Rheology of polymer carbon nanotubes composites
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Invited Review Article
Rheology of Polymer Carbon Nanotubes Composites
Tirtha Chatterjee1, Ramanan Krishnamoorti
2
1Core R&D, The Dow Chemical Company, Midland, MI 48667.
2Department of Chemical and Biomolecular Engineering, University of Houston, Houston, TX
77204–4004.
Abstract
In this review paper the rheology of polymer nanocomposites with dispersed carbon nanotubes is
presented. The major factors controlling the rheology of these nanocomposites are the overall
concentration of the nanotubes and their state of dispersion. Percolation of anisotropic nanotubes and the
transition from isotropic to nematic structures bound the range of concentrations over which the
rheological properties of these nanocomposites is dominated by the meso-scale structure and dispersion
and are of significance to the processing of nanotube based polymer nanocomposites. The percolation
threshold and the concentration for the isotropic to nematic transition are strong functions of the inverse
of the effective aspect ratio of the dispersed nanotubes and therefore restrict the range of concentrations
over which such nanocomposites can be deployed. In this review we briefly describe the rheology in the
dilute regime, where especially for the case of polymer nanocomposites the rheology is dominated by that
of the polymer. Subsequently, the percolation phenomenon and rheological significances are presented.
Finally, both linear and non-linear rheologies of semi-dilute dispersions with random orientation of
nanotubes are discussed in detail. Where possible, the rheological responses are contextualized through
the underlying structure of the nanocomposites and interplay of different forces.
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INTRODUCTION
Polymer nanocomposites are at the forefront of advanced materials applications, where the
incorporation of dispersed nanoparticles into a polymer matrix provides materials with tailored and
controlled properties without compromising the processing ease of the parent polymer.1 Polymer
nanocomposites are distinguished from traditional composites by the significantly increased interfacial
surface area between dispersed nanoparticles and polymer chains in the case of nanocomposites. In
addition, the relatively low concentration of anisotropic nanoparticles at which percolation occurs and the
orientational and positional correlation between particles even at low volume fractions of nanoparticles
have profound influences on the rheological properties and processing of these materials.2 Therefore,
rheology or flow-properties of polymer nanocomposites are extremely rich and diverse and lie between
those of the pure polymer (melt) rheology and the rheology of colloidal suspensions. A monograph on
the broad topic of the rheological properties of polymer nanocomposite can be found elsewhere.3
Broadly, the phase behavior of macroscopically unoriented carbon nanotubes (CNTs) dispersed in
a polymer matrix can be classified into three regimes on the basis of the concentration of nanotubes and
the orientational or structural correlations between CNTs. At low concentration or in the dilute regime,
dispersed CNTs exist and behave as individual tubes or as small dispersed bundles. In this regime, some
short-range inter-tube interactions are present but long-range interactions are largely absent. Therefore the
structural reinforcement in such dispersions can be conceived to occur due to the influence of individual
nanotubes or small bundles of tubes and their coupling with the properties of the polymer matrix.
Transition from dilute to semi-dilute regime coincides with a percolation event (i.e., the formation of a
matrix spanning backbone path/network made of nanotubes). With improved dispersion the percolation
transition occurs at lower concentrations and the variation of the percolation threshold with aspect ratio
can be computed numerically.4 Beyond the percolation threshold, and especially in close proximity to the
threshold value, dramatic changes in rheological properties are observed and are conjectured to arise from
the inter-tube interaction. In the semi-dilute regime, the state of dispersion of the nanotubes or their small
bundles and their interaction with each other control the overall rheological behavior of the
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nanocomposite. At concentrations significantly larger than the percolation concentration, a concentrated
regime is observed and the rheological properties tend to reach asymptotic values. Additionally for well
dispersed CNTs, at high concentration, excluded-volume interactions lead to an isotropic to nematic
transition.5
The two dominant factors that govern the rheology of polymer CNT nanocomposites are the
concentration of nanotubes and their state of dispersion. For purposes of structural reinforcement, the
most significant advantage of one-dimensional CNTs is their high aspect ratio (α = L/d, where L and d are
nanotube length and diameter, respectively). Often the length of nanotubes is on the order of several
micrometers and when well dispersed should present an effective aspect ratio of ~ 103
,which is
significantly larger than three dimensional nanospheres (α=1) or two dimensional clay nanoparticles (α ~
102).
6 Additionally, the percolation threshold (ϕc, typically expressed as a volume fraction or volume
percentage) decreases with an increase in aspect ratio, which suggests the formation of a matrix spanning
path at a low or modest CNTs loading. However, the effective aspect ratio is a direct outcome of the
dispersion state of CNTs in polymer matrices that is largely controlled by the enthalpy of mixing and is
seldom driven by entropic forces alone. For CNTs, van der Waals attraction is dominant and is directly
proportional to the diameter of the particles and inversely related to the inter-particle distance.7 Note that
both the dispersion state and concentration of CNTs control the inter-particle distances. Therefore, even at
a modest concentration, the inter-tube attraction yields formation of aggregates or bundles which
diminishes the dimensional advantages of the nanotubes and poses a significant challenge in realizing the
theoretically predicted property enhancement. The energy penalty for CNT dispersion is extremely high.
Moreover, due to the high anisotropy of dispersed CNTs an isotropic to nematic transition occurs at a
CNT concentration of ~ 1/α.8 It is worthwhile to note that CNTs are extremely difficult to disperse in
aqueous as well as organic solvents.9 In conclusion since both the percolation and the isotropic to nematic
transition occur at lower concentration for particles with higher effective aspect ratio, there is only a
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limited concentration window where the advantages of isotropic 1-D high aspect ratio CNT dispersion
and their effect on rheological/structural properties are realized.
Considering the significance of the concentration of nanotubes and their state of dispersion in the
polymer matrix on the rheology of such nanocomposites, we organize this review article based on these
two factors. We will discuss the rheology in the dilute state followed by a brief discussion on different
approaches for successful dispersion and the percolation event. To emphasize the relationship between
structure and properties, the hierarchical structure of the nanocomposites close to and above percolation
will be presented. Based on the underlying structure, the linear and non-linear rheology in semi-dilute
concentration regime will be discussed. Finally, the effect of flow to induce nanotube directionality in
polymer matrices and their rheological consequences will be discussed.
DILUTE REGIME: BELOW THE PERCOLATION THRESHOLD
An understanding of the phase behavior of rod-like molecules was initially developed by
Onsager,10
modified by Flory11
and subsequently by Doi and Edwards.12
Briefly, in this formalism, the
dilute regime is loosely defined as where ν < L-3
, where ν is the number density of rods. In terms of a
volume fraction (ϕ/100) of nanotubes, the dilute regime is defined as: νL3 ≈ (4/π)(L/d)
2ϕ/100 ≤ 1.0.
