Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes
A simplified theory of the telescopic oscillations in multi-walled carbon nanotubes is developed. The explicit expressions for the telescopic force constants (longitudinal rigidity) and the frequencies of telescopic oscillations are derived. The contribution of small-amplitude telescopic oscillation...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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nasplib_isofts_kiev_ua-123456789-1185392025-02-09T15:07:50Z Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes Zavalniuk, V. Marchenko, S. Наноструктуры при низких температурах A simplified theory of the telescopic oscillations in multi-walled carbon nanotubes is developed. The explicit expressions for the telescopic force constants (longitudinal rigidity) and the frequencies of telescopic oscillations are derived. The contribution of small-amplitude telescopic oscillations to the nanotubes low temperature specific heat is estimated. Authors are grateful to Prof. V.M. Adamyan for the significant help. 2011 Article Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes / V. Zavalniuk, S. Marchenko // Физика низких температур. — 2011. — Т. 37, № 4. — С. 432–437. — Бібліогр.: 18 назв. — англ. 0132-6414 PACS: 61.46.Fg, 62.25.–g, 65.80.–g https://nasplib.isofts.kiev.ua/handle/123456789/118539 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Наноструктуры при низких температурах Наноструктуры при низких температурах |
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Наноструктуры при низких температурах Наноструктуры при низких температурах Zavalniuk, V. Marchenko, S. Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes Физика низких температур |
| description |
A simplified theory of the telescopic oscillations in multi-walled carbon nanotubes is developed. The explicit expressions for the telescopic force constants (longitudinal rigidity) and the frequencies of telescopic oscillations are derived. The contribution of small-amplitude telescopic oscillations to the nanotubes low temperature specific heat is estimated. |
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Zavalniuk, V. Marchenko, S. |
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Zavalniuk, V. Marchenko, S. |
| author_sort |
Zavalniuk, V. |
| title |
Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes |
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Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes |
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Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes |
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Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes |
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Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes |
| title_sort |
theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2011 |
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Наноструктуры при низких температурах |
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https://nasplib.isofts.kiev.ua/handle/123456789/118539 |
| citation_txt |
Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes / V. Zavalniuk, S. Marchenko // Физика низких температур. — 2011. — Т. 37, № 4. — С. 432–437. — Бібліогр.: 18 назв. — англ. |
| series |
Физика низких температур |
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AT zavalniukv theoreticalanalysisoftelescopicoscillationsinmultiwalledcarbonnanotubes AT marchenkos theoreticalanalysisoftelescopicoscillationsinmultiwalledcarbonnanotubes |
| first_indexed |
2025-11-27T04:18:30Z |
| last_indexed |
2025-11-27T04:18:30Z |
| _version_ |
1849915726970748928 |
| fulltext |
© Vladimir Zavalniuk and Sergey Marchenko, 2011
Fizika Nizkikh Temperatur, 2011, v. 37, No. 4, p. 432–437
Theoretical analysis of telescopic oscillations
in multi-walled carbon nanotubes
Vladimir Zavalniuk and Sergey Marchenko
Department of Theoretical Physics, Odessa I.I. Mechnikov National University,
2 Dvoryanska Str., Odessa 65026, Ukraine
E-mail: VZavalnyuk@onu.edu.ua
Sergey.Marchenko@onu.edu.ua
Received June 30, 2010, revised September 10, 2010
A simplified theory of the telescopic oscillations in multi-walled carbon nanotubes is developed. The explicit
expressions for the telescopic force constants (longitudinal rigidity) and the frequencies of telescopic oscillations
are derived. The contribution of small-amplitude telescopic oscillations to the nanotubes low temperature speci-
fic heat is estimated.
PACS: 61.46.Fg Nanotubes;
62.25.–g Mechanical properties of nanoscale systems;
65.80.–g Thermal properties of small particles, nanocrystals, nanotubes, and other related systems.
Keywords: carbon, nanotube, MWCNT, telescopic oscillations, rigidity, specific heat.
1. Introduction
Multi-walled carbon nanotubes (MWCNTs) are the first
discovered nanoscopic quasi-1D nanostructures [1]. Each
MWCNT consists of some nested single-walled nanotubes
(shells) held mostly by van der Waals forces [2].
