Molecular dynamics simulation studies of mechanical properties of different carbon nanotube systems

Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=gmos20
Download by: [Nanyang Technological University] Date: 26 August 2016, At: 06:30
Molecular Simulation
ISSN: 0892-7022 (Print) 1029-0435 (Online) Journal homepage: http://www.tandfonline.com/loi/gmos20
Molecular dynamics simulation studies of
mechanical properties of different carbon
nanotube systems
Z. K. J. Kok & C. H. Wong
To cite this article: Z. K. J. Kok & C. H. Wong (2016) Molecular dynamics simulation studies of
mechanical properties of different carbon nanotube systems, Molecular Simulation, 42:15,
1274-1280, DOI: 10.1080/08927022.2016.1185790
To link to this article: http://dx.doi.org/10.1080/08927022.2016.1185790
Published online: 22 Jun 2016.
Submit your article to this journal
Article views: 36
View related articles
View Crossmark data

Molecular Simulation, 2016
VOL. 42, NO. 15, 1274–1280
http://dx.doi.org/10.1080/08927022.2016.1185790
Molecular dynamics simulation studies of mechanical properties of different carbon
nanotube systems
Z. K. J. Kok and C. H. Wong
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore
ABSTRACT
Various mechanical properties of single-walled carbon nanotubes (SWCNT) and double-walled carbon
nanotubes (DWCNT) are evaluated using molecular dynamics (MD) simulations. A tensioning process was
firstperformedonaSWCNTwhoseinteractionisbasedontheBrenner’s‘secondgeneration’potentialunder
varying length–diameter ratios and strain rates, in order to understand the SWCNT’s behaviour under axial
tension. The results showed an increase in the SWCNT’s ultimate tensile strength and a decrease in critical
strain given the conditions of increasing strain rate and a decreasing length–diameter ratio. Comparison
was done with previous studies on axial tensioning of SWCNT to validate the results obtained from the
set-up, based on the general stress–strain relationship and key mechanical properties such as the strain
at failure and the Young’s modulus. A DWCNT was then constructed, and Lennard-Jones ‘12-6’ potential
was used to describe the energy present between the nanotube layers. Extraction of the inner tube in a
DWCNT was performed using two inner wall tubings of different diameters to draw comparison to the
energies needed to separate fully the outer and inner tubing. Finally, a bending test was performed on
two DWCNTs with different intertube separations. Insights into the entire bending process were obtained
through analyses of the variations in the strain energy characteristic of the surface atoms near the bending
site, as the DWCNT is gradually bent until failure.
1. Introduction
Carbon nanotubes (CNTs) are well known for their excellent
mechanical properties. They are able to withstand high amounts
of tension and compression under a wide range of temperature.
With the advent of new production techniques, such as elec-
tric-arc discharge [1] proposed by Journet et al. and laser ablation
[2] proposed by Maser et al. aimed at lowering cost of produc-
tion for a single nanotube, attention has been given to possible
applications of CNTs in the field of aeronautics, biomechanics
or as composite-strengthening materials.
Investigations conducted on single strains of single-walled
carbon nanotubes (SWCNT) and double-walled carbon nano-
tubes (DWCNT) have been mostly confined to the use of com-
putational modelling and simulations owing to the limitations
in obtaining them physically. Therefore, most experiments
were conducted on CNTs aggregated as large systems. Selzer
and Friedrich [3] investigated the effects of moisture on car-
bon fibre-reinforced polymer composites subjected to a tensile,
compression and fatigue loading. Yu et al. [4] performed tensile
loading on 15 SWCNT bundled into ropes of varying diameters
under a LEO 1530 scanning electron microscope to determine
each sample’s breaking strength and Young’s modulus. However,
the use of theoretical modelling such as molecular dynamics
(MD) simulations has proven to generate reliable results in
analysis of smaller CNT systems. Liew et al. [5] employed MD
to obtain the stress–strain responses of SWCNT and DWCNT
subjected to axial tension, with results matching the ab initio
computed values obtained by Kudin and Scuseria.[6] Wong
[7] introduced defects on various configurations of SWCNT to
determine the relation between ultimate tensile strength and
the amount of defects present. Chen and Lusk [8] constructed a
system of defect-free nano-rings and applied loading at opposite
ends of each link to study the deformation of the rings as well as
their force–strain behaviour. Liew et al. [9,10] investigated the
buckling of SWCNTs and multi-walled CNTs (MWCNTs) under
compression as well as the tension and compression of CNT
bundles to understand the effects intertube van der Waals forces
have on the entire system. The use of MD is hence a popular
tool to understand CNT behaviour subjected to user-defined
conditions.
