APPLIED PHYSICS LETTERS 94, 101917 共2009兲
Continuum modeling of van der Waals interactions between carbon
nanotube walls
W. B. Lu,1 B. Liu,1,a兲 J. Wu,1 J. Xiao,2 K. C. Hwang,1 S. Y. Fu,3 and Y. Huang2,4,a兲
1
FML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084,
People’s Republic of China
2
Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 61801, USA
3
Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190,
People’s Republic of China
4
Department of Civil and Environmental Engineering, Northwestern University, Evanston,
Illinois 61801, USA
共Received 19 January 2009; accepted 24 February 2009; published online 13 March 2009兲
Prior continuum models of van der Waals force between carbon nanotube walls assume that the
pressures be either the same on the walls or inversely proportional to wall radius. A new continuum
model is obtained analytically from the Lennard-Jones potential for van der Waals force, without the
above assumptions. Buckling of a double-wall carbon nanotube under external pressure is studied,
and the critical buckling pressure is much smaller than those models involving the above
assumptions. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3099023兴
In order to overcome limitations of atomistic simulations, continuum models have been developed for carbon
nanotubes 共CNT兲, such as the linear elastic shell theory,1,2
finite-deformation membrane theory,3 and shell theory4 based
on interatomic potentials. For multiwall CNTs, the interaction between walls is governed by the van der Waals force
共vdW兲, which is characterized by the Lennard-Jones 6-12
potential, V共d兲 = 4␧关共␴ / d兲12 − 共␴ / d兲6兴, where d is the distance
between a pair of atoms and ␴ = 0.3415 nm and ␧
= 0.002 39 ev for carbon.5 Its derivative gives the force between two atoms, F = V⬘共d兲 共positive for attractive force兲.
There are two types of continuum models to represent
the vdW force between CNT walls. One assumes the pressure to be the same pin = pout, between two adjacent CNT
walls,6 where pin and pout are pressures on the inner and outer
walls. The other assumes the pressure to be inversely proportional to the wall radius,7,8 i.e., pinRin = poutRout, where Rin and
Rout are the inner and outer wall radii 共Fig. 1兲.
The purpose of this letter is to avoid the above assumptions and obtain the pressures on inner and outer walls analytically from the Lennard-Jones 6-12 potential. Such analytical expressions will be useful in the continuum modeling
of multiwall CNTs and their composite materials. Analytical
expressions are also obtained for the effective pressures on
the inner and outer walls of electrically charged double-wall
CNTs.
Cylindrical coordinates 共R , ␪ , z兲 are used for the doublewall CNT with the inner and outer radii Rin and Rout shown
in Fig. 1. Without losing generality, the distance between an
atom on the inner wall 共Rin , 0 , 0兲 and an atom on the outer
2
− 2RinRout cos ␪ + z2. The
wall 共Rout , ␪ , z兲 is d = 冑R2in + Rout
forces between two atoms are equal and opposite and are
obtained from the Lennard-Jones potential as F = V⬘共d兲 共Fig.
1兲. Their projections along the normal direction give the normal forces as FRin = F共Rout cos ␪ − Rin兲 / d and FRout = F共Rout
− Rin cos ␪兲 / d, which are not equal anymore 共Fig. 1兲.
The carbon atoms on each wall are represented by
their area density ␳c = 4 / 共3冑3l20兲 such that the number
of carbon atoms is ␳cdA over the area dA, where l0 is the
equilibrium bond length 共of graphene兲. This accurately
represents the average behavior of vdW interactions
between atoms from adjacent CNT walls.9 For an infinitesimal area dAin on the inner wall, the net force in the normal
⬁
direction is ␳cdAin兰AFRin␳cdAc = ␳2c dAin兰−⬁
dz兰20␲F共Rout cos ␪
− Rin兲d−1Routd␪, which gives the pressure on the inner wall as
⬁
dz兰20␲F共Rout cos ␪ − Rin兲d−1d␪ 共positive for
pin = −␳2c Rout兰−⬁
compression兲. Similarly, the pressure on the outer wall is
⬁
pout = −␳2c Rin兰−⬁
dz兰20␲F共Rout − Rin cos ␪兲d−1d␪.
For the Lennard-Jones 6-12 potential, the above pressures are obtained analytically as
pin =
冋 冉 冊冉
冊 冉 冊冉
3␲␧␴␳2c Rout
␴
231
32Rin
Rin + Rout
− E11 − 160
␴
Rin + Rout
5
11
Rout − Rin
E13
Rout + Rin
Rout − Rin
E7 − E5
Rout + Rin
冊册
,
共1兲
and
a兲
Author to whom correspondence should be addressed. Electronic addresses: y-huang@northwestern.edu and liubin@tsinghua.edu.cn.
0003-6951/2009/94共10兲/101917/3/$25.00
94, 101917-1
FIG. 1. A schematic diagram of double-wall CNT.
© 2009 American Institute of Physics
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Appl. Phys. Lett. 94, 101917 共2009兲
Lu et al.
