Yellampalli S. (ed.) Carbon Nanotubes - Synthesis, Characterization, Applications
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Study of Carbon Nanotubes Based on Higher Order Cauchy-Born Rule
237
0.0
500.0
1000.0
1500.0
2000.0
2500.0
3000.0
3500.0
4000.0
4500.0
0.00.20.40.60.81.0
Tube Radius(nm)
Bending Stiffness(eVnm
)
zigzag
armchair
chiral
Eq.(47)
Fig. 11. Variation of bending stiffness with tube radius for different chiral SWCNTs
6. Conclusion
In this charpter, a higher order Cauchy-Born rule has been constructed for studying
mechanical properties of graphene sheet and carbon nanotubes. In the present model, by
including the second order deformation gradient tensor in the kinematic description, we can
alleviate the limitation of the standard Cauchy-Born rule for the modeling of nanoscale
crystalline films with less computational efforts. Based on the established relationship
between the atomic potential and the macroscopic continuum strain energy, analytical
expressions for the tangent modulus tensors are derived. From these expressions, the hyper-
elastic constitutive law for this generalized continuum can be obtained.
With the use of this constitutive model and the Tersoff-Brenner atomic potential for carbon,
the size and chirality dependent mechanical properties (including strain energy, Young’s
modulus, Poisson’s ratio, shear modulus, bending stiffness) of graphene sheet and carbon
nanotube are predicted systematically. The present investigation shows that except for
Poisson’s ratio other mechanical properties (such as Young’s modulus, shear modulus,
bending stiffness and so on) for graphene sheet and SWCNTs are size-dependent and their
chirality-dependence is not significant. With increasing of tube radius, Young’s modulus
and shear modulus of SWCNTs increase and converge to the corresponding limit values of
graphene sheet. As for Poisson’s ratio, it can be found that it is very sensitive to the radius
and the chirality of SWCNTs when the tube diameter is less than 1.3 nm. The present results
agree well with those obtained by other experimental, atomic modeling and continuum
concept based studies.
Besides, the present work also discusses some basic problems on the study of the bending
stiffness of CNTs. It is pointed out that the bending stiffness of a flat graphite sheet and that
of CNTs are two different concepts. The former is an intrinsic material property while the
later is a structural one. Since the smeared-out model of CNTs is a generalized continuum
with microstructure, the effective bending stiffness of it should be regarded as an
independent structural rigidity parameter which can not be determined simply by
employing the classic formula in beam theory. It is hoped that the above findings may be
helpful to clarify some obscure issues on the study of the mechanical properties of CNTs
both theoretically and experimentally.
Carbon Nanotubes - Synthesis, Characterization, Applications
238
It should be pointed out that the present method is not limited to a specific interatomic
potential and the study of SWCNTs. It can also be applied to calculate the mechanical
response of MWCNTs. The proposed model can be further applied to other nano-film
materials. The key point is to view them as generalized continuum with microstructures.
7. Acknowledgment
This work was supported by the National Natural Science Foundation of China (10802076),
the Nature Science Foundation of Zhejiang province (Y6090543), China Postdoctoral Science
Foundation (20100470072) and the Scientific Research Foundation of Zhejiang Ocean
University.
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12
In-Situ Structural Characterization
of SWCNTs in Dispersion
Zhiwei Xiao, Sida Luo and Tao Liu
Florida State University
United States
1. Introduction
Owing to its excellent mechanical robustness – high strength, stiffness, toughness (Saito et
al., 1998; Baughman et al., 2002), excellent electrical and thermal conductivity and
piezoresistivity (Cao et al., 2003; Grow et al., 2005; Skakalova et al., 2006), and versatile
spectroscopic and optoelectronic properties (Burghard, 2005; Dresselhaus et al., 2005;
Dresselhaus et al., 2007; Avouris et al., 2008), single-walled carbon nanotubes (SWCNTs)
offer a great promise as the building blocks for the development of multi-functional
nanocomposites (Hussain et al., 2006; Moniruzzaman & Winey, 2006; Green et al., 2009;
Chou et al., 2010; Sahoo et al., 2010). To fabricate the SWCNT based multi-functional
nanocomposites, one of the most used approaches is through solution or melt processing of
SWCNT dispersions in various polymer matrices (Hilding et al., 2003; Moniruzzaman &
Winey, 2006; Schaefer & Justice, 2007; Grady, 2009). In addition, the SWCNT dispersions in
different liquid media of small molecules, e.g., water or organic solvents, were also proved
to be useful for cost-effective processing of SWCNT thin film based novel applications (Cao
& Rogers, 2009), e.g., CNT film strain sensors (Li et al., 2004), high mobility CNT thin film
transistors (Snow et al., 2005), SWNT thin film field effect electron sources (Bonard et al.,
1998) and various CNT film-based transparent electronics (Gruner, 2006). To fully explore
the use of SWCNT dispersions for various technologically important applications, it is
critical to have a good understanding of the processing-structure relationship of SWCNT
dispersions processed by different techniques and methods (Luo et al., 2010).
