Rheology of polymer carbon nanotubes composites

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1

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.

Page 2 of 43Soft Matter

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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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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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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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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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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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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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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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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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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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REFERENCES

1.) a S. K. Kumar and R. Krishnamoorti, Annual Review of Chemical and Biomolecular

Engineering, Vol 1, 2010, 1, 37-58; b K. I. Winey and R. A. Vaia, MRS Bulletin, 2007, 32,

314-319; c A. C. Balazs, T. Emrick and T. P. Russell, Science, 2006, 314, 1107-1110.;

2.) R. Krishnamoorti and R. A. Vaia, Journal of Polymer Science Part B: Polymer Physics,

2007, 45, 3252-3256.;

3.) R. Krishanamoorti and T. Chatterjee, in Applied Polymer Rheology: Polymeric Fluids with

Industrial Applications, ed. M. Kontopoulou, Wiley, 1st edn., 2012, pp. 153-178.;

4.) a J. Bicerano, J. F. Douglas and D. A. Brune, Journal of Macromolecular Science-Reviews in

Macromolecular Chemistry and Physics, 1999, C39, 561-642; b E. J. Garboczi, K. A.

Snyder, J. F. Douglas and M. F. Thorpe, Physical Review E, 1995, 52, 819-828.;

5.) a M. F. Islam, A. M. Alsayed, Z. Dogic, J. Zhang, T. C. Lubensky and A. G. Yodh, Physical

Review Letters, 2004, 92, 088303; b S. J. Zhang and S. Kumar, Small, 2008, 4, 1270-

1283.;

6.) It is possible to get higher aspect ratio from two-dimesnsional structure. For example, in

principle mica platelets could have aspect ratio well above 1000.;

7.) J. Israelachvili, Intermolecular and Surface Forces, Elsevier, 1992.;

8.) R. G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press,

New York, 1999.;

9.) a K. D. Ausman, R. Piner, O. Lourie, R. S. Ruoff and M. Korobov, Journal of Physical

Chemistry B, 2000, 104, 8911-8915; b M. D. Clark, S. Subramanian and R.

Krishnamoorti, Journal of Colloid and Interface Science, 2011, 354, 144-151.;

10.) L. Onsager, Annals of New York Academy of Sciences, 1949 51, 627-659.;

11.) P. J. Flory, Proceedings of the Royal Society of London Series a-Mathematical and

Physical Sciences, 1956, 234, 73-89.;


So

ft M

atte

r A

ccep

ted

Man

usc

rip

t

Publ

ishe

d on

28

Aug

ust 2

013.

Dow

nloa

ded

by Q

ueen

s U

nive

rsity

- K

ings

ton

on 2

8/08

/201

3 12

:29:

31.



34

12.) M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press,

Oxford, 1986.;

13.) a D. Fry, B. Langhorst, H. Wang, M. L. Becker, B. J. Bauer, E. A. Grulke and E. K. Hobbie,

Journal of Chemical Physics, 2006, 124, 054703; b D. Fry, B. Langhorst, H. Kim, E.

Grulke, H. Wang and E. K. Hobbie, Physical Review Letters, 2005, 95, 038304.;

14.) V. A. Davis, L. M. Ericson, A. N. G. Parra-Vasquez, H. Fan, Y. H. Wang, P. V, J. A. Longoria,

S. Ramesh, R. K. Saini, C. Kittrell, W. E. Billups, W. W. Adams, R. H. Hauge, R. E. Smalley

and M. Pasquali, Macromolecules, 2004, 37, 154-160.;

15.) V. A. Davis, A. N. G. Parra-Vasquez, M. J. Green, P. K. Rai, N. Behabtu, V. Prieto, R. D.

Booker, J. Schmidt, E. Kesselman, W. Zhou, H. Fan, W. W. Adams, R. H. Hauge, J. E.

Fischer, Y. Cohen, Y. Talmon, R. E. Smalley and M. Pasquali, Nature Nanotechnology,

2009, 4, 830-834.;

16.) a N. Fakhri, D. A. Tsyboulski, L. Cognet, R. B. Weisman and M. Pasquali, Proceedings of the

National Academy of Sciences of the United States of America, 2009, 106, 14219-14223;

b W. Zhou, M. F. Islam, H. Wang, D. L. Ho, A. G. Yodh, K. I. Winey and J. E. Fischer,

Chemical Physics Letters, 2004, 384, 185-189; c R. Duggal and M. Pasquali, Physical

Review Letters, 2006, 96, 246104; d L. A. Hough, M. F. Islam, P. A. Janmey and A. G.