Physically, in this regime, dispersed CNTs can be treated as isolated particles. Interestingly, for isolated
axisymmetric particles (i.e., in dilute condition), in the absence of Brownian motion, the angle of the axis
of symmetry with the flow direction is time-periodic. Therefore, particles with finite aspect ratio rotate
slowly when its long axis is nearly parallel to the flow direction, and rapidly otherwise. This rotation is
called as a Jeffrey orbit which has practical implications in field-flow fractionation of CNTs by length or
pressure driven Poiseuille flow in a narrow channel (microfluidic devices).13
Unfortunately, there is almost no literature available focusing on the dilute regime rheology of
CNTs dispersed in polymer matrix and possibly because the viscoelastic response of the polymer would
far exceed that of the dispersed CNTs. So far, the most complete study is dilute dispersions of CNTs in
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superacids (102% H2SO4).14
For such dilute dispersions the reduced viscosity depends linearly on CNT
concentration and the relaxation time is independent of CNT concentration, both of which are in
accordance with the theory developed for dynamics of dilute Brownian Rods.14
With increasing
concentration of CNTs at constant solvent quality, a transition occurs from an isotropic phase (random
tube orientation) to a biphasic system where the isotropic phase is in equilibrium with a liquid-crystalline
phase.14-15
At even higher concentrations a single liquid-crystalline phase with randomly oriented domains
is observed. It has been proposed that long range attraction pulls a tube out from the isotropic state
whereas; short range (electrostatic) repulsion prevents their collapse into an aggregate. More importantly,
for solvent mediated interactions, Pasquali and coworkers concluded that the nature of solvent strongly
influences the phase transitions.15
Controlling liquid-crystalline ordering has significant impact in fiber
spinning and nanotube mat preparation.15
At this point it is relevant to discuss briefly the nature of individual CNTs dispersed in a solvent
or a polymer matrix. The characteristic persistence length is defined as the ratio of bending stiffness (κ) to
thermal energy (LP = κ/kBT, where kBT is thermal energy). In a viscous medium, Brownian forces tend to
bend a 1-D object whereas restoring elastic forces try to minimize the curvature. Therefore, if the length
of a nanotube inside a network is shorter than LP, then it essentially appears as rigid rod; whereas, if the
length of the nanotube segment in the network is longer than LP, then it behaves as a semi-flexible rod.
This is important since in semi-flexible particle networks, the elastic energy is stored primarily in
mechanical bending and stretching of the particles. In contrast, for rigid rods the enthalpic effect on the
network elasticity is also important. There is considerable debate on the values of LP for isolated CNTs as
well as when in networks of nanotubes.16
Further, the nature of CNTs (e.g., multi-walled carbon
nanotubes, MWNTs, are found to exhibit smaller values of LP compared to SWNTs),17
variation in CNTs
quality with preparation method and source, state of dispersion and the selection of dispersing agent,
different purification methodologies, and, finally, error associated with measurement techniques to obtain
LP values, make reaching a single conclusion difficult. In this manuscript, individual (single walled) CNTs
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inside the network (i.e., semi-dilute suspensions) are considered as semi-flexible rods. On the other hand,
using linear rheological data collected on MWNTs dispersed in polydimethylsiloxane (PDMS) as a
function of concentration, Cassagnau and coworkers concluded that those MWNTs are Brownian (or rigid
rod) for concentrations ranging between the dilute and semi-dilute regime and are non-Brownian in the
semi-dilute to concentrated regime.18
We note that for asymptotic scaling behavior in the slender body
limit requires L >> d (or, α>>1.0). However, the expansion parameter in the slender body theory scales
as ln(L/d) and asymptotic behavior is only expected at aspect ratios of e20
≈ 109.19
Before entering this
asymptotic scaling regime, rod-like materials should exhibit properties of semi-flexible rods.
SEMI-DILUTE REGIME: DISPERSION AND PERCOLATION
The most significant challenge towards the dispersion of CNTs in a polymer matrix is the
overcoming of strong inter-tube attractive forces. Many of the strategies developed for the dispersion of
CNTs are largely borrowed from the colloidal dispersion literature.20
For the case of a polymer matrix
with no specific interactions with CNTs, the driving force for dispersion emerges principally from the
gain in translational entropy which is typically small for high molecular weight polymers and anisotropic
nanoparticles and therefore leads to de-mixing and poor dispersion.8 On the other hand, for cases where
the polymer and nanoparticle are mutually attractive resulting in polymer adsorption on to the particle, the
effective inter-particle attractions is strongly attractive at low polymer concentration and low to moderate
adsorption strengths and is short-range repulsive at high concentrations and high adsorption strength. For
the case where polymer chains are grafted on to the particle surface, the excluded volume repulsion
between tethered chains becomes dominant and stabilizes the particle dispersion.21
For the dispersion of
CNTs in a polymer, a variety of approaches such as surfactant adsorption,22
covalent modification of the
tube surface including the end-cap modification,23
block copolymer grafting/wrapping,24
and non-specific
adsorption of polymer have been successfully developed. Amongst these routes, surfactant aided
dispersion is particularly popular where surfactant molecules are randomly adsorbed22c
onto the CNT and
act as a bridge between the particle and polymer matrix.22a, 22c
Adsorption of small molecules or polymers
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and grafting of polymers on CNTs induce short-range repulsion so as to prevent their aggregation and
facilitate dispersion. Finally, while these chemical and physical approaches are successful at low and
intermediate CNT concentration regime, at sufficiently high concentration, successful dispersion is still a
challenging problem.
The state of dispersion can be quantitatively measured through the percolation threshold (ϕc, in
volume %). For instance, ϕc is measured via the concentration of CNTs where linear viscoelastic
properties of the nanocomposite change from liquid-like to solid-like. Structurally, at this concentration,
the earliest network spanning path (or backbone connectivity) is developed. For example, for a polymer
melt or a liquid, under application of small-amplitude linear oscillatory shear,25
the low frequency (ω)
storage and loss modulus (G’ and G”, respectively) exhibits terminal behavior with G’~ ω2 and G” ~ ω,
respectively. Equivalently, the magnitude of the complex viscosity (|η*|) is independent of frequency (i.e.,
Newtonian behavior). Incorporation of CNTs in the polymer gradually transforms the liquid-like terminal
behavior to solid-like non-terminal behavior (i.e. both G’ and G” are independent of ω as ω→0) (Figure
1a, in this specific example CNTs are dispersed in polycarbonate, an engineering polymer).26
Equivalently, the low frequency complex viscosity diverges with |η*|~ 1/ω.
The evolution of structural properties of the nanocomposites as a function of CNT concentration
follows a typical sigmoidal dependence.22b, 26
As an example, the |η*| as a function of nanotube content
for a series of polycarbonate – CNT nanocomposites is presented in Figure 1b where the different regimes
are apparent.26
At low CNT concentration (i.e., ϕ < ϕc), individual CNTs act as dispersed, isolated
objects and the mechanical properties (in this case |η*|) are expressed as a perturbation due to the
dispersed objects of the matrix properties (described by Guth’s modification of Einstein’s viscosity
relationship).27
Close to and beyond the geometrical percolation of the nanotubes, ϕ ≥ ϕc, the
development of a percolative network and the network superstructure dominates the mechanical response
and structural properties follow typical power-law scaling with concentration (ϕ - ϕc). At high
concentrations, on the other hand, the addition of nanotubes results in aggregation of the tubes and
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weakens the composition dependence of the reinforcement. Similarly, typical polymer nanocomposites
act as insulators so long as CNTs are dispersed as individual elements or as isolated small clusters.
