz 2)1/2. By using the
substitution zZl tan j we obtain
ð p ð p=2 A cos4 j B cos10 j
K
C
dj dq:
ð2:5Þ
E Z ang
l5
l11
Kp Kp=2
Proc. R. Soc. A (2007)
481
Mechanics of fullerenes and nanotubes II
0.20
(6, 6)
(10, 10)
potential, E (eV)
0.15
0.10
0.05
0.00
1
2
3
4
– 0.05
– 0.10
– 0.15
offset from tube axis, e (Å)
Figure 2. The potential energy of an offset atom inside a (6, 6) and a (10, 10) carbon nanotube, with
respect to the radial distance 3 from the tube axis.
For a positive integer m, we know
ð p=2
ð2mK1Þ!!p
cos2m j dj Z
;
ð2mÞ!!
Kp=2
ð2:6Þ
where !! is the double factorial operator, defined as ð2mÞ!!Z 2$4$6$.$ð2mK2Þ$2m
and ð2mK1Þ!!Z 1$3$5$.$ð2mK3Þ$ð2mK1Þ, which upon using equation (2.5)
becomes
ð 3pang p
A
21B
EZ
K 5C
dq:
ð2:7Þ
8
l
32l11
Kp
Further, if we define the integrals Jn as
ðp
dq
Jn Z
;
nC1=2
Kp ðaKb cos qÞ
ð2:8Þ
where n is a positive integer, aZa2C32 and bZ2a3, noting that since aKbZ(aK3)2
and aO3 then aObR0, and equation (2.7) becomes
3pang
21B
ð2:9Þ
J5 :
EZ
KAJ2 C
32
8
In appendix A, we show that the integrals Jn can be evaluated either in terms of
elliptic integrals or in terms of hypergeometric functions.
In figure 2, we plot the potential energy E as given by equation (2.9) with
respect to 3 for an atom inside the carbon nanotube (6, 6) (aZ4.071 Å) and
(10, 10) (aZ6.784 Å). It can be seen that the minimum energy of the atom inside
(6, 6) occurs at 3Z0 and 3.291 Å for (10, 10) measured from the tube axis. This
equates to a distance between the atom and the wall of 4.071 and 3.494 Å,
respectively. We observe numerically that as the tube radius gets larger, the atom
is likely to be closer to the tube wall. We note that the constants used in the
numerical calculations throughout this paper are given in table 1, and A and B are
the approximate values arising from those for the C60 and graphene interaction.
Proc. R. Soc. A (2007)
482
B. J. Cox et al.
Table 1. Constants used in the model.
radius of (6, 6) (Å)
radius of (10, 10) (Å)
radius of (16, 16) (Å)
radius of C60 (Å)
carbon–carbon bond length (Å)
pffiffiffi
mean surface density—graphene ½4 3=ð9s2 Þ (ÅK2)
mean surface density—fullerene [60/(4pb2)] (ÅK2)
mass of a single carbon atom (kg)
mass of a single C60 fullerene [60m 0] (kg)
length of carbon nanotubes (Å)
attractive constant (eV!Å6)
repulsive constant (eV!Å12)
spatial period of the interatomic locking (Å)
sliding resistance strength (MPa)
interaction angle
aZ4.071
aZ6.784
aZ10.86
bZ3.55
sZ1.421
ngZ0.3812
nfZ0.3789
m0Z1.993!10K26
mfZ1.196!10K24
2LZ129
AZ17.4
BZ29!103
pffiffiffi
[ Z 3s
tsZ0.48
q0Zp/2
x
b
e
a
r
b e
a
z
Figure 3. An offset buckyball inside a single-walled carbon nanotube.
