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Proc. R. Soc. A (2008) 464, 1423–1438
doi:10.1098/rspa.2007.0349
Published online 19 February 2008
Group velocity of wave propagation
in carbon nanotubes
B Y L IFENG W ANG 1,2, * , W ANLIN G UO 1
AND
H AIYAN H U 2
1
Institute of Nanoscience, and 2MOE Key Lab of Structural Mechanics
and Control of Aircraft, Nanjing University Aeronautics and Astronautics,
210016 Nanjing, China
The group velocities of longitudinal and flexural wave propagations in single- and multiwalled carbon nanotubes are studied in the frame of continuum mechanics. The dispersion
relations between the group velocity and the wavenumber for flexural and longitudinal waves,
described by a beam model and a cylindrical shell model, are established for both single- and
multi-walled carbon nanotubes. The effect of microstructures in carbon nanotubes on the
wave dispersion is revealed through the non-local elastic models of a beam and a cylindrical
shell, including the second-order gradient of strain and a parameter of microstructure. It is
shown that the microstructures in the carbon nanotubes play an important role in the
dispersion of both longitudinal and flexural waves. In addition, the non-local elastic models
predict that the cut-off wavenumber of the dispersion relation between the group velocity and
the wavenumber is approximately 2!1010 mK1 for the longitudinal and flexural wave
propagations in both single- and multi-walled carbon nanotubes. This may explain why the
direct molecular dynamics simulation cannot give a proper dispersion relation between the
phase velocity and the wavenumber when the wavenumber approaches approximately
2!1010 mK1, much lower than the cut-off wavenumber for the dispersion relation between
the phase velocity and the wavenumber predicted by continuum mechanics.
Keywords: carbon nanotube; non-local elasticity; group velocity; Timoshenko beam;
cylindrical shell
1. Introduction
Interest in carbon nanotubes has grown rapidly since their discovery (Iijima 1991).
Recent studies have indicated that carbon nanotubes exhibit superior mechanical
and electronic properties over any known materials, and hold substantial promise for
new super-strong composite materials, among others. For instance, carbon
nanotubes have an exceptionally high elastic modulus (Treacy et al. 1996), and
sustain large elastic and failure strains (Yakobson et al. 1996; Wong et al. 1997).
Apart from an extensive experimental study to characterize the mechanical
behaviour of carbon nanotubes, theoretical or computational modelling of carbon
nanotubes has received considerable attention. The current computational
modelling approaches include both atomistic and continuum modelling.
* Author and address for correspondence: MOE Key Lab of Structural Mechanics and Control
of Aircraft, Nanjing University Aeronautics and Astronautics, 210016 Nanjing, China
([email protected]).
Received 28 November 2007
Accepted 28 January 2008
1423
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1424
L. F. Wang et al.
Among the research of continuum modelling of carbon nanotubes, numerous
studies have concentrated on the static mechanical behaviour, such as buckling,
of carbon nanotubes using the elastic models of a beam and a cylindrical shell
(Qian et al. 2002). Several research teams implemented the elastic models of
beam to study the dynamic problems, such as vibration and wave propagation
(Poncharal et al. 1999; Popov et al. 2000; Yoon et al. 2002, 2003a,b, 2004), of
carbon nanotubes. Some other teams used the elastic models of a cylindrical shell
to study the vibration and wave dispersion relations of carbon nanotubes
(Natsuki et al. 2005, 2006; Wang et al. 2005; Dong & Wang 2006). Furthermore,
a multi-walled carbon nanotube was modelled as an assemblage of cylindrical
shell elements connected throughout their lengths by distributed springs to
investigate the elastic waves of very high frequency in carbon nanotubes
(Chakraborty 2006), and to give the dispersion relation between the group
velocity and the wavenumber. Their studies showed that both elastic models of a
beam and a cylindrical shell are valid to describe the vibration or wave
propagation of carbon nanotubes in a relatively low-frequency range.
Recognizing such a limit mainly coming from the microstructures in carbon
nanotubes, several researchers (Sudak 2003; Zhang et al. 2004, 2005; Wang
2005; Wang et al. 2006a; Xie et al. 2006) established non-local elastic models with
the second-order gradient of stress taken into account so as to describe the
vibration and wave propagation of both single- and multi-walled carbon nanotubes, in a higher frequency range. The molecular dynamics simulations showed
that these models worked much better than the elastic models in a relatively
high-frequency range. However, the molecular dynamics simulations also showed
that the microstructures of a carbon nanotube might have a very significant
influence on the high frequency waves such that non-elastic models could not
predict the wave dispersion well (Wang & Hu 2005; Wang et al. 2006b).