Beyond the electrical percolation threshold (ϕc) a continuous conducting path spanning the matrix is
developed and the nanocomposites turn into conductors. Therefore, insulator to conductor transition is
also an alternative way to track the percolation event. An excellent review article on percolation modeling
of dispersion viscosity at different particle concentration regimes can be found elsewhere.4a
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Figure 1: (a) Storage modulus, G’ of nanotube filled polycarbonate at 260
oC. (Adapted with permission
from Ref. 26. Pϕtschke, P.; Fornes, T.D..; Paul, D.R, Polymer, 43 (11), 3247, 2002.Copyright 2002
Elsevier) (b) Complex viscosity versus nanotube content at different frequencies (Adapted with
permission from Ref. 26. Pϕtschke, P.; Fornes, T.D..; Paul, D.R, Polymer, 43 (11), 3247, 2002.Copyright
2002 Elsevier) (Inset) Schematic of (left) isolated nanotube dispersion below the percolation threshold,
(center) onset of percolation, the matrix spanning backbone connectivity is marked red, and (right) fully
grown network. (c) Schematic of CNTs/polymer nanocomposites in which the nanotube bundles have
isotropic orientation. (Top) At low nanotube concentrations, the rheological and electrical properties of
the composite are comparable to those of the host polymer. (Middle) The onset of solid-like viscoelastic
behavior occurs when the size of the polymer chain is somewhat large to the separation between the
nanotube bundles. (Bottom) The onset of electrical conductivity is observed when the nanotube bundles
are sufficiently close to one another to form a percolating conductive path along the nanotubes. (Adapted
with permission from Ref. 28b. Du, F. M.; Scogna, R. C.; Zhou, W.; Brand, S.; Fischer, J. E.; Winey, K.
I., Macromolecules, 37 (24), 9048, 2004.Copyright 2004 American Chemical Society).
The effective aspect ratio (α) and the percolation threshold (ϕc) are coupled parameters and
inversely related to each other. Considering two independent contributions to the linear viscoleastic
properties that account for the dilute regime (through the modified Einstein equation27
) and the network
superstructure beyond the percolation (through a power-law scaling), both ϕc and α can be
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simultaneously calculated from concentration dependent linear viscoelastic properties.22b
Alternatively,
ϕc alone can be calculated from the electrical properties (conductivity measurements).28
We note that
even while ϕc is a parameter that describes the onset of a percolation transition, it is somewhat dependent
on the system properties (such as viscoelastic or electrical) which has been used to calculate its value.
Typically structural/viscoelastic properties, unlike the electrical properties, depend on rheological
manifestation of percolation which does not require absolute connectivity between the CNTs (Figure
1c).28b
Therefore, for the three dimensional isotropic/random dispersion, the rigidity percolation precedes
the connectivity percolation or the electrical percolation threshold is somewhat greater than the
corresponding geometrical threshold. Using experiment and Monte Carlo simulations, Winey and
coworkers have shown that, at a fixed CNTs loading in oriented nanocomposites (PMMA matrix),
electrical conductivity parallel to the orientation direction is also a critical phenomenon and follows a
power law scaling with the degree of orientation.28a
Under highly oriented condition, the CNTs are not in
contact with each other and the conductive path is broken. However, beyond a critical orientation
condition (relatively poor) nanotubes touch each-other to form a conduction pathway. For increased
nanotube loading, the critical orientation condition for percolation parallel to alignment is decreased.28a
Therefore, percolation threshold (parallel to the alignment) appears under more anisotropic conditions as
CNTs concentration increases.
In addition to the concentration of CNTs and their state of dispersion, other factors which
influence the percolation threshold are nanotube polydispersity and local clustering. Besides some effort
with dynamic light scattering based particle size analysis and microscopy based visual characterization,
experimental investigation of the polydispersity of the dispersed CNTs has been mostly ignored.29
There
are no simple experimental methods available to measure size polydispersity directly and accurately in 3D
macroscopic samples. Therefore, in most of the cases the size polydispersity is ignored and the mean
values of ϕc or α are used as the dispersion state descriptor. Theoretical calculations have revealed that for
length or diameter distribution with an upper cutoff (i.e., L<Lmax or d<dmax), the percolation threshold is
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always lower for a polydisperse than for a monodisperse system.30
In contrast, for length or diameter
distribution with a lower cutoff (i.e., L>Lmax or d>dmax), the percolation threshold is always larger for a
polydisperse than for a monodisperse system.30
For a fixed volume fraction of randomly oriented
particles, formation of local clustering expectedly yields an increase in the percolation threshold.31
Finally, characterization of the dispersion state of CNT in a polymer matrix is challenged by the hierarchy
of structure present in the system and the absence of any single method/measurement/instrument that can
evaluate the dispersion state over a wide length scale.20
NANOCOMPOSITE STRUCTURE: NEAR PERCOLATION & SEMI-DILUTE
Even at a modest aspect ratio (α=L/d) of the CNTs, the overlap concentration is quite small and
for most practical applications suspensions of CNTs in a polymer matrix are in a semi-dilute state (L-3
<<
ν << d-1
L-2
). The semi-dilute regime is distinguished from the concentrated regime in that in the former a
new particle can be randomly inserted into the system with negligible probability of it being correlated
with other particles.8 In a random network of semi-flexible rods, the network elements retain their
identity through network junctions (equivalent to cross links) since the tangent vector on a given rod
remains correlated over distances much longer than the network mesh size (ζ).32
Therefore, elasticity
associated with the CNT network depends on both the mesh size and length of the semi-flexible rods.
Hence, we briefly discuss the network structure of the CNTs at concentrations close to and above their
percolation threshold.
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Figure 2: (a) A schematic of the hierarchical network structure showing different length scales. The
major characteristic length scales are the floc size (R) and mesh size (ζ). (b) A representative model fit
(solid red line) to the smeared scattering data. The solid vertical lines represent scattering vector q values
associated with different network length scales (= 2π/q). The intensity mismatch is due to instrument
smearing. (c) Optical microscopic image of a representative SWNTs network (ϕ/ϕc = 4.0, in PEO)
verifies the presence of micron-sized flocs as obtained from scattering data fitting. (Adapted with
permission from Ref. 35. Chatterjee, T.; Jackson, A.; Krishnamoorti, R., Journal of American Chemical
Society, 130 (22), 6934, 2008. Copyright 2008 American Chemical Society).
In aqueous solutions, both single and multi walled CNTs are found to develop mass fractal
networks (with fractal dimensions between 2 and 3) and hierarchical morphologies.33
Similar mass
fractal based network structures have been observed for semi-dilute dispersion of CNTs in polymer
matrices, where the matrix offers a kinetic i.e., viscoelastic, barrier to dispersion. For CNTs in a polymer
matrix the true semi-dilute regime is bounded by the percolation event (at the lower limit) and the
isotropic-nematic transition (at the upper limit).8, 34
A schematic of the network structure conjectured and
verified by experiments in semi-dilute dispersions of CNTs in a polymer is shown in Figure 2a. A matrix
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13
spanning network is formed which consists of a number of aggregated clusters or flocs which are in
dynamic equilibrium. The associated length scales are the average density correlation length between two
clusters and the average cluster/floc size (R), measured through either x-ray or neutron scattering
methods. Inside the floc, individual or bundle of a small number of nanotubes overlap each other and the
average distance between two adjacent nanotubes is the network mesh size (ζ).35
At even smaller length
scale individual tubes should display rigid rod scattering (I(q) ~ q-1
) which was not found for SWNT-PEO
system (Figure 2b). This deviation from q-1
scaling may stem from the presence of semi-flexible tubes or
small tube bundles even at the highest q value probed.33a, 33b, 35
Mass fractal structures exhibit signature power-law scattering at low q, i.e., ( )fd-
q~ I(q) , where
df is the fractal dimension. For the case of mass fractals, df lies between 2 and 3, and for the case of
nanotube dispersions can be qualitatively related to the dispersion state. For relatively poor dispersion,
CNTs remain in an agglomerated state and for such compact structure df approaches a value of 3. The
average floc size (R) is found to be the order of a few microns (typically <10µm) and largely independent
of the dispersion state, particle concentration, and dispersing medium. The presence of such micron sized
flocs is also confirmed by the optical microscope image (Figure 2c).35
The other prominent length scale of
the network is the mesh size (ζ) and is a strong function of CNT concentration. With increasing loading, ζ
decreases following a power-law (ζ ~ ϕ-β
) where β is ~ 0.5 and scales in a manner that is consistent with a
diffusion limited concentration dependence calculated for the semi-flexible rods in semi dilute
concentration regime.36
On the other hand, the number of flocs in the nanocomposite grows roughly
linearly with CNT concentration. This indicates that beyond the percolation threshold, with increasing
CNT loading in polymer matrix, the mesh size decreases, the floc size remains unchanged (diffusion
limited) and the network primarily grows through the formation of new flocs.35
Further, in the following
sections we will correlate the CNT network structure with the linear and non-linear viscoelasticity
exhibited by their polymer composites.
