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Proc. R. Soc. A (2007) 463, 461–476
doi:10.1098/rspa.2006.1771
Published online 11 October 2006
Mechanics of atoms and fullerenes
in single-walled carbon nanotubes.
I. Acceptance and suction energies
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
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 nanoantennae. 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 C60
fullerenes could feasibly operate at these high frequencies. This paper investigates the suction
force experienced by either an atom or a C60 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 C60 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.
Keywords: carbon nanotubes; fullerenes C60; gigahertz oscillators;
Lennard-Jones potential
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
* Author for correspondence ([email protected]).
Received 29 March 2006
Accepted 17 August 2006
461
This journal is q 2006 The Royal Society
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462
B. J. Cox et al.
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 C60 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 C60 fullerene molecule. These studies show that a C60
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 (Wa) 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 C60 fullerene molecules. The calculation of
the atom–nanotube interaction is provided as an exposition of the method in the
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Mechanics of fullerenes and nanotubes I
463
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 C60 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,
XX
EZ
Fðrij Þ;
ð2:1Þ
i
j
where F(rij) is a potential function for atoms i and j at distance rij 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
ðð
E Z n1 n2
FðrÞdS1 dS2 ;
ð2:2Þ
where n1 and n2 represent the mean surface density of atoms on each molecule
and r denotes the distance between two typical surface elements dS1 and dS2 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
A
B
FðrÞ ZK 6 C 12 ;
ð2:3Þ
r
r
where A and B are the attractive and the repulsive constants, respectively.
Equation (2.3) can be written in the form
s 6 s 12
;
ð2:4Þ
C
FðrÞ Z 4e K
r
r
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B. J. Cox et al.
Table 1. Lennard-Jones constants in graphitic systems (Girifalco et al. 2000).
graphene–graphene
C60–C60
C60–graphene
A (eV!Å6)
B (eV!Å12)
r0 (Å)
jej (meV)
15.2
20.0
17.4
24.1!103
34.8!103
29.0!103
3.83
3.89
3.86
2.39
2.86
2.62
where s is the van der Waals diameter. The equilibrium distance r0 is given by
1=6
2B
1=6
r0 Z 2 s Z
;
ð2:5Þ
A
and the well depth, eZA2/(4B). 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 C60 molecule (both
inside and outside the tube) (Girifalco et al. 2000) and between two C60
molecules (Girifalco 1992). The values of interaction Lennard-Jones constants
for atoms in graphene–graphene, C60–C60 and C60–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 zR0. As shown in figure 1, the distance r between the
atom and a typical surface element of the tube is given by r2Za2C(ZKz)2.
Owing to the symmetry of the problem, we are only concerned with the force in
the axial direction, Fz Z FvdW ðZ KzÞ=r, where FvdW is the van der Waals
interaction force defined by
dF
6A 12B
FvdW ZK
ZK 7 C 13 :
dr
r
r
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Mechanics of fullerenes and nanotubes I
x
r
a
z
Z
Figure 1. Geometry of the single atom entering a carbon nanotube.
Consequently, the interaction force between an atom located on the z-axis and all
the atoms of the carbon nanotube is given by
ðN
dF ðZ KzÞ
Fztot ðZÞ ZK2pang
dz;
ð3:2Þ
r
0 dr
where ng is the uniform surface density of carbon atoms in a graphene structure such
as a carbon nanotube. Since r2 Z a2 C ðZ KzÞ2 and drZK½ðZ KzÞ=rdz, equation
(3.2) becomes
ðN
dF
A
B
Fztot ðZÞ Z 2pang pffiffiffiffiffiffiffiffiffiffi
:
ð3:3Þ
K
dr Z 2pang
ða2 C Z 2 Þ3 ða2 C Z 2 Þ6
a2CZ 2 dr
We note that Fztot ðZÞ is a continuous function with zeros at ZZGZ0, where
"
Z0 Z a
B
Aa6
#1=2
1=3
K1
;
ð3:4Þ
pffiffiffiffiffiffiffiffiffiffi
and Z0 is real only when a%a 0, where a 0 Z 6 B=A. In figure 2, we plot Fztot ðZÞ for
carbon nanotubes of various radii, which illustrates that as the radius of the
nanotube increases beyond a 0 (in this case a 0z3.443), the value of Fztot ðZÞ
remains positive for all values of Z.