(b ) An offset C60 molecule inside a single-walled carbon nanotube
In this section, we determine the location of the minimum potential energy of a
C60 molecule with reference to the cross-section of a carbon nanotube. In axially
symmetric cylindrical polar coordinates, we assume the buckyball of radius b is
located at (3, 0, 0), as shown in figure 3, and in a carbon nanotube of infinite extent
with a parametric equation (a cos q, a sin q, z). We note that 3 is the distance
between the centre of the offset molecule and the central axis of the tube, a is the
tube radius, Kp%q%p and KN!z!N. From figure 3, the distance from the
centre of C60 molecule to the wall of carbon nanotube is given by
ð2:10Þ
r Z ða2 C 32 K2a3 cos q C z 2 Þ1=2 :
The interaction between the C60 molecule and the carbon nanotube in the
continuum approximation is obtained by averaging over the surface of each entity.
By performing the surface integral of the Lennard-Jones potential over the sphere,
we find that the potential energy for a typical surface element on the tube
interacting with the entire spherical C60 molecule of radius b is given by
!
(
)
2n f pb A
1
1
B
1
1
K
K
K
PZ
; ð2:11Þ
r
4 ðr C bÞ4 ðrKbÞ4
10 ðr C bÞ10 ðrKbÞ10
Proc. R. Soc. A (2007)
potential, E (eV)
Mechanics of fullerenes and nanotubes II
1.5
1.0
0.5
0
– 0.5
–1.0
–1.5
–2.0
–2.5
–3.0
–3.5
1
2
3
4
483
5
(10, 10)
(16, 16)
offset from tube axis, e (Å)
Figure 4. The interaction potential of an offset C60 molecule inside a (10, 10) and a (16, 16) carbon
nanotube with respect to the radial distance 3 from the tube axis.
where r is the distance between the tube surface element and the centre of the
buckyball, as shown in figure 3. The detail of the derivation of equation (2.11) is
given in appendix A of part I. Thus, the potential energy between the C60 molecule
and the entire carbon nanotube is obtained by performing the surface integral of
equation (2.11) over the carbon nanotube, thus
ð p ðN
E Z ang
P dz dq:
ð2:12Þ
Kp KN
Here, we rewrite equation (2.11) in the form
B
5
80b2
336b4
512b6
256b8
2
P Z 4pb n f
C
C
C
C
5 ðl2 C z 2 Þ6 ðl2 C z 2 Þ7 ðl2 C z 2 Þ8 ðl2 C z 2 Þ9 ðl2 C z 2 Þ10
"
#)
1
2b2
ð2:13Þ
;
C
KA 2
ðl C z 2 Þ3 ðl2 C z 2 Þ4
where l2 Z a2 C 32 K2a3 cos qKb2 , which gives r2 Z l2 C b2 C z 2 . Again, we employ
the substitution zZl tan j, so that with the use of
ðN
ð p=2
dz
1
ð2nK1Þ!!p
Z 2nC1
cos2n j dj Z
;
ð2:14Þ
2
2 nC1
l
ð2nÞ!!l2nC1
KN ðl C z Þ
Kp=2
and equation (2.8), where aZ a 2 C 32 Kb2 and bZ2a3, the potential energy (2.12)
becomes
B 105
1155b2
9009b4
6435b6
12155b8
2 2
J C
J6 C
J C
J8 C
J9
E Z 4p ab n f ng
5 128 5
64
128 7
64
256
A
K ð3J2 C 5b2 J3 Þ ;
ð2:15Þ
8
where Jn are the integrals defined by equation (2.8) and evaluated in appendix A.
As shown in figure 4, the preferred location of the C60 molecule inside the
carbon nanotube (10, 10) is where the centre of the buckyball lies on the tube
axis. In the case of (16, 16) (aZ10.856 Å), we obtain 3Z4.314 Å (see the dash
line in figure 4). These results are equivalent to the distance between the centre
of the buckyball and wall of the nanotube of 6.784 and 6.542 Å, respectively.
Proc. R. Soc. A (2007)
484
B. J. Cox et al.
x
r
a
Z
z
L
Figure 5. Geometry of the single atom oscillation.
Furthermore, we observe that as the radius of the tube gets larger, the location
where the minimum energy occurs, tends to be closer to the nanotube wall. These
results agree with the work of Girifalco et al. (2000).
