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This is the submitted version of a paper published in Journal of Applied Physics.
Citation for the original published paper (version of record):
Jackman, H., Krakhmalev, P., Svensson, K. (2015)
Mechanical behavior of carbon nanotubes in the rippled and buckled phase.
Journal of Applied Physics, 117(8): 084318
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Mechanical behavior of carbon nanotubes in the rippled and buckled phase
H. Jackman,
1
P. Krakhmalev,
1
and K. Svensson
1, a)
Department of Engineering and Physics, Karlstad University, SE-651 88 Karlstad,
Sweden
(Dated: 7 March 2015)
We have studied the mechanical behavior of multi-walled carbon nanotubes for bending strains beyond the
onset for rippling and buckling. We found a characteristic drop in the bending stiffness at the rippling and
buckling onset and the relative retained stiffness was dependent on the nanotube dimensions and crystallinity.
Thin tubes are more prone to buckle, where some lose all of their bending stiffness, while thicker tubes are
more prone to ripple and on average retain about 20% of their bending stiffness. In defect rich tubes the
bending stiffness is very low prior to rippling but these tubes retain up to 70% of their initial bending stiffness.
I. INTRODUCTION
Carbon nanotubes (CNTs) have a high mechan-
ical stiffness, high electrical conductivity and are
lightweight. These properties make CNTs ideal for high
frequency nanoelectromechanical applications, such as
nanorelays,
1,2
and nanoresonators.
3–5
Carbon nanotubes
can be thought of as rolled up layers of graphene sheets,
and the high in-plane stiffness of graphene gives the tubes
an axial Young’s modulus on the order of 1TPa.
6
In bend-
ing configurations, the high axial Young’s modulus pro-
vides a high bending stiffness of the tubes, but only up
to a critical point where tube walls start to ripple or
buckle.
7–9
These deformations are similar to the behavior
seen in macroscopic tubes (e.g. drinking straws) and it
results in a significant reduction of the bending stiffness.
At the very onset of rippling, the changes in morphol-
ogy are very subtle,
10
and the onset is best detected by
the abrupt drop in bending stiffness.
11
In the rippled and
buckled phase there is also a concomitant reduction in the
electrical conductivity.
12,13
Thus, if a nanorelay is oper-
ated into the rippled regime, it can cause problems both
with stiction, due to a low restoring force, and a low elec-
trical conductivity in the on-state. On the other hand,
the deformation can be used for intentional modification
of the electron transport, and it has enabled the construc-
tion of room-temperature single electron transistors.
14
As
the rippling and buckling are reversible processes, they
can also be exploited in energy absorbing materials.
15
The rippling and buckling behavior of carbon nan-
otubes have been modelled extensively (for recent re-
views see e.g. Ref. 16 and 17), but measurements are
very challenging and the experimental characterizations
are still at an infant stage.
16
The rippling pattern it-
self has been studied for embedded tubes
9,18
by using
transmission electron microscopy (TEM), and force mea-
surements have also been done for tubes supported on a
substrate
19,20
by using atomic force microscopy (AFM).
Ideally these two methods should be combined to enable
studies of freestanding structures, and early attempts
have been made by using different spring elements and
a)
Electronic mail: krister.s@kau.se
20 nm
20 nm
0 10 20 30 40 50 60 70 80 90
0
1
2
3
4
5
6
δ
piezo
[nm]
F [nN]
k
tot
i
k
tot
r
forward
backward
linear fit
2
1
(a)
(b)
(c)
FIG. 1. TEM-images of a freestanding MWCNT, (a) when
unloaded and (b) when bent. (c) A typical force (F ) versus
deflection (δ) curve obtained for a MWCNT.
post image analysis to estimate forces
21,22
. Only recently
has the use of non-optical force sensors
23
provided a true
joining of AFM with regular electron microscopy instru-
ments, and this has enabled detailed studies of the crit-
ical bending-strain for buckling and rippling in carbon
nanotubes.
11,24
These measurements have shown that
both the initial stiffness and the rippling onset are very
sensitive to the amount of defects in the nanotubes.
24
Tubes with a large defect density, grown by chemical
vapour deposition (CVD) techniques, are much softer
and ripple at larger strains compared to highly crystalline
2
tubes grown by arc-discharge techniques. The rippling
onset is also dependent on the number of walls in the
tubes, as the inner tubes support the outermost tube,
where the rippling first will appear. Such a supporting
effect was first suggested from modeling.
25
The actual ef-
fect is larger than predicted though, and measurements
have shown a variation by a factor of three in the critical
strain, for CNTs with the same outer diameter.
