Axial Young’s modulus prediction of single-walled carbon nanotube arrays with
diameters from nanometer to meter scales
C. H. Sun, F. Li, H. M. Cheng, and G. Q. Lu
Citation: Applied Physics Letters 87, 193101 (2005); doi: 10.1063/1.2119409
View online: http://dx.doi.org/10.1063/1.2119409
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APPLIED PHYSICS LETTERS 87, 193101 共2005兲
Axial Young’s modulus prediction of single-walled carbon nanotube arrays
with diameters from nanometer to meter scales
C. H. Sun, F. Li, and H. M. Chenga兲
Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy
of Sciences, Shenyang 110016, China
G. Q. Lu
ARC Center for Functional Nanomaterials, The University of Queensland, Brisbane, QLD 4072, Australia
共Received 2 May 2005; accepted 1 September 2005; published online 31 October 2005兲
Based on a self-similar array model, we systematically investigated the axial Young’s modulus
共Y axis兲 of single-walled carbon nanotube 共SWNT兲 arrays with diameters from nanometer to meter
scales by an analytical approach. The results show that the Y axis of SWNT arrays decreases
dramatically with the increases of their hierarchy number 共s兲 and is not sensitive to the specific size
s
is independent of the
and constitution when s is the same, and the specific Young’s modulus Y axis
packing configuration of SWNTs. Our calculations also show that the Y axis of SWNT arrays with
diameters of several micrometers is close to that of commercial high performance carbon fibers
s
of SWNT arrays is much better than that of high performance CFs. © 2005
共CFs兲, but the Y axis
American Institute of Physics. 关DOI: 10.1063/1.2119409兴
Theoretical and experimental results have shown that individual single-walled carbon nanotubes 共SWNTs兲 have extremely high elastic modulus, greater than 1 T Pa,1,2 which
has stimulated great efforts to synthesize or fabricate large
SWNT arrays with a macrosize.3–10 Unfortunately, even
the best macroscopic fibers made up of aligned SWNTs fall
far short of the expected values.8,9,11–13 Moreover, it is
widely observed that the axial mechanical performance of
SWNT bundles strongly decreases with the increase of their
diameters.11–14 To explain this phenomenon, Salvetat suggested that it is due to the possibility that larger ropes contain
more imperfections than small ones; moreover, the absence
of registry between neighboring tubes induced by variations
in tube diameter and helicity.11 Also Yu et al. proposed that
only the perimeter SWNTs act as the load-carrying
elements.13 From the earlier work, it is concluded that the
fundamental problem in translating the wonderful mechanical properties of individual SWNT into larger bundles may
be achieved by strengthening intertube bonding,15 which has
been realized by Kis et al. through irradiation of SWNT
bundles with high-energy electrons,14 and a high bending
modulus is achieved. Ajayan commented that this work
paves the way towards realizing stronger nanotube fibers in
the future.15 Behind the impressive work reported by Kis,14
however, there is still another basic obstacle, the packing of
SWNTs. It has been found that SWNTs are apt to aggregate
into bundles, bundles into bundle arrays, and so
on.6,7,10 Ebbesen et al. figured out a schematic diagram of
this hierarchy packing pattern of SWNTs, from large bundles
down to individual carbon nanotubes.16 Different from the
close packing, hierarchy SWNT arrays are much looser, as
observed in experiments.6,7,10 Therefore, estimations of the
mechanical properties of SWNT arrays are required to take
the hierarchical structure of SWNT arrays into account. Previously, we have proposed a self-similar array model to
quantitatively describe the hierarchy structures in SWNT
arrays.17 Using this model, SWNT arrays, including close
and hierarchy packing patterns, are universally described.17
Figure 1 presented a schematic view of hierarchy structures,
from isolated SWNT to SWNT bundle and bundle arrays,
according to the self-similar array model.17 And the goal of
the present work is to study the effect of the hierarchical
structure of SWNT arrays on their mechanical properties
based on the self-similar array model of SWNTs, and a systematical estimation of the axial Yong’s modulus, Y axis, is
carried out for SWNT arrays with diameters from nanometer
to meter scales.
To estimate the best performance of SWNT arrays at
different scales, those adverse factors, such as impurity and
irregularity of SWNTs, are not taken into account here.