The telescopic motion ability of inner shells [3] and
their unique mechanical properties [4] permit to use multi-
walled nanotubes as main movable arms in coming nano-
mechanical devices. The variety of gadgets of this kind
was already suggested such as a possible mechanical giga-
hertz oscillator (linear bearing) [3,5], nanoswitch [6], na-
norelay and nanogear [7], nanorail, reciprocating nanoen-
gine [8]. Therefore the analysis of mechanical characte-
ristics of MWCNT is an important objective of study. The
present work is devoted to a simplified continuum version
of this problem. The continuum model for telescopic oscil-
lations, in which each shell of MWCNT is considered as
continuous infinitesimally thin cylinder is described in the
next section. The third section is devoted to the description
of the small (thermal) and large-amplitude oscillations for
double-walled carbon nanotubes (DWCNT) and MWCNT
in the framework of proposed model. Note that the similar
continuum model was used recently for the investigation of
the suction energy and large amplitude telescopic oscilla-
tions in DWCNT [9,10]. The contribution of temperature-
induced oscillations into the tubes heat capacity within
Debye model is also discussed. In the last section the ob-
tained results are compared with the available experimental
data [3].
2. Intertube interaction in MWCNT within continuum
model
The interaction energy of two shells of the multi-walled
tube is modelled as the sum of pair interaction potentials of
atoms from different shells. In doing so we took for the
potential energy of two atoms at the distance l the Len-
nard-Jones potential
6 12
6 12( ) = ,LJE l
l l
γ γ
− +
with attractive and repulsive constants 24 6
6 = 2.43·10 J·nmγ −
and 27 12
12 = 3.859·10 J·nmγ − borrowed from [2]. In ac-
cordance with this approximation the total intertube inter-
action energy takes the form
1 2
6 12
6 12
1, 2, 1, 2,=1 =1
= ,
( ) ( )
N N
i j i ji j
U
γ γ
r r r r
⎛ ⎞
− +⎜ ⎟⎜ ⎟− −⎝ ⎠
∑∑ (1)
where 1,ir and 2, jr are radii vectores of the inner and out-
er tube's atoms, respectively.
As in [2] we used instead of (1) the continuum model,
for which
Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes
Fizika Nizkikh Temperatur, 2011, v. 37, No. 4 433
2 2 2
2
1 2 1 2 1
0 0 2 1
2
12
2 2 2 2 6
1 2 1 2 1 2 1 20
6
2 2 2 3
1 2 1 2 1 2 1 2
( ) =
[ 2 cos ( ) ( ) ]
,
[ 2 cos ( ) ( ) ]
L z
L L z
L
U z r r d d dz
dz
r r r r z z
r r r r z z
Δπ π
Δ
Δ σ θ θ
γ
θ θ
γ
θ θ
+
− +
×
⎧⎪× −⎨
+ − − + −⎪⎩
⎫⎪− ⎬
+ − − + − ⎪⎭
∫ ∫ ∫
∫
(2)
where 1r and 2r are inner and outer tubes radii, 1L and 2L
are their lengths (from now on we assume that 1 2 )L L≤
and zΔ is the distance between tubes outer edges and σ is
the surface density of carbon atoms in graphene, which is
almost independent on the tube chirality,
–2
2
4= = 38.2 nm ,
3 3 b
σ
where = 0.142 nmb is the interatomic distance in gra-
phene. Note that expression (2) governs any one of coaxial
DWCNT configurations, but for stable natural multi-wal-
led nanotubes the interlayer distance d ranges from 0.342
to 0.375 nm, and that it is a function of the curvature [11].
The integration over variables 1z and 2z can be easily
carried out analytically, but obtained expressions are too
cumbersome to be presented here.
It's clear, that the system energy is minimal when the
inner tube is completely retracted into the outer tube. In
terms of hypergeometric functions the minimum interac-
tion energy is given by expression
3 2
min 1 2 1 2
1 2 1 2
2 1 2 12 2
1 2 1 2
12 611 5
1 2 1 2
3= min ( , )
2
4 41 11 1 5, ,1, , ,1,
2 2 2 2( ) ( )21 ,
32 ( ) ( )
U r r L L
r r r rF F
r r r r
r r r r
π σ
γ γ
×
⎡ ⎤⎛ ⎞ ⎛ ⎞
⎢ ⎥⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠⎢ ⎥× −⎢ ⎥+ +⎢ ⎥
⎢ ⎥⎣ ⎦
(3)
where
2
1 2 1 2
2 1 2
1 2
2 2
1 2 1 2
4 ( )1 , ,1, =
2 2( )
.