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14
SEMI-DILUTE REGIME: LINEAR VISCOELASTICITY
As described in the previous sections (Figure 1a and b) it has been observed that with increasing
CNTs in a polymer matrix, a liquid to solid-like transition occurs that is demarcated by the percolation
concentration. It is conjectured that at the percolation concentration, a matrix spanning network is formed
that grows in a self-similar manner (Figure 2a and b) with increased CNT concentration. Near and above
the percolation concentration (i.e. in the semi-dilute and concentrated regime), the rheological behavior is
dominated by the mesoscale superstructure. Moreover, the impact of this network superstructure of CNTs
on the elastic modulus and electrical conductivity are well documented for different polymer matrices.26,
37 It is noteworthy to mention that the absolute value of the modulus or conductivity depends on a number
of factors including the choice of polymer, source and nature of CNTs,37a
method of dispersion and the
choice of dispersing agents among others. Briefly we note that there has been effort to model the elastic
properties using short fiber theory. However, the mechanical reinforcement due to presence of CNTs
cannot be explained using traditional Halpin-Tsai model (multi-fiber interaction).38
The failure of the
traditional fiber theory stems from treating CNTs as fibers of finite length where, for a well dispersed
nanocomposites system, excellent stress transfer can take place from polymer to tube under deformation.
Close to and above the percolation threshold, formation of the CNT network has been widely
reported.16d, 18, 28b, 34, 39
Interestingly, in this concentration regime (i.e., semi-dilute to concentrated),
beyond the typical time-temperature superposition, the CNT based polymer nanocomposites demonstrate
time-temperature-composition superposition which is a characteristic of the weakly attracting particles
such as anisotropic carbon black fillers.40
A representative example of time-temperature-composition
superpositioning is presented in Figure 3a-b. One should note that the shifts along the time axis are
performed by superpositioning of tanδ (the phase angle between the applied strain in linear oscillatory
shear measurement25
and the evolved stress, defined as G”/G’), while the shifts along the modulus axis
are obtained by examination of the components of the complex viscosity. The superpositioning of
viscosity is only dependent on the modulus shift factors and that of tanδ only dependent on the time shift
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15
factors. Further, for a true linear viscoelastic materials, the viscoelastic response in time and frequency
domains are identical (i.e., the two point collocation method proposed by Ninomiya and Ferry is valid).41
Therefore, the scaling factors for concentration (ϕ) dependent linear stress relaxation (i.e., in the time
domain) and the modulus shift factors used to obtain mastercurves (Figure 3a-b) for the linear dynamic
oscillatory shear (i.e., in frequency domain) are comparable.34
For nanocomposites with nanoparticles
concentrations just in excess of percolation, superpositioning fails because of a combination of the
competing magnitudes of the viscoelasticity of the polymer and the fractal network and the changing
nature of the nanotube network superstructure at concentrations close to the percolation threshold. On the
other hand, at much higher concentrations, the dispersed anisotropic nanotubes exhibit a tendency to form
nematically ordered structures and do not allow for a comparison with the fractal networks. Simulation
studies reported that by varying the polymer/CNT adsorption strength, the isotropic/nematic transition can
be controlled where the stronger adsorbing polymer shifts the transition at higher particle concentration.42
Further, for high nanotube loading samples, alignment of the nanotubes in response to the handling
(compressive strains etc.) becomes a significant experimental issue and disorientation kinetics in such
nanocomposites have been shown to be extremely slow.43
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16
Figure 3: (a) Frequency dependence of the phase angle, tanδ from linear dynamic oscillatory shear
measurements. For ϕ ≥ 0.3, a composition invariant behavior is observed at low frequencies. (b) Time-
temperature-composition superposed mastercurves for different linear dynamic rheological properties.
(Reprinted with permission from Ref 34: Chatterjee, T.; Krishnamoorti, R., Physical Review E, 75(5),
050403, 2007. Copyright (2007) by the American Physical Society.)
The linear viscoelastic data for a network of dispersed CNTs in a polymer matrix, in general, are
representative of a solid-like material indicating that the superstructure of the nanotube dominates the
viscoelastic response, as has been observed in general soft-glassy materials.44
Further, the
superpositioning of the linear viscoelastic response with composition indicates that the superstructure
responsible for the dominant viscoelastic behavior in these nanocomposites is self-similar. Using an
elegant combination of experiment and simulation, Yodh and coworkers39a
have shown that the intertube
bonding is the dominant contributor to the network elasticity. For the same system (Sodium
dodecylbenzenesulfonate, NaDDBS assisted SWNTs dispersion in water), Hough et. al.16d
concluded that
it is the bonding of the rods and not the bending or stretching of the individual tube is the origin of the
elasticity. However, both studies were performed in aqueous medium and not in a polymeric matrix.
Remarkably even in aqueous dispersion, analogous to epoxy curing, SWNTs form a network at t>t*,
where t* is the critical gelation time. Beyond the t*, G’>G” at low frequencies and the data corrected for
their crossover values (i.e. G’/Gc or G”/Gc vs. ω/ωc) forms a mastercurve.39a
The crossover of G’ and G”
on ω axis represents the change in mode of relaxation with a characteristic time scale ~ 2π/ωc which is
10-1
100
101
10-3
10-1
101
tan
δ
aTω (rad/s)
(b)
0.1
0.3-1.00.2
0.15
10-1
100
101
10-3
10-1
101
tan
δ
aTω (rad/s)
(b)
0.1
0.3-1.00.2
0.15
104
106
10-1
100
101
10-3
10-1
101
103
0.30.5
0.71.0
bp
os
cG
* (d
yn
es
/cm
2)
tanδ
ηsω/κ (rad/s)
(a)
G"
G'
p
tanδ10
4
106
10-1
100
101
10-3
10-1
101
103
0.30.5
0.71.0
bp
os
cG
* (d
yn
es
/cm
2)
tanδ
ηsω/κ (rad/s)
(a)
G"
G'
p
tanδ
!
(a) (b)
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17
controlled by the intertube bonding (for t>t*, as t increases, the number of bonds increases and ωc
decreases).
For concentrations only slightly exceeding the percolation concentration, the elastic shear
modulus demonstrates the scaling relation: G0’ ~ (ϕ-ϕc)µ where µ is the percolation exponent. The
magnitude of the percolation exponent is sensitive to the mode of deformation. For example, a percolation
exponent of 2.1±0.2 has been reported for simulation of percolating bonds that resist stretching but are
free to rotate.45
In contrast, for bonds which resist both the stretching and rotating, µ=3.75±0.11.45
For
polymer bridged gels (or energetic gels), from simulation studies µ has been found to be 1.8.46
In the CNT
literature a wide range of µ values are reported (from 2.0-7.0) which signifies that accommodation of
strain in microscopic level depends on several factors including CNT dispersion state, aspect ratio,39b
choice of solvent (or polymer matrix) and dispersant22b
among others.