The integral of Fztot ðZÞ 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 KN to KZ0 needs to be greater than that which is lost
when the van der Waals force is negative (i.e. in the region KZ0!Z!Z0). We
term this the acceptance energy (Wa), which allows us to write the acceptance
condition as
m 0 v02
C Wa O 0;
2
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B. J. Cox et al.
0.03
Fztot (a) (eV/Å)
0.02
0.01
0.00
– 0.01
F (3.32)
F (3.443)
F (3.739)
– 0.02
– 0.03
–10
–5
0
5
10
Z (Å)
Figure 2. Force experienced by an atom due to van der Waals interaction with a semi-infinite
carbon nanotube.
where m0 is the mass of the atom and v0 is its initial velocity, and
ð Z0 A
B
dZ:
K 2
Wa Z 2pang
2
2 3
ða C Z 2 Þ6
KN ða C Z Þ
ð3:6Þ
Employing the substitution ZZa tan j, this integral is changed into the
following form:
ð
2png j0
B
4
10
Wa Z 4
A cos jK 6 cos j dj;
ð3:7Þ
a
a
Kp=2
where j0 Z tanK1 f½B=ðAa6 Þ1=3 K1g1=2 . Evaluation of this integral gives the
acceptance energy in explicit form as
png
p
3
Wa Z
32A sin j0 ð2 cos j0 C 3 cos j0 Þ C 3 j0 C
2
128a4
B sin j0
ð128 cos9 j0 C 144 cos7 j0 C 168 cos5 j0 C 210 cos3 j0
K 6
5
a
p
ð3:8Þ
:
C 315 cos j0 Þ C 63 j0 C
2
Assuming that the atom is initially at rest, the acceptance condition becomes
simply WaO0. 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 Z0 as graphed in figure 4. We comment that the acceptance energy is positive
for tubes of radius aO3.276 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
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Mechanics of fullerenes and nanotubes I
Table 2. Constants used in the model.
radius of (5, 5) (Å)
radius of (8, 8) (Å)
radius of (10, 10) (Å)
radius of C60 (Å)
carbon–carbon bond length (Å)
mean surface density for fullerene [60/(4pb2)] (ÅK2)
pffiffiffi
mean surface density for grapheme ½4 3=ð9s2 Þ (ÅK2)
mass of a single carbon atom (kg)
mass of a single C60 fullerene [60m 0] (kg)
attractive constant (eV!Å6)
repulsive constant (eV!Å12)
aZ3.392
aZ5.428
aZ6.784
bZ3.55
sZ1.421
n fZ0.3789
n gZ0.3812
m 0Z1.993!10K26
m fZ1.196!10K24
AZ17.4
BZ29!103
acceptance energy, Wa (eV)
0.05
0.00
3.15
3.20
3.25
3.30
3.35
3.40
– 0.05
– 0.10
nanotube radius, a (Å)
Figure 3. Acceptance energy threshold for an atom to be sucked into a carbon nanotube.
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 aZ3.206 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 aOa 0, the force graph does not cross the axis and therefore Z0 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
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B. J. Cox et al.
acceptance energy limit, Z0 (Å)
1.5
1.0
0.5
0.0
3.15 3.20 3.25 3.30 3.35 3.40 3.45
nanotube radius, a (Å)
Figure 4. Upper limit of integration Z0 used to determine the acceptance energy for an atom and
carbon nanotube.