( 2 cos )
J
J
r r r rF J
r r
d
r r r r
π
π
π
θ
θ−
⎛ ⎞ + ×⎜ ⎟
+⎝ ⎠
×
+ −∫ (4)
3. Telescopic oscillations in DWCNT
If the outer tube is rigidly mounted, then the longitudin-
al motion of the internal tube is described by Newton equa-
tion:
1 ( )( ) = ,z
U za z
m z
Δ
Δ
∂−
∂
(5)
where a is the acceleration of the inner tube with mass m .
We ignore here the contribution of some defect-induced
dissipative forces since for high-quality nanotubes they are
by several orders lower than the retraction force due to
self-healing mechanism [3,5,12].
By (5) the motion of inner tube is cyclic with the period
0
0
02 1 0
( )( ) = 2 ,
( )
z
L L z
d zE mL
E U z
Δ
Δ
Δ
τ
Δ− − −∫ (6)
where maximal displacement 0zΔ is determined by the
equation 0 2 1 0 0( ) ( ) =U z U L L z EΔ Δ≡ − − .
Due to the special form of potential (2) we can separate
out two limiting forms of motion (Fig. 1):
1) steady movement for 0z bΔ while the potential is
linear in zΔ ;
2) small oscillations when 0z bΔ and the potential is
quadratic in zΔ .
It is obvious that for real DWCNTs the interaction
energy and force are affected by the atomic structure of its
shells. As a result the interaction energy is modulated [13]
with period defined by the lattice parameters of both shells.
The amplitude of energy modulation can reach a value
of 1000 K for zigzag@zigzag and 60 100− K for arm-
chair@armchair DWCNTs (for 5 nm length inner shell)
and is linear in length.
On the other hand due to incommensurability of atomic
lattices for most chiral nanotubes as well as arm-
chair@chiral or zigzag@chiral pairs the modulation period
can be much bigger than the whole DWCNT length. This
means that impact of the shells structure substantially re-
duces as the smaller nanotube length increases.
Actually the interaction between two (or more) nano-
tubes of different length is well-described by the conti-
nuum model if the oscillation energy is much higher than
1000 K which corresponds to the great amplitude telescop-
ic motion ( 0z bΔ ). The small amplitude oscillations
also can be considered within the continuum model for
most cases of incommensurate nanotubes for which the
energy modulation amplitude varies between 210− and 10 K.
Fig. 1. The intertube interaction energy (a) and longitudinal inter-
tube interaction force (b) for the (5,5)@(17,1) DWCNT with 10
and 20 nm lengths.
–20 –10 0 10
–7.78
E, 10 J
–18
Δz, nm
–20 –10 0 10
–0.778
0.778
Δz, nm
F, nN
a
b
Vladimir Zavalniuk and Sergey Marchenko
434 Fizika Nizkikh Temperatur, 2011, v. 37, No. 4
Furthermore, for DWCNTs with shell of equal length the
effect of lattice structure on the intertube interaction energy
is negligible compared to that of nanotube edges. As a result
if 2 1 < 0.4L L− nm (where 0.4 nm is the van der Waals
force saturation displacement) the continuum model is valid
regardless of temperature and shells structure.
3.1. Large-amplitude oscillations in DWCNT
When the displacement 0zΔ is greater than few nano-
meters the potential energy is linear on zΔ except small-
displacement region (with quadratic potential energy)
which can be neglected. In such a case the period of oscil-
lation can be derived from simple formulas for the steady
and uniformly accelerated motion. For equal-lengths tubes
the period takes the form
0 min0 min
0
2 ( )2( )
( ) = 4 = 4 ,
z z z
m E UE U
E
a F F
τ
−−
(7)
where zF is the longitudinal component of retraction force
3 2
1 2 1 2
1 2 1 2
2 1 2 12 2
1 2 1 2
12 611 5
1 2 1 2
3( , ) =
2
4 41 11 1 5, ,1, , ,1,
2 2 2 2( ) ( )21 .