In this study, MD simulation is employed to study specific
behaviours for SWCNT and DWCNT. Analysis at the atomic
level allows us to design nanotube units that are foundational
to the entire geometry of physically fabricated SWCNT and
DWCNT. This gives us valuable insights on the characteristic
responses of these units, of which we can easily scale up to larger
models. Pristine armchair CNTs were modelled and the simula-
tions were performed without consideration of the environmen-
tal temperature. Our choice of timestep for each experiment is
critical in ensuring a balance between reasonable computational
© 2016 Informa UK Limited, trading as Taylor & Francis Group
KEYWORDS
Molecular dynamics
simulation; carbon
nanotubes; mechanical
properties; bending;
extraction
ARTICLE HISTORY
Received 15 October 2015
Accepted 30 April 2016
CONTACT C. H. Wong chwong@ntu.edu.sg School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue,
Singapore 639798

Molecular Simulation 1275
Lennard-Jones potential
(
ELJ
)
, as well as torsional potential
(
Etors
)
due to dihedral angles. The REBO potential factors in the short-
ranged attractive (VA
) and repulsive (VR
) bond terms due to the
interaction between two atom types are separated by a distance
rij. The REBO potential is defined as
where VA
and VR
are both functions of rij
, and bij
is the REBO
between the atoms.
The Lennard-Jones ‘12-6 potential’ (ELJ
) is described by Wu
et al. [14] as a pairwise attractive and repulsive field that extends
across a spatial domain which gradually tapers off to 0 J when 2
atoms are infinitely far apart from each other. It is computed as
where ɛ is the well depth parameter taken as 𝜀 = 4.55meV,
𝜎ij = 0.3367nm from [10] which is the distance at which the
potential energy is 0 J, and rij
is the distance of separation
between 2 carbon atoms.
The overall expression for the AIREBO force field is a sum-
mation of the three potentials given as
Since our simulations involve perturbing nanotubes along their
axial direction without the explicit twisting of the C–C bonds,
we can neglect torsional energy in our evaluation of the AIREBO
force field for a given nanotube structure. We also focus our
scope on nanotubes free of defects and without a predefined
temperature. The SWCNT is modelled as a perfect (5,5) armchair
system of fixed diameter but varying lengths. In actual experi-
ments, defects can significantly alter the tensile strength of the
nanotubes as observed by Wong.[7] One common defect is the
Stone–Wales defect [15,16] arising from the rotation of a C–C
bond by 90° due to structural changes in sp2
bonded carbon
nano-systems. In this work, CNT atoms at each end are fixed and
displaced at a rate of 5, 10, 20 and 30 m/s opposite to each other
per given timestep to simulate a tensioning process in the axial
direction. A timestep of 1 fs is used throughout the entire simula-
tion. The remaining inner atoms are free to rearrange themselves
under an NVE ensemble constraint. The entire system is first set
to minimise using a conjugate gradient minimisation technique.
Thereafter, the SWCNT is stretched by a given distance during
each iteration, with a duration of 1200 timesteps allocated for
relaxation in between. Finally, CNT models with length–diame-
ter ratios of 7.07, 8.89 and 10.88 are constructed using nanotube
lengths of 47.96, 60.26 and 73.79 Å, respectively. Axial tension is
fixed at 20 m/s at both ends of the nanotubes to investigate the
significance of the length–diameter ratio in affecting the prop-
erties of SWCNT.
3. Validation of MD simulation
The stress along the SWCNT can be determined from the strain
energy–strain relationship during the tensioning process through
its first derivative multiplied by 1
V0
whereV0 is the original volume
(1)ERebo =
[
VR
− bij
(
VA
)]
,
(2)ELJ = 4𝜀
⎡
⎢
⎢
⎣
�
𝜎ij
rij
�12
−
�
𝜎ij
rij
�6
⎤
⎥
⎥
⎦
,
(3)E = ERebo + ELJ + Etors
run time, as well as the accuracy of the results from the sim-
ulation. In numerous MD analysis of CNT systems subjected
to a force oriented axially or radially, the choice of timestep
smaller or equivalent to 1 fs has been proven to be adequate, as
seen from the works of Liew et al. [10]. We follow likewise for
all our simulations, while ensuring that each CNT system has
been given sufficient time to relax before reapplying the force.