(pext )cr (GPa)
101917-2
6
(8,8)@(13,13)
5
Assumption pin= pout
4
Assumption pinRin= poutRout
3
2
Taylor series expansion
0
Exact solution
Single-wall CNT
1
4
6
8
10
12
14
16
18
20
L/Rout
FIG. 2. The normalized pressure and pressure multiplied by the wall radius
R on the inner and outer walls of a double-wall CNT vs the normalized sum
of inner and outer wall radii. Results of Ref. 8 are also shown for
comparison.
pout =
冋 冉
3␲␧␴␳2c Rin
␴
231
32Rout
Rin + Rout
冊 冉
+ E11 − 160
␴
Rin + Rout
冊冉
5
冊冉
11
Rout − Rin
E13
Rout + Rin
Rout − Rin
E7 + E5
Rout + Rin
冊册
,
共2兲
where Em = 兰0␲/2共1 − k2 sin2 ␪兲−m/2d␪ and k2 = 4RinRout / 共Rin
+ Rout兲2. Em satisfies the recurring relation 共m − 2兲共1 − k2兲Em
= 共m − 3兲共2 − k2兲Em−2 − 共m − 4兲Em−4, and E1 = K共k兲 = 兰0␲/2共1
− k2 sin2 ␪兲−1/2d␪ and E−1 = E共k兲 = 兰0␲/2冑1 − k2 sin2 ␪d␪ are the
complete elliptic integrals of the first and second kind,
respectively.
Figure 2 shows the normalized pressure p / 共8␲␧␴␳2c 兲
versus normalized radius 共Rin + Rout兲 / ␴ for the interwall spacing Rout − Rin = ␴, which is the equilibrium spacing due to
vdW force between graphenes.9 It is observed that pin and
pout have different signs, and therefore pin ⫽ pout. The result
⬁
dz兰20␲FRoutd␪, also shown in Fig. 2,
of He et al.,8 pin = −␳2c 兰−⬁
is 共in absolute value兲 much larger than Eq. 共1兲 because the
force F was not projected to the normal direction, and therefore misses the factor 共Rout cos ␪ − Rin兲 / d inside the integration. For small interwall spacing Rout − Rin = ␴ / 2 共not shown
in Fig. 2兲, all pressures are positive 共i.e., repulsive force兲,
while they become negative 共i.e., attractive force兲 for large
interwall spacing Rout − Rin = 2␴. Figure 2 also shows the normalized product of pressure and radius pR / 共8␲␧␴2␳2c 兲,
which has different signs for the inner and outer walls and
therefore pinRin ⫽ poutRout.
There exist other potentials that are more accurate than
the Lennard-Jones 6-12 potential to represent the C–C repulsive term, such as a registry dependent potential to describe
the corrugation in interlayer interactions in graphene nanostructures 共Kolmogorov and Crespi10兲 or a modified Morse
potential based on the ab initio local density approximation
共LDA兲 calculations 共Wang et al.11兲 that has been verified for
graphite subject to high pressure.12 We used the Morse-type
potential11 to calculate the pressures pin and pout of a doublewall CNT with an inner wall radius 0.35 nm and the interwall spacing 0.3 nm. The resulting pressures are pin = 35.0
and pout = 6.7 GPa, which give pin ⫽ pout and pinRin
FIG. 3. The critical external pressure at the onset of buckling 共pext兲cr vs the
length to outer radius ratio L / Rout for a 共8,8兲@共13,13兲 double-wall CNT.
Results for the 共13,13兲 single-wall CNT and those based on prior assumptions are also shown.
⫽ poutRout. These conclusions also hold, in general, for other
potentials.
Equations 共1兲 and 共2兲 are used to study buckling of a
double-wall CNT subjected to external pressure pext. The deformation is uniform prior to buckling but its rate becomes
nonuniform at the onset of buckling. The analysis is similar
to that for a single-wall CNT 共Ref. 13兲 except
共i兲
共ii兲
共iii兲
the outer wall is subjected to a net external pressure
pext − pout,
the inner wall is subjected to an external pressure pin,
and
pin and pout are related to the inner and outer wall radii
Rin and Rout via Eqs. 共1兲 and 共2兲.
Figure 3 shows the critical external pressure at the onset
of buckling 共pext兲cr, versus L / Rout for a 共8,8兲@共13,13兲
double-wall CNT, where L and Rout are the length and outer
radius. 共pext兲cr is much larger than that for the outer wall
关共13,13兲 single-wall CNT兴,13 almost four times for long
CNTs. This reflects the resistance of vdW force between
walls against buckling. Results based on prior assumptions
overestimate the critical external pressure for buckling; the
assumption pin = pout overestimates by 75% for long CNTs
and pinRin = poutRout overestimates by 26%.