3. Oscillation of an atom or a C60 fullerene inside
a single-walled carbon nanotube
Here, we adopt Newton’s second law to describe the oscillatory motion of a
particle (an atom or a fullerene C60) in a carbon nanotube, namely
d2 Z
Z FvdW ðZÞKFr ðZÞ;
ð3:1Þ
dt 2
where Z is the distance between the centres of the particle and the carbon
nanotube; M, the mass of the particle; FvdW(Z ) and Fr(Z ), the restoring force
and the frictional force, respectively. In part I of this paper, we determine
the suction force that is sufficient to attract the particle (an atom or a C60
molecule) into the tube and begin oscillating, and we also prescribe the magnitude
of the energy imparted to the particle by this force. Section 3a illustrates the
analysis necessary in the case of the oscillating atom. Subsequently, we employ
similar techniques to study the oscillation of a fullerene C60 inside a single-walled
carbon nanotube.
M
(a ) Oscillation of a single atom inside a single-walled carbon nanotube
In an axially symmetric cylindrical polar coordinate system (r, z), an atom is
assumed to be located at (0, Z ) inside a carbon nanotube of length 2L, centred
around the z-axis and of radius a, as shown in figure 5. Here, we assume that the
atom oscillates along the z-axis. This assumption is valid for the carbon nanotube
(6, 6), where the atom is likely to be on the z-axis owing to the minimum
potential energy, as shown in §2a. From the symmetry of the problem, we are
only concerned with the force in the axial direction, and on neglecting the
frictional force Fr(Z ), we have from equation (3.1)
m0
Proc. R. Soc. A (2007)
d2 Z
Z Fztot ðZÞ;
dt 2
ð3:2Þ
485
Mechanics of fullerenes and nanotubes II
0.02
Fztot
0.01
–100
0
–50
Z
100
50
– 0.01
– 0.02
Figure 6. Plot of Fztot ðZÞ as given by equation (3.3) for the atom oscillating inside the carbon
nanotube (6, 6).
x
r
a
Z
b
z
L
Figure 7. Geometry for the fullerene C60 oscillation.
where m0 is the mass of a single atom and Fztot ðZÞ is the total axial van der Waals
interaction force between the atom and the carbon nanotube length 2L, given by
A
B
Fztot ðZÞ Z 2pang K 2
C 2
2 3
ða C ðZ KLÞ Þ
ða C ðZ KLÞ2 Þ6
ð3:3Þ
A
B
;
K
C 2
ða C ðZ C LÞ2 Þ3 ða 2 C ðZ C LÞ2 Þ6
and we refer the reader to part I for details of this derivation. In figure 6,
we plot Fztot ðZÞ as given by equation (3.3), for the case of the carbon nanotube
(6, 6) (aZ4.071 Å). We note that (6, 6) with aZ4.071 Å satisfies the condition
stated in part I of this paper, where both ends of the tube can generate the
necessary attractive force to suck the atom inside, and the atom oscillates and
never escapes the carbon nanotube. Throughout this paper, we note that the unit
of potential energy is given by electron volt (1.602!10K19 N m), the unit of
length and force are Å and eV/ÅZ1.602!10K9 N, respectively.
(b ) Oscillation of a fullerene C60 inside a single-walled carbon nanotube
In an axially symmetric cylindrical polar coordinate system (r, z), we assume a
fullerene C60 is located inside a carbon nanotube of length 2L, centred around the
Proc. R. Soc. A (2007)
486
B. J. Cox et al.
Fztot
0.3
0.2
0.1
–100
–50
0
50
Z
100
– 0.1
– 0.2
– 0.3
Figure 8. Plot of Fztot ðZÞ as given by equation (3.5) for the buckyball oscillating inside the carbon
nanotube (10, 10).
z-axis and of radius a. As shown in figure 7, we also assume that the centre of
the C60 molecule is in the z-axis. Again, this is justified for the carbon nanotube
(10, 10), as previously described in §2b.