Regardless of the dispersion processing methods, it has been recognized that, to disperse
SWCNTs at a molecular level in either small molecule solvent or polymer solution or melt is
extremely difficult (Moniruzzaman & Winey, 2006; Schaefer & Justice, 2007; Mac Kernan &
Blau, 2008). The fundamental reasons for such difficulties are threefold. First, the one
dimensional tubular structure of SWCNTs imparts this novel species of very high rigidity.
When mixed with the solvent of small molecules or flexible chain polymers, the highly rigid
nature of SWCNTs as well as its long aspect ratio character (typically >100) results in a
competition between the orientational entropy and the packing entropy that drives the
mixture towards phase separation (Onsager, 1949; Flory, 1978; Fakhri et al., 2009). The
persistence length is a physical measure of the rigidity of a chain-like or worm-like molecule
(Tracy & Pecora, 1992; Teraoka, 2002). Depending upon the tube diameter, the theoretically
estimated persistence length for an individual SWCNT is as high as of 30 – 1000 µm
(Yakobson & Couchman, 2006). This result has been confirmed by the experimental studies
Carbon Nanotubes - Synthesis, Characterization, Applications
242
of SWCNT dynamics in aqueous suspension (Duggal & Pasquali, 2006; Fakhri et al., 2009).
For comparison, the persistence length of a few widely studied stiff particles/molecules is:
300 nm for the tobacco mosaic virus (TMV), 80 nm for poly (-benzyl L-glutamate) (PBLG),
and 50 nm for double-stranded DNA (Vroege & Lekkerkerker, 1992). Second, the intertube
van der Waals interaction of SWCNTs is very strong. The cohesive energy for a pair of
parallel arranged SWCNTs at equilibrium is greater than 2.0 eV/nm (Girifalco et al., 2000).
For this reason, one often finds that the SWCNTs organize into a rope or bundle structure in
the as-produced materials (Thess et al., 1996; Salvetat et al., 1999). To disperse SWCNTs in a
given medium at the molecular level or to exfoliate the SWCNT bundles into individual
tubes, the strong intertube cohesive energy has to be overcome. This proved to be a difficult
task (O'Connell et al., 2002; Islam et al., 2003; Moore et al., 2003; Zheng et al., 2003; Cotiuga
et al., 2006; Giordani et al., 2006; Bergin et al., 2007; Liu et al., 2007; Liu et al., 2009). Lastly,
the difficulty to disperse SWCNTs is also attributed to the topological entanglement or
enmeshment of long aspect ratio SWCNTs, which could result kinetically quenched fractal
structures or aggregates.
