Computational Materials Science 23 (2002) 166–174
www.elsevier.com/locate/commatsci
Molecular dynamic simulations on tensile mechanical
properties of single-walled carbon nanotubes with and
without hydrogen storage
L.G. Zhou a, S.Q. Shi
b,*
a
b
Institute of Metal Research, Chinese Academy of Sciences, Shenyang, 110016, China
Department of Mechanical Engineering, Hong Kong Polytechnic University, Hong Kong
Accepted 1 June 2001
Abstract
Using a bond order potential, molecular dynamics (MD) simulations have been performed to study the mechanical
properties of single-walled carbon nanotubes (SWNTs) under tensile loading with and without hydrogen storage.
ð10; 10Þ armchair and ð17; 0Þ zigzag carbon nanotubes have been studied. Up to the necking point of the armchair
carbon nanotube, two deformation stages were identified. In the first stage, the elongation of the nanotube was primarily due to the altering of angles between two neighbor carbon bonds. Young’s Modulus observed in this stage was
comparable with experiments. In the second stage, the lengths of carbon bonds are extended up to the point of fracture.
The tensile strength in this stage was higher than that observed in the first stage. Similar results were also found for the
zigzag carbon nanotube with a lower tensile strength. Hydrogen molecules stored in the nanotubes reduced the
maximum tensile strength of the carbon nanotubes, especially for the armchair type. The effect may be attributed to
the competitive formation between the hydrogen–carbon and the carbon–carbon bonds. Ó 2002 Elsevier Science
B.V. All rights reserved.
Keywords: Hydrogen effect; Carbon nanotubes; Molecular dynamics simulation
1. Introduction
A single-walled carbon nanotube (SWNT) can
be described as a grapheme sheet rolled into a
cylindrical shell and is a quasi one-dimensional
system [1]. Since its discovery by Iijima and Ichlhashi [2] and Bethune et al. [3], research activities
*
Corresponding author.
E-mail address: [email protected] (S.Q. Shi).
in its synthesis, characterization and nanoscale
applications have grown exponentially. Apart
from the well-known electrical properties [4,5] of
SWNT, its remarkable mechanical properties and
potentially high hydrogen-storage capacities are
also intensively studied. SWNTs have very high
strength so that the ropes of SWNTs are promising
as super strong fibers. Some recent experimental
and theoretical results show that Young’s Modulus of SWNTs can be up to the order of 1 TPa [6–
9]. This is comparable to that of diamond. Recent
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 2 3 3 - 6
L.G. Zhou, S.Q. Shi / Computational Materials Science 23 (2002) 166–174
progresses on hydrogen storage in SWNTs have
been reviewed by Dresselhuas et al. [10]. The high
capacity (4–10 wt% of hydrogen) and reversibility of hydrogen storage in ropes of SWNTs are
likely to bring about significant advance for the
use of hydrogen as an environmental friendly fuel
[10–13].
Due to the nanosize of SWNT, a direct measurement of its mechanical properties is rather
difficult. However, this extreme size is very suitable
for performing atomistic simulations. Yakobson
et al. [14] performed a molecular dynamics (MD)
simulation to study the high strain rate fracture
in nanotubes. They proposed that nanotubes
have an extremely large breaking strain. More recently, tight-binding electronic calculations on the
mechanical properties of SWNT are reported by
Ozaki et al. [15]. Their results revealed that under
large strain (30%), a zigzag nanotube and an
armchair nanotube are the stiffest under elongation and compression deformation, respectively.
Currently, the atomistic simulations on hydrogen
in SWNTs are mainly focused on two issues: how
much hydrogen can be stored and where the atoms
are stored. It is known that in some metal-based
materials, the absorption of hydrogen often embrittles the materials, or even generates metalhydrides which are brittle in most cases. An
interesting question is, since carbon nanotubes are
both promising for use in structure and hydrogen
storage, what is the effect of hydrogen storage on
mechanical properties of carbon nanotubes? In
this paper, we report our MD simulations on
SWNTs under a tensile loading, with and without
hydrogen storage.
2. Methodology
Brenner’s hydrocarbon potential, a bond order
potential, is used in this simulation. The first version of this potential was originally developed to
model the vapor deposition of diamond films [16],
and has been widely used in modeling fullerenes
[17–19]. Here, we use its improved new version [20]
which, as declared by Brenner, yields a better fit to
solid-state elastic properties, molecular bond energies and distances, and barriers for rotation
167
about dihedral angles. 1 Nanotubes in this paper
are ð10; 10Þ 100u armchair type and ð17; 0Þ 58u
zigzag type, respectively, with and without hydrogen storage. The subscript u denotes a repeat
unit of SWNT along the axial direction. These
units were chosen so that the two types of nanotubes have a similar diameter and tube length.