seen in figure 2, the force is only appreciable within a few tube radii either side of
the tube end (jZj(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 Fztot ðZÞ
from KN to N 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 (j0) replaced with p/2 and therefore by evaluation of this integral we have
3p2 ng
21B
32AK
:
ð3:9Þ
WZ
a6
128a4
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
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3ng
p
21B
vZ 2
ð3:10Þ
32AK 6 :
m0
8a
a
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 amax which will give the
maximum suction energy and therefore the maximum velocity on entering the
tube. This occurs for a value of radius amax which is given by
rffiffiffiffiffiffiffiffiffiffiffiffi
6 105B
;
ð3:11Þ
a max Z
64A
and for our values of A and B, we have a maxz3.739 Å. We also comment that both
(6, 5) and (9, 1) nanotubes have a radius of aZ3.737 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 aZa max. We also note that W is
positive for any value of radius aO3.210 Å, which means that there is a range
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Mechanics of fullerenes and nanotubes I
0.20
suction energy, W (eV)
0.15
0.10
0.05
0.003
– 0.05
4
5
6
7
8
– 0.10
– 0.15
– 0.20
nanotube radius, a (Å)
Figure 5. Suction energy for an atom entering a carbon nanotube.
of nanotube radii 3.210!a!3.276 Å for which W is positive but Wa 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
PðrÞ ZKQ6 ðrÞ C Q12 ðrÞ;
ð4:1Þ
where the derivation of Qn is given in appendix A with coefficients C6ZA and
C12ZB, and r 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
#
( "
)
n f pb A
1
1
B
1
1
PðrÞ Z
K
K
K
;
ð4:2Þ
r
2 ðr C bÞ4 ðrKbÞ4
5 ðr C bÞ10 ðrKbÞ10
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 FvdWZKVP, and therefore, we have the
axial force
ðZ KzÞ dP
Fz ZK
:
ð4:3Þ
r
dr
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B. J. Cox et al.
x
r
a
b
Z
z
Figure 6. Geometry of a fullerene molecule entering a carbon nanotube.
As a result, the total axial force between the entire carbon nanotube and the
fullerene sphere is given by
ðN
dP ðZ KzÞ
dz;
ð4:4Þ
Fztot ðZÞ ZK2pang
r
0 dr
and since r2 Z a 2 C ðZ KzÞ2 , we have drZK½ðZ KzÞ=rdz. Thus, equation (4.4) can
be simplified to give
ðN
dP
Fztot ðZÞ Z 2pang pffiffiffiffiffiffiffiffiffiffi
dr
a2CZ 2 dr
"
!
1
1
2
2 A
Z K2p n f ng ab
K
2rb ðr C bÞ4 ðrKbÞ4
B
1
1
K
K
:
5rb ðr C bÞ10 ðrKbÞ10 rZpffiffiffiffiffiffiffiffiffiffi
ð4:5Þ
a2CZ 2
Now by placing the fractions over common denominators, expanding and reducing
to fractions in terms of powers of (r2Kb2), it can be shown that
!
!
A
1
1
1
2b2
ZK4A
;
ð4:6Þ
K
C
2rb ðr C bÞ4 ðrKbÞ4
ðr2 Kb2 Þ3 ðr2 Kb2 Þ4
B
1
1
4B
5
80b2
336b4
K
C
C
ZK
5rb ðr C bÞ10 ðrKbÞ10
5 ðr2 Kb2 Þ6 ðr2 Kb2 Þ7 ðr2 Kb2 Þ8
512b6
256b8
C
C 2
:
ðr Kb2 Þ9 ðr2 Kb2 Þ10
ð4:7Þ
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
8p2 n f ng a
2
B
80 336 512 256
tot
Fz ðZÞ Z
; ð4:8Þ
A 1C K 6 3 5C C 2 C 3 C 4
l
l
b4 l3
5b l
l
l
l
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Mechanics of fullerenes and nanotubes I
0.4
0.3
Fztot (a) (eV/Å)
0.2
0.1
0.0
– 0.1
F (6.4)
F (6.509)
F (6.784)
– 0.2
– 0.3
– 0.4
–15
–10
–5
0
Z (Å)
5
10
15
Figure 7. Force experienced by a C60 fullerene owing to van der Waals interaction with a
semi-infinite carbon nanotube.