The concept of group velocity may be useful in understanding the dynamics of
carbon nanotubes, since it is related to the energy transportation of wave
propagation. To the best knowledge of the authors, however, there is little
work on the group velocity of wave propagation in carbon nanotubes with the
effect of microstructures taken into consideration. The primary objective of this
work is to study the group velocity of the longitudinal and flexural wave
propagations in carbon nanotubes, so as to examine the effect of microstructures
of a carbon nanotube on the wave dispersion from the viewpoint of group velocity
or energy transportation.
This work deals with the dispersion relations between the group velocity and
the wavenumber for both longitudinal and flexural waves in single- and multiwalled carbon nanotubes via the non-local elastic models of both Timoshenko
beams and cylindrical shells. The study focuses on the comparison between the
non-local elastic and the elastic models in predicting the dispersion relation of
the group velocity with respect to the wavenumber. For this purpose, §2 presents
the dispersion relation between the group velocity and the wavenumber of
longitudinal waves in a single-walled carbon nanotube from a non-local elastic
model of cylindrical shell, which includes the second-order gradient of strain in
order to characterize the microstructures of the carbon nanotube. Section 3 gives
the dispersion relation between the group velocity and the wavenumber of
flexural waves in a single-walled carbon nanotube using a non-local elastic model
of a Timoshenko beam. Section 4 turns to the dispersion relation between the
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Group velocity of waves in CNT
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group velocity and the wavenumber of longitudinal waves in a multi-walled
carbon nanotube on the basis of a non-local elastic model of multi-cylindrical
shells, which also takes the second-order gradient of strain into account.
Similarly, §5 gives the dispersion relation between the group velocity and the
wavenumber of flexural waves in a multi-walled carbon nanotube from a nonlocal elastic model of multi-Timoshenko beams. Finally, in §6, the paper ends
with some concluding remarks.
2. Group velocity of a longitudinal wave in a single-walled
carbon nanotube
To describe the effect of microstructures of a carbon nanotube on its mechanical
properties, it is assumed that the model of the carbon nanotube is made of a kind
of non-local elastic material, where the stress state at a given reference location
depends not only on the strain of this location but also on the higher order
gradient of strain, so as to take the influence of the microstructures into account.
The simplest constitutive law to characterize the non-local elastic material in the
one-dimensional case reads (Askes et al. 2002)
2 2 v 3x
sx Z E 3x C r
;
ð2:1Þ
vx 2
where E is Young’s modulus;
pffiffiffiffiffisx is the axial stress; and 3x is the axial strain. The
material parameter r Z d= 12 characterizes the influence of microstructures on
the constitutive law of the non-local elastic materials, and d describes the interparticle distance (Askes et al. 2002) and is the axial distance between two
neighbouring rings of carbon atoms when the single-walled carbon nanotube is
modelled as a non-local elastic cylindrical shell.
The single-walled carbon nanotube can be considered as a thin cylindrical
shell, where the bending moments can be neglected for simplicity. The non-local
elastic model of a cylindrical shell with the second-order gradient of strain taken
into consideration is as follows (Wang et al. 2006b):
4
3 rð1Ky2 Þ v2 u v2 u
y vw
2v u
2v w
Cr
K
Kr
K
Z 0;
ð2:2aÞ
E
vt 2 vx 2
vx 3
vx 4 R vx
4
2rð1 C yÞ vv 2 v2 v
2v v
K
Kr
Z 0;
E
vt 2 vx 2
vx 4
2 3 rð1Ky2 Þ v2 w
1
y vu
2v w
2v u
C 2 w Cr
C
Z 0;
Cr
E
R vx
vt 2
R
vx 2
vx 3
ð2:2bÞ
ð2:2cÞ
where x is the coordinate in the longitudinal direction; u, v and w are the
displacement components in the longitudinal, tangential and radial directions,
respectively; R is the radius of the cylindrical shell; r is the mass density; and y is
Poisson’s ratio. It is interesting that equation (2.2b) is not coupled with
equations (2.2a) and (2.2c) such that the torsional waves in the cylindrical shell
are independent of the longitudinal and radial waves.
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L. F. Wang et al.