32 ( ) ( )
zF r r r r
r r r rF F
r r r r
r r r r
π σ
γ γ
×
⎛ ⎞⎛ ⎞ ⎛ ⎞
⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠⎜ ⎟× −⎜ ⎟+ +⎜ ⎟
⎜ ⎟⎝ ⎠
(8)
Taking into account that 2 1=r r d+ for natural DWCNTs,
expression (8) may be written as
1 2
1 2 0 1 1 0 2 2( , ) ( ) ( ) ,z z zF r r F r r F r r≡ − ≡ − (9)
where 1
0 1( )zF r and 2
0 2( )zF r are approximately constant for
tubes of rather large radii and their asymptotic value 0zF
is about 1.54 nN/nm (Fig. 2).
In terms of maximal displacement 0 0 min= ( )z E UΔ − ×
1| |zF −× the period can be rewritten as follows:
0 02 2
= 4 = 4 .
| | | |z z
z m z
a F
Δ Δ
τ (10)
If 2 1>L L then the region of steady motion also contri-
butes to (6):
0 min
accelerated steady
02 1 2 1
0 min 0
2 ( )
= = 4
| |
2| |1 = 4 1 .
4 | | 4
z
z
z
m E U
F
m zF L L L L
E U F z
τ τ τ
Δ
Δ
−
+ ×
⎛ ⎞ ⎛ ⎞− −× + +⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠
(11)
Actually the oscillatory period does not depend on inner
tube radius 1r (if it is sufficiently large) since both the in-
ner tube mass m and retraction force zF are linear with 1r .
If the outer tube is also mobile, then the above expres-
sions for periods remain to be valid with m replaced by
the reduced mass
1 2
1 2
= .m mM
m m+
Note that the interaction energy of atoms forming the tubes
rapidly decreases with the interatomic distance. Therefore
it is enough to consider only interaction of adjacent tubes
in MWCNT. Since some adjacent shells of MWCNT can
be rigidly glued by defects, then glued tubes should be
considered as double-sided shells with integrated masses.
3.2. Thermal oscillations of DWCNT
For low temperatures the telescopic oscillations are the
smallest frequency 1D modes in DWCNT. Therefore for
0T → by Boltzmann theorem their mean energy is
= .BE k T
For small oscillations the maximal potential energy of
DWCNT coincides with E :
2
0
max
( )
= = ,
2
k z
U E
Δ (12)
where k is the rigidity parameter. Taking into account that
in harmonic approximation the rigidity is the second deriv-
ative of potential energy on the inner tube longitudinal
displacement and assuming the tube radius is much smaller
of its length we obtain for 1 2=L L the following expres-
sion:
2
1 2 1 2
2
6 12
2 2 3 2 2 6
1 2 1 2 1 2 1 20
( , ) = 4
.
( 2 cos ) ( 2 cos )
k r r r r
d
r r r r r r r r
π
πσ
γ γ
θ
θ θ
×
⎛ ⎞
× −⎜ ⎟
+ − + −⎝ ⎠∫
(13)
The harmonic oscillations frequency for this 1 2( , )k r r is
0
1= ,
2
k
m
ω
π
(14)
and the amplitude of longitudinal thermal oscillations can
be estimated using the next relation:
30 2 nm= 6 10 .
K
Bz k
kT
Δ −≈ ⋅ (15) Fig. 2. The external shell radius dependence of 2
0zF for the natu-
ral DWCNT with interlayer distance = 0.34d nm.
,
n
N
/n
m
F
2 z
0
0 2 4 6 8 10
1.5
1.0
0.5
r
2
, nm
F
z0
Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes
Fizika Nizkikh Temperatur, 2011, v. 37, No. 4 435
It can be shown that 0zΔ is few times smaller than the
graphene lattice parameter even for 300T ∼ K.