where lZ ða 2 Kb2 C Z 2 Þ=b2 . This is the corresponding expression for the sphere as
equation (3.3) is for the atom. However, in this case, determining the roots of
Fztot ðZÞ 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 ZZGZ0 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 C60 fullerene, if bZ3.55 Å, then a 0z6.509 Å. As in
the previous section, the integral of Fztot ðZÞ represents the work imparted to the
fullerene and equates directly to the kinetic energy. Therefore (as before), the
integral of equation (4.8) from KN to Z0 represents the acceptance energy (Wa) 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.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiTo calculate
this acceptance energy, we make
the
change
of
variable
Z
Z
a 2 Kb2 tan j. Then,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
2
2
2
2
sec j dj, and the limits of the integration
lZ ða Kb Þsec j=b and dZ Z a Kb
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K1
2
change to Kp/2 and j0 Z tan ðZ0 = a Kb2 Þ which yields
8p2 n f ng a
B
Wa Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðJ2 C 2J3 ÞK 6 ð5J5 C 80J6 C 336J7 C 512J8 C 256J9 Þ ;
5b
b2 a2 Kb2
ð4:9Þ
Ð
j
0
where Jn Z b2n ða2 Kb2 ÞKn Kp=2
cos2n j dj. However, in the case of the sphere, a value
of Z0 cannot be specified explicitly and must be determined numerically. Once
determined, it can be substituted in the expression for Wa for any value of
parameters where a!a 0. In figure 8, we graph the acceptance energy for a C60
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472
B. J. Cox et al.
acceptance energy, Wa (eV)
2
1
0
6.10 6.15 6.20 6.25 6.30 6.35 6.40 6.45 6.50
–1
–2
–3
–4
–5
–6
nanotube radius, a (Å)
Figure 8. Acceptance energy threshold for a C60 fullerene to be sucked into a carbon nanotube.
acceptance energy limit, Z 0 (Å)
2.5
2.0
1.5
1.0
0.5
0.0
6.1
6.2
6.3
6.4
6.5
nanotube radius, a (Å)
6.6
Figure 9. Upper limit of integration Z0 used to determine the acceptance energy for a C60 fullerene
and carbon nanotube.
fullerene and a nanotube of radii in the range 6.1!a!6.5 Å, using the values of Z0 as
graphed in figure 9. We comment that WaZ0 when az6.338 Å and nanotubes
which are smaller than this will not accept C60 fullerenes by suction force alone.
Therefore, this model predicts that a (10, 10) nanotube (aZ6.784 Å) will accept a
C60 fullerene from rest; however, a (9, 9) nanotube (aZ6.106 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 C60 molecules, and
Okada et al. (2001) who interpolated a value of approximately 6.4 Å as the
minimum radius for a nanotube to encapsulate C60 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 C60 molecules from rest but (10, 10) nanotubes
will. Qian et al. (2001) report that firing C60 molecules at speeds of up to 1600 m sK1
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 aZ5.428 Å, the
acceptance energy predicted by our model is WaZK252 eV. This equates to firing
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Mechanics of fullerenes and nanotubes I
473
4
suction energy, W (eV)
3
2
1
0
6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0
–1
–2
–3
–4
nanotube radius, a (Å)
Figure 10. Suction energy for a C60 fullerene entering a carbon nanotube.
the C60 fullerene at the unlikely speed of more than 8200 m sK1. 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 aOa 0,
the force graph does not cross the axis and, therefore, Z0 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
j0 to p/2. In this case, the value for Jn becomes
b 2n
ð2nK1Þ!!
Jn Z 2
p;
ð4:10Þ
2 n
ð2nÞ!!
ða Kb Þ
where !! represents the double factorial notation such that ð2nK1Þ!!Z ð2nK1Þ!