Now consider the motions governed by the coupled dynamic equations in
u and w, and let
~ KutÞÞ and w Z w
~ KutÞÞ;
^ exp ðiðkx
u Z u^ exp ðiðkx
ð2:3Þ
pffiffiffiffiffiffiffi
^ is the
where i h K1; u^ is the amplitude of longitudinal vibration; and w
amplitude of radial vibration. Furthermore, u is the angular frequency and k~ is
the wavenumber related to the wavelength l via lk~Z 2p. Substituting equation
(2.3) into equations (2.2a) and (2.2c) yields
3
2 2
u
2
2 ~4
2 ~3 y
~
~
iðk Kr k Þ
" #
6 c2 Kk C r k
R 7
7 u^
6 p
7
6
ð2:4Þ
7 ^ Z 0:
6
2
u
1
3 y
2 5 w
4
2
2
Kiðk~Kr k~ Þ
K
ð1Kr k~ Þ
R cp2 R2
The determinant of the coefficient matrix of equation (2.4) gives
2
cp4 ð1Ky2 Þk~
1
2
2
2 ~2
2
~
u
ð1Kr 2 k~ Þ2 Z 0;
ð2:5Þ
k C 2 ð1Kr k Þu C
2
R
R
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where cp h E=ðð1Ky2 ÞrÞ. Solving equation (2.5) gives the two branches of the
wave dispersion relation as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
u
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
u 2
2 Þk
~
2
ð1Ky
u k~ C 12 H
k~ C R12 K4 R2
t
R
2
ð2:6Þ
u Z cp
ð1Kr 2 k~ Þ:
2
4
Kcp2
Then, the group velocity reads
cp
du
1
1Ky2
2
~
~
~
~
cg Z
2kH 2k k C 2 K4k
Z pffiffiffi
R
R2
2 2
dk~
1
"
#
2 Kð1=2Þ
1 2
ð1Ky2 Þk~
2
~
Að1Kr 2 k~2 Þ
! k C 2 K4
2
R
R
9
2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
2
2
=
2
~
1
1
ð1Ky Þk 5
2
2
2~
C4 k~ C 2 H
k~ C 2 K4
ðK
2r
kÞ
;
R
R
R2
82
9 Kð1=2Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
2
< =
2 ~2
1
1
ð1Ky
Þ
k
2
2
5ð1Kr 2 k~2 Þ
k~ C 2 K4
:
! 4 k~ C 2 H
:
;
R
R
R2
ð2:7Þ
Figure 1 shows the dispersion relation between the group velocity and
the wavenumber of longitudinal waves in an armchair (5, 5) and (10, 10) singlewalled carbon nanotubes. Now, the product of Young’s modulus E and the
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Group velocity of waves in CNT
group velocity (×103 m s–1)
(a)
25
lower branches
20
upper branches
15
10
5
0
group velocity (×103 m s–1)
(b)
25
lower branches
20
upper branches
15
10
5
0
0.5
1.0
wavenumber
1.5
2.0
(×1010 m–1)
Figure 1. Dispersion relations between the group velocity and the wavenumber of longitudinal
waves in armchair (a) (5, 5) and (b) (10, 10) single-walled carbon nanotubes. Solid lines, non-local
theory; dashed lines, classical theory.
wall thickness h is EhZ346.8 Pa m, and Poisson’s ratio is yZ0.20 for a (5, 5)
and a (10, 10) single-walled carbon nanotube. In addition, one has rZ0.0355 nm
when the axial distance between two neighbouring rings of atoms is
dZ0.123 nm. The product of the mass density r and the wall thickness h yields
rhz760.5 kg mK3 nm. The radii of (5, 5) and (10, 10) single-walled carbon
nanotubes are 0.34 and 0.68 nm, respectively. There is a slight difference between
the non-local theory and the classical theory of elasticity for the lower branches.
The group velocity decreases rapidly with an increase in the wavenumber. The
group velocity goes to zero for the (5, 5) and (10, 10) single-walled carbon
nanotubes when the wavenumber approaches to k~Z 5 !109 and 3 !109 m K1 or
so, respectively. This may explain the difficulty that the direct molecular
dynamics simulation can only offer the dispersion relation between the phase
velocity and the wavenumber up to k~ z5 !109 and 3 !109 m K1 for the (5, 5)
and (10, 10) single-walled carbon nanotubes, respectively. For the upper
branches of the dispersion relation, the difference can hardly be identified
when the wavenumber is lower. However, the results of the elastic cylindrical
shells remarkably deviate from those of the non-local elastic cylindrical
shells with an increase in the wavenumber. Figure 1a,b shows the intrinsic
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L. F. Wang et al.
pffiffiffiffiffi
limit k~! 2 !1010 m K1 , instead of k~! 12=d z2:82 !1010 m K1 for the maximum
wavenumber owing to the microstructures. This can explain the contradiction
that the cut-off
pffiffiffiffiffilongitudinal wave predicted by the non-local elastic cylindrical
shell is k~! 12=d z2:82 !1010 m K1 , but the molecular dynamics simulation
offers the dispersion relation up to the wavenumber k~ z2 !1010 m K1 only
(Wang et al. 2006b).
