ACS Nano
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Supporting Online Material
Higher Recovery and Better Energy Dissipation at Fa
ster
Strain Rates in Carbon Nanotube Bundles: an
in-situ
Study.
Siddhartha Pathak
1
∗
, Ee J. Lim
1
, Parisa Pour Shahid Saeed Abadi
2
, Samuel Graham
2,3
,
Baratunde A. Cola
2,3
, Julia R. Greer
1
1
Materials Science, California Institute of Technolo
gy (Caltech), Pasadena, CA, USA
2
George W. Woodruff School of Mechanical Engineering
, Georgia Institute of
Technology, Atlanta, GA, USA.
3
School of Materials Science and Engineering, Georgi
a Institute of Technology, Atlanta,
GA, USA.
∗
Contact author: - Materials Science, California Ins
titute of Technology (Caltech), 1200 E. California
Blvd., MC 309-81, Pasadena, CA 91125-8100, Phone: +
1 (626) 395-8165, Fax: +1 (626) 395-8868, E-
mail: pathak@caltech.edu, siddharthapathak@gmail.co
m
ACS Nano
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Viscoelastic Characterization
In addition to their distinctive buckling behavior
and their ability to recover from
large deformations, VACNTs have also been reported
to demonstrate a unique
viscoelastic response, which was reported by a subs
et of the current authors for two
separate VACNT systems
1, 2
. The remarkable nature of the viscoelastic behavio
r of
VACNTs was further demonstrated in
3
, where the material was found to exhibit this
behavior for a wide temperature range from -196
o
C to 1000
o
C – something no other
material had shown previously. In the research cond
ucted in our group viscoelastic
behavior including storage and loss moduli over a r
ange of frequencies was reported by
Hutchens, et al
1
. Below we describe the results of our additional v
iscoelastic experiments
performed on the described VACNT micro-pillars.
The viscoelastic properties of the VACNT film were
measured using flat punch
indentation following the procedure outlined in Ref
1
. In this method the indenter is
loaded into the sample at a constant displacement r
ate of 10 nm/s up to a specified strain,
where the indenter head is oscillated at ~8 nm ampli
tude across a range of frequencies
from 1 to 50 Hz. This cut-off frequency (50 Hz) is
an instrument limitation as detailed in
Ref
1, 4
. The procedure was repeated at four different stra
ins:
ε
= 0.18, 0.4, 0.62 and 0.84,
which ensured visco-elastic characterization of the
samples in both their pre and post-
densification regimes.
Viscoelastic materials are commonly characterized b
y their storage (
E
′
) and loss
(
E
′
′
) moduli, where the former represents the stored en
ergy or the elastic response, and
the latter corresponds to the amount of energy diss
ipated or the viscous response, as well
as their ratio –
δ
tan
. Assuming linear viscoelastic behavior, these term
s can be computed
following the calculations described in Refs
4-7
as follows:
A
k
E
s
2
1
2
ν
β
π
−
′
=
′
,
air
air
u
F
u
F
k
φ
φ
cos
cos
0
0
0
0
−
=
′
,
A
k
E
s
2
1
2
ν
β
π
−
′′
=
′′
,
air
air
u
F
u
F
k
φ
φ
sin
sin
0
0
0
0
−
=
′′
,
E
E
′
′
′
=
δ
tan
(1)
Here
k
′
and
k
′
′
are the storage and loss stiffnesses of the sample,
obtained by finding the
real and complex parts, respectively, of the stiffn
ess differences between oscillating the
indenter head on the sample at a fixed displacement
and in air at the same raw
displacement,
β
is a constant (=1 for a flat punch indenter),
0
F
and
0
u
are the load and
displacement oscillation amplitudes respectively, a
nd
φ
is the phase angle between the
load and displacement oscillations. The accuracy in
the values of
E
′
and
E
′
′
in Eq. (1)
can be affected by a couple of factors: uncertainti
es in the value of the Poisson ratio
(since the Poisson’s ratio can also be frequency de
pendent), and accuracy in the value of
the contact area, especially at lower indentation d
epth where full contact may not have
been established. On the other hand, calculation of
δ
tan
is independent of these
parameters, and thus is ideally suited as a measure
of the viscoelasticity of the indented
material
2, 8
.
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The viscoelastic indentation response of the VACNT
film, in terms of the
measured values of their storage modulus (
E
′
), loss modulus (
E
′
′
) and
δ
tan
values, is
shown below (Figure S1). Similar to the % recovery
results discussed in the main
manuscript, two distinct responses are seen in this
figure depending on if the indenter is
oscillated at levels corresponding to the (i) pre-d
ensification regime (open symbols in
Figure S1) or (ii) post-densification regime (fille
d symbols). While both
E
′
and
E
′
′
are
seen to increase by around 2-3 times after densific
ation (Figures S1a and b), there is no
difference in the values of
δ
tan
between the pre- pand post-densification regimes
(Figure S1c). This indicates that both the values o
f
E
′
and
E
′
′
have increased in equal
proportions after densification.