ð2nK3Þ.3$1 and ð2nÞ!!Z ð2nÞð2nK2Þ.4$2. Substitution and simplification
gives
p3 n f ng ab 2
Bð315C4620mC18018m2 C25740m3 C12155m4 Þ
WZ 2
Að3C5mÞK
;
160ða2 Kb2 Þ3
ða Kb 2 Þ5=2
ð4:11Þ
where mZb =ða Kb Þ. In figure 10, we plot the suction energy W for a C60 fullerene
entering a nanotube with radii in the range 6!a!10 Å. We note that W is positive
whenever aO6.27 Å and has a maximum value of WZ3.242 eV when aZa maxZ
6.783 Å. We also comment that a (10, 10) carbon nanotube with az6.784 Å
is almost exactly the optimal size to maximise W and therefore have a C60 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 C60–(10, 10) oscillator, we calculate
the suction energy to be 3.243 eV. This corresponds to the C60 molecule being
accelerated to a velocity of 932 m sK1. This agrees reasonably well with Qian
et al. (2001) whose molecular dynamical simulation demonstrates an initial
velocity of around 840 m sK1. For a C60–(11, 11) oscillator, our model predicts an
initial velocity of 798 m sK1. This is in contrast to the molecular dynamical
simulation of Liu et al. (2005) who report speeds in excess of 1200 m sK1. While
2
2
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2
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474
B. J. Cox et al.
Table 3. Suction energy and velocity for various oscillator configurations.
oscillator configuration
tube radius a (Å)
energy W (eV)
velocity v (m sK1)
C60–(10, 10)
C60–(11, 11)
C60–(12, 12)
C60–(13, 13)
6.784
7.463
8.141
8.820
3.243
2.379
1.512
0.982
932
798
636
513
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 C60
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 C60–(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.
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.
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Mechanics of fullerenes and nanotubes I
f
b
475
Q
p
b sinf
r
P
Figure 11. Geometry of the problem where C60 interacts with an atom on a tube.
Appendix A. Derivation of equation (4.2)
Here, we show in detail the derivation of equation (4.2), based on Mahanty &
Ninham (1976, p. 15) and Ruoff & Hickman (1993). As shown in figure 11, the
distance p between atoms at the points P and Q is given by p2 Z b 2 C r2 K2br cos f.
At the point Q, we generate a ring of radius b sin f, we find that the potential energy
for an atom on the tube interacting with all atoms of the sphere radius b is given by
PðrÞZKQ6 ðrÞC Q12 ðrÞ where Qn(r) is defined by
ð
ðp
1
2pb sin f
Qn ðrÞ Z Cn n f
b df
n dS Z Cn n f
p
pn
0
S
ð
2pb 2 p
2br sin f
df
Z Cn n f
2
2
2br 0 ðb C r K2br cos fÞn=2
"
#p
pb ðb 2 C r2 K2br cos fÞ1Kn=2
Z Cn n f
r
ð1Kn=2Þ
0
ip
2Cn n f pb h 2
ðb C r2 K2br cos fÞð2KnÞ=2
0
rð2KnÞ
2Cn n f pb
1
1
;
Z
K
rð2KnÞ ðr C bÞnK2 ðrKbÞnK2
Z
ðA 1Þ
noting that r is the distance of the tube surface element from the centre of the
buckyball. From equation (A 1), we obtain the potential P(r), as given in equation
(4.2), where C6ZA and C12ZB. We comment that we may use this formal device to
determine the Lennard-Jones interaction for any shaped nanostructure which is
interacting with a spherical molecule.
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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
Proc. R. Soc. A (2007) 463, 3395–3396
Errata
Proc. R. Soc. A 463, 461–476 (11 October 2006) (doi:10.1098/rspa.2006.1771)
Mechanics of atoms and fullerenes
in single-walled carbon nanotubes.
I. Acceptance and suction energies
B Y B ARRY J. C OX , N GAMTA T HAMWATTANA
AND
J AMES M. H ILL
The following equation contains typographical errors that has no consequence for
any other equation or result in the above paper.
In this case, the value for Jn becomes
b 2n
ð2nK1Þ!!
p:
Jn Z 2
n
2
ð2nÞ!!
ða Kb Þ
ð4:10Þ
Proc. R. Soc. A 463, 477–494 (11 October 2006) (doi:10.1098/rspa.2006.1772)
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
The following equation contains typographical errors that has no consequence for
any other equation or result in the above paper.
B 315
1155b2
9009b4
6435b6
12155b8
J C
J6 C
J C
J8 C
J9
E Z 4p ab n f ng
5 256 5
64
128 7
64
256
A
K ð3J2 C 5b2 J3 Þ ;
ð2:15Þ
8
2
2
3395
This journal is q 2007 The Royal Society