Yodh, Physical Review Letters, 2004, 93, 168102.;

17.) H. S. Lee and C. H. Yun, Journal of Physical Chemistry C, 2008, 112, 10653-10658.;

18.) S. Marceau, P. Dubois, R. Fulchiron and P. Cassagnau, Macromolecules, 2009, 42, 1433-

1438.;

19.) M. L. Mansfield and J. F. Douglas, Macromolecules, 2008, 41, 5422-5432.;

20.) R. Krishnamoorti, MRS Bulletin, 2007, 32, 341-347.;

21.) a C. A. Mitchell and R. Krishnamoorti, Polymer, 2005, 46, 8796 - 8804; b C. A. Mitchell

and R. Krishnamoorti, Macromolecules, 2007, 40, 1538-1545.;

22.) a T. Chatterjee, A. T. Lorenzo and R. Krishnamoorti, Polymer, 2011, 52, 4938-4946; b

T. Chatterjee, K. Yurekli, V. G. Hadjiev and R. Krishnamoorti, Advanced Functional


So

ft M

atte

r A

ccep

ted

Man

usc

rip

t

Publ

ishe

d on

28

Aug

ust 2

013.

Dow

nloa

ded

by Q

ueen

s U

nive

rsity

- K

ings

ton

on 2

8/08

/201

3 12

:29:

31.



35

Materials, 2005, 15, 1832-1838; c K. Yurekli, C. A. Mitchell and R. Krishnamoorti,

Journal of the American Chemical Society, 2004, 126, 9902-9903.;

23.) S. Banerjee, T. Hemraj-Benny and S. S. Wong, Advanced Materials, 2005, 17, 17-29.;

24.) a F. Meyer, J. M. Raquez, P. Verge, I. M. de Arenaza, B. Coto, P. Van Der Voort, E. Meaurio,

B. Dervaux, J. R. Sarasua, F. Du Prez and P. Dubois, Biomacromolecules, 2011, 12, 4086-

4094; b C. Semaan, M. Schappacher and A. Soum, Polymer Composites, 2012, 33,

1-9.;

25.) In linear frequency sweep measurements a small amplitude oscillatory strain, γ(t) of the

form γ(t) =γ0sin(ωt), where γ0 is the strain amplitude, is applied. Values of γ0 are kept as

low as possible to apply a minimal deformation as well as not to change the quiescent

state microstructure of the nanocomposites. The resulting time dependent linear shear

stress,σ(t), is recorded and interpreted as,σ(t)=γ0[G'sin(ωt)+G"cos(ωt)], where G’ and

G” are the storage and loss modulus respectively. The other rheological properties

derived include the complex modulus G*= (G'2+G"

2)(1/2)

and the complex viscosity

(η∗=G*/ω).;

26.) P. Pӧtschke, T. D. Fornes and D. R. Paul, Polymer, 2002, 43, 3247-3255.;

27.) E. J. Guth, Journal of Applied Physics, 1945, 16, 20-25.;

28.) a F. M. Du, J. E. Fischer and K. I. Winey, Physical Review B, 2005, 72, 121404(R); b F. M.

Du, R. C. Scogna, W. Zhou, S. Brand, J. E. Fischer and K. I. Winey, Macromolecules, 2004,

37, 9048-9055.;

29.) L. Vaisman, G. Marom and H. D. Wagner, Advanced Functional Materials, 2006, 16, 357-

363.;

30.) A. P. Chatterjee, Journal of Physics-Condensed Matter, 2008, 20, 255250.;

31.) A. P. Chatterjee, Journal of Physics-Condensed Matter, 2011, 23, 375101.;

32.) D. A. Head, A. J. Levine and F. C. MacKintosh, Physical Review E, 2003, 68, 061907.;


So

ft M

atte

r A

ccep

ted

Man

usc

rip

t

Publ

ishe

d on

28

Aug

ust 2

013.

Dow

nloa

ded

by Q

ueen

s U

nive

rsity

- K

ings

ton

on 2

8/08

/201

3 12

:29:

31.



36

33.) a J. M. Brown, D. P. Anderson, R. S. Justice, K. Lafdi, M. Belfor, K. L. Strong and D. W.

Schaefer, Polymer, 2005, 46, 10854-10865; b D. Schaefer, J. M. Brown, D. P.