24
The
bending stiffness in the rippled phase has also been sug-
gested to depend on the number of walls in a multi-walled
carbon nanotube (MWCNT), but the modeling results
vary from a rather weak,
26
to a very strong dependence.
27
There have not yet been any quantitative experimental
data available, but the behavior will be important for the
exploitation of strains beyond the initial linear regime in
carbon nanotubes.
Here we report studies of the mechanical behavior of
carbon nanotubes beyond the rippling and buckling on-
set. The tubes are mounted rigidly at one end while
the free end is bent using a piezoresistive force sensor.
The force response from each individual tube has been
monitored continuously as a function of deflection, and
provides a measure of the bending stiffness in the rippled
phase. Beyond the critical strain, we find a near linear
relation between force and deflection, albeit with a re-
duced bending stiffness compared to the initial stiffness.
The relative retained stiffness has been analyzed with re-
spect to a normalized thickness of the tubes. We find
that highly crystalline tubes have a low relative retained
stiffness after the rippling onset, while defects increase
the retained stiffness, in reasonable agreement with pre-
vious modeling studies. In highly crystalline tubes, the
number of walls will affect both the deformation mecha-
nism and the relative retained stiffness.
II. EXPERIMENTAL
Measurements were performed on MWCNTs grown
by CVD and the arc-discharge method. The CVD-
grown MWCNTs were synthesized by Nanocyl (NC2100
and NC2101), and the tubes grown by the arc-discharge
method were obtained from Professor Hui-Ming Cheng
(at the Institute of Metal Research, Chinese Academy
of Sciences). Arc-discharge grown CNTs have a much
higher crystallinity compared to CVD-grown ones. This
is fairly obvious from TEM images, where the walls of
arc-discharge tubes are much straighter, but it is diffi-
cult to quantify the amount of defects.
28
By using both
materials we are able to observe the influence of defects
on the material properties qualitatively.
Individual MWCNTs were pushed against a piezore-
sistive force sensor, in a cantilever-to-cantilever fashion,
inside a TEM and inside a scanning electron microscope
(SEM). The in situ instrument, enabling manipulation
and force measurements, was obtained from Nanofactory
Instruments AB for the TEM and custom made for the
SEM using the same type of manipulator and force de-
tection system. Measurements on tubes grown by arc-
discharge where performed inside a JEOL (JEM 2100)
TEM, equipped with a LaB
6
cathode and a digital cam-
era from Gatan (SC1000 Orius). The sample was inserted
and left in the chamber for at least 10 hours before the
measurements, reaching a pressure of about 7×10
−8
Torr
(with the use of a liquid nitrogen cooling trap). This
was done in order to minimize any risk of amorphous
carbon build up when exposing the carbon nanotubes to
the electron beam. The acceleration voltage was set at 80
kV, which is below the threshold for knock-on damage in
graphene.
29
To further reduce the risk for beam induced
effects, the electron beam was deflected away from the
sample and force sensor during the force measurements
since damages can occur even at 80kV in strained sp
2
-
carbon bonds,
30
due to ionization effects.
31
Measurements on the CVD-grown tubes were per-
formed inside a LEO 1530 FEG-SEM, operated at an
acceleration voltage of 12 kV and a chamber pressure
of about 5 × 10
−7
Torr. The beam was deflected away
from sample and sensor during measurements, in order to
avoid beam induced effects. The experimental set-up and
further details on how the measurements were performed,
and calibrated, have been described elsewhere.
11,24
III. DROP IN BENDING STIFFNESS
A typical measurement is shown in Fig. 1, where a
MWCNT has been imaged (a) before, and (b) during
force measurements. Fig. 1 (c) shows a typical force, F ,
versus deflection, δ, curve, where at first the force is zero
until the CNT comes in contact with the force sensor
at point 1 indicated in the figure. From this point and
on the force increases linearly with the deflection, having
an initial spring constant of k
tot
i
, until at point 2 where
the stiffness abruptly decreases. After point 2 the force
continues to increase approximately linearly but with a
lower rippling phase spring constant k
tot
r
. The measured
spring constants, k
tot
i,r
, are related to the spring constants
of the sensor, k
sens
, and that of the CNT, k
i,r
, through:
k
i,r
=
k
sens
k
tot
i,r
k
sens
− k
tot
i,r
(1)
By assuming the CNTs to be cantilevered beams with a
hollow circular cross-section, the Young’s modulus can be
obtained using the spring constants and the dimensions
of the CNT.