3. Group velocity of a flexural wave in a single-walled carbon nanotube
This section starts with the dynamic equation of a non-local elastic model of a
Timoshenko beam of infinite length and uniform cross section placed along the x
direction in the frame of coordinates ðx; y; zÞ, with wðx; tÞ being the displacement
of a section x of the beam in the y direction at the moment t (Wang & Hu 2005)
8
"
!
!#
2
2
3
4
>
v
w
v4
v
w
v
4
v
w
>
2
>
Z 0;
r
C bG
K 4
K 2 Cr
>
>
< vt 2
vx
vx
vx 3
vx
!
"
!#
!
2
2
3
2
4
>
>
v
4
vw
v
4
v
w
v
4
v
4
2
> rI
>
KEI
C r 2 4 Z 0;
>
: vt 2 C bAG 4K vx C r vx 2 K vx 3
vx 2
vx
ð3:1Þ
where G is the shear modulus; 4 is the slope of the deflection curve Ðwhen the
shearing force is neglected; A is the cross-sectional area of the beam; I Z y 2 dA is
the moment of inertia for the cross section; b is the form factor of shear depending
on the shape of the cross section; and bZ0.5 holds for the circular tube of the thin
wall (Timoshenko & Gere 1972). The other parameters are the same as those
defined in §2.
To study the flexural wave propagation in an infinitely long beam, let the
dynamic deflection and slope be given by
~ KutÞÞ and 4ðx; tÞ Z 4
~ KutÞÞ;
^ exp ðiðkx
^ exp ðiðkx
wðx; tÞ Z w
ð3:2Þ
^ represents the amplitude of deflection of the beam and 4
^ is the amplitude
where w
of the slope of the beam due to the bending deformation alone. As defined in
§2, u is the angular frequency of wave and k~ is the wavenumber related to the
wavelength l via lk~Z 2p. Substituting equation (3.2) into equation (3.1) yields
8
2
2
4
< ðKibAG k~ C ibAGr 2 k~3 Þw
^ C ðKrI u2 C bAGKbAGr 2 k~ C EI k~ KEIr 2 k~ Þ4
^ Z 0;
:
2
4
3
^ C ðibG k~K ibGr 2 k~ Þ4
^ Z 0:
ðKru2 C bG k~ KbGr 2 k~ Þw
ð3:3Þ
^ 4Þ
^ of equation (3.3), one
If there exists at least one non-zero solution ðw;
arrives at
r2 I 4
E ~2
2
4
2
ð3:4Þ
u K rA C rI 1 C
k ð1Kr 2 k~ Þu2 C EI k~ ð1Kr 2 k~ Þ2 Z 0:
bG
bG
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Group velocity of waves in CNT
Solving equation (3.4) for the angular frequency u gives two branches of the
wave dispersion relation
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Kb1 G b21 K4a 1 c1
uZ
;
ð3:5Þ
2a 1
2
2
4
2
where a 1 Z r2 I =bG, b1 Z ½KrAKrI ð1C E=bGÞk~ ð1Kr 2 k~ Þ and c1 Z EI k~ ð1Kr 2 k~ Þ2 .
The group velocity reads
0 db db
Kð1=2Þ 1
dc1 2
1
1
K
G
b
K2a
K4a
c
b
1
1
1
1
1
du
1 @ dk~
dk~
dk~
A;
Z
ð3:6Þ
cg Z
~
2u
2a 1
dk
where
db1
~ ð1 C E=bGÞð1Kr 2 k~2 ÞK2r 2 kðK
~ rAKrI ð1 C E=bGÞk~2 Þ
ZK2krI
dk
ð3:7Þ
and
dc1
3
2
5
2
Z 4EI k~ ð1Kr 2 k~ Þ2 K4EIr 2 k~ ð1Kr 2 k~ Þ:
dk
ð3:8Þ
Figure 2 shows the dispersion relations between the group velocity and the
wavenumber of flexural waves in an armchair (5, 5) and (10, 10) single-walled
carbon nanotubes. The product of Young’s modulus and the wall thickness is EhZ 346:8 Pa m and Poisson’s ratio is yZ 0:20, and it follows that
G Z E=ð2ð1C yÞÞ. In addition, the material parameter rZ0.0355 nm. The product
of the mass density and the wall thickness yields rhz760.5 kg mK3 nm. For the
(5, 5) single-walled carbon nanotube, the product of the mass density and the
section area yields rAZ 1:625 !10K15 kg m K1 , the product of the mass density and
the moment of inertia for the cross section yields rI Z 3:736 !10K35 kg m, and it
follows that EI Z 1:704 !10 K26 Pa m4 . For the (10, 10) single-walled carbon
nanotube, the product of the mass density and the section area yields
rAZ 3:25 !10K15 kg m K1 , the product of the mass density and the moment of
inertia for the cross section yields rI Z 2:541 !10 K34 kg m, and it follows that
EI Z 1:159 !10K25 Pa m4 . For both lower and upper branches of the dispersion
relation, the results of the elastic Timoshenko beam remarkably deviate from those
of the non-local elastic Timoshenko beam with an increase in the wavenumber.