Associated with the threefold difficulty to disperse SWCNTs is their hierarchical structures
that one may encounter in the dispersion. As schematically shown in Fig. 1, these structures
include: 1) the individual tubes with different molecular structure as specified by the rolling
or chiral vector (n, m) (Saito et al., 1998); 2) the SWCNT bundles that is composed of
multiple individual tubes approximately organized into a 2D hexagonal lattice with their
long axis parallel to each other (Thess et al., 1996; Salvetat et al., 1999); 3) the SWCNT
aggregates formed by the topological entanglement or enmeshment of individual tubes
and/or SWCNT bundles; and 4) the SWCNT networks that span the entire dispersion
sample, which may occur as a result of inter-tube, inter-bundle and inter-aggregate
connection when the SWCNT loading in the dispersion is high. In a given SWCNT
dispersion, the diameter and length of the individual tubes and the SWCNT bundles, the
radius of gyration of the SWCNT aggregates, as well as the relative amount of the
hierarchical structures of the SWCNTs could be subject to random variations. This brings
out the length-scale related polydispersity issues. The length scales of the hierarchical
SWCNT structures vary from ~ 10
0
nm for the diameter of individual tube, ~10
1
nm for the
diameter of SWCNT bundles, ~ 10
2
– 10
3
nm for the length of SWCNT tubes and bundles,
~10
4
– 10
5
nm for the size of SWCNT aggregates, and up to the macroscopic sample size for
the SWCNT networks. Given such a broad range of length scales involved in the hierarchical
structures of SWCNTs possibly encountered in the dispersion, one can expect that, to
quantitatively characterize the structures of SWCNT dispersion and establish the related
dispersion processing-structure relationship, a multi-scale characterization approach should
be utilized.
The past decades witnessed significant progress being made toward qualitative and
quantitative characterization of the SWCNT dispersions by various experimental
techniques. Among the different techniques, the microscopy based methods, e.g., optical
microscopy (OM), electron microscopy (SEM and TEM) and atomic force microscopy
(AFM), have been routinely used for characterizing the SWCNT structures to provide
valuable information regarding the diameter, length, and the overall morphology for a
given SWCNT sample. However, when applied to characterizing the SWCNT dispersions,
the microscopy techniques typically require a sample preparation protocol that converts the
dispersion sample from a liquid state to solid state. This may cause the structural changes of
the SWCNTs during the sample preparation and thus fail to faithfully provide the desired
In-Situ Structural Characterization of SWCNTs in Dispersion
243
Fig. 1. Schematic SWCNT structures at different length scales
in-situ structural information of the SWCNTs in the dispersion. For this reason, the
microscopy technique will not be considered as suitable methods for in-situ structural
characterization of SWCNT dispersions. In addition to the microscopy techniques, a few
other conventional or non-conventional techniques emerge to show great promise for the in-
situ structural characterization of SWCNT dispersions. These emerging techniques include:
1. Viscosity and rheological measurements;
2. Scattering based techniques, e.g., elastic and quasi-elastic light scattering (SLS and
DLS), small-angle X-ray and neutron scattering (SAXS and SANS);
3. Sedimentation methods, e.g., analytical and preparative ultracentrifuge method; and
4. Spectroscopic techniques, e.g., simultaneous Raman scattering and photoluminescence
spectroscopy.
To facilitate a multi-scale characterization approach for a better understanding of the in-situ
SWCNT structures in the dispersion, the above listed experimental techniques and methods
will be reviewed in this chapter. For each of the methods, the underlying physical principles
and their applications for the in-situ structural characterization of SWCNT dispersions are
discussed in the subsequent sections.