To simulate the tubes under tensile loading, we
followed two steps. First, the tubes were annealed
at simulation temperature for 5000 MD steps. The
time interval between two MD steps is 0.5 fs.
Then, the tube was pulled in axial direction (i.e., zdirection) with a strain of 5 104 . Following each
step of pulling, some additional MD steps were
used to relax the structure. A periodic boundary
condition has been used along the axial direction
and the simulations were performed at temperatures either 300, or 600 K.
3. Armchair ð10; 10Þ and zigzag ð17; 0Þ carbon
nanotubes under tensile loading
Unlike the continuum shell model, SWNT is
constructed by hexangular carbon rings so that its
mechanical properties are strongly depended on its
chiral directions. Bond angle and bond length are
the two crucial factors that control the deformation. For armchair SWNT, the elongation of tube
due to the altering of bond angles can be up to
15% if the C–C bonds are assumed to be rigid.
While for zigzag SWNT, one-third of the C–C
bonds are parallel to the axis. The loading force is
then directly acting on these bonds so that they are
easy to break. Fig. 1 shows the tensile force ðFz Þ
along the axial direction of the armchair ð10; 10Þ
and zigzag ð17; 0Þ SWNTs as a function of strain
(Fz e curve). The tensile force Fz is defined as
follows:
*
+
n
X
1 X
Fz ¼
mi Vzi Vzi þ
Fzij Zij ;
ð1Þ
L i¼1
i>j
where L is the length of nanotube, which is increased at a constant strain rate; n is the total
1
http://www.mse.ncsu.edu/CompMatSci/Brenner/brennercv.
PDF
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L.G. Zhou, S.Q. Shi / Computational Materials Science 23 (2002) 166–174
Fig. 1. Tensile force along axes of the armchair ð10; 10Þ and
zigzag ð17; 0Þ SWNT as a function of strain. The numbers 1, 2
and 3 denote the three deformation stages of armchair SWNT
under tensile loading.
number of atoms and m is the atomic mass. Vz ; Fz
and Z represent the z-component (axial direction)
of the velocity of an atom, force and distance between two atoms, respectively. The subscripts i, j
are the atom indices. The two other components of
the force, Fx and Fy can also be calculated in a
similar way. Their values were found to fluctuate
around zero and can be neglected. Eq. (1) is
modified from the one that was deduced earlier for
calculation of inner stress [21]. The volume of the
MD simulation cell appeared in the earlier work
[21] has been changed to the tube length here. By
doing so, we avoid the description of the crosssectional area of a nanotube, which is somewhat
ambiguous in definition [1], while still keep the
properties to represent the strength of the tube.
Young’s modulus of the tubes can be directly
evaluated from the figure from the liner region in
Stage 1, which gives about 750 GPa. This value is
comparable with the recently experimental results
[6] (at the order of 1 TPa, an exact value is still not
agreed upon). Three typical structures of ð10; 10Þ
SWNT corresponding to the stages at 1, 2 and 3 in
Fig. 1 are shown in Fig. 2(a)–(c). A brighter color
denotes a higher kinetic energy of the atoms. Fig.
2(d) represents the tube at the point right before
breakage. It is interesting to note that some atoms
Fig. 2. Typical snapshots of the armchair ð10; 10Þ SWNT without hydrogen during tensile deformation. Brighter color denotes higher
kinetic energy of the carbon atoms.
L.G. Zhou, S.Q. Shi / Computational Materials Science 23 (2002) 166–174
are linearly queued one by one, which connects the
tubes on its ends. The ends of the tube at breakage
were automatically closed up during the necking of
the tube. Comparing the structures in Fig. 2(a)–
(c) and the Fz e curve in Fig. 1, we may propose
that the elongation of the ð10; 10Þ nanotube is
initially due to the altering of bond angles (Stage
1). Under further pulling, the contribution from
the elongation of the C–C bonds becomes significant and plays the main role (Stage 2). When the
strain is up to a critical level, some groups of the
C–C bonds are broken. Then, the tube starts
necking and the force Fz decreases dramatically
(Stage 3). The breaking and the reconstruction of
carbon bonds in local areas are shown in Fig. 3. In
order to present a better view, we have unrolled
the tube into a plane. The distributions of the
distance between two carbon atoms in the ð10; 10Þ
SWNT during deformation are displayed in Fig. 4.
169
Under loading, some of the C–C bonds are elongated. However, those bonds that are vertical to
the tube axis experiencelittle change in their length.
It should be noted that, when the tube is about to
break, the stress in the tube is reduced so much
that the average length of the C–C bonds is recovered.