Axial tensioning under varying strain rates is applied to different
SWCNTs with varying length–diameter ratios to compute their
stress–strain relationships. Brenner’s ‘second generation’ reactive
empirical bond-order (or REBO) force field [11] is employed to
describe the intratube and short-range interactions between the
carbon atoms, while the van der Waals force (or Lennard-Jones
‘12-6’ potential) is adopted to describe the intratube and long-
range interactions between the carbon atoms whose distance is
greater than 2 Å. The use of both short- and long-ranged force
fields constitutes the adaptive intermolecular reactive empirical
bond-order (or AIREBO) potential. The AIREBO potential was
also implemented in the work by Liew et al. [10] after a study
showing that the long-range Lennard-Jones force field plays a
significant role in MD simulations where atoms influence each
other across the entire simulation domain. Since then, the choice
of using the AIREBO potential has become increasingly popular
for nanotube MD simulations akin to our model. Nevertheless,
one could consider the choice of other potential fields, such as
the reactive force field (or ReaxFF). The ReaxFF potential goes
deeper in analysing the energies of the C–C bond at the various
orbital levels. It has been employed in MD simulations by [8]
and Pigos et al. [12].
Thereafter, experiments conducted on DWCNTs involved the
extraction of the inner tube relative to a stationary outer tube.
This is an extension of Li and Chou [13] who incorporated a truss
and frame structure to depict the short- and long-ranged bonds
present in a MWCNT. Their conclusion was that the presence of
interlayer van der Waals forces would influence the mechanical
properties of the MWCNT. To the best of knowledge, few studies
have been made focusing on the direct extraction of the inner
tube and how changing inner tube diameter would influence the
extraction process, which is part of this study. To account for
the forces between the outer and inner SWCNTs, an intertube
potential was incorporated into the DWCNT using the long-
range Lennard-Jones ‘12-6’ potential.
Finally, a bending test is conducted on a DWCNT fixed at
both ends. The force was applied normal to the axial direction
of the DWCNT and was recorded against the strain. Analysis of
the strain energy variations of the surface atoms near the bend-
ing site gave insights into the cause of failure of the DWCNT
when critical strain is reached. It was observed that the DWCNT
failure first occurred with a tear at the outer tube and thereafter
the inner tube, similar to observations from Ref. [5]. It was only
when the tear began to propagate to the innermost layer did the
entire nanotube structure begin to undergo catastrophical failure,
suggesting that the inner layer acted as an additional support to
reinforce the DWCNT during a bending process.
2. Modelling and simulation of SWCNT system
The AIREBO force field takes into account the short-range
REBO potential
(
ERebo
)
that defines covalent bonds, long-range

1276 Z. K. J. Kok and C. H. Wong
of the SWCNT. Mathematically, it is expressed as 𝜎 = 1
V0
dU
d𝜀
. U
denotes the strain energy stored in the carbon atoms that are
subjected to axial loading. Strain is defined as 𝜀 =
L−L0
L0
where L is
the stretched length and L0
the initial length of the nanotube. V0
is calculated using the formula V0
= L0
AC
, where AC
is the initial
cross-sectional area of the SWCNT. For an armchair SWCNT, the
diameter d is determined by d = 2.46
𝜋
√
n2
+ mn + m2
, where m
and n are indices representing the CNT configuration. AC
is the
cross-sectional area and is defined as AC
= πdh, where h is the
thickness of the nanotube layer and is taken to be3.4Å.[17,18] In
this section, a (5,5) armchair SWCNT was chosen, and hence, the
values of d and AC
are 6.781 Å and 7.243 × 10−19
m2
, respectively.
A strain rate of 20 m/s is applied on a SWCNT of 610 C atoms
with length–diameter ratio of 10.88 and length of 73.79 Å. The
strain energy per atom was measured against the strain of
the nanotube. The results shown in Figure 1 match closely with
the work of Liew et al. [9] who performed an axial compression
test on a SWCNT of similar characteristics and at a rate of 20 m/s
to obtain an independent strain energy–strain relationship.
The results showed that critical strain of the SWCNT was
approximately 0.225 which matches closely with the work
reported in Ref. [19]. The Young’s modulus was also calculated
to be 1.1 TPa which agrees well with previously computed values
from Ref [5]. Finally, the ultimate tensile strength of the nano-
tube was calculated to be 1.43 × 105
MPa. Having similar results
validates the use of the MD simulation for further experiments,
and the slight discrepancies can be attributed to the different
SWCNT systems modelled and simulated.