2. Viscosity and rheological measurements
Suspensions or dispersions, in which the microscopically visible solid particles or fillers are
dispersed in a continuous phase like water, organic solvent or polymer solutions, find
themselves a great technical importance in many different areas, e.g., biotechnology, cement
and concrete technology, ceramic processing, coating and pigment technology, etc. The
rheological behavior of a two-phase suspension system has received a great attention and
been studied for many years (Batchelor, 1974; Jeffrey & Acrivos, 1976; Russel, 1980; Metzner,
1985; Bicerano et al., 1999; Hornsby, 1999; Larson, 1999; Petrie, 1999). One area concerning
Molecular structure of
individual SWNCT tube
SWCNT bundles formed
due to the strong inter-tube
van der Waals interaction
SWCNT raw materials –
entangled network of
SWCNT bundles
SWCNT Dispersions
Individual
tubes
Bundles
A
gg
re
g
ates
Carbon Nanotubes - Synthesis, Characterization, Applications
244
the rheological behavior of a suspension system is to understand the shear viscosity of a
suspension. To this effect, hundreds of empirical, semi-empirical, and theoretical
relationships have been developed for relating the dispersion viscosity,
, with respect to
the volume fraction, the shape of the fillers, and the shear rate under which the viscosity is
measured (Bicerano et al., 1999; Hornsby, 1999; Shenoy, 1999). At very low filler volume
fraction and zero shear rate, the viscosity
of a suspension or dispersion is given by:
0
1
(1)
where
0
is the viscosity of the liquid medium,
is the volume fraction of the fillers, and [
]
is termed as the intrinsic viscosity and it is a dimensionless, scale-invariant functional of the
shape of the filler particle. By applying the numerical path integration technique, Douglas et
al (Mansfield & Douglas, 2008) presented an accurate expression for the intrinsic viscosity of
cylinders applicable to a broad range of aspect ratios (2.72 < A < ∞), which is:
1
2 1.86 6.28 12.67
25/12 3.83 3.83 12.07
8 4 1 1.178 1.233 1.925 0.625
ln
45
1 1.094 0.757 1.344 1.978
AA tt t t
etttt
(2a)
L
A
d
(2b)
1
ln
t
A
(2c)
where
A is the aspect ratio of the cylinder and equals to the ratio of the cylinder length L to
its diameter
d. By taking advantage of the rigid rod nature of SWCNTs (Yakobson &
Couchman, 2006; Duggal & Pasquali, 2006; Fakhri et al., 2009) and on the basis of Eq. (1) and
(2), the aspect ratio of SWCNTs in the dispersion might be determined by the viscosity
measurement technique, e.g., the steady-state simple shear experiments.
By following this line of thought, a few studies were carried out for determining the aspect
ratio of SWCNT particles in superacid (Davis et al., 2004) and aqueous dispersions (Parra-
Vasquez et al., 2007), where the volume fraction of SWCNTs is a few factor of 10
-5
. The
experimentally determined intrinsic viscosity of SWCNTs in superacid (8300
830) and in
aqueous dispersion (7350
750) respectively lead to the estimated aspect ratio of SWCNTs
to be 470
30 and 505 35. The similar and very large aspect ratio of the SWCNTs in these
two distinctly different dispersion systems indicate the dominant structures of SWCNTs in
the dispersion are the individual tubes and/or the SWCNT bundles. It is noted that, the
intrinsic viscosity relationship used in these studies is based on a formula given by
Batchelor (Batchelor, 1974), which is:
2
2
8 1 0.64
1.659
45 1 1.5
A
(3a)
1
ln2A
(3b)
In-Situ Structural Characterization of SWCNTs in Dispersion
245
When compared to the accurate expression given by Eq. (2) (Mansfield & Douglas, 2008), the
relationship given by Eq. (3) overestimates the intrinsic viscosity by 12% or more for the
rods with aspect ratio below 100.
In order to appropriately use Eq. (1) and (2) for estimating the aspect ratio of SWCNTs in the
dispersion, the volume fraction of the filler particles has to be kept low. In addition to this
requirement, the viscosity measurements also need to be done at a relatively low shear rate.
Otherwise, the slender particles with large aspect ratio, e.g., SWCNTs, can be aligned along
the flow direction to cause the shear-thinning effect and thus result in a shear-rate
dependent intrinsic viscosity, which is not taken into account by Eq. (1) and (2). The Peclet
number (Bicerano et al., 1999; Larson, 1999) (
Pe), defined by the ratio of the characteristic
experimental shear rate to the rotational diffusion coefficient of the filler particle,
r
Pe
D
(4a)
3
0
311
ln 2ln 2
6
B
r
kT
DA
L
(4b)
0
ln 2ln 2 1
3
B
t
kT
DA
L
(4c)
can be used as a criterion to select appropriate experimental conditions to avoid the
complication caused by the shear-thinning effect. When
Pe is smaller than 1, the rotational
Brownian motion of the slender particle is able to overcome the shear-field induced
alignment and randomize the particle orientation to minimize the shear-thinning effect. In
Eq. (4),
k
B
is the Boltzmann constant and T is the temperature. The rotational and
translational diffusion coefficient,
D
r
and D
t
, is taken from the work by Bonet Avalos
(Avalos et al., 1993) and Yamakawa (Yamakawa, 1975). D
t
is given here for completeness
and convenience and will be used for a later discussion on the dynamic light scattering
technique for characterizing the SWCNT structures.