Compared with the ð10; 10Þ SWNT, the ð17; 0Þ
zigzag SWNT has significantly smaller maximum
strain and maximum tensile force. Here, we define
the maximum strain (MS) and the maximum tensile force Fz (MTF) at the turning point on Fz strain curve that has the highest value of Fz . Due to
the nature of the hexagonal carbon ring, pulling
the zigzag tube along its axial direction would
cause some second nearest neighbor C–C atoms to
become closer and to form new C–C bonds (Fig.
3(c)). In the local regions of these newly formed
bonds, some old carbon–carbon bonds have to
Fig. 3. The breaking and reconstruction of the C–C bonds without hydrogen. (a) and (b) are the snapshots of the (10,10) armchair
SWNT at 9.27 and 9.30 ps, respectively. (c) and (d) are the snapshots of the ð17; 0Þ zigzag SWNT at 6.75 and 6.80 ps, respectively. The
tubes have been unrolled into planes in order to achieve a better view.
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L.G. Zhou, S.Q. Shi / Computational Materials Science 23 (2002) 166–174
Fig. 4. The length distribution of the carbon–carbon atoms as a function of MD simulation steps. The first nearest neighbor is
corresponding to the C–C bond length. Brighter color denotes higher probability.
Fig. 5. Self-vibration of the ð10; 10Þ armchair SWNT under tensile loading. ‘‘D’’ in the vertical coordinate is the average diameter of
the SWNT. In the parenthesis of the legends, the first column and second column represent for the strain of the tube in one pulling step
and the number of relaxation steps after pulling, respectively. The last column is a measure of strain rate.
break due to the saturation of covalent bonds.
This would lead to the necking and breakage of
the zigzag SWNT.
Another interesting phenomenon we observed
in simulation is the self-vibration of SWNT in
radial direction. Fig. 5 shows the average radius of
L.G. Zhou, S.Q. Shi / Computational Materials Science 23 (2002) 166–174
the ð10; 10Þ SWNT as a function of MD simulation steps. It was identified as the self-vibration
because no matter how we change the strain rate,
the frequency is kept constant. The frequency
ofvibration can be evaluated from the figure, i.e.,
157 cm1 (or 4:7 103 GHz), which is very closed
to the value obtained from experiment ð165 cm1 Þ
[22].
4. Hydrogen effects
A range of 4–10 wt% of hydrogen stored in
SWNT were reported [13]. In the following simulations, the armchair ð10; 10Þ and zigzag ð17; 0Þ
SWNTs are used again for the convenience of
comparison. In our simulation, these SWNTs were
pre-stored with 4.17 or 8.34 wt% of hydrogen gas
(H2 ), both are in the reported range. The absorption of hydrogen in carbon nanotubes has been
theoretically studied [10,11]. In the current simulation, however, the details of the absorption procedure are not of primary concern. We place H2
molecules into tubes directly. The initial positions
and the orientations of H2 molecules in the tubes
are chosen randomly. To avoid the overlap of
atom positions, atomistic relaxation is performed.
When this is done, same to the simulation procedure in Section 3, 5000 MD steps are used to anneal the structures of carbon and hydrogen atoms
at simulation temperatures and the tensile loading are then carried out. The results show that
the maximum tensile force and the maximum
strain both decreased due to the storage of H2 , see
Table 1.
171
The effect of stored molecular hydrogen on
mechanical properties of SWNT is found to
strongly depend on temperature. It can be seen
from Table 1 that the reductions of MS and MTF
caused by hydrogen storage at 600 K are much
larger than that at 300 K. As proposed in the
following text, the competition between the carbon–carbon bonds and the carbon–hydrogen
bonds are believed to be the main reason that
causes the reductions in MS and MTF.
Another effect caused by H2 storage is on
the vibration properties. As we have described in
the early section, there is an intrinsic vibration
frequency of SWNT. For the ð10; 10Þ SWNT,
its value was found to be at about 157 cm1 .
When hydrogen is stored in the tube, this intrinsic vibration frequency does not seem to change
(Fig. 6). However, the amplitude of vibration is
remarkably reduced. The gas of H2 in tubes
plays the role of resistance to the vibration of
SWNT. The average radius of the tube is slightly
enlarged due to the internal pressure of hydrogen
gas.
The effect of H2 on the zigzag ð17; 0Þ SWNT
seems not as significant as on the armchair ð10; 10Þ
one. The loss in the maximum tensile force for the
and for
tube with 4.17 wt% stored H2 is 2.5 eV/A
(Table 1).