Hence, identical, infinitely long and straight SWNTs 关for instance, 共12, 12兲 nanotubes with a diameter D = 1.62 nm and a
van der Waals gap G = 0.3162 nm兴18 are employed. To further simplify our discussion but without loss generality, only
three kinds of SWNT arrays are investigated: 共cm, 1兲 for one
共isolated兲 bundles consisting of cm SWNTs 共specially, cm = 1
denotes an isolated SWNT兲, 共cm , cn兲 for the arrays of cn
bundles and each bundle consisting of cm SWNTs, and
a兲
FIG. 1. Schematic image of hierarchy structures of SWNT arrays, from
isolated SWNT to SWNT bundle and to bundle array.
Author to whom correspondence should be addressed; electronic mail:
[email protected]
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87, 193101-1
© 2005 American Institute of Physics
23:12:55
193101-2
Appl. Phys. Lett. 87, 193101 共2005兲
Sun et al.
共cm , cn , cl兲 for a superstructure consisting of cl bundle arrays
共cm , cn兲. Also we define a concept of hierarchy number, s, for
SWNT arrays, and s labels the times of self-similar transformation under the self-similar array model.17 For instance, 1,
2, and 3 times of self-similar transformations are needed to
be carried out for the construction of hierarchy structures of
共cm , 1兲cm⫽1 , 共cm , cn兲cn⫽1, and 共cm , cn , cl兲cl⫽1 under the selfsimilar array model, respectively, hence, s = 1, 2, and 3. Specially, for isolated SWNT marked as 共1兲, no packing is carried out and no self-similar transformation is performed,
hence, s = 0. Examples of isolated SWNT, SWNT bundle
共7, 1兲 and bundle array 共7,7兲 are presented in Fig. 1. Details
and these notations can be found elsewhere.17 Since our estimation is carried out from several nanometers to meters, an
analytical calculation method,19 rather than atomistic simulations, is employed
Y axis =
⳵ 2E
⫻ L ⫻ A,
⳵ ␧2
共1兲
where ⳵2E / ⳵␧2 is the second derivative of energy with respect to strain along the tubule axis, and the value of
58.2 eV atom−1 was used for ⳵2E / ⳵␧2 independent of the tuFIG. 2. 共a兲 Calculated Young’s modulus, Y axis, for 14 SWNT arrays with
diameters from 1 nm to 1 m. The point labeled by 䊏 corresponds to isolated
bule diameter and packing configuration of SWNTs.20 L is
共12, 12兲 SWNT. 共b兲 Measured Y axis for SWNT arrays at nanometer and
the number of atoms per tubule per unit length and A is the
micrometer dimensions are compared with the calculated results and those
number of tubules per unit area in the cross section. Clearly,
for commercial carbon fibers. Experimental data are reported by Salvetat et
for different packing patterns and configurations, the
al. 共see Ref. 11兲 and Li et al. 共see Ref. 12兲. Dada of commercial carbon
difference comes from A. Based on the self-similar array
fibers are taken from Torayca Carbon Fibers America, Inc. for their products
of T300, T800H, M40, and M60J. Values of 5 nm and 1 ␮m are taken for
model, A = 8cm / 3冑3D21, 8cmcn / 3冑3D22, and 8cmcncl / 3冑3D23
the scale increments before and after the break of the horizontal axis for a
for 共cm, 1兲, 共cm , cn兲, and 共cm , cn , cl兲, respectively,
better legibility.
where D1 = 共2m + 1兲D + 2mG, D2 = 共2n + 1兲D1 + 2nG, and
D3 = 共2l + 1兲D2 + 2lG.
those experimental and calculated data of SWNT arrays with
Figure 2共a兲 shows how the axial Young’s modulus of
a diameter of several micrometers with those of commercial
SWNT arrays from Eq. 共1兲 varies with packing configuracarbon fibers 共CFs兲, as shown in Fig. 2共b兲. Data of the comtions. We can see: 共1兲 the calculated Y axis for an isolated
mercial carbon fibers are taken from Torayca Carbon Fibers
共12, 12兲 is highly consistent with the experimental result
America, Inc. for their products of T300, T800H, M40, and
cited in Ref. 11 共but this result was given by Chopra et al.2兲;
M60J. The comparison in Fig. 2共b兲 suggests that, even
共2兲 Y 共1兲 ⬎ Y 共cm,1兲 ⬎ Y 共cm,cn兲 ⬎ Y 共cm,cn,cl兲. Obviously, the Y axis deSWNT arrays at the micrometer scale have a perfect struccreases rapidly with the increase of the hierarchy number s
ture and packing density 共for s = 1兲, their axial Young’s
of SWNT arrays; hence, decreasing the hierearchy of SWNT
modulus is just similar to that of high modulus CFs. Morearrays is an effective way to improve their Young’s modulus;
over, in our estimation, some complex structural factors are
共3兲 for SWNT arrays with the same s, the Y axis is not sensinot considered in the model yet, such as the nonuniformity of
tive to their sizes. Using 共37,91, cl兲 for instance, when its
SWNT diameters, packing density, and length distribution,
diameter changes from 1 ␮m to 1 m, the Y axis only decreases
the distance between SWNT bundles or ropes, and these facfrom 318.7 to 316.4 GPa. 共4兲 The best Y axis of SWNT arrays
tors normally can worsen the axial mechanical performance
can be expected in SWNT bundles 共cm, 1兲, about 550–600
of SWNT arrays. Therefore, SWNT arrays with diameters of
GPa at the scale of micrometers to meters.