Figure 2a,b shows
limit of the wavenumber k~! 2 !1010 m K1 ,
pffiffiffiffiffi again the intrinsic
10
K1
instead of k~! 12=d z2:82 !10 m . This fact explains why the cut-off flexural
pffiffiffiffiffi
wave predicted by the non-local elastic cylindrical shell is k~! 12=
d z2:82 !1010 m K1 , but the direct molecular dynamics simulation only gives the
dispersion relation up to the wavenumber k~ z2 !1010 m K1 (Wang & Hu 2005).
4. Group velocity of a longitudinal wave in a multi-walled
carbon nanotube
This section deals with the propagation of longitudinal waves in a multi-walled
carbon nanotube via a non-local elastic model of multi-cylindrical shells,
including the second-order gradient of strain. The dynamic equations of the
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L. F. Wang et al.
group velocity (×103 m s–1)
(a) 25
20
15
10
5
0
group velocity (×103 m s–1)
(b) 25
20
15
10
5
0
0.5
1.0
wavenumber
1.5
(×1010
2.0
m–1)
Figure 2. Dispersion relations between the group velocity and the wavenumber of flexural waves in
armchair (a) (5, 5) and (b) (10, 10) single-walled carbon nanotubes. Other descriptions are the same
as given in the legend of figure 1.
kth layer for an N-walled carbon nanotube are (L. F. Wang et al. 2007,
unpublished data)
4
rð1Ky2 Þ v2 uk v2 uk
y
2 v uk
K
Kr
K
E
Rk
vt 2
vx 2
vx 4
vwk
v 3 wk
Cr2
vx
vx 3
4
2rð1 C yÞ v2 vk v2 vk
2 v vk
K
Kr
Z 0;
E
vt 2
vx 2
vx 4
Z 0;
ð4:1aÞ
ð4:1bÞ
2
rð1Ky2 Þ v2 wk
1
y vuk v3 uk
qk ð1Ky2 Þ
2 v wk
C
w
C
r
C
Z
; ð4:1cÞ
C
k
E
Rk vx
Eh
R2k
vt 2
vx 2
vx 3
where x is the coordinate in longitudinal direction; u k, vk and wk are the
displacement components in the longitudinal, tangential and radial directions of
the kth tube, respectively; and R k is the radius of the kth tube. Other parameters
are those defined in §§2 and 3.
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Group velocity of waves in CNT
As only infinitesimal vibration is considered, the net pressure due to the van
der Waals interaction is assumed to be linearly proportional to the deflection
between two layers, i.e.
qk Z
N
X
ckj ðwk K wj Þ Z wk
kZ1
N
X
N
X
ckj K
ckj wj ;
jZ1
jZ1
ð4:2Þ
where N is the total number of layers of the multi-walled carbon nanotube and ckj
is the van der Waals interaction coefficient, which can be determined through the
Lennard-Jones pair potential (Jones 1924)
12 6 s
s
V ð
r Þ Z 4
3
K
;
ð4:3Þ
r
r
where 3Z 2:968 !10 K3 eV, sZ3.407 Å and r is the distance between two
interacting atoms. In this study, the coefficient of the van der Waals interaction
comes from the Lennard-Jones pair potential suggested by He et al. (2005)
1001p3s12 13 1120p3s6 7
ckj ZK
E kj K
E kj Rj ;
ð4:4Þ
3a4
9a4
where
Em
kj
Z ðRk C Rj Þ
ð p=2
dq
0
½1K Kkj cos2 qm=2
Km
ð4:5Þ
and
Kkj Z
4Rk Rj
ðRk C Rj Þ2
:
ð4:6Þ
Consider the motions governed by the coupled dynamic equations in u k and wk,
and let
~ KutÞÞ;
uk Z u^k exp ðiðkx
~ KutÞÞ;
^ k exp ðiðkx
wk Z w
k Z 1; 2; .; N ; ð4:7Þ
^ k is the amplitude of radial
where u^k is the amplitude of longitudinal vibration and w
vibration. As defined in §2, u is the angular frequency of the wave and k~ is the
wavenumber related to the wavelength l via lk~Z 2p. Substituting equation (4.7)
into equations (4.1a) and (4.1c) yields
2
h
i
u
2
4
2
~ C r k~ u^k C iðk~Kr 2 k~3 Þ y w
^ Z 0;
K
k
ð4:8aÞ
R k
cp2
h
!