The steady-state simple shear experiments for the SWCNT dispersion at relatively low
particle volume fraction allow one to determine the intrinsic viscosity of SWCNTs and thus
infer the particle aspect ratio. In addition to this, the unsteady-state simple shear
experiments, e.g., small-amplitude oscillatory flow, also enable one to study the viscoelastic
behavior of SWCNT dispersions at relatively high particle volume fraction. Hough et al.
(Hough et al., 2004) investigated the dynamic mechanical properties of SWCNT aqueous
dispersions with particle volume fraction greater than 10
-3
. The observed oscillation
frequency independent storage modulus G’ and loss modulus G’’ allow the author to infer
the presence of SWCNT network structures in the dispersion. The network structure is
formed by the physical association of the SWCNT rods, and the bonding energy responsible
for the association is as high as ~ 40 k
B
T. The similar viscoelastic behavior studies were
performed for the SWCNT dispersion in epoxy (Ma et al., 2009) and in unsaturated
polyester (Kayatin & Davis, 2009). These polymeric resin based dispersion system presents a
strong elastic response at relatively high volume fraction of SWCNTs, which also signifies
the formation of SWCNT networks.
In brief, the viscosity and rheological measurements are capable of providing the in-situ
structural information of SWCNTs in different dispersing media. The SWCNT structures
Carbon Nanotubes - Synthesis, Characterization, Applications
246
being probed include the aspect ratio of the individual tubes or SWCNT bundles as well as
the network formation of SWCNTs.
3. Scattering techniques
For a long time, the elastic scattering techniques, e.g., static light scattering (SLS), small
angle X-ray (SAXS) and neutron scattering (SANS) have been widely used for obtaining the
structural information of materials of many kinds (Guinier & Fournet, 1955; Glatter &
Kratky, 1982; Feigin & Svergun, 1987; Chu, 1991; Higgins & Benoit, 1994). In a typical elastic
scattering experiment, a collimated beam of probe particles, e.g., photons in SLS and SAXS,
neutrons in SANS, interacts with a sample system that is composed of many scattering units
or scatterers. The interaction between the probe beam and the scatterer at position
i
r
produces a spherical scattered wave propagating outwardly from
i
r
toward the detector.
The scattering beam intensity recorded by the detector,
I
D
, is a result of the superposition of
the multiple spherical scattered waves originated from the many scatterers that are bathed
in an illuminated volume
V defined by the incident probe beam and the detection optics. I
D
is related to the differential scattering cross section
d
/d
and given by (Graessley, 2004):
2
0
11
exp
nn
DD
jk j k
jk
dIr
bb iq r r
dVI
(5a)
0
2
D
qss
(5b)
4
sin
2
qq
(5c)
Normalized by the incident flux of the probe particles, which is the number of the particles
impinging on a unit area of the sample per unit time, the differential scattering cross-section
d
/d
is defined as the number of scattered particles generated per unit time per unit
volume of the sample within a unit solid angle subtended by the detector. In Eq. (5), b
j
is the
scattering length of the scatterer j, a quantity to measure the scattering power of a given
species that depends on the details of the probe/scatterer interaction; I
0
is the incident beam
intensity; r
D
is the distance from the scatterer to the detector; q
is the scattering vector and
defined by the difference between the propagation vector of the incident beam (2
0
s
/) and
that of the scattered beam (2
D
s
/); the scattering angle formed by the incident beam and
the scattered beam is
; and
is the wavelength of the incident beam. As noted in Eq. (5), the
scattered beam intensity contains the relative spatial position (
j
r
-
k
r
) of the scatterers, which
forms the basis of using the elastic scattering techniques for characterizing the structures of
suspension or dispersions. For a dispersion system of monodispersed particles with random
orientation, the generalized differential scattering cross section given by Eq. (5a) can be
simplified to (Ballauff et al., 1996; Pedersen, 1997; Peterlik & Fratzl, 2006):
22
()()
d
nP
q
S
q
d
(6)