From the symmetry of the problem, we only need to consider the force in the
axial direction. As a result, from Newton’s second law, again neglecting the
frictional force, we have
d2 Z
Z Fztot ðZÞ;
ð3:4Þ
dt 2
where mf is the total mass of a C60 molecule and Fztot ðZÞ is the total axial van der
Waals interaction force between the C60 molecule and the carbon nanotube,
given by
ð3:5Þ
Fztot ðZÞ Z 2pang ½Pðr2 ÞKPðr1 Þ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where P(r) is the potential function given by equation (2.11), r1 Z a 2 C ðZ C LÞ2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
and r2 Z a 2 C ðZ KLÞ2 . Again we refer the reader to part I for details of this
mf
derivation. Here, we assume that the effect of the frictional force may be
neglected, which is reasonable for certain chiralities and diameters of the tube.
For example, the preferred position of the C60 molecule inside the carbon
nanotube (10, 10) is where the centre is on the z-axis. Thus, the molecule tends to
move along the axial direction and not to suffer a rocking motion. However, in §4
the frictional force is included in this model.
In figure 8, we plot Fztot ðZÞ as given by equation (3.5), for the case of the
carbon nanotube (10, 10) and we observe that the force is very close to zero
everywhere except at both ends of the tube, where there is a pulse-like force
which attracts the buckyball back towards the centre of the tube. For b!a/2L,
we find that Fztot ðZÞ can be estimated using the Dirac delta function and thus,
equation (3.4) reduces to give
mf
Proc. R. Soc. A (2007)
d2 Z
Z W ½dðZ C LÞKdðZ KLÞ;
dt 2
ð3:6Þ
Mechanics of fullerenes and nanotubes II
487
whereÐ W is pulse strength
Ð or the work (energy) of the C60 molecule, given by
0
W Z KN
Fztot ðZÞdZ ZK 0N Fztot ðZÞdZ. Therefore, from equation (3.5), we find
p3 n f ng ab2
Bð315C4620mC18018m2 C25740m3 C12155m4 Þ
WZ 2
Að3 C5mÞK
;
160ða 2 Kb2 Þ3
ða Kb2 Þ5=2
ð3:7Þ
where mZ b2 =ða 2 Kb2 Þ. We refer the reader to part I of this paper for the derivation
of equation (3.7) (see eqn (4.11) in part I). For a double-walled carbon nanotube
oscillator, we note that as the inner tube gets smaller to the order of the diameter of a
buckyball, the van der Waals interaction force as shown in the molecular dynamics
simulation of Legoas et al. (2003) becomes a peak-like force operating at both ends of
the outer tube. This is similar to the behaviour of the C60–nanotube oscillator as
shown here and this similarity is also observed by Baowan & Hill (in press).
Now, we consider equation (3.6) and multiplying dZ/dt on both sides of
equation (3.6), we obtain
d2 Z dZ
dZ
mf 2
Z W ½dðZ C LÞKdðZ KLÞ
:
ð3:8Þ
dt
dt dt
From dH(x)/dxZd(x), where H(x) is the usual Heaviside step function, equation
(3.8) can be written as
m f d dZ 2
d
dZ
ZW
fH ðZ C LÞKH ðZ KLÞg
:
ð3:9Þ
dZ
dt
2 dt dt
By integrating both sides of equation (3.9) with respect to t, from tZ0 (assuming
the buckyball is at infinity) to any time t, where the ball is located at Z, and upon
using that H(ZCL)KH(ZKL)Z1 for KL%Z%L and zero elsewhere, we obtain
m f dZ 2
m
Z W fH ðZ C LÞKH ðZ KLÞg C f v02 ;
ð3:10Þ
dt
2
2
where v0 is the initial velocity that the ball is fired on the z-axis towards the open
end of the carbon nanotube in the positive z-direction. We note that the initial
velocity v0 is introduced for the case, where the C60 molecule is not sucked into
the carbon nanotube owing to the strong repulsion force. From equation (3.10)
for KL%Z%L, we find
m f dZ 2
m
Z W C f v02 ;
ð3:11Þ
dt
2
2
which implies that the buckyball travels inside the carbon nanotube at the
constant speed dZ=dtZ vZ ð2W =m f C v02 Þ1=2 . Alternatively, equation (3.11) can
also be formally obtained using the Lorentzian limit as shown in appendix B.