Anderson, J. Zhao, K. Chokalingam, D. Tomlin and J. Ilavsky, Journal of Applied

Crystallography, 2003, 36, 553-557; c D. W. Schaefer and R. S. Justice,

Macromolecules, 2007, 40, 8501-8517.;

34.) T. Chatterjee and R. Krishnamoorti, Physical Review E, 2007, 75, 050403.;

35.) T. Chatterjee, A. Jackson and R. Krishnamoorti, Journal Of The American Chemical

Society, 2008, 130, 6934-6935.;

36.) C. F. Schmidt, M. Barmann, G. Isenberg and E. Sackmann, Macromolecules, 1989, 22,

3638-3649.;

37.) a M. N. Tchoul, W. T. Ford, M. L. P. Ha, I. Chavez-Sumarriva, B. P. Grady, G. L. Lolli, D. E.

Resasco and S. Arepalli, Chemistry of Materials, 2008, 20, 3120-3126; b F. Buffa, G. A.

Abraham, B. P. Grady and D. Resasco, Journal of Polymer Science Part B-Polymer Physics,

2007, 45, 490-501; c Y. Y. Huang, S. V. Ahir and E. M. Terentjev, Physical Review B,

2006, 73, 125422; d Q. H. Zhang, D. R. Lippits and S. Rastogi, Macromolecules, 2006,

39, 658-666; e Q. H. Zhang, S. Rastogi, D. J. Chen, D. Lippits and P. J. Lemstra, Carbon,

2006, 44, 778-785; f Z. H. Fan and S. G. Advani, Journal of Rheology, 2007, 51, 585-604;

g P. Pӧtschke, M. Abdel-Goad, I. Alig, S. Dudkin and D. Lellinger, Polymer, 2004, 45,

8863-8870.;

38.) K. W. Putz, C. A. Mitchell, R. Krishnamoorti and P. F. Green, Journal of Polymer Science

Part B-Polymer Physics, 2004, 42, 2286-2293.;

39.) a D. T. N. Chen, K. Chen, L. A. Hough, M. F. Islam and A. G. Yodh, Macromolecules, 2010,

43, 2048-2053; b S. S. Rahatekar, K. K. Koziol, S. R. Kline, E. K. Hobbie, J. W. Gilman

and A. H. Windle, Advanced Materials, 2009, 21, 874-878; c E. K. Hobbie and D. J.

Fry, Physical Review Letters, 2006, 97, 036101; d L. Moreira, R. Fulchiron, G. Seytre, P.

Dubois and P. Cassagnau, Macromolecules, 2010, 43, 1467-1472.;


So

ft M

atte

r A

ccep

ted

Man

usc

rip

t

Publ

ishe

d on

28

Aug

ust 2

013.

Dow

nloa

ded

by Q

ueen

s U

nive

rsity

- K

ings

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on 2

8/08

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3 12

:29:

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37

40.) a V. Prasad, V. Trappe, A. D. Dinsmore, P. N. Segre, L. Cipelletti and D. A. Weitz, Faraday

Discussion, 2003, 123, 1-12; b V. Trappe and D. A. Weitz, Physical Review Letters,

2000, 85, 449-452.;

41.) K. Ninomiya and J. D. Ferry, Journal of Colloid Science, 1959, 14.;

42.) M. Surve, V. Pryamitsyn and V. Ganesan, Macromolecules, 2007, 40, 344-354.;

43.) a T. Chatterjee, C. A. Mitchell, V. G. Hadjiev and R. Krishnamoorti, Advanced Materials,

2007, 19, 3850-3853; b T. Chatterjee, C. A. Mitchell, V. G. Hadjiev and R.

Krishnamoorti, Macromolecules, 2012, 45, 9357-9363; c J. X. Ren and R.

Krishnamoorti, Macromolecules, 2003, 36, 4443-4451.;

44.) a P. Sollich, Physical Review E, 1998, 58, 738-759; b P. Sollich, F. Lequeux, P. Hebraud

and M. E. Cates, Physical Review Letters, 1997, 78, 2020-2023; c L. Berthier, Journal of

Physics-Condensed Matter, 2003, 15, S933-S943; d D. Bonn, P. Coussot, H. T. Huynh, F.

Bertrand and G. Debregeas, Europhysics Letters, 2002, 59, 786-792; e D. Bonn, H.