To model the intertube interaction force
1 2 1 2( , , , , )F r r L L zΔ in the case of 1 2L L≠ depending on
the inner tube edge position zΔ let us assume that the axis
of outer tube coincides with the interval 2[0, ]L of real axis
and introduce two parameters:
____________________________________________________
1 2 1 2
2 1 2 12 23 2
1 2 1 2
0 1 2 1 2 1 2 2 1 12 611 5
1 2 1 2
4 41 11 1 5, ,1, , ,1,
2 2 2 2( ) ( )3 21( , , , ) = sgn ( ) ,
4 32 ( ) ( )
r r r rF F
r r r r
F r r L L r r L L
r r r r
π σ
γ γ
⎡ ⎤⎛ ⎞ ⎛ ⎞
⎢ ⎥⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠⎢ ⎥− −⎢ ⎥+ +⎢ ⎥
⎢ ⎥⎣ ⎦
(16)
1 2 1 2
2 1 2 12 2 2
1 2 1 2 1 22 2
1 2 1 2 1 2 12 12 2 2 6
1 2 1 2 1 2
1 2
2 1 2 12
1 2
6 6
1 2
4 41 12 1 12, ,1, , ,1,
2 2 2 2( ) ( ) ( )
( , , , ) = 4
( ) [( ) ( ) ]
41 6 1 6, ,1, ,
2 2 2 2( )
( )
r r r rF F
r r r r L L
k r r L L r r
r r r r L L
r rF F
r r
r r
π σ γ
γ
⎧ ⎡ ⎤⎛ ⎞ ⎛ ⎞
⎪ ⎢ ⎥⎜ ⎟ ⎜ ⎟+ + + −⎪ ⎝ ⎠ ⎝ ⎠⎢ ⎥+ −⎨ ⎢ ⎥+ + + −⎪ ⎢ ⎥
⎪ ⎢ ⎥⎣ ⎦⎩
⎛ ⎞
⎜ ⎟+⎝ ⎠
− +
+
1 2
2 2
1 2 1 2
0 22 2 3
1 2 1 2
4,1,
( ) ( )
= ,
[( ) ( ) ]
r r
r r L L
k r
r r L L
⎫⎡ ⎤⎛ ⎞
⎪⎢ ⎥⎜ ⎟+ + − ⎪⎝ ⎠⎢ ⎥
⎬⎢ ⎥+ + − ⎪⎢ ⎥
⎪⎢ ⎥⎣ ⎦⎭
(17)
where 0k is almost independent of 1 2,r r for rather large values of these parameters. For 1 > 10r nm we have
2
0 3.7 nN/nmk ≈ − .
For small maximum retractions of inner tube ( < 0.3 nm)zΔ 1 2 1 2( , , , , )F r r L L zΔ can be written as follows:
0 0
1 2 1 2 0 2 1 0
0 2 1 2 1 0 2 1
, 0.3 nm ,
( , , , , ) 0, ,
( ), 0.3 nm.
F k z z x
F r r L L z x z L L x
F k L L z L L x z L L
Δ Δ
Δ Δ
Δ Δ
+ < <⎧
⎪= < < − −⎨
⎪− + − − − − < < − +⎩
(18)
Here 0 0= /x F k is the displacement, which makes the longitudinal retraction force equals to zero, 1 2 1 2( , , , )k r r L L is the
DWCNT longitudinal rigidity and 0 1 2 1 2 1 2 1 2( , , , ) = ( , , , ,0)F r r L L F r r L L .
In case of tubes with significantly different lengths 1 2(| | 1L L− nm) the expression (17) takes the form
1 2 1 2
2 1 2 12 2
1 2 1 22 2
1 2 1 2 12 612 61 2
1 2 1 2
4 41 12 1 6, ,1, , ,1,
2 2 2 2( ) ( )
( , ) = 4 ,
( ) ( )L L
r r r rF F
r r r r
k r r r r
r r r r
π σ γ γ≠
⎡ ⎤⎛ ⎞ ⎛ ⎞
⎢ ⎥⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠⎢ ⎥−⎢ ⎥+ +⎢ ⎥
⎢ ⎥⎣ ⎦
(19)
and in the case of equal lengths = 1 2 1 21 2 1 2( , ) = 2 ( , )L L L Lk r r k r r≠ .