several micrometers do not present much advantage with reFigure 2共b兲 presents the experimental results of axial
spect to commercial high performance carbon fibers if only
Young’s modulus of SWNT arrays at the nanometer11 and
the Young’s modulus is considered. However, with the incremicrometer dimensions.12 According to the description in the
ment of diameters and hierarchy number s, the density of
earlier references, we select two possible hierarchy strucSWNT arrays decreases, suggesting the specific Young’s
tures, 共cm, 1兲 and 共37, cn , cl兲, to describe the samples emmodulus of SWNT arrays has a different changing tendency
ployed in Refs. 11 and 12, respectively. The parameters cm,
from the Y axis. Using the continuum model of SWNTs, the
cn, and cl are determined by the formula of Di earlier, and
specific density of SWNT arrays can be described by
experimental values are taken for Di; for 共37, cn , cl兲, an assumption of cn = cl is made since the Y axis is not sensitive to
their values. By these simplifications, a rough estimation is
␳ = ␲A ⫻ D ⫻ ␪ ⫻ M C ,
共2兲
carried out, as shown in Fig. 2共b兲. Considering the model and
where ␪ labels the surface density of carbon atoms, ␪
method used here for so complex structures, the calculated
= 4 / 共冑3a2兲, a = 2.49 Å is the lattice constant of carbon
results are well consistent with the experimental data, which
suggest that the earlier analysis based on the self-similar arnanotubes.21 M C is the atomic mass of carbon. Based on Eq.
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ray model of SWNTs is reliable. Hence, we can compare
共1兲, the specific Young’s modulu of SWNT arrays is
23:12:55
193101-3
Appl. Phys. Lett. 87, 193101 共2005兲
Sun et al.
⳵ 2E
⫻L
⳵ ␧2
Y axis
s
Y axis =
=
.
␳
␲D ⫻ ␪ ⫻ M C
共3兲
s
Y axis
Obviously, the
is independent of the packing configuras
of individual
tion of SWNTs, which suggests that the Y axis
SWNTs can be theoretically realized at a macrodimension.
s
= 469 GPa,
For 共12,12兲 SWNT arrays, for instance, the Y axis
which is much higher than those of commercial CFs selected
above, 131, 162, 305, 217 GPa for T300, T800H, M60J, and
M40, respectively. Hence, SWNT arrays present significant
s
,
importance for those applications in which the Y axis
rather than the Y axis, is strictly required, such as the space
elevator proposed by Edwards.22
In summary, the theoretical Young’s modulus of SWNT
arrays with diameter from 1 nm to 1 m have been calculated
based on the self-similar array model of SWNTs by an analytical approach. The agreement between experimental data
and calculated results suggests that the present analysis based
on the self-similar array model is reliable. The results show
that the axial Young’s modulus of SWNT arrays, Y axis,
decreases dramatically with the increment of their hierarchy
number s and is not sensitive to the specific size and constitution when s is the same, which suggests that, to improve
the axial mechanical properties of SWNT arrays, it is necessary to reduce the hierarchy of SWNT arrays. Our calculation also indicates that the high mechanical properties of
individual SWNTs are possibly realized in a macroscale dimension if SWNT bundles with low hierarchy numbers can
be closely packed, but at the micrometer dimension, the Y axis
of SWNT arrays with diameters of several micrometers is
close to that of commercial high performance carbon fibers.
s
is independent
However, our calculations show that the Y axis
of the packing configuration of SWNTs and, to fabricate
s
much higher than those of high
SWNT array with the Y axis
performance CFs, is theoretically possible.
The authors would like to thank Dr. A. Kis and Professor
L. Forrófor for their kind help. This work was supported by
NSFC 共50328204, 90406012兲 and the State Major Basic
Research Project of MOST 共G2000026403兲. The support
from CAS Centre for Interfacial Materials, China and the
ARC Centre for Functional Nanomaterials, Australia to this
work are also acknowledged.
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