On using equation (3.7) and the constants given in table 1, we obtain the velocity
vZ932 m sK1 for the case when the C60 molecule is initially at rest outside the
carbon nanotube (10, 10) and the molecule gets sucked into the tube owing to the
attractive force. This gives rise to the frequency fZv/(4L)Z36.13 GHz. In figure 9,
we plot the oscillatory frequency with respect to the length of the nanotube. The
result obtained agrees with the molecular dynamics study of Liu et al. (2005). It
confirms their finding that the shorter the nanotube, the higher the oscillatory
frequency. Further, we consider the case where the C60 molecule is fired on the tube
axis towards the open end of the carbon nanotube of radius a!6.338 Å, which does
Proc. R. Soc. A (2007)
488
B. J. Cox et al.
frequency, f (GHz)
90
80
70
60
50
40
30
50
75
100
125
nanotube length 2L (Å)
150
Figure 9. The variation of the oscillatory frequency of the buckyball with respect to the length of
the carbon nanotube (10, 10).
not accept a C60 molecule by suction forces alone owing to the strong repulsive force
of the carbon nanotube (see part I of this paper). For the carbon nanotube (9, 9)
(aZ6.106 Å), the initial velocity v0 needs to be approximately 1268 m sK1 for the
C60 molecule to penetrate into the tube. A result similar to that given by Qian et al.
(2001) is also found here, where a C60 molecule cannot penetrate into either of (8, 8),
(7, 7), (6, 6), (5, 5) even though it is fired into the tube with an initial velocity as
high as 1600 m sK1. In addition, we find for (8, 8) with aZ5.428 Å from our model
that the minimum initial velocity must be approximately 8210 m sK1 for the C60
molecule to penetrate into the tube.
4. Frictional force
Although an ultra low-frictional effect in nano-oscillators has been observed
through experiments and molecular dynamics studies, we still need to properly
understand the frictional behaviour at the molecular level. There are a number of
studies on the frictional effect of the sliding of the inner shell inside the outer shell of
double- and multi-walled carbon nanotubes. Rivera et al. (2003) and Servantie &
Gaspard (2006) assume that the friction is dependent on the sliding velocity. The
latter authors also consider friction to depend on the position of the inner shell with
respect to the outer shell, and they find that this dependence gives rise to a relatively
small effect. While Zheng & Jiang (2002) neglect the friction effect between the
inner and outer shells in their calculation of the oscillatory frequency, they
introduce an inter-atomic locking force which acts against the intershell sliding.
This force is also incorporated in the model of Zheng et al. (2002). Further, Zheng &
Jiang (2002) state that the frictional force resulting from inter-atomic locking is a
potential force and therefore is non-dissipative. This approach is different from
Zhao et al. (2003) and Ma et al. (2005), where the frictional effect is assumed to be a
result of a dissipative energy during an off-axial rocking motion of the inner tube and
which in consequence gives rise to a wavy deformation on the outer shell.
In contrast to the double- and multi-walled carbon nanotube oscillators, there
are no proposed frictional force models for a C60 molecule oscillating inside a
single-walled carbon nanotube. For a C60 spherical fullerene, the tendency for
Proc. R. Soc. A (2007)
Mechanics of fullerenes and nanotubes II
489
a rocking motion is considerably reduced. Accordingly, we propose here a model
for the frictional term for the spherical C60 as that arising from a ring of contact
of a certain prescribed length. Following Zheng et al. (2002), we assume a
periodic inter-atomic locking force
2pZ
;
ð4:1Þ
Fr ðZÞ Z k0 sin
[
where [ is the spatial period of the inter-atomic locking, typically for an armchair or
(n, n) carbon nanotube, where n is a positive integer, it is the distance between
opposite bonds of the carbon ring and k0Ztsa, where ts denotes the resistance
strength and a denotes the area of a ring of contact of a certain prescribed length of
the sphere, and which is given by aZ 4pb2 sinðq0 =2Þ, for a certain angle q0. For
multi-walled carbon nanotube oscillators, Zheng et al. (2002) state that the spatial
period is dependent on the helicities of both tubes in the intershell sliding, and that
the resistance force increases with the degree of commensurability of the two shells.