Figure 2 illustrates the morphological changes in the (5, 5)
armchair SWCNT when subjected to axial tensioning to the
point at which critical strain is reached. The nanotube first under-
goes thinning concentrated at the centre (Figure 2(a)) due to
its geometrical symmetry and identical stress applied to both
ends of the nanotube in opposite directions. Failure happens
at the centre where the stress concentration reaches the highest
and where AC
is the smallest. Figure 2(b) illustrates the onset of
nanotube failure when the nanotube begins to split at its geomet-
ric centre.
4. Mechanical properties of SWCNT subjected to
varying strain rates
As per previous simulation, the mechanical properties of the
same (5, 5) armchair SWCNT are investigated by applying dif-
ferent axial strain rates of 5, 10, 20 and 30 m/s at both ends
of the nanotube until it undergoes mechanical failure. The
length–diameter ratio was maintained at 10.88. The stress was
then obtained through numerical differentiation of the strain
energy–strain relationship and using a modified logistic function
coupled with the constraint of 0 Pa stress when the strain is 0 (i.e.
when the CNT has not been stretched). The function is defined as
where A, B, C are variable parameters in the curve fitting process.
Thereafter, the stress–strain relationship of the nanotube
under the specified strain rate was recorded. It should be
noted that at higher strain rates, failure occurs nearer to the
tensioned ends of the nanotube due to the lack of time given
for the tensile force to be transmitted along the nanotube to
its centre. Figure 3(a) and (b) illustrates the nanotube failure
due to thinning at the ends which maximises the stress present
at the tip of the nanotube system. This occurred at a higher
strain rate of 30 m/s.
This phenomenon causes the nanotube’s performance to
worsen as the applied axial force does not have time to transmit
throughout the nanotube system, inducing failure to the nano-
tube closer to the ends at which the force was applied. Figure 4
shows the stress–strain relationship of the nanotube at different
strain rates (5, 10, 20 and 30 m/s).
Conclusions were made from Figure 4 that the SWCNT
becomes more resistant to axial tension at low values of strain
rates (5, 10 and 20 m/s), exhibiting lower critical strain and
higher ultimate tensile strength as the strain rate increases.
(4)𝜎(𝜀) =
A
1 + e−B(𝜀+C)
−
A
1 + e−BC
Figure 1. Strain energy per atom of a (5,5) armchair SWCNT subjected to tensioning
at 20 m/s.
Figure 2. A (5,5) SWCNT subjected to axial tension. Thinning was observed at the
centre (a) before it fails catastrophically (b). Balls in white represent the C atoms at
the geometric centre of the SWCNT.

5. Mechanical properties of SWCNT subjected to
varying length–diameter ratios
The influence of the length–diameter ratio is investigated on a
(5,5) SWCNT. As per previous simulations above, the nanotube
was tensioned axially at both its ends under a fixed strain rate
and the strain energy–strain curve of was recorded for a given
length–diameter ratio. This allows the derivation of the stress–
strain curve through the first derivative. The length–diameter
ratio was varied by keeping the configuration of the SWCNT the
same and varying the length of the nanotube. Length–diameter
ratios of 7.07, 8.89 and 10.88 were chosen under a uniform biaxial
strain rate of 5 m/s. The respective nanotube lengths are 47.96,
60.26 and 73.79 Å, respectively.
It was observed that a decrease in the length–diameter ratio
causes a compromise between the SWCNT’s ultimate tensile
strength and its critical strain. Critical strain decreases, and in
return, the SWCNT becomes harder to tension, thus causing it
to fail at a higher stress. This result is captured in Table 1, a sum-
mary of the respective calculated properties of the nanotube for
the specified length–diameter ratio. In transiting from a length–
diameter ratio of 10.88, 8.89 to 7.07, the Young’s modulus of the
SWCNT increased from 0.768 TPa, 0.874 TPa to 0.979 TPa. This
implies an increase in difficulty to subject SWCNT’s with lower
length–diameter ratios to axial tension. An explanation for this
phenomena is because the shorter the nanotube, the more signif-
icant are the effects of fixed strain rate on the nanotube. This is
akin to the results obtained when a SWCNT of fixed dimensions
is subjected to the increase in strain rate.
6. Nanotube separation of DWCNT
Kuang et al. [20] investigated the effects of intertube van der
Waals forces on the stability of functionalised MWCNTs.