E
i,r
=
64
3π
k
i,r
l
3
(d
4
o
− d
4
i
)
(2)
Where l is the length, and d
o
and d
i
are the outer and
inner diameter of the CNT. The rippling phase Young’s
modulus, E
r
, should only be seen as an apparent param-
eter, as the cross-section of the CNT is altered after the
rippling onset. Further on, E
r
is only used in comparison
with E
i
and not as an absolute measure.
3
0 5 10 15 20 25 30 35
0
500
1000
1500
d
o
[nm]
E [GPa]
(b)
(a)
0 5 10 15 20 25 30 35
0
50
100
150
200
250
300
d
o
[nm]
E [GPa]
E
i
rippled
E
i
buckled
E
r
rippled
E
r
buckled
E
i
E
r
250
750
1250
E
i
mean
=780 GPa
E
r
mean
=140 GPa
E
i
mean
=80 GPa
E
r
mean
=40 GPa
FIG. 2. Young’s modulus (E) vs. outer diameter (d
o
) be-
fore and after the rippling onset, for (a) tubes grown by arc-
discharge and (b) CVD-grown tubes. The horizontal lines
indicate the average values for each phase.
Since the CVD-grown tubes were measured inside a
SEM, the inner diameter could not easily be measured.
However most tubes had a large normalized thickness
(t
N
=
d
o
−d
i
d
o
> 0.5) making the effect from the inner
diameter on E
i
negligible.
11
To get a good accuracy in
measuring the outer diameter, the second derivative of
the integrated intensity profile was used.
32
Multiplying
Eq. 2 with the ratio k
r
/k
i
yields the Young’s modulus
after the rippling onset. The initial linear regime and the
very onset of localized deformations have been analyzed
in detail before.
11,24
Here we are focussing on the behav-
ior after the very onset of buckling and rippling. This
is most often referred to as a non-linear regime, but this
can be a little misleading as we find an approximately
linear dependence also above the critical strain. Instead
we here refer to this strain regime as a rippled phase of
the nanotube. To analyze the mechanical behavior in the
rippled phase, we have plotted Young’s modulus before,
E
i
, and after, E
r
, the rippling onset in Fig. 2, for (a)
tubes grown by arc-discharge and (b) CVD-grown tubes.
The arc-discharge tubes are, to begin with, much stiffer
than the CVD tubes as a result of their higher degree
of crystallinity.
24
Moreover, the relative drop in E for
the arc-discharge tubes are much larger than the drop
for the CVD tubes as is shown in Fig. 2. For the tubes
grown by arc-discharge we have also distinguished be-
tween the tubes that buckle and those that ripple. Here
we define buckling as a deformation where only one kink
were observed, as is seen in Fig. 1 (b). Deformation
where more than one kink were observed is referred to as
rippling. From this definition it can be seen, in Fig. 2
(a), that the Young’s modulus after the critical strain is
much smaller for the buckled tubes compared to the rip-
pled tubes. Some buckled tubes have even lost all their
bending stiffness, as one might expect for thin walled
tubes.
8,33
From Fig. 2 there seems to be a trend for CNTs with
a large E
i
, i.e. tubes with a high crystallinity, to have
a larger drop in their stiffness compared to tubes with a
smaller E
i
. To investigate this further, we have plotted
the relative retained stiffness, k
r
/k
i
, versus the initial
Young’s modulus, E
i
, in Fig. 3. In this plot there is a
general trend that tubes with a high crystallinity (high
E
i
) have a smaller relative retained stiffness, compared to
defect rich ones (low E
i
). This also indicates that some
of the tubes grown by arc-discharge are not defect-free
(i.e. tubes with E
i
< 500 GPa).
The smaller relative reduction in the bending stiff-
ness for CVD-grown tubes is consistent with simulations
where it was found that the existence of interlayer bridges
in MWCNTs will give a higher post-rippling stiffness.
34
CVD-grown tubes contain many types of defects, and
these might have different effects on the mechanical prop-
erties. Vacancies might contribute mainly to a reduced
axial stiffness, while inter-wall bridges may improve the
post-rippling stiffness. It is worth noting that although
the relative retained stiffness is higher for CVD-grown
tubes, the value of E
r
is still lower in these tubes com-
pared to the ones grown by arc discharge.
In Fig. 4 the relative retained stiffness, k
r
/k
i
, is plot-
ted versus the normalized thickness, t
N
. Here we have
only included data-points from tubes with E
i
> 500
GPa, in order to ensure a high crystallinity of the tubes.