_______________________________________________
The oscillation cycle can be considered as
2 1 0
steady accelerated
max
| | 2
= = 2 2 ,
L L x m
V k
τ τ τ π
− −
+ + (20)
where
max 0 min
2= [ ( ) ]V E z U
m
Δ −
is the maximum inner tube velocity and
2 1
min
| |=
2
L LU U −⎛ ⎞−⎜ ⎟⎝ ⎠
is the system minimum potential energy defined by (3).
With an accuracy of several percent previous equation can
by approximated by
2 1 0
0 0
| | 2
= 2 .
| |
L L xm
k z x
τ π
Δ
⎛ ⎞− −
+⎜ ⎟+⎝ ⎠
(21)
Since the longitudinal rigidity depends only on the
tubes radii and the inner tube mass is proportional to the
product of its length and radius, then the harmonic oscilla-
tion frequency scales on the length as 1/2
0 .Lω −∼
Vladimir Zavalniuk and Sergey Marchenko
436 Fizika Nizkikh Temperatur, 2011, v. 37, No. 4
From the above discussion it is clear that for 300T K
the continuum model is valid for the DWCNTs with in-
commensurate shells but ceases to be true for commensu-
rate (armchair@armchair, zigzag@zigzag) and some qua-
si-commensurate configurations. As an example, for the
majority of zigzag@chiral and armchair@chiral DWCNTs
the considered thermal oscillations are possible for tempe-
ratures higher than 0.01–1 K. As for the chiral@chiral con-
figurations, in some cases the shells atomic structure im-
pact may becomes negligible even for 310 KT −∼ .
3.3. Thermal oscillation frequencies in multi-walled CNTs
Considering the long-amplitude oscillations of multi-
walled nanotube we assumed that some nanotube's shells
can be bounded by the defects, but in the case of thermal
oscillations we should take into account the motion of all
shells because amplitudes of their oscillations are of the
same order and much lesser then interatomic distance. For
simplicity assume that all MWCNT's shells are equal in
length.
We obtained the MWCNT's thermal oscillations fre-
quencies by solving the system of equations for longitu-
dinal displacements of tubes with forces defined by the
expression above for the intertube potential. In the conti-
nuum model arbitrary multi-walled nanotube can be cha-
racterized by the inner shell radius 0r , the number of shells
n under consideration and the constant distance between
adjacent shells ( = 0.34d nm).
Considering only adjacent shells interaction we find ex-
plicit values of the consequent MWCNT eigenfrequencies
, = 1...i i nω (Fig. 3). The smallest eigenvalue is always
equal to zero corresponding to the whole nanotube transla-
tional motion. The analysis shows that the maximal fre-
quency depends on the MWCNT characteristics but in case
of tube with large number of shells ( 10n ) it tends to the
asymptotic value which depends only on the tubes length
(Fig. 4):
max
280= GHz .
L
ω (22)
The obtained value of maximal frequency is underesti-
mated for real tubes because defects may increase the lon-
gitudinal rigidity of MWCNT. The minimal oscillation
frequency strongly depends on the number of shells (as a
result of increasing of the outer shell mass) and can be
found using the following interpolation formula (Fig. 4):
11 0.0128
min 0.9365
4.46·10 e= GHz .
n
L n
ω
−
(23)
Using obtained frequencies the contribution of tube's te-
lescopic oscillations to the individual MWCNT internal
energy is calculated
0
1 1( ) ,
2
exp 1
i
ii
B
E T
k T
ω
ω
ω≠
⎡ ⎤
⎢ ⎥
⎢ ⎥= +⎢ ⎥⎛ ⎞
−⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
∑ (24)
where Bk is the Boltzmann constant and T is an absolute
temperature, and specific heat
2
0
1
2( )( ) = = .
1sinh
2
i
B
v B
ii
B
k TE TC T k
T
k T
ω
ω
ω≠
⎡ ⎤
⎢ ⎥∂ ⎢ ⎥
⎢ ⎥∂ ⎛ ⎞
⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
∑ (25)
By (25) if minBk T ω , then
2
min min( ) expv B
B B
C T k
k T k T
ω ω⎛ ⎞ ⎛ ⎞
−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
(26)
while ( )v BC T nk if maxBk T ω .