2
n
n
i
2
X
X
y
u
1
1Kv
2
2
~
^k
^ j Z 0:
^k C
w
k Þ u^k C 2 K 2 ð1Kr k~ Þ w
KiðkKr
ckj K
ckj w
R
Eh
cp R
jZ1
jZ1
2 ~3
ð4:8bÞ
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L. F. Wang et al.
The dispersion relation can be obtained by solving the following eigenvalue
equation:
3
2
u^1
7
6
7
6w
6 ^1 7
7
6
u^2 7
2
6
7
6
u
7
6
ð4:9Þ
I
CH
^
w
7 Z 0;
6
2N!2N
2N!2N
6 27
cp2
6 « 7
7
6
7
6
6 u^ 7
4 N5
^N
w
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
where cp h E=ðð1Ky ÞrÞ and I 2N!2N is an identity matrix and the entries in the
H 2N!2N matrix are
H ð2k; 2jÞ ZK
ð1Ky2 Þckj
; k sj;
Eh
2
4
H ð2k K1; 2k K 1Þ ZKk~ Cr 2 k~ ;
2 ~3 y
~
H ð2k K1; 2kÞ Z iðkKr
k Þ ;
Rk
2 ~3 y
~
H ð2k; 2k K 1Þ ZKiðkKr
k Þ ;
Rk
H ð2k; 2kÞ ZK
n
1
1Kv 2 X
2 ~2
ð1Kr
Þ
C
c :
k
Eh jZ1 kj
R2k
ð4:10aÞ
ð4:10bÞ
ð4:10cÞ
ð4:10dÞ
ð4:10eÞ
From equation (4.9), the dispersion relation between the group velocity and the
wavenumber of longitudinal waves in the multi-walled carbon nanotube can be
numerically determined by
du
cg Z
:
ð4:11Þ
dk~
Figure 3 presents the dispersion relation between the group velocity and the
wavenumber of longitudinal waves in a double-walled carbon nanotube with
radii R1 Z0:34 nm and R2 Z0:68 nm, and in a four-walled carbon nanotube
with radii R1 Z0:34 nm, R2 Z0:68 nm, R3 Z1:02 nm and R4 Z1:36 nm. The product
of Young’s modulus and the wall thickness is EhZ346:8 Pa m, and Poisson’s ratio is
yZ0:20 for both the (5, 5) and (10, 10) single carbon nanotubes. In addition, one
has r Z0:0355 nm when the axial distance between two neighbouring rings of
atoms is dZ0:123 nm. The product of the mass density and the wall thickness
yields rhz760.5 kg mK3 nm. Now, there is a slight difference between the theory
of non-local elasticity and the classical theory of elasticity for the lower branches.
The group velocity decreases rapidly with an increase in wavenumber. When the
9
~
wavenumber approaches approximately kZ5!10
m K1 , the group velocity of all
these lower branches tends to zero. For the upper branches of the dispersion
relation, the difference is almost invisible when the wavenumber is lower.
However, the results of the elastic cylindrical shells remarkably deviate from
Proc. R. Soc. A (2008)
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1433
Group velocity of waves in CNT
group velocity (×103 m s–1)
(a) 25
20
upper branches
15
10
lower branches
5
0
group velocity (×103 m s–1)
(b) 25
20
upper branches
15
10
lower branches
5
0
0.5
1.0
1.5
2.0
wavenumber (×1010 m–1)
Figure 3. Dispersion relation between the group velocity and the wavenumber of a longitudinal
wave in multi-walled carbon nanotubes. (a) Double-walled carbon nanotube with radii R1Z0.34
and R2Z0.68 nm. (b) Four-walled carbon nanotube with radii R1Z0.34, R2Z0.68, R3Z1.02 and
R4Z1.36 nm. Other descriptions are the same as given in the legend of figure 1.
those of non-local elastic cylindrical shells with an increase in the wavenumber.