Conclusions were made that inner tubes offered additional stabil-
ity for the outer nanotube, at the same time, improving nanotube
characteristics such as the critical strain. This section focuses
on the significance of intertube forces based on a DWCNT by
analysing the variation in the total potential energy between the
inner and outer tubes as they are separated. In addition, a study
was carried out to understand the effects that the different inter-
spatial distances between the outer and inner tubes would have
on the force and the stability of the DWCNT during the process
of extraction. Two different armchair DWCNT models were con-
structed with configurations (7,7) @ (12,12) and (9,9) @ (12,12)
with a total of 760 and 840 C atoms, respectively (see Figure 5).
The length of each DWCNT was fixed at 23.37 Å. There are two
regions in the nanotube that are governed by different types of
potential function. The first pertains to the carbon atoms existing
within each tube. The AIREBO potential was incorporated to
describe the interactions within these intratube carbon atoms,
without consideration of torsional energy. The second pertains
However, at strain rate of 30 m/s and beyond, the critical
strain drastically decreased to 0.165. The carbon atoms in
the SWCNT did not have sufficient time to relax before the
system was subjected to further tension, resulting in a com-
putational instability which caused the SWCNT’s failure near
the site of tension. Physically, this translates to the nanotube’s
inability to respond effectively to critically high strain rates.
Hence, for SWCNT systems in general, attention has to be
given to the axial strain rate applied. Beyond a critical strain
rate, premature failure of the nanotube occurs, resulting in
the sharp decline in the SWCNT’s performance.
Figure 3. Illustration of (5,5) nanotube failure at strain rate of 30 m/s. Failure
happens closer to the ends as the rate of tensioning increases, represented by the
C atoms in white.
Figure 4. Stress–strain relationship of a (5,5) SWCNT at varying strain rates.
Table 1. Ultimate tensile strength, critical strain and Young’s modulus at different length–diameter ratios.
Length–diameter ratio Ultimate tensile strength (MPa) Critical strain Young’s modulus (TPa)
7.07 1.88 × 105
0.224 0.979
8.89 1.43 × 105
0.231 0.874
10.88 1.35 × 105
0.245 0.768

calculated to be 3.40 Å, the theoretical intertube distance which
corresponds to the interlayer distance of graphite. By decreasing
the intertube distance to 2.04 Å through using a (9,9) @ (12,12)
DWCNT, the overall nanotube system becomes unbalanced due
to increased repulsive forces felt by the inner tube since it is closer
to the outer tube.
Extraction of the inner tube was conducted through fixing the
entire outer tube as well as the first two rings at both ends of the
inner nanotube, to ensure that the atoms are rigidly locked in place
during relaxation. The remaining inner tube atoms are subjected to
an NVE ensemble constraint. The DWCNT system was allowed to
minimise before the fixed atoms in the inner tube were displaced
by 41.7 m/s to simulate extraction. The timestep used throughout
was 0.1 fs. The free carbon atoms in the inner nanotube were then
allowed to relax through 1200 timesteps at each instant the fixed
atoms of the inner tube were displaced. This process is captured in
Figure6.TheDWCNTsbeginatrestwiththesame(12,12)armchair
nanotube fixed as the outer ring. Since the Lennard-Jones potential
isinverselyrelatedtothedistanceofseparation,itisanticipatedthat
the magnitude of the intertube potential would decrease gradually
tillitreachesalimitof0 Jastheinnertubeisseparatedfarawayfrom
the outer tube. Moreover, since the (9,9) @ (12,12) DWCNT has a
smaller intertube distance, the energy within the nanotube layers
is expected to be higher due to closer proximity between the tubes.
Such were the observations from Figure 7 which depicts a plot of
the intertube potential against extraction distance.
Sharp irregularities observed in the plot for the (9,9) @ (12,12)
DWCNT in Figure 7 corresponds to a highly unstable nanotube
extraction process. This is affirmed from the simulation in Figure
6 by an increased degree or disorder in the (9,9) nanotube relative
to the (7,7) nanotube at any instance of extraction. The strong
repulsive forces presented by the outer SWCNT towards the
inner SWCNT create a compressive effect that tends to squeeze
the diameter of the inner SWCNT inwards. Hence, additional
displacements of the inner SWCNT perturb the system’s state of
equilibrium to a large degree. Subjected to the constraint of hav-
ing a fixed displacement rate, the inner SWCNT carbon atoms
do not have sufficient time to reach a minimum energy state
during relaxation. This results in an erratic response from the
inner SWCNT carbon atoms during the separation process. In
contrast, Figure 6 illustrates a smooth extraction of the (7,7) @
(12,12) DWCNT configuration as seen from the lack of promi-
nent fluctuations in the graph. It should be noted that extraction
of the inner tube in the (7,7) @ (12,12) DWCNT system requires
less force as compared to the (9,9) @ (12,12) DWCNT system
owing to two factors. The first is the slower rate of decline of
intertube potential as well as less dynamic friction present due
to smaller contact area between the (7,7) and (12,12) nanotube.