It is well known that under the Debye temperature the
bulk solid specific heat decreases as a cubic function of the
temperature and for the one dimensional structures such
decreasing is given by the linear function. In the case of
MWCNTs the telescopic oscillation induced specific heat
decreases exponentially in the range of small temperatures
and the corresponding Debye-like temperature is Fig. 3. Frequencies of small telescopic oscillations for the 50 nm-
length MWCNT with 20 shells.
40
30
20
10
0 2 4 6 8 10 12 14 16 18 20
ω
,
G
H
z
i
Fig. 4. The maximal ( ) and minimal ( ) oscillation frequency
for 30 nm-length MWCNT for different number of shells.
50
40
30
20
10
0 10 20 30 40
ω
,
G
H
z
n
Theoretical analysis of telescopic oscillations in multi-walled carbon nanotubes
Fizika Nizkikh Temperatur, 2011, v. 37, No. 4 437
0.0128
min
0.9365
1.7 e= = K,
2
n
D
B
T
k L n
ω −
(27)
where minω is given by (23). As a result of exponential
decreasing MWCNT's specific heat may be several orders
higher than that of environment for DT T while for
DT T these values change over. For the natural carbon
nanotubes DT varies in the interval 310 –1 K− .
If MWCNT's shells have different lengths the hamper-
ing of low-energy telescopic motion by the lattice structure
causes the abrupt increase of the oscillation frequency and
also leads to the specific heat exponential decreasing.
Moreover for most MWCNTs the thermal oscillations
freezing-out as a result of both processes takes place within
the same temperature range from 310− to 1 10− K.
For the temperature = 1 KT (assuming that > DT T )
the telescopic oscillations contribution to the total nano-
tube's specific heat may run as high as 50% for the rela-
tively small double-walled nanotubes ( = 20 30L − nm,
2 ~ 1r nm). For the ten-walled MWCNTs of length =L
50 nm= and external radius = 2.5r nm the telescopic
oscillation specific heat is about 0.025 J/(kg·K) and pho-
non contribution is in the range 0.2 to 0.3 J/(kg·K) [14].
The maximal electronic contribution for metallic SWCNT
is estimated to be ten times smaller than that of lattice os-
cillations [14]. As for all semiconducting nanotubes (which
are the majority of natural MWCNTs) the electronic spe-
cific heat is negligible. Taking into account the 1D struc-
ture phonon specific heat linear decreasing in the consi-
dered temperature region it is reasonable to expect that for
110T − K (if it is higher than DT ) the telescopic oscilla-
tion contribution may become dominant.
4. Summary
The explicit expressions for longitudinal rigidities and
frequencies of small and large-amplitude telescopic oscil-
lations of DWCNT and MWCNT were deduced in the
framework of continuum Lennard-Jones model borrowed
from [2]. Besides the obtained frequencies of telescopic
oscillations of MWCNT are in good agreement with avail-
able experimental data [3] and results of numerical simula-
tions [15–17].
For example, the thermal oscillation frequency of
12.21 nm (7,0)@(9,9) DWCNT obtained in Ref. 15 is
(75 8) GHz± while the considered model gives 62 GHz.
The retraction force zF for the (9,0)@(12,0) DWCNT ob-
tained in [16,17] is 1.6 nN and using Lennard-Jones para-
meters from [16] it yields = 1.54zF nN. For (5,5)@(10,10)
and (10,10)@(15,15) DWCNTs the maximum retraction
forces ratio is 1.67 by our model and 1.7 in [18]. So, the
difference between frequencies calculated by our analytical
formulas and those found by numerical methods with ac-
count of discrete structure of nanotubes lies within the 5%-
range.
It is worth to be mentioned that investigation of multi-
walled nanotube oscillations by using exact expression for
the two-shell retraction force and longitudinal rigidity is
not computationally intensive in contrast to the molecular
dynamics simulations. This permits us easily to calculate
(within the bounds of continuum model) all oscillation
frequencies and corresponding parameters (such as specific
heat) of any MWCNT regardless of number of shells and
their configuration.
Therefore the considered Lennard-Jones continuum
model is seemingly well suited for description of telescop-
ic trembling of MWCNT.
Authors are grateful to Prof. V.M. Adamyan for the
significant help.
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