SEMI-DILUTE REGIME: NONLINEAR VISCOELASTICITY
Shear stress relaxation at large deformation:
For semi-dilute dispersions of CNTs in a polymer, the stress relaxation behavior in response to a
step shear strain as a function of applied strain amplitude can be categorized into three regimes:34
(i) For
small deformation, the response is linear and the time-dependent relaxation modulus, G(t), is independent
of strain amplitude (γ). The maximum value of the strain amplitude for the observation of a linear
response is referred to as a critical strain (γcritical) and is inversely related to CNT loading; (ii) For
deformations beyond the linear region (γ > γcritical), the stress relaxation behavior exhibits a strain-
softening with the shape of the relaxation spectrum being preserved and suggesting the applicability of
time-strain separability (i.e., G(t,γ) = ( ) ( )⊗G t h γ , where G(t) is the linear relaxation modulus and h(γ) is
the damping function); (c) For still higher deformations (γ >> γcritical), the shape of the relaxation spectrum
is no longer preserved and time-strain superposability is no longer valid. A representative strain amplitude
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18
dependent relaxation curve for SWNTs based PEO nanocomposites (ϕSWNTs = 0.7 %, ϕ/ϕc~ 7.0, PEO
MW=8K) is presented in Figure 4a.34
Beyond percolation i.e., ϕ>ϕc, the elastic shear modulus increases with increasing CNT loading:
G0’~ϕµ. Concurrently, the shear sensitivity of the stress relaxation behavior and the underlying shear
sensitivity of the structural elements increase with increasing CNT loading with the critical strain for the
onset of non-linear behavior scales as: γcritical ~ ϕ-δ
. These observations suggest that with increasing CNT
loading, the polymer nanocomposite gets stiffer and more fragile.34, 39b, 39c, 47
Such behavior is typical of
fractal networks such as those of colloidal gels, layered silicates, and flocculated silica spheres.40a, 48
For
dispersions of CNTs in PEO, 3 ≤ ϕ/ϕc ≤ 10, Chatterjee and Krishnamoorti34
reported, µ = 4.3 ± 0.6 and δ
= 2.3 ± 0.2. Similar scaling is anticipated from theoretical efforts49
that examined the three-dimensional
percolation of random percolating elements and from computer simulations45
that considered resistance of
individual bonds to bending and stretching deformations. For fractal networks well above the percolation
threshold (ϕ>>ϕc) where the interactions between the flocs dominate over those within a floc (i.e., the
strong link regime) the concentration scaling exponents for the elasticity and critical strain can be
expressed as: µ= (D+db)/(D-df) and δ= (1+db)/(D-df), where db and df are the backbone and fractal
dimensions of the network, respectively and D is the value of the Euclidian dimension.48b
The values of
the backbone dimension range from 1 to the fractal dimension of the network; however the value cannot
exceed 5/3. For many CNT dispersions, db is found to be ~ 1.0, indicating that CNTs are rod-like objects,
at least on a local length scale, in these nanocomposites.34, 39c, 47
The fractal dimension (df) of the nanotube
network deduced from the rheological measurements confirms mass fractal network and is in good
agreement with those obtained from independent neutron scattering measurements.34-35
These internally
consistent scaling of G0’ and γcritical with nanotube concentrations indicate that, (a) the weak and relatively
short-range interactions between nanotubes and multiple pathways between percolating paths dominate
the network properties, and (b) the tube-tube bonding, rather than bending or stretching, is the origin of
network elasticity observed in such nanocomposites.
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19
On the other hand, the shear sensitivity of the network structure captured through the damping
function, h(γ), when scaled by the concentration shift factor,50
f(ϕ), collapse onto a single master curve
(Figure 4b-c) [γlocal = γbulk*f(ϕ)]. The concentration shift factor is defined as: f(ϕ) = G’(ω, ϕ)/(G’(ω,
ϕ=0) = G”(ω, ϕ)/(G”(ω, ϕ=0) = 1+0.67(αϕ/100) + 1.62 (αϕ/100)2. Such a form for the concentration
dependent shift factor explicitly accounts for the change in the bulk “linear” viscosity due to the
dispersion of anisotropic objects. However, extension of this notion to describe the scaling of the non-
linear deformation in such nanocomposites suggests that the effective deformation of the suspension of
particles in this intermediate regime of strain amplitudes is “affine”. Mathematically it suggests that for
ϕ>>ϕc, concentration scaling of any linear memory function can be written as: M0(t, ϕ) = f(ϕ).M0(t,
ϕ=0). Therefore, the factorized non-linear memory function for self-similar particle network appears to
be consistent with the concentration scaling of the material function.51
In fact, such a local strain
controlled deformation is valid for fractal systems dominated by weak short-range interactions34, 43c, 51-52
whereas it fails where long-range interactions53
(due to ionic, H-bonding and polymer bridged gels)
dominate the development of structure. Finally, the time-temperature-composition superposition suggests
that above the percolation threshold, both the linear and non-linear viscoelasticity are dominated by the
network superstructure. In fact, the non-linear viscoelastic regime can be broadly divided into two
regimes. In regime 1, the network deformation is reversible and the superposition principle holds. In
contrast, in regime 2, the deformation is irreversible or permanent and the recovery process is extremely
sluggish.
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Figure 4 (a) Representative stress relaxation behavior for CNTs loading ϕ =0.7 % in PEO (Mw = 8000
Da) as a function of the applied bulk strain amplitude. For low-amplitude strain (γ ≤ 0.003 where γcritical =
0.003), linear behavior is observed followed by a time-strain superposable zone (blue curves, 0.003 ≤ γ ≤
0.03). At higher strain amplitude (γ > 0.03), time-strain superposability is violated (red curves). (b)
Damping function h(γ) required for the time-strain superposition for different nanocomposites is plotted
against the applied or bulk strain (γbulk). Deviation from h(γ) = 1.0 marks the onset of nonlinearity. With
increasing nanotube loading an earlier onset of non-linear response (i.e. lower γcritical) is observed. (c) The
local strain dependence of h(γ). The onset of the shear thinning is observed at γlocal ~ 0.1 and is similar to
other nanocomposite systems with short-range interactions. Therefore at and around 10% deformation the
nanocomposite network starts to flow. (Reprinted with permission from Ref 34: Chatterjee, T.;
Krishnamoorti, R., Physical Review E, 75(5), 050403, 2007. Copyright (2007) by the American Physical
Society.)