One can see that thin tubes (with t
N
< 0.6) preferably
buckle while for thick tubes (with t
N
> 0.8) there is only
rippling deformation. There is however a fairly large
range in the t
N
values where tubes can both ripple or
buckle, which means that the deformation mechanism is
not solely determined by the value of t
N
. Our definition
of rippling and buckling is also rather crude due to the
bending geometry with cantilevered tubes, as the bend-
ing moment varies linearly along the length of the tubes
and has a maximum at the attachment point. Hence,
the rippling deformations will have tendency to become
localized near the attachment point at low loads, and the
distinction between the two deformation mechanisms is
4
0 500 1000 1500
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
E
i
[GPa]
k
r
/k
i
arc rippled
arc buckled
CVD
FIG. 3. Retained relative bending stiffness, k
r
/k
i
, versus
Young’s modulus in the initial linear phase, E
i
.
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t
N
k
r
/k
i
Chang
Nikiforov
Ripple d
Linear fit
Linear fit
Linear fit
Arroyo
Buckled
FIG. 4. Retained relative bending stiffness, k
r
/k
i
, ver-
sus normalized wall thickness, t
N
, for arc-discharge grown
tubes (crosses). Triangles indicate results from modelling,
Nikiforov,
27
Chang,
26
and Arroyo.
10
not as well defined as in modeling studies using a con-
stant bending moment.
26,27
A linear fit to all the experimental data points in Fig.
4 (rippled and buckled), results in a weakly increasing
function, with values of k
r
/k
i
ranging from about 0.12 to
0.17 (as t
N
goes from 0 to 1). This behavior is similar to
what was found from modeling in Ref. 26, where a weak
dependence of k
r
/k
i
on t
N
was found, but it is in sharp
contrast to the strong dependence found from modeling
in Ref. 27, as illustrated in Fig. 4. In both these modeling
studies the application of the load differs from our exper-
imental work, as the models used an evenly distributed
bending moment along the MWCNTs. In Ref. 10 a can-
tilevered MWCNT was modelled and a value for k
r
/k
i
of about 0.23 was obtained, in close agreement with the
values obtained for rippled tubes in this study.
IV. GRADUAL DEFORMATION
For short and thick tubes, i.e. with small l and large
d
o
and t
N
, the F − δ curves reveal some interesting fea-
tures, as can be seen in Fig. 5. At point 1 there is a
sudden drop in stiffness, as most simulations have pre-
dicted. In contrast to most simulations, a drop in the
force is observed at point 2. After this drop the force
continues to increase linearly, albeit with a slightly lower
stiffness than between point 1 and 2. Upon retraction
the F − δ curve is linear until point 3 where there is a
small increase in the force as the tube returns to the ini-
tial linear response. This behavior is reproducible during
cyclic motion, indicative of purely elastic deformations.
The drop in force at point 2, which is beyond the criti-
cal strain, suggests that there is more than one critical
point where the mechanical behavior is changed. Due
to the high sensitivity of strained sp
2
-carbon bonds to
electron irradiation, we have not attempted any further
TEM analysis of the rippling regime.
Simulations of MWCNTs have found a similar abrupt
drop in the force,
26
but the mechanism behind the drop
was not discussed, and no cyclic loading was studied.
Modeling of SWNTs have predicted a similar hysteretic
behavior.
33
The effect was attributed to a gradual buck-
ling (during loading), combined with van der Waals in-
teractions between opposing walls in the fully collapsed
tube, thus delaying the relaxation point during unload-
ing. The tubes where we have clearly observed this be-
havior all have large t
N
, and we suggest that at point
1 the outermost wall ripples and between point 1 and 2
the rippling deformation spreads to the inner walls of the
CNT. Finally at point 2 the innermost wall ripples and
there is a drop in the force. A similar behavior is likely
present in all tubes, but tubes with a smaller t
N
will
display a shorter distance between point 1 and 2. The
behavior is then easily obscured, especially in measure-
ments having a low signal-to-noise ratio.
Going back to Fig. 1 (c), there seems to be a drop in
the force around the critical strain for rippling, but the
signal-to-noise ratio prohibits this from showing conclu-
sively. Also the extent of the regime just after the change
in bending stiffness and before the drop in force is much
smaller, compared to Fig. 5, as a result of the smaller
value of t
N
.
The behavior described above will give rise to hys-
teretic damping, and affect the performance of nanore-
lays and nanoresonators if these are oscillated beyond the
critical strain for rippling, i.e. beyond point 2 in Fig. 5.
V. CONCLUSIONS
We find that the relative retained stiffness in the rip-
pled and buckled state is highly dependent on the crys-
tallinity. For highly crystalline tubes the relative re-
tained stiffness varies between 0 and about 30%, depend-
ing mostly on the deformation mechanism. Thin tubes,