[21] This is evident from the much gentler slope exhibited by the
plot of (7,7) @ (12,12) DWCNT system as compared to the plot of
(9,9) @ (12,12) DWCNT system across the domain in which the
inner nanotube is present within the outer nanotube. Therefore,
a clear relationship exists between the ease of separation of the
nanotubes and the overall nanotube stability. By choosing a com-
bination of nanotubes with intertube separation less than the
stable value of 3.40 Å, it becomes increasingly difficult to separate
the inner tube from the outer tube as this introduces additional
instability into the overall system.
to the long-ranged attractive and repulsive forces that the outer
and inner tubes exert on each other, which was described by
the Lennard-Jones ‘12-6’ potential. We use the term ‘intertube
potential’ to describe the aggregated sum of the Lennard-Jones
interaction between the inner and outer nanotubes across all
atoms. The intertube distance for the (7,7) @ (12,12) system was
Figure 5. Cross-sectional views of (7,7) @ (12,12) DWCNT (a) and (9,9) @ (12,12)
DWCNT (b).
Figure 6. Simulation of a DWCNT system consisting of a (12,12) armchair nanotube
held fixed on the exterior while a (7,7) inner nanotube is extracted through
displacements at both ends in (a). The same (12,12) nanotube is held fixed relative
to a (9,9) inner nanotube undergoing the same displacements in (b).The snapshots
are taken at the instant in which the inner nanotubes have been displaced by a
distance of 16.65 Å.
Figure 7. Plots of intertube potential against extraction distance of a (7,7) @ (12,12)
and (9,9) @ (12,12) DWCNT.

Thereafter, a point load was applied at the centre of the nano-
tube to push the DWCNT downwards. This was done through
the displacement of 6 central carbon atoms to maintain the
symmetry of loading. The remaining carbon atoms were free to
move subjected to an NVE ensemble constraint. As per previous
experiment concerning DWCNT, a combination of the AIREBO
(without consideration for torsional energy) and the Lennard-
Jones ‘12-6’ potential was employed to describe the intratube
and intertube interactions. Figure 8 illustrates the entire bending
process of the DWCNT, subjected to a vertical displacement rate
of 41.67 m/s. Each simulation timestep is 0.1 fs.
Morphological deformation first occurs for both the inner
and outer nanotubes in which the central portion begins to thin
during the process of loading (Figure 8(a)). In the initial stages
of bending, the DWCNT deforms through compression of the
outer and inner nanotubes at the middle without any strain along
the lengths of the nanotube. The surrounding C atoms near the
point of contact of the external loading gradually redistribute
themselves into a ‘bean-like’ oval geometry to achieve a new
state of equilibrium, based on the energy imparted by the load,
while maintaining a constant inter tube distance throughout.
Upon reaching a final geometry in Figure 8(b) where no further
internal distribution of the atoms is possible, the DWCNT then
begins to deform radially, creating strain along the length of the
nanotube. At the point where the outer nanotube reaches its crit-
ical bending moment (Figure 8(c)), failure occurs but the entire
DWCNT structure is maintained because the inner tube remains
intact and bears the load that is applied. The hexagonal gap seen
in Figure 8(c) is developed through the overlapping of the carbon
orbitals with their neighbour atoms brought closer towards each
other through the bending, causing them to form covalent bonds.
Through further loading, critical bending moment of the inner
nanotube is reached, causing it to undergo failure (Figure 8(d)).
At this point, the entire nanotube structure collapses and is no
longer capable of supporting the load. Such observations suggest
a critical role the inner nanotube plays on a DWCNT subjected
to bending, as it provides additional structural support to bend-
ing as well as a fail-safe mechanism in case the outer nanotube
collapses first. This is complementary to the observations from
Liew et al. [5] who noted that the inner nanotube reinforced the
outer nanotube during axial tension, preventing the early onset
of failure.