Steady Shear Properties:
Elastic networks such as those formed by CNT dispersions in a polymer matrix are expected to
exhibit a yield stress and best evidenced through their steady shear behavior (Figure 5a).54
For CNT
concentrations much in excess of the percolation threshold (ϕ>>ϕc), application of steady shear (from
rest) typically results in a stress overshoot (independent of that of the polymer matrix itself) which
102
104
106
10-2
100
102
104
0.005 0.02 0.05
0.01 0.03
G(t
) (d
yn
es
/cm
2)
T im e (s)
1.000.80
0.50
0.300.150.10
T = 70o
C
γ<=0.003
10-2
10-1
100
10-4
10-3
10-2
10-1
100
0.30.50.71.0
h(γ
)
γbulk
vol % SWNT (φ)
10-2
10-1
100
10-2
10-1
100
101
0.3
0.5
0.7
1.0h
(γ)
γlocal
vol % SWNT(φ)
h(γ) = 1/(1+2.27*γlocal
)
(a)
(b)
(c)
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equilibrates to a steady state value (σ∞) at long times. Note that overshoot responses of the stress,
resulting from the elastic network of the CNTs, are absent when ϕ < ϕc. In fact, this claim is further
supported by a calculation of a non-dimensional Peclet number that is greater than one, and indicates that
convective transport dominates the dynamic processes and that shear rate controls the structure. In
contrast, for dilute dispersions of CNTs (specifically SWNTs) in superacid (102% H2SO4), at low shear
rates (where the Peclet number is significantly smaller than one), Brownian motion dominates the
relaxation process and the resulting rotation and tumbling of anisotropic particles lead to the observation
of time-dependent oscillating shear and normal stresses in start-up measurements.14
These CNTs suspensions in solvent14
or in a polymer matrix3 do not obey the Cox-Merz rule
55
(i.e., the steady state shear and dynamic viscosities at comparable shear rates / frequencies do not
quantitatively agree) (Figure 5b). The linear complex dynamic viscosity, η*, represents the quiescent or
near-quiescent state structure. In contrast, the steady shear viscosity, η, represents the steady state
structure at a fixed shear rate during shear flow. While the failure of the Cox-Merz rule has been observed
for many systems, including in some cases monodisperse polymers, the breakdown of the Cox-Merz rule
for these nanocomposites (while being obeyed for the polymer itself) suggests a breakdown of the
superstructure when large displacements are imposed on them during steady shear flow.3, 14
It is hypothesized that the network superstructure, under steady shear, initially, rearranges locally
to accommodate the displacement.54
The stress increase is a manifestation of the structural changes that
result from the aggregation due to collisions of clusters and formation of new bonds across clusters.54
For
times greater than the time corresponding to the maximum in the stress, the fractal network breaks and the
effective stress supported by the network reduces. Thus, the overshoot stress, or the maximum stress
(σmax) is analogous to a yield stress beyond which the network starts to flow. This idea is further
supported by optical observation of CNTs aggregation under flow39d
and theoretical treatment of semi-
dilute CNTs suspension response to steady shear.56
The time required to attain the maximum stress (tmax)
is independent of CNTs loading and only a function of shear rate.54
Therefore, the shear stress data, when
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scaled in terms of the network yield stress and plotted against a dimensionless shear strain collapse onto a
mastercurve.39b
For strains beyond the maximum in shear stress, the fractal network breaks up (bonds are
broken) and the system flows until a steady shear viscosity is obtained. The steady state flow behavior is
governed by an establishment of an equilibrium between bond-breaking and bond-formation processes.57
For dense colloidal suspensions Silbert et al. found that under continuous strain only a fraction of
the initial (quiescent state) populations of clusters essentially bears the stress and controls the response.58
On the other hand, non-equilibrium molecular dynamics simulation studies of polymer nanocomposites
have identified the elastic stretching of the particle-polymer network as the primary cause for the stress
overshoot.59
Additionally MD simulations also revealed an inhomogeneity in stress distribution and only
a small number of bonds actually carry the excess stress developed. The current understanding is that
under steady shear, the flocs locally rearrange to accommodate the applied deformation and this results in
collisions of clusters and jamming of the network elements that gives rise to the observed stress
overshoot. When the applied local stress exceeds the yield stress, the network bonds break and stress
dissipation leads to flow until a final steady state is reached where equilibrium between bond-formation
and bond-breaking occurs. Finally, the power-law scaling of steady shear viscosity to the applied shear
rate is independent of the concentration of CNT with a power-law exponent –(0.7±0.01).54
Based on bond
formation and bond breaking argument, it can be inferred that about 1/3rd
of the network junction that
resist deformation are eliminated and these bonds are presumably those with principle directions along the
flow direction.54
Further insight into the structural aggregation and relaxation under shear is obtained by following
the characteristic time scales associated with the process. At short times after start-up, the material,
behaving like an elastic solid, generates a stress that is roughly proportional to the total strain. On the
other hand, similar to shear rejuvenation, under strain the nanoparticle network rearranges and results in
dissipation of the stress. Therefore, there are two major characteristic time scales associated: (a)
aggregation time (tagg) and (b) relaxation time (trlx) which can be extracted from fitting the time dependent
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shear stress data to a cluster aggregation – breakdown model.54, 57
It is found that both of these
characteristic time scales are independent of the CNT concentration with tagg < trlx. This suggests that
under shear, structural aggregation precedes the relaxation process and both of these processes are
primarily dominated by the applied shear rate only.54
This behavior can theoretically be explained using
the fractal cluster dynamics model in presence of continuous shear as well.54, 57, 60
Figure 5 (a) Representative transient shear stress response obtained during start-up of steady shear
measurements for a SWNTs -PEO dispersion (ϕ/ϕc = 5.0). For all shear rates, the stress data exhibit an
initial overshoot arising from the shear-induced cluster aggregation, and in the long time, the network
breaks to reach a steady state. Solid lines are model fits to the experimental data as described in the
reference 53. (Adapted with permission from Ref. 54. Chatterjee, T.; Krishnamoorti, R., Macromolecules
41 (14), 5333, 2008. Copyright 2008 American Chemical Society) (b) Comparison of the complex and
steady shear viscosities as a test for the Cox-Merz rule. The nanocomposites fail to obey the Cox-Merz
rule presumably because of an alteration in the mesoscale structure during steady flow.
Beyond the yield stress, Hobbie and coworkers47
have identified a second critical stress referred
to as the critical stress for homogenization. According to their argument, the shear thinning behavior
arises from breaking up of the network superstructure into smaller fragments. These fragments become
gradually smaller under increasing shear rate. The stress of homogenization is the critical stress beyond
which these fragments eventually lose their identity and approaches the scale of an individual nanotube.47
This idea is captured into a non-equilibrium phase diagram (Figure 6). At low stress an isotropic
0
1000
2000
3000
10-2
100
102
0.1
0.3
1.0
3.0
5.0
σ (
dy
ne
s/c
m2
)
Time (s)
T = 70 o
C p/pc = 5.0
( )1sγ −•
(a)(b)
)(sγ -1&
102
105
108
10-4
10-2
100
102
η∗
η
η∗
or η
(P
ois
e)
ω or
T = 70 oC
~ (ω)-0.9 ± 0.03
φ/φc = 7.0
0.01-0.7γ ±&
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superstructure is present which with increasing stress progressively moves through vorticity banding,
broken/cavitated network and finally to the isolated aggregates till the stress of homogenization is
reached. Beyond this stress for any shear limit (or any h/R0, where h is the gap and R0 is the mean
aggregate size) a nematic phase is observed. The extent of ordering or nematic order parameter (S or
structural anisotropy) bears a logarithmic scaling with the shear rate or shear stress39c, 47, 61
which is a
deviation from Doi-Edwards theory.12
It has been argued that •
γdS/d should scale with time and the only
characteristic time scale of the system is the Jeffrey time scale (Period ~ 1/shear rate). Therefore, the
shear alignment and logarithmic scaling of anisotropy with shear rate are perhaps consistent with rigid rod
signature of nanotubes.47
Figure 6: Scaled ‘phase diagram’ describing the evolution from a solid-like disordered network (I) to a
flowing nematic (N) for MWNTs suspended in a low molecular weight polymer solvent, where
concentration increases from top to bottom. Open circles are (para)nematic, closed circles are isolated
aggregates, open squares are vorticity bands, and closed squares are cavitated networks. The vertical
dashed line marks the stability limit. Colors denote MWNT mass concentration; blue (0.025%), brown
(0.1%), pink (0.4%), green (0.85%), purple (1.7%), and magenta (3%). The parameter h denotes gap and
R0 represents the mean aggregate size in the limit of large gap and weak shear. (Reprinted with
permission from Ref 39c: Hobbie, E.K.; Fry, D.J., Physical Review Letters, 97(3), 036101, 2006.