Here, we assume high commensurability between a C
p60
ffiffiffi molecule and a single-walled
carbon nanotube, which then leads to assuming [ Z 3s (Zheng et al. 2002), where s
is the bond length. By introducing equation (4.1) into equation (3.6), we obtain
d2 Z
2pZ
:
ð4:2Þ
m f 2 Z W ½dðZ C LÞKdðZ KLÞKk0 sin
[
dt
Again, by multiplying dZ/dt on both sides of equation (4.2), we deduce
m f d dZ 2
d
dZ k0 [
ZW
fH ðZ C LÞKH ðZ KLÞg
C
dZ
dt
2 dt dt
2p
d
2pZ dZ
!
cos
;
ð4:3Þ
dZ
dt
[
which upon integrating with respect to t, applying the zero initial condition, and
assuming that there is no frictional force at tZ0 since the C60 molecule is assumed
outside the nanotube, we find
m f dZ 2
k0 [
2pZ
;
ð4:4Þ
Z W ½H ðZ C LÞKH ðZ KLÞ C
cos
dt
2
2p
[
which for the buckyball inside the carbon nanotube (KL%Z%L), equation (4.4)
gives rise to
m f dZ 2
k0 [
2pZ
ZW C
;
ð4:5Þ
cos
dt
2
2p
[
or
ðm f pÞ1=2 dZ
:
ð4:6Þ
dt Z
f2W p C k0 [ cosð2pZ=[Þg1=2
Upon integrating equation (4.6), we have the time T where the buckyball travels
from KL to L as
ðL
dZ
T Z ðm f pÞ1=2
:
ð4:7Þ
1=2
KL f2W p C k0 [ K2k0 [ sin2 ðpZ=[Þg
From equation (4.7), on making the substitution xZ pZ=[, we obtain
1=2 ð pL=[
2m f [k 2
dx
;
ð4:8Þ
TZ
2
pk0
0
ð1Kk sin2 xÞ1=2
Proc. R. Soc. A (2007)
490
B. J. Cox et al.
1.0 t /T
0.8
0.6
0.4
0.2
Z
– 60 – 40 –20
0
20
40
60
Figure 10. Plot of equation (4.13) for the carbon nanotube (10, 10) from KL to L.
where the modulus k2 is defined by
1 W p K1
:
ð4:9Þ
C
k Z
2
k0 [
The integral appearing in equation (4.8) is the normal elliptic integral of the first
kind, usually denoted by F(f, k), thus
ðf
dx
;
ð4:10Þ
Fðf; kÞ Z
1=2
2
0 ð1Kk sin2 xÞ
and equation (4.8) becomes
1=2
2m f [k 2
TZ
FðpL=[; kÞ:
ð4:11Þ
pk0
Generally, the position of the buckyball is determined from
1=2 ð pL=[
m f [k 2
dx
tZ
;
ð4:12Þ
1=2
2
2k0 p
KpZ=[ ð1Kk sin2 xÞ
2
which on non-dimensionalizing by T we obtain
t
1
FðKpZ=[; kÞ
Z
1K
;
T
2
FðpL=[; kÞ
ð4:13Þ
which is shown graphically in figure 10, noting that it is almost a straight line owing
to the small value of k2.
5. Conclusions
In part I of this paper, the condition for a C60 fullerene initially outside the
carbon nanotube to be sucked into the tube is given. Here in part II, we assume
that the C60 molecule is sucked in and oscillates inside the carbon nanotube of a
finite length. For simplicity, we first consider the ideal situation of an oscillating
single atom and a similar but more involved method can be employed in the case
of C60–nanotube oscillator.