8.34 wt%, is 3.3 eV/A
Up to now, we had assumed that H2 are stored
inside of the tubes. However, the interstitial
channels between adjacent nanotubes in a rope of
SWNTs are also the possible sites to accommodate
H2 , although there still exists some debate at present on this issue. Motivated by this possibility, we
also studied the case in which H2 is stored in
Table 1
MS and MTF of SWNT ð10; 10Þ and ð17; 0Þ at different weight ratios of hydrogen and different temperatures
Tube type
wt% of Hydrogen
Temperature (K)
MS (%)
)
MTF (eV/A
ð10; 10Þ
ð10; 10Þ
ð10; 10Þ
0.00
4.17
8.34
300
300
300
39.9
39.0
37.9
221.3
209.8
195.8
ð10; 10Þ
ð10; 10Þ
0.00
4.17
600
600
38.6
34.6
203.2
149.7
ð17; 0Þ
ð17; 0Þ
ð17; 0Þ
0.00
4.17
8.34
300
300
300
23.4
23.1
22.9
111.6
109.1
108.3
Hydrogen molecules were stored inside the nanotubes.
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L.G. Zhou, S.Q. Shi / Computational Materials Science 23 (2002) 166–174
Fig. 6. The effect of hydrogen storage on self-vibration of the ð10; 10Þ SWNT.
outside surface of the tubes. The simulation procedure is as follows. A ð10; 10Þ 100u SWNT was
placed in the center of a box of 3 nm 3 nm in
length and width. The height of the box was set
equal to the length of the SWNT. H2 molecules
were stored in the box, while the space inside the
tube was kept empty. In order to compare to those
tubes with H2 stored inside, the molar fractions of
H2 were chosen to satisfy that the pressure of the
hydrogen gas outside the tubes is equal to that of
those tubes with H2 stored inside (4.17 and 8.34
wt%). Periodic boundary conditions were used in
the X, Y and Z directions of the box. The simulation temperature was 300 K. Compared to the
hydrogen-free SWNT, the reductions in the max for 4.17 and
imum force are 37.7 and 82.7 eV/A
8.34 wt% hydrogen storage, respectively. The increase in reduction of MTF might be due to the
increase in effective contact area between H2 and
the nanotube surface, because the outer surface
Fig. 7. The variation in the number of the C–H bonds in the ð10; 10Þ armchair SWNT stored with 4.17 wt% hydrogen during tensile
deformation.
L.G. Zhou, S.Q. Shi / Computational Materials Science 23 (2002) 166–174
173
Fig. 8. Snapshots of the ð10; 10Þ SWNT with 4.17 wt% hydrogen storage: (a) a cross-section at 4.7 ps; (b) the same cross-section at 7.6
ps; and (c) a view at the point of fracture. Small white balls are hydrogen molecules or atoms.
area of the nanotube is larger than the inner surface area.
There may be a couple of reasons that had
caused the reduction of strength of nanotubes by
hydrogen storage. One of the reasons might be the
high pressure of H2 acting on the wall of the tubes.
Take the ð10; 10Þ armchair tube with 4.17 wt%
stored hydrogen as an example. The atomic ratio
of H/C is 1:2. If we assume that all hydrogen
atoms stored inside the tube are in the gas phase,
the pressure of gas would be 95.2 MPa at 300 K.
Such high pressure of hydrogen gas acting on the
wall of tubes would change the loading mode of
the tube. In addition, under the high-strain tensile
loading, some bonds in the carbon rings were
broken and this created local defect regions on the
wall (Fig. 3(a)). The pressure of hydrogen gas
would drive molecular hydrogen passing through
these defect regions and cause the regions to extend into holes. Another important reason comes
from the competition in formations of the hydrogen–carbon and the carbon–carbon bonds. Fig. 7
shows the total number of the C–H bonds during
the tensile deformation. In this figure one can find
that the number of the C–H bonds is significantly
increased when the tube necking starts. As we have
showed in Fig. 4, under the tensile loading, some
C–C bonds were elongated and broken. Hydrogen
atoms, in this stage, were likely to ‘‘catch’’ some of
the free carbon bonds to generate the C–H bonds
(see Fig. 8), so that the fracture of SWNT was then
accelerated.
5. Conclusions
MD simulations based on a new bond order
potential have been performed to study the mechanical properties of SWNT under tensile loading
with and without hydrogen storage. The results
show that:
1. The tensile deformation of SWNTs up to the
point of necking experiences two stages, controlled by altering of the C–C bond angle and
the C–C bond length, respectively;
2. Hydrogen storage in SWNT reduces the maximum tensile strength of the tube as well as the
maximum tensile strain; and
3. The competition between the formations of the
H–C and the C–C bonds may be responsible for
the reduction in the mechanical strength of the
SWNT with H2 storage.
Acknowledgements
The MD program used in current simulations
was modified from that of Brenner’s. The authors
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L.G. Zhou, S.Q. Shi / Computational Materials Science 23 (2002) 166–174
are grateful to Prof. Donald W. Brenner for allowing us to share his program. This work was
supported by a central research grant from the
Hong Kong Polytechnic University (A-PC04).
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