Copyright (2006) by the American Physical Society.)
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Flow induced alignments and consequences:
Finally, we focus on the flow induced structure of CNT dispersions. There is substantial literature
available on alignment of nanotubes under extensional flow62
and the CNTs directed hierarchical
alignment of polymer (from unit cell to lamellar ordering).43a, 43b
In the case of nanotube directed
controlled crystallization, ordering of polymer crystals are realized where SWNTs are oriented by a shear
or elongational flow fields or electrical fields.43a, 63
However, in this review we limit our discussion on
orientation of CNTs or CNT flocs under shear flow field only and their impact on different rheological
properties such as shear stress, normal stress etc.
In semi-dilute concentration regime, under weak shear, CNTs aggregate along the vorticity
direction (Figure 7a).64
Previously, alignment along the vorticity direction has been reported for various
systems including attractive emulsions near their colloidal glass transition,65
carbon black in tetradecane,66
and polymeric emulsions (polyethylene/polystyrene)67
. Such aggregation in the vorticity direction is
observed when inter-floc or inter-particle attractions are comparable to hydrodynamic forces. Therefore,
there is a finite range of shear rates over which the vorticity alignment of CNT flocs is realized. If the
shear rate is too high, then the CNT aggregates break up and align along the flow direction. For too low a
shear rate, Brownian dynamics dominates and non preferred orientation is observed. Recent rheo-optical
measurements suggest that the origin of cylindrical flocs and their vorticity alignment are primarily a
mechanical and geometrical phenomenon.64b
The gap height (h) at which shear is applied influences both
the formation kinetics and the final diameter of the cylindrical structure. The time scales over which such
anisotropic structures form depends on both the shear rate and gap height (h) where small h and high •
γ
facilitate their formation. A schematic of the growth mechanism of vorticity band structure, as proposed
by Ma et. al., is presented in Figure 7b.64b
A faster moving aggregate interlocks with a slower moving
aggregate/particle and forms a nucleus. As individual or small bundles of CNTs tumble in the fluid and
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collide with rotating nucleus (note shear flow is a combination of rotation and symmetric elongation flow)
the aggregate grows in the vorticity direction akin to orthokinetic aggregation mechanism proposed by
Smoluchowski.68
While the gap between the plates controls the diameter of the cylindrical structure (for a
fixed shear rate, the diameter increases with increasing h), the length is presumably controlled by the
matrix viscosity. As an alternative, Hobbie and coworkers proposed that inelastic instability of nanotubes
are responsible for vorticity alignment.64a
Analogous to Weissenberg’s rod-climbing effect,69
for soft
aggregates surrounded by a less viscous fluid in flow-gradient plane, hoop stress leads to elongation in the
vorticity direction along with somewhat compression in the radial direction of the cylinder. However, this
argument does not provide any insight into the role of confinement on the growth of the structure and
dependence of structural dimensions on the gap between the plates.
Figure 7: (a) The formation of cylindrical flocs aligned along the vorticity direction. The above
micrograph was collected for shear rate = 0.5 s−1
, gap = 180 µm and time = 600s. Direction of flow is
vertical as indicated. For this optical micrograph Vorticity alignment of CNT flocs is clearly visible. (b)
Schematic diagram of the growth mechanism. A nucleus rotates within the steady shear and captures
initially isotropic aggregates of nanotubes. The nanotubes are then wound helically to form a cylinder
with long axis perpendicular to the direction of flow. (Adapted with permission from Ref. 64b. Ma, A. W.
K.; Mackley, M. R.; Rahatekar, S. S., Rheologica Acta, 46 (7), 979, 2007. Copyright 2007 Springer)
A direct consequence of the vorticity elongation of aggregated cylindrical structures is their
negative first normal stress difference (N1). The first and second normal stress differences are defined as:
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27
N1 = σ11 – σ22, N2 = σ22 – σ33, where σii are the normal stresses (components of the stress tensor) and 1, 2
and 3 are flow, gradient and vorticity directions, respectively. As discussed before, the vorticity
alignment of CNT flocs is associated with a negative normal stress difference where the effect has been
explained through a competition between flow-induced orientation and thermodynamically driven
nematic state.64a
Further, for CNT dispersions in low viscosity media, there is a time scale associated with
the –∆N development (equivalent to vorticity elongation time scale) which decreases with increasing
shear rate and gap (h) between the plates.64b
In contrast, there is an instantaneous development of –∆N in
MWNTs based polypropylene (PP) nanocomposites under shear flow.70
Note that below percolation, ϕ <
ϕc, a positive ∆N is observed and it becomes large and negative for ϕ > ϕc. Further, measurement of
normal force using a cone-and-plate geometry revealed that N1 < 0.70
Based on parallel plate and cone-
and-plate measurement, CNTs based nanocomposites70a
exhibits |N2| << |N1|, consistent with pure
polymer melts and polymer solutions also.71
While the underlying causes behind the observation of –∆N
in MWNT/PP is not completely clear, it is assumed that both the large-scale deformation of the network
(ϕ > ϕc) and the local scale deformation of individual tube are responsible. Physically, the network
structure gets distorted under shear flow where the network tilts rather than elongates due to presence of
stiff nanotubes (Figure 8).70b
As a result, the average mesh size increases in the flow direction and
decreases along the gradient direction giving rise to a negative ∆N or N1. In contrast, for semi-flexible
biopolymer network,72
filament stretching has been claimed to be responsible for negative ∆N. It has been
reported that the sign of ∆N depends on the CNT aspect ratio in CNT/PP nanocomposites. For low aspect
ratio CNTs, the volume fraction for network formation is high and in that case the mesh size (ζ ~ ϕ1/2
) is
too small to accommodate rotational distortion.70b
Therefore, -∆N is only observed for tubes with high
aspect ratio or a good state of dispersion.
One remarkable consequence of negative ∆N is the impact on nanocomposite processing.
Extrusion of high molecular weight polymers results in die-swelling which is an outcome of their positive
normal stress difference. On the contrary, CNT based polymer nanocomposites with an overall negative
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∆N exhibit die contraction behavior (Figure 8). While the reduction in extrudate swell is well known for
polymers with inorganic fillers the specific importance in CNT based polymer nanocomposites lies in the
filler loading content. Generally, for spherical particles, this is observed at a high volumetric filler
loading (>0.3).73
In contrast, such a reduction is observed for CNT based polymer nanocomposites at
much lower loading (< 10% volume fraction) due to high interfacial area between well-dispersed
anisotropic nanotubes and polymer chains. Additionally, polypropylene is known for its shape-distortion
instability during extrusion which can also be minimized by incorporating CNTs into the matrix.
Therefore CNT based polymer nanocomposites are promising materials for controlling dimensional
characteristics and surface distortion in the composite manufacturing.