The strain energy per surface carbon atom was obtained for
both the inner and outer nanotubes by analysing the surface
atoms near the bending site during bending. Through each
timestep, the nanotube is bent by a fixed distance and the bend-
ing moments created by the external loading attribute to the
increased strain energy of the surface atoms. Figure 9 is a plot of
average strain energy per surface carbon atom against the dura-
tion of bending. Initially, up till a duration of 10−11
s, the effect
of the external loading is felt only by the outer SWCNT whose
graph (plotted in red) showed a steady increase in steepness. A
simultaneous decrease from 10−11
s thereafter in the steepness of
the outer SWCNT’s graph, together with an increase in steepness
of the inner SWCNT’s graph (plotted in blue), indicates the shar-
ing of the external load between the outer and inner SWCNT.
Also from the simulation, it was observed that the outer nano-
tube fails after 3.6 × 10−11
s. However, the graphs still continue
to exhibit a smooth increase till a duration of 4.2 × 10−11
s, at
7. Bending test on DWCNT
In this section, a bending test was conducted on a (7,7) @ (12,12)
DWCNT whose length is 97.15 Å and comprises 3040 carbon
atoms. In order to ensure that the inner (7,7) nanotube is not dis-
placed freely during the process, 10% of the carbon atoms at both
ends of the nanotube are constrained, keeping them stationary.
Figure 8. An illustration of the bending simulation conducted on an armchair
(7,7) @ (12,12) DWCNT together with the corresponding cross-sectional view
of the nanotube system at a given instant. Initial phases of the bending cause
compression of both the outer and inner nanotube systems in (a). Upon achieving
a ‘bean-like’ geometry in (b) of which no further deformation of the nanotubes
radially is possible, the nanotube begins to strain along its length. (c) depicts the
onset of outer nanotube failure with hexagonal defects circled in red. C atoms in
blue denote the point of application of the load. (d) depicts the complete failure
of the DWCNT.
Figure 9. Plot of the average strain energy of the surface atoms near the bending
site, for both the inner and outer nanotubes, against timesteps. Failure of the outer
nanotube (red) occurs after 36 ps, while the inner nanotube (blue) fails after 42 ps,
when the entire DWCNT also fails.

References
[1] Journet C, Maser W, Loiseaut A, et al. Large-scale production of
single-walled carbon nanotubes by the electric-arc technique.
Nature. 1997;388:756–758.
[2] Maser W, Muñoz E, Benito A, et al. Production of high-density
single-walled nanotube material by a simple laser-ablation method.
Chem. Phys. Lett. 1998;292:587–593.
[3] Selzer R, Friedrich K. Mechanical properties and failure behaviour
of carbon fibre-reinforced polymer composites under the influence
of moisture. Composites Part A. 1996;28A:595–604.
[4] Yu M, Files B, Arepalli S, et al. Tensile loading of ropes of single wall
carbon nanotubes and their mechanical properties. Phys. Rev. Lett.
2000;84:5552–5555.
[5] Liew K, He X, Wong C. On the study of elastic and plastic
properties of multi-walled carbon nanotubes under axial
tension using molecular dynamics simulation. Acta Mater.
2004;52:2521–2527.
[6] Kudin K, Scuseria G, Yakobson B. C2
F, BN, and C nanoshell
elasticity from ab initio computations. Phys. Rev. B. 2001;64:235406.
[7] Wong C. Elastic properties of imperfect single-walled carbon
nanotubes under axial tension. Comput. Mater. Sci. 2010;49:143–
147.
[8] Chen N, Lusk M, Duin A, et al. Mechanical properties of connected
carbon nanorings via molecular dynamics simulation. Phys. Rev. B.
2005;72:085416.
[9] Liew K, Wong C, He X, et al. Nanomechanics of single and
multiwalled carbon nanotubes. Phys. Rev. B. 2003;69:115429.
[10] Liew K, Wong C, Tan M. Tensile and compressive properties of
carbon nanotube bundles. Acta Mater. 2005;54:225–231.
[11] Stuart S, Tutein A, Harrison J. A reactive potential for hydrocarbons
with intermolecular interactions. J. Chem. Phys. 2000;112:6472–
6486.
[12] PigosE,PenevE,RibasM,etal.Carbonnanotubenucleationdrivenby
catalystmorphologydynamics.ACSNano.2011;5:10096–10101.doi:
http://dx.doi.org/10.1021/nn2040457
[13] Li C, Chou T. Elastic moduli of multi-walled carbon nanotubes
and the effect of van der Waals forces. Compos. Sci. Technol.