Proc. R. Soc. A (2007)
Mechanics of fullerenes and nanotubes II
491
For both the atom and the C60 molecule, using the Lennard-Jones potential,
we determine its most stable positions with reference to the cross-section of the
carbon nanotube. The preferred location is where the potential interaction
energy between the particle and the nanotube is a minimum. Generally, we find
that inside the carbon nanotube, the atom or the C60 molecule is at an interatomic distance from the tube wall. However, in particular, inside the carbon
nanotube (6, 6), the atom is most probably at the centre of the cross-section of
the carbon nanotube and inside the carbon nanotube (10, 10), the centre of C60
molecule is most probably at the centre of the cross-section of the tube.
For both an oscillating single atom and an oscillating buckyball C60 within the
interior of a carbon nanotube, we use the Lennard-Jones potential to calculate
the van der Waals restoring force. Owing to the symmetry of the problem, only
the force in the axial direction needs to be considered. In both cases, we
demonstrate that the resultant van der Waals axial force can be approximated
by two equal and opposite Dirac delta functions operating at the two extremities
of the carbon nanotubes. Assuming zero friction, this model implies that the
atom or buckyball C60 oscillates at constant velocity and the instantaneous
forces at the extremities serve only to change direction. Our model also predicts
the oscillating frequency of the C60 molecule to be in the gigahertz range, which
is in agreement with molecular dynamical simulations.
The authors are grateful to the Australian Research Council for their support through the
Discovery Project Scheme and the provision of an Australian Professorial Fellowship for J.M.H.
The authors also wish to acknowledge Professor Julian Gale of Curtin University of Technology for
his many helpful comments and discussions on this and other related work, and for suggesting the
calculations given in §2.
Appendix A. Evaluation of the integral Jn defined by equation (2.8)
The integral equation (2.8) may be evaluated either in terms of elliptic integrals
or using hypergeometric functions. We first present the evaluation of equation
(2.8) in terms of elliptic integrals. Bisecting the interval of the integral Jn in
equation (2.8) and reversing the sign of one of them and combining gives
ðp
dq
Jn Z 2
:
ðA 1Þ
nC1=2
0 ðaKb cos qÞ
Now, we make the following standard substitutions
sn2 u Z
bð1Kcos qÞ
;
cos qÞ
k 2 ðaKb
k2 Z
2b
;
a Cb
2
g Z pffiffiffiffiffiffiffiffiffiffiffiffi ;
a Cb
ðA 2Þ
then it follows that
dq Z g
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aKb cos qdu;
ðA 3Þ
and also
dn2 u Z 1Kk 2 sn2 u Z
Proc. R. Soc. A (2007)
aKb
:
aKb cos q
ðA 4Þ
492
B. J. Cox et al.
Upon substituting equations (A 2)–(A 4) into equation (A 1), we deduce
ð KðkÞ
2g
dn2n u du;
ðA 5Þ
Jn Z
ðaKbÞn 0
where K denotes the complete elliptic function of the first kind. From Byrd &
Friedman (1971), the following are given for integrals of the Jacobian elliptic
delta amplitude function raised to a power
ð u1
G0 Z
du Z u1 Z Fðf; kÞ;
0
ð u1
G2 Z
G2mC2 Z
0
dn2 u du Z Eðu1 Þ Z Eðf; kÞ;
ðA 6Þ
k 2 dn 2mK1 u snu cnu C ð1K2mÞk 02 G2mK2 C 2mð1 C k 02 ÞG2m
;
2m C 1
0
where cn2uZ1Ksn2u, k 2Z1Kk2, F is the incomplete elliptic integral of the first
kind and E is the complete elliptic function of the second kind. For our purposes,
as we deal with complete integrals, the term involving snu cnu disappears, and
thus there remains
G0 Z KðkÞ;
G2 Z EðkÞ;
ðA 7Þ
ð1K2mÞk 02 G2mK2 C 2mð1 C k 02 ÞG2m
:
2m C 1
By using these values and the recurrence formula, it is possible to express Gn as a
simple linear combination of K and E, and thereby using these functions to
evaluate the definite integral Jn to any n and to any degree of accuracy required.