~
Figure 3a,b shows again pthe
of wavenumber yielding k!2
ffiffiffiffiffi intrinsic limit
10
K1
10
K1
~
!10 m , instead of k! 12=d z2:82!10 m .
5. Group velocity of flexural waves in a multi-walled carbon nanotube
This section starts with the dynamic equation of a non-local elastic model of
multi-Timoshenko beams of infinite length and uniform cross section placed
along the x direction in the frame of coordinates ðx; y; zÞ. The dynamics
equations of the kth tube for an N-walled carbon nanotube are
v2 w
rAk 2k CbAk G
vt
2
n
n
3
4
X
X
v4k
v wk
2 v 4k
2 v wk
Cr
Cr
C
K
Ckj wj ;
K
Zw
k
kj
vx
vx 3
vx 2
vx 4
jZ1
jZ1
ð5:1aÞ
Proc. R. Soc. A (2008)
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1434
v2 4
rIk 2k CbAk G
vt
L. F. Wang et al.
2
2
4
vwk
v3 w k
v 4k
2 v 4k
2 v 4k
4k K
K 3
Cr
KEIk
Z 0;
Cr
vx
vx 2
vx
vx 2
vx 4
ð5:1bÞ
where wk ðx;tÞ is the displacement of section x of the kth tube in the y direction at
the moment t; 4k is the slope of the deflection curve of the kth tube when the
shearing
Ð 2 force is neglected; Ak is the cross-sectional area of the kth tube;
Ik Z y dA is the moment of inertia for the cross section of the kth tube; b is the
form factor of shear depending on the shape of the cross section; and bZ0.5 holds
for the circular tube of the thin wall (Timoshenko & Gere 1972). Cjk is the van der
Waals interaction coefficient for the interaction pressure per unit axial length and
is estimated based on an effective interaction width (Ru 2000)
ð5:2Þ
Ckj Z2Rk ckj ;
where ckj can be obtained by equation (4.4).
To study the flexural wave propagation, consider the deflection and the slope
given by
~ KutÞÞ;
^ k exp ðiðkx
wk Z w
~ KutÞÞ;
^k exp ðiðkx
4k Z 4
k Z 1; .; N ;
ð5:3Þ
^k is the
^ k represents the amplitude of deflection of the kth tube and 4
where w
amplitude of the slope of the kth tube due to bending deformation alone. As in §2,
u is the angular frequency of wave and k~ is the wavenumber related to the
wavelength l via lk~Z 2p. Substituting equation (5.3) into equation (5.1a) and
(5.1b) yields
8
>
>
>
>
<
n
X
Ckj
n
X
Ckj
jZ1
2
4
2 ~3
~
^ Z0;
^ k KðibG kKibGr
^k
w
k Þ^
ðru2 KbG k~ CbGr 2 k~ Þw
4k C w
K
Ak
Ak j
>
>
jZ1
>
>
:
2
2
4
2 ~3
~
^ k CðrIk u2 KbAk GCbAk Gr 2 k~ KEIk k~ CEIk r 2 k~ Þ^
ðibAk G kKibA
4k Z0:
k Gr k Þw
ð5:4Þ
^ k ;4
^ k Þ of
Under the assumption that there exists at least one non-zero solution ðw
equation (5.4), one arrives at
2 3
^1
w
6 7
7
64
6 ^1 7
6 7
6w
^27
7
2
6
6 7
ð5:5Þ
ru I 2n!2n CH 2n!2n 6 4
^ 2 7 Z0;
6 7
6 « 7
6 7
6 7
7
6w
4 ^n5
^n
4
Proc. R. Soc. A (2008)
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1435
Group velocity of waves in CNT
group velocity (×103 m s–1)
(a) 25
20
15
10
5
0
group velocity (×103 m s–1)
(b) 25
20
15
10
5
0
0.5
1.0
1.5
wavenumber (×1010 m–1)
2.0
Figure 4. Dispersion relations between the group velocity and the wavenumber of flexural wave in
multi-walled carbon nanotubes. (a) Double-walled carbon nanotube with the radii 0.34 and
0.68 nm. (b) Four-walled carbon nanotube with the radii 0.34, 0.68, 1.02 and 1.36 nm. Other
descriptions are the same as given in figure legend 1.