The storage modulus values were found to be frequen
cy independent over the
range of frequencies used in this work. On the othe
r hand the loss modulus (and the
δ
tan
) values are strongly affected by the frequency, an
d they generally increase with
increasing frequency, although a couple of local mi
nimas can be indentified at 30 and 50
Hz respectively. Unfortunately the cut-off frequenc
y (50 Hz) of our instrument prevents
further study of this behavior at higher frequencie
s.
As shown in the figures of the main manuscript (Fig
ures 1 and 2), the VACNT
microstructure starts getting densified at strain l
evels of
ǫ
≥
0.65-0.7. Thus the higher
values of
E
′
obtained in the post-densification regime indicate
that the material is capable
of storing a higher amount of energy in this state,
with the excess energy presumably
being stored in the buckled/densified regions. A co
rrespondingly higher proportion of
energy is also dissipated in this regime, as indica
ted by a similar increase in the values of
E
′
′
. However, the overall visco-elastic response of th
e VACNT material still remains
unaffected, as seen in Figure S1c. These results ma
tch well with the loss coefficient
measurements shown in Figure 6 (main manuscript), w
hich also show similar values of
loss coefficient between the pre- and post densific
ation regimes. An analogous response
was also noted by Xu et al. over a much wider tempe
rature range of -196
o
C to 1000
o
C
3
.
In addition the unloading modulus values shown in F
igure 3 (main manuscript) are also
are also found to match well with that of
E
′
in Figure S1a.
Characterization of Material Damping Response
The loss coefficient,
η
, (a dimensionless quantity) measures the degree to
which a
material dissipates energy and is calculated in two
different ways as shown below
9
1
2
U
U
i
π
η
=
,
∫
=
max
d
U
σ
ε
σ
0
,
∫
=
ε
σ
d
U
(2)
i
i
U
U
π
η
2
=
,
∫
=
max
d
U
σ
ε
σ
0
,
∫
=
ε
σ
d
U
(3)
where
1
U
and
i
U
are the elastic energy stored in the material when
it is loaded elastically
to a stress
max
σ
in the 1
st
and the i
th
cycle respectively, and
i
U
is the energy dissipated
in the i
th
load-unload cycle (see Fig. S2). The main differenc
e between the above two
equations is that in Equation (2) the normalization
is done with respect to the area of the
1
st
loading cycle, while in Equation (3) it is done w.
r.t. the area of the i
th
cycle
ACS Nano
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respectively. In other words, the denominator in Eq
uation (2) is a constant while that of
Equation (3) changes with every cycle. Each definit
ion shows a different trend in the
values of
η
as shown in the figure below.
As seen from the figure below, use of Equation (2)
results in sharp drop in the
values of
η
after the 1
st
cycle (Figure S2a), but there is no such drop when
using Equation
(3). Note that the values of
η
for the 1
st
cycle are exactly the same in both figures (Figure
S2a and S2b). After the 1
st
cycle, both figures show a similar drop in the val
ues of
η
for
increasing cycle number. In both cases,
η
also appears to be strain dependent, and is
maximized at the fastest 1000 nm/sec rate, similar
to the trends noted for modulus and
recovery. As in the case of the tan
δ
values described earlier, no significant different
difference is seen between the pre- and post-densif
ication regimes.
1.
Hutchens, S. B.; Hall, L. J.; Greer, J. R., In s
itu Mechanical Testing Reveals
Periodic Buckle Nucleation and Propagation in Carbo
n Nanotube Bundles.
Advanced
Functional Materials
2010, 20, 2338-2346.
2.
Pathak, S.; Cambaz, Z. G.; Kalidindi, S. R.; Swa
dener, J. G.; Gogotsi, Y.,
Viscoelasticity and high buckling stress of dense c
arbon nanotube brushes.
Carbon
2009,
47, 1969-1976.
3.
Xu, M.; Futaba, D. N.; Yamada, T.; Yumura, M.; H
ata, K., Carbon Nanotubes
with Temperature-Invariant Viscoelasticity from-196
degrees to 1000 degrees C.
Science
2010, 330, 1364-1368.
4.
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oindentation and the dynamic
characterization of viscoelastic solids.
Journal of Physics D: Applied Physics
2008, 41,
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5.
Herbert, E. G.; Oliver, W. C.; Lumsdaine, A.; Ph
arr, G. M., Measuring the
constitutive behavior of viscoelastic solids in the
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punch nanoindentation.
Journal of Materials Research
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6.
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proved analysis for viscoelastic
damping in dynamic nanoindentation.
International Journal of Surface Science and
Engineering
2007, 1, 274-92.
7.
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Storage and loss stiffnesses and moduli as
determined by dynamic nanoindentation
, 506 Keystone Drive, Warrendale, PA 15086,
United States, Materials Research Society: 506 Keys
tone Drive, Warrendale, PA 15086,
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Pathak, S.; Gregory Swadener, J.; Kalidindi, S.
R.; Courtland, H.-W.; Jepsen, K.
J.; Goldman, H. M., Measuring the dynamic mechanica
l response of hydrated mouse
bone by nanoindentation.
Journal of the Mechanical Behavior of Biomedical Ma
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2011, 4, 34-43.
9.
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Materials Selection in Mechanical Design
. Butterworth-
Heinemann; : 2005 (3 ed.).