2003;63:1517–1524.
[14] Wu X, Sun Y, Li C, et al. Parametric effects of the potential energy
function on the geometrical features of ternary Lennard-Jones
clusters. J. Phys. Chem. A. 2012;116:8218–8225.
[15] Stone A, Wales D. Theoretical studies of icosahedral C60 and some
related species. Chem. Phys. Lett. 1986;128:501–503.
[16] Ma J, Alfè D, Michaelides A, et al. Stone–Wales defects in
graphene and other planar sp2
-bonded materials. Phys. Rev. B.
2009;80:033407.
[17] Huang Y, Wu J, Hwang K. Thickness of graphene and single-wall
carbon nanotubes. Phys. Rev. B. 2006;74:245413.
[18] Gupta S, Bosco F, Batra R. Wall thickness and elastic moduli
of single-walled carbon nanotubes from frequencies of axial,
torsional and inextensional modes of vibration. Comput. Mater. Sci.
2009;47:1049–1059.
[19] Belytschko T, Xiao S, Schatz G, et al. Atomistic simulations of
nanotube fracture. Phys. Rev. B. 2002;65:235430.
[20] Kuang Y, Shi S, Chan P, et al. The effect of intertube van der
Waals interaction on the stability of pristine and functionalized
carbon nanotubes under compression. Nanotechnology.
2010;21:125704.
[21] Kis A, Jensen K, Aloni S, et al. Interlayer forces and ultralow
sliding friction in multiwalled carbon nanotubes. Phys. Rev. Lett.
2006;97:1–4. doi:10.1103/PhysRevLett. 97.025501
which the inner SWCNT begins to fail. Thereafter, the entire
nanotube fails under the loading. The results thus highlight the
integral role of the inner SWCNT in providing a support for load
distribution and sustenance of the DWCNT, thereby ensuring
that the DWCNT does not fail that easily.
8. Conclusion
This paper analysed the various mechanical properties of a
SWCNT and DWCNT. For the SWCNT, an understanding on
how the strain rate as well as the length–diameter characteristic
can influence key properties such as ultimate tensile stress, crit-
ical strain and the Young’s modulus was performed. An increase
in the strain rate had the effect of causing the nanotube to be gen-
erally less elastic. This relationship holds until a critical strain rate
is reached in which the nanotube is unable to effectively distrib-
ute the force across its system and breaks prematurely at the ends.
Also, an increase in the length–diameter ratio revealed a trade-off
between critical strain and ultimate tensile stress. SWCNTs with
lower length–diameter ratio exhibited more resistance to stretch-
ing via axial tension. Tensile force applied is generally higher but
the SWCNT’s critical strain value is lower.
For the DWCNT, a study was conducted to investigate the
variations in intertube potential when the inner SWCNT is
extracted out from the outer SWCNT, based on the variation of
intertube distance. This was carried out by choosing two different
DWCNTs that have an intertube separation of 3.40 Å (stable) and
2.04 Å (non-stable). The results obtained showed that it becomes
increasingly difficult to extract the inner tube in a non-stable
configuration. Slight displacements of the inner SWCNT add
a large degree of disturbance to the DWCNT’s original state of
equilibrium. This presents strong inertia to oppose the extraction
process. Finally, a bending test was performed to understand
behaviour of the DWCNT through a point load at its centre.
The results highlighted the importance of the inner nanotube as
structural support for the outer nanotube as it is able to withstand
the point loading even after the outer tube has failed, delaying
the nanotube system from collapsing early.
The work presented by this paper offers numerical and analyt-
ical insights into SWCNT and DWCNT systems when subjected
to perturbations at the atomic level. These insights can be applied
to the design of larger and aggregated nanotube systems that have
their practical applications. Further work can be conducted to tie
the results obtained from our MD simulations to experimental
results and observations as a further proof of concept. In doing
so, one has to be mindful to consider defects and temperature
of the experimental set-up which would inevitably influence the
results obtained.
Disclosure statement
No potential conflict of interest was reported by the authors.

Molecular dynamics simulation studies of mechanical properties of different carbon nanotube systems

More Related Content

What's hot (18)

Viewers also liked (20)

Similar to Molecular dynamics simulation studies of mechanical properties of different carbon nanotube systems (20)

Molecular dynamics simulation studies of mechanical properties of different carbon nanotube systems