Alternatively, equation (2.8) can be evaluated using hypergeometric functions.
Upon introducing xZq/2 and mZnC1/2, equation (A 1) becomes
ð p=2
dx
Jn Z 4
;
ðA 8Þ
ðg
C
u
sin2 xÞm
0
G2mC2 Z
where gZaKb and uZ2b. By making the substitution tZcot x into equation
(A 8), we obtain
ðN
4
ð1 C t 2 ÞmK1
Jn Z
dt;
ðA 9Þ
m
ðg C uÞ 0 ð1 C kt 2 Þm
where kZg/(gCu). Therefore, equation (A 9) can be rewritten in the form
ðN
4
1
dt
;
ðA 10Þ
Jn Z
m
m
2
2
ðg C uÞ 0 ½1 C ðkK1Þt =ð1 C t Þ ð1 C t 2 Þ
which on making the substitutions z Z t=ð1C t 2 Þ1=2 and uZz 2, we find
ð 1 K1=2
2
u
ð1KuÞK1=2
Jn Z
du:
ðg C uÞm 0 ½1Kð1KkÞum
Proc. R. Soc. A (2007)
ðA 11Þ
Mechanics of fullerenes and nanotubes II
493
From Gradshteyn & Ryzhik (2000, p. 995), we have
Jn Z
2p
Fðm; 1=2; 1; 1KkÞ;
ðg C uÞm
ðA 12Þ
where F(p,q;s;z)
denotes the usual hypergeometric function and we have used
pffiffiffi
Gð1=2ÞZ p. Further, from Gradshteyn & Ryzhik (2000, p. 998), we find that
equation (A 12) can be written in the form
Jn Z
2p
Fðm; 1=2; 1; Ku=gÞ;
gm
ðA 13Þ
which is not a terminating hypergeometric series since mZnC1/2 and therefore
the parameters p and q are both non-integer.
Appendix B. Alternative derivation of equation (3.11)
Equation (3.11) can be formally obtained using the Lorentzian limit
dðxÞ Z lim
e/0
1
e
;
2
p e C x2
so that from equation (3.6), we have in the limit e tending to zero
d2 Z
We
1
1
:
K
mf 2 Z
p e2 C ðZ C LÞ2 e2 C ðZ KLÞ2
dt
By multiplying both sides of equation (B 2) by dZ/dt, we deduce
m f d dZ 2 W e
1
1
dZ
Z
K 2
;
2
2
2
p e C ðZ C LÞ
dt
2 dt dt
e C ðZ KLÞ
ðB 1Þ
ðB 2Þ
ðB 3Þ
and upon integrating with respect to t and applying the initial condition tZ0
(assuming the buckyball is at infinity), we obtain
ð m f dZ 2 W e Z
1
1
Z
K
dx C C ;
ðB 4Þ
dt
p N e2 C ðx C LÞ2 e2 C ðxKLÞ2
2
where C Z m f v02 =2. On substitution of xZKLC e tan j and xZ LC e tan j into
the first and the second terms, respectively, of equation (B 4), we have
ð tanK1 ððZKLÞ=eÞ !
ð tanK1 ððZCLÞ=eÞ
m f dZ 2 W
Z
djK
dj C C ;
ðB 5Þ
dt
p
2
p=2
p=2
which simply gives rise to
m f dZ 2 W
K1 Z C L
K1 Z KL
Z
tan
Ktan
C C:
dt
p
e
e
2
Proc. R. Soc. A (2007)
ðB 6Þ
494
Since
B. J. Cox et al.
K1
tan tan
Z CL
2Le
K1 Z KL
;
Ktan
Z 2
e
e
e C Z 2 KL2
so that as e tends to zero, we have tanK1 ððZ C LÞ=eÞKtanK1 ððZ KLÞ=eÞZ p and as
such equation (B 6) becomes equation (3.11).
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