Equations 共1兲 and 共2兲 can be expressed in the Taylor
series of the small parameter ␰ = 共Rout − Rin兲 / 共Rout + Rin兲 as
再冉 冊 冉 冊
冉 冊 冉 冊 冋冉
冉 冊册 冎
␴
Rout − Rin
p = − 8␲␧␴␳2c
−
−
␴
Rout − Rin
11
6
␴
5 Rout − Rin
1+
5
1
1 + ␰2
8
7 2
3
␰ ⫾ ␰
20
4
␴
Rout − Rin
冊
5
11
+ O共␰3兲 ,
共3兲
where + terms are for pin and ⫺ terms for pout. Similarly,
pinRin and poutRout are given by
−
+
再冉
冊冉 冊
冉 冊冉 冊 冋 冉
冉 冊册 冎
pR = − 8␲␧␴2␳2c
␴
Rout − Rin
␴
Rout − Rin
10
1
␴
20 Rout − Rin
4
5
1
− ␰
2␰ 16
1
1 11
␴
− ␰ ⫾ −
8 Rout − Rin
2␰ 40
冊
4
10
+ O共␰2兲 ,
共4兲
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101917-3
Appl. Phys. Lett. 94, 101917 共2009兲
Lu et al.
where + terms are for pinRin and ⫺ terms for poutRout. For the
interwall spacing Rout − Rin = ␴ as in Fig. 2 pin, pout
= 0.6␲␧␴␳2c 共⫾2␰ + 3␰2兲 共and therefore pin ⫽ pout兲, and pinRin,
poutRout = 0.3␲␧␴2␳2c 共⫾2 + ␰兲 共and therefore pinRin ⫽ poutRout兲.
Figure 3 also shows the critical external pressure at the onset
of buckling 共pext兲cr, based on the Taylor series expansion in
Eqs. 共3兲 and 共4兲, which agrees very well with the exact solution based on Eqs. 共1兲 and 共2兲.
The last example to illustrate pin ⫽ pout and pinRin
⫽ poutRout is for two concentric tubes with electrical
charges, which are characterized by the electrostatic
potential V共d兲 ⬃ d−1 and force F = V⬘共d兲 ⬃ d−2, where d
2
= 冑R2in + Rout
− 2RinRout cos ␪ + z2 is the distance between electrical charges. The electrical charges on the inner and outer
tubes are represented by their area densities 共number of
charges per unit area兲 ␳in and ␳out. The pressures on the
inner and outer walls are then given by pin
⬁
= ␳in␳outRout兰−⬁
dz兰20␲F共Rout cos ␪ − Rin兲d−1d␪
and
pout
⬁
2␲
−1
= ␳in␳outRin兰−⬁dz兰0 F共Rout − Rin cos ␪兲d d␪. For F = V⬘共d兲
⬃ d−2, it can be shown analytically that the pressure on the
inner wall is identically zero, pin ⬅ 0, while the pressure on
the outer wall is not pout ⫽ 0, such that pin ⫽ pout and pinRin
⫽ poutRout.
In summary, continuum models for vdW force or electrostatic force between CNT walls are obtained analytically.
They bypass assumptions in prior studies, and have been
applied to determine the critical external pressure for buckling of double-wall CNTs.
Y.H. acknowledges the support from the NSF 共Grant
No. CMMI 0800417兲 and ONR Composites for Marine
Structures Program 共Grant No. N00014-01-1-0205, Program
Manager Dr. Y.D.S. Rajapakse兲. B.L. acknowledges the
supports from the NSFC 共Grant Nos. 10702034, 10732050,
and 90816006兲. K.C.H. also acknowledges the support
from NSFC 共Grant No. 10772089兲 and National Basic
Research Program of China 共973 Program, Grant No.
2007CB936803兲. S.Y.F. acknowledges the support from CAS
共Grant No. 2005-2-1兲 and NSFC 共Grant No. 10672161兲.
1
B. I. Yakobson, C. J. Brabec, and J. Bernholc, Phys. Rev. Lett. 76, 2511
共1996兲.
2
Y. Huang, J. Wu, and K. C. Hwang, Phys. Rev. B 74, 245413 共2006兲.
3
P. Zhang, Y. Huang, H. Gao, and K. C. Hwang, ASME J. Appl. Mech. 69,
454 共2002兲.
4
J. Wu, K. C. Hwang, and Y. Huang, J. Mech. Phys. Solids 56, 279 共2008兲.
5
L. A. Girifalco, M. Hodak, and R. S. Lee, Phys. Rev. B 62, 13104 共2000兲.
6
A. Pantano, D. M. Parks, and M. C. Boyce, J. Mech. Phys. Solids 52, 789
共2004兲.
7
C. Q. Ru, J. Mech. Phys. Solids 49, 1265 共2001兲.
8
X. Q. He, S. Kitipornchai, and K. M. Liew, J. Mech. Phys. Solids 53, 303
共2005兲.
9
W. B. Lu, J. Wu, L. Y. Jiang, Y. Huang, K. C. Hwang, and B. Liu, Philos.
Mag. 87, 2221 共2007兲.
10
A. K. Kolmogorov and V. H. Crespi, Phys. Rev. B 71, 235415 共2005兲.
11
Y. Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. B 44, 6562 共1991兲.
12
D. Qian, W. K. Liu, and R. S. Ruoff, J. Phys. Chem. B 105, 10753 共2001兲.
13
J. Wu, K. C. Hwang, J. Song, and Y. Huang, ASME J. Appl. Mech. 75,
061006 共2008兲.
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