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Figure 8: (A) Photos of extrudates of pure iPP and CNT/iPP melt (7.4% mass fraction) at 210 °C under
different shear rates. The diameter and length of capillary die were 1 and 32 mm, respectively. iPP at 100
s-1
, mean diameter of 1.34 mm (a), iPP at 500 s-1
, mean diameter of 1.46 mm (b), iPP at 1000 s-1
, mean
diameter of 1.67 mm (c), iPP at 2000 s-1
, mean diameter of 1.82 mm (d), 7.4% mass fraction CNT/iPP at
100 s-1
, mean diameter of 1.18 mm (e), 7.4% mass fraction CNT/iPP at 500 s-1
, mean diameter of 1.38
mm (f), 7.4% mass fraction CNT/iPP at 1000 s-1
, mean diameter of 1.54 mm (g), 7.4% mass fraction
CNT/iPP at 2000 s-1
, mean diameter of 1.66 mm (h). The red circle in the figure shows the size of the
capillary die for comparison. The minimum scale of the ruler at the downside of the figure is 1 mm. The
relative measurement uncertainty of the extrudate diameter was estimated to be about 1%. (Adapted with
permission from Ref. 70b. Xu, D. H.; Wang, Z. G.; Douglas, J. F., Macromolecules, 41 (3), 815, 2008.
Copyright 2008 American Chemical Society) (B) Models about deformations of low aspect ratio
CNT/iPP and high aspect ratio CNT/iPP networks under steady shear. (Adapted with permission from
Ref. 70b. Xu, D. H.; Wang, Z. G.; Douglas, J. F., Macromolecules, 41 (3), 815, 2008. Copyright 2008
American Chemical Society)
SUMMARY
In this review paper we have briefly discussed the major rheological properties of CNT based
polymer nanocomposites and attempted to explain those results in terms of the underlying structure. The
two most important factors which control the rheology of carbon nanotube based nanocomposites are
their dispersion state and concentration of CNTs. As a function of increasing concentration, nanotube
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dispersed in polymer goes from dilute to semi-dilute to concentrated to nematic phases. A good state of
dispersion of carbon nanotubes in polymer matrix is required to improve overall nanocomposite
properties (mechanical, electrical etc.). At the same time, in a well dispersed state, the effective nanotube
aspect ratio is high which results in excluded volume induced isotropic-nematic transition at lower CNT
loading. Therefore, a small concentration window is available for semi-dilute isotropic CNT dispersion in
polymer, a region which is rheologically significant from both the academic and industrial point of view.
At low CNT concentration in polymer, the nanotubes are dispersed as individual anisotropic
particles and the overall nanocomposite properties are a combination of CNT contribution and their
coupling with the matrix. Unfortunately, there is a significant gap in reported literature about the dilute
regime rheology of CNT based polymer nanocomposites. In the dilute regime, the observed linear relation
between the viscosity and concentration14
can be exploited as a model system to study nanotube
alignment and their transport properties under shear. A detailed understanding of the flow induced
alignment is essential to prepare large scale nanocomposites with directional properties.
Close to and beyond the percolation threshold the CNT network controls both, the linear and non-
linear rheology. The nanotube network is a self-similar mass fractal system which grows almost linearly
with concentration and has a profound influence on semi-dilute regime rheology. Beyond the percolation
threshold, CNTs display a time-temperature-composition superposition of linear rheological properties
which is a signature of short range attraction. Moreover, the nonlinear deformation of the network
exhibits a concentration independent scaling with the local strain experienced by the individual tubes
inside the self-similar network. Therefore, in the superposable non-linear regime, changes in the network
structure are reversible and beyond that deformation results in irreversible or permanent damage to the
structure. Overall, the CNT network is elastic in nature which originates from the inter-tube bonding.
Therefore, under strain individual tubes resist both stretching and rotation and local scale deformation of
tubes leads to non-linear response.
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Under continuous shear, the CNT network flows only beyond the yield stress. At earlier time
(before reaching the yield stress) the nanotube network bears the stress through structural rearrangement.
Beyond the yield stress the network structure breaks down and flows like a power-law fluid until a steady
state viscosity is reached. The steady state has been conceived as a condition where an equilibrium
between shear induced bond-formation and bond-breaking occurs. Undoubtedly, steady shear brings
irreversible changes in the network structure as the nanocomposite fails to obey the Cox-Merz rule.
Additionally, steady state viscosity shows a concentration independent scaling with shear rate. A shear
thinning nature of steady-shear viscosity with scaling exponent ~ -0.7 suggests that at steady state inter-
tube bonds in the flow direction are presumably removed. Alternatively, a three-dimensional network
converts into a two-dimensional network. There is even a higher critical stress of homogenization where
all the network elements are completely broken and the nanotubes exist as individual entity. Beyond this
stress limit an oriented nematic phase is observed.
At well dispersed state in semi-dilute concentration regime and under weak shear, nanotubes in a
low viscous medium form cylindrical flocs with the long axis parallel to the vorticity direction. It is still
debatable whether such vorticity alignment is purely an outcome of mechanical and geometrical
(confinement) effect or has as its origin the Weissenberg effect (inelastic instability). One of the major
consequences of such directional structure is a negative first normal stress difference.
Finally, we conclude that while significant amounts of experimental research are available in
literature, this field still needs further theoretical and simulation effort to understand those
properties/observation. Particular focus should be directed in finding newer or more convenient routes of
CNT dispersion in polymer, exploring the concentration dependent phases and ways to shift the isotropic-
nematic transition at higher volume fraction, and the origin of negative normal stress under steady shear.
It is expected that the polymer polarity and other physical properties can be exploited to tune the liquid
crystalline state. Flow-processing of liquid crystalline CNTs can be utilized to form high performance
fiber and CNT embedded polymer mats. Detailed understandings of the linear and non-linear rheological
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properties and their relation with the underlying structure are essential for industrial success of polymer
nanocomposites manufacturing and applications.
ACKNOWLEDGEMENTS
TC acknowledges Drs. Valeriy V. Ginzburg, Greg F. Meyers, and Wayde Konze for their detailed and
insightful comments. RK acknowledges funding provided by DOE/NETL/RPSEA (Project 10121-4501-
01). Sponsors are not responsible for any of the findings in this study.
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34
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Tirtha Chatterjee is an Associate Research Scientist in Core R&D at The Dow Chemical Company. He
received his bachelor and masters degree in Chemical Engineering from India. In 2003 he moved to the
US to pursue graduate studies at the University of Houston under the direction of Professor Ramanan
Krishnamoorti; he received his PhD degree in Chemical Engineering in 2008. From 2008 to 2010 he was
a post-doctoral fellow in Professor Edward Kramer's group at the Materials Research Laboratory at the
University of California at Santa Barbara. He joined The Dow Chemical Company in 2010. His research
interests are polymer nanocomposites, associative polymers and their interaction with colloidal particles,
solution and gel properties of biopolymers among others.
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Ramanan Krishnamoorti is a Professor of Chemical and Biomolecular Engineering at the University of
Houston with affiliated appointments as Professor of Petroleum Engineering and Professor of Chemistry
at the University of Houston. Dr. Krishnamoorti obtained his bachelors degree in Chemical Engineering
from IIT Madras in India and doctoral degree in Chemical Engineering from Princeton University in
1994. After two one year post-doctoral positions at Caltech and Cornell University, he joined UH as an
Assistant Professor of Chemical Engineering at UH. He has subsequently served as Associate Dean for
Research in the College of Engineering, Chair of the Department of Chemical and Biomolecular
Engineering and currently as the Chief Energy Officer at UH. His research has focused on understanding
the structure – processing – property relations for multicomponent polymers and other classes of soft
materials including lipid membranes.
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Structure
Dominant
rheological
signature
Dilute Isotropic
network Nematic
Onset of
percolation
concentration
Matrix
controlled
Dilute
Brownian
rod
dynamics
Onset of
Network
Dynamics
Highly
Sensitive to
Conc. &
Orientation
CNT
Network
dominated
Time-
Temp-Conc.-
Strain
master
curves
Liquid-
crystalline
Vorticity
alignment
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