where I 2n!2n is an identity matrix and the entries in the H 2n!2n matrix are
H ð2k K 1;2j K 1ÞZK
Ckj
ðjskÞ;
Ak
ð5:6aÞ
n
P
Ckj
jZ1
2
2 ~4
~
H ð2k K1;2k K 1ÞZKbG k CbGr k C
;
Ak
ð5:6bÞ
2 ~3
~
H ð2k K1;2kÞZKðibG kKibGr
k Þ;
ð5:6cÞ
H ð2k;2k K1ÞZ
2 ~3
~
ibAk G kKibA
k Gr k
;
Ik
2
H ð2k;2kÞZ
Proc. R. Soc. A (2008)
2
ð5:6dÞ
4
KbAk GCbAk Gr 2 k~ KEIk k~ CEIk r 2 k~
:
Ik
ð5:6eÞ
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1436
L. F. Wang et al.
Table 1. Parameters of double-walled carbon nanotubes.
no. of tube k
(radius in nm)
rAk (kg mK1)
rIk (kg m)
EIk (Pa m4)
kZ1 (0.34)
kZ2 (0.68)
1.625!10K15
3.250!10K15
3.737!10K35
2.541!10K34
1.704!10K26
1.158!10K25
Table 2. Parameters of four-walled carbon nanotubes.
no. of tube k
(radius in nm)
rAk (kg mK1)
rIk (kg m)
EIk (Pa m4)
kZ1
kZ2
kZ3
kZ4
1.625!10K15
3.250!10K15
4.874!10K15
6.466!10K15
3.737!10K35
2.541!10K34
8.296!10K34
1.943!10K33
1.704!10K26
1.158!10K25
3.782!10K25
8.860!10K25
(0.34)
(0.68)
(1.02)
(1.36)
From equation (5.5), the dispersion relation between the group velocity and the
wavenumber of flexural waves in a multi-walled carbon nanotube can be
~
numerically obtained using cg Zdu=dk.
Figure 4 presents the dispersion relation between the group velocity and the
wavenumber of flexural waves in a double-walled carbon nanotube with radii
R1 Z 0:34 nm and R2 Z 0:68 nm, and in a four-walled carbon nanotube with radii
R1 Z 0:34 nm, R2 Z 0:68 nm, R3 Z 1:02 nm and R4 Z 1:36 nm. The product of
Young’s modulus and the wall thickness is EhZ 346:8 Pa m, and Poisson’s ratio
is yZ 0:20 and it follows that G Z E=ð2ð1C yÞÞ. The product of the mass density
and wall thickness yields rhz760.5 kg mK3 nm, and the form factor of shear
bZ 0:5. Table 1 shows the parameters rAk, rIk and EIk of the double-walled
carbon nanotube. Table 2 gives the parameters rAk, rIk and EIk of the fourwalled carbon nanotube. For the lower and upper branches of the dispersion
relation, the difference is almost invisible when the wavenumber is lower.
However, the results of the elastic model of Timoshenko beams remarkably
deviate from those of the non-local elastic model of Timoshenko beams with an
increase in the wavenumber. Figure
pffiffiffiffiffi 4a,b shows again the intrinsic limit
k~! 2 !1010 m K1 , rather than k~! 12=d z2:82 !1010 m K1 for the maximum
wavenumber owing to the microstructures.
6. Concluding remarks
The paper presents a detailed study on the dispersion relation between the group
velocity and the wavenumber for the propagation of longitudinal and flexural
waves in single- and multi-walled carbon nanotubes, on the basis of elastic and
non-local elastic cylindrical shells and Timoshenko beams. The study indicates
that both elastic and non-local elastic models can offer the correct prediction
when the wavenumber is lower. However, the results of the elastic model
remarkably deviate from those given by the non-local elastic model with an
Proc. R. Soc. A (2008)
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Group velocity of waves in CNT
1437
increase in the wavenumber. As a result, the microstructures play an important
role in the dispersion of both longitudinal and flexural waves in both single- and
multi-walled carbon nanotubes.
The cut-off wavenumber of the dispersion relation between the group velocity
and the wavenumber is approximately 2!1010 mK1 for both longitudinal and
flexural waves in both single- and multi-walled carbon nanotubes. This
contradiction is interpreted to show that the direct molecular dynamics simulation cannot give the dispersion relation between the phase velocity and the
wavenumber when the wavenumber approaches approximately 2!1010 mK1,
which is much lower than the cut-off wavenumber of the dispersion relation
between the phase velocity and the wavenumber predicted by the continuum
mechanics.
This work was supported by the National Natural Science Foundation of China under grants
10702026 and 10732040, the National Postdoctoral Foundation of China, the Postdoctoral
Foundation of Jiangsu Provincial Government and the Ministry of Education of China under
grants 705021 and IRT0534.
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