Tanaka, P. Coussot and J. Meunier, Journal Of Physics-Condensed Matter, 2004, 16,

S4987-S4992; f D. Bonn, S. Tanase, B. Abou, H. Tanaka and J. Meunier, Physical

Review Letters, 2002, 89, 015701.;

45.) M. Sahimi and S. Arbabi, Physical Review B, 1993, 47, 703-712.;

46.) M. Surve, V. Pryamitsyn and V. Ganesan, Physical Review Letters, 2006, 96, 177805.;

47.) E. K. Hobbie and D. J. Fry, Journal of Chemical Physics, 2007, 126, 124907.;

48.) a M. Chen and W. B. Russel, Journal of Colloid and Interface Science, 1991, 141, 564-577; b

W. H. Shih, W. Y. Shih, S. I. Kim, J. Liu and I. A. Aksay, Physical Review A, 1990, 42,

4772-4779.;

49.) a S. Feng, P. N. Sen, B. I. Halperin and C. J. Lobb, Physical Review B, 1984, 30, 5386-5389; b

Y. Kantor and I. Webman, Physical Review Letters, 1984, 52, 1891-1894.;

50.) W. Gleissle and B. Hochstein, Journal of Rheology, 2003, 47, 897-910.;

51.) A. Mongruel and M. Cartault, Journal of Rheology, 2006, 50, 115-135.;


So

ft M

atte

r A

ccep

ted

Man

usc

rip

t

Publ

ishe

d on

28

Aug

ust 2

013.

Dow

nloa

ded

by Q

ueen

s U

nive

rsity

- K

ings

ton

on 2

8/08

/201

3 12

:29:

31.



38

52.) Y. Aoki, A. Hatano, T. Tanaka and H. Watanabe, Macromolecules, 2001, 34, 3100-3107.;

53.) a G. Schmidt, A. I. Nakatani and C. C. Han, Rheologica Acta, 2002, 41, 45-54; b K.

Yurekli, R. Krishnamoorti, M. F. Tse, K. O. McElrath, A. H. Tsou and H. C. Wang, Journal of

Polymer Science Part B-Polymer Physics, 2001, 39, 256-275; c Q. Zhang and L. A.

Archer, Langmuir, 2002, 18, 10435-10442.;

54.) T. Chatterjee and R. Krishnamoorti, Macromolecules, 2008, 41, 5333-5338.;

55.) W. P. Cox and E. H. Merz, Journal of Polymer Science, 1958, 28, 619-622.;

56.) W. K. A. Ma, F. Chinesta, A. Ammar and M. R. Mackley, Journal of Rheology, 2008, 52,

1311-1330.;

57.) M. Whittle and E. Dickinson, Journal of Chemical Physics, 1997, 107, 10191-10200.;

58.) L. E. Silbert, R. S. Farr, J. R. Melrose and R. C. Ball, Journal of Chemical Physics, 1999, 111,

4780-4789.;

59.) J. D. Thomin, P. Keblinski and S. K. Kmnar, Macromolecules, 2008, 41, 5988-5991.;

60.) S. D. T. Axford, Proceedings of the Royal Society of London Series A-Mathematical

Physical And Engineering Sciences, 1996, 452, 2355-2368.;

61.) S. Pujari, S. S. Rahatekar, J. W. Gilman, K. K. Koziol, A. H. Windle and W. R. Burghardt,

Journal of Chemical Physics, 2009, 130, 214903.;

62.) L. Jin, C. Bower and O. Zhou, Applied Physics Letters, 1998, 73, 1197-1199.;

63.) a M. L. Minus, H. G. Chae and S. Kumar, Acs Applied Materials & Interfaces, 2012, 4, 326-

330; b C. Lin and J. W. Shan, Physics of Fluids, 2007, 19; c Y. D. Liu and H. J. Choi, Soft

Matter, 2012, 8, 11961-11978.;

64.) a S. Lin-Gibson, J. A. Pathak, E. A. Grulke, H. Wang and E. K. Hobbie, Physical Review

Letters, 2004, 92; b A. W. K. Ma, M. R. Mackley and S. S. Rahatekar, Rheologica Acta,

2007, 46, 979-987.;

65.) A. Montesi, A. A. Pena and M. Pasquali, Physical Review Letters, 2004, 92.;

66.) C. O. Osuji and D. A. Weitz, Soft Matter, 2008, 4, 1388-1392.;


So

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atte

r A

ccep

ted

Man

usc

rip

t

Publ

ishe

d on

28

Aug

ust 2

013.

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ded

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ueen

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rsity

- K

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8/08

/201

3 12

:29:

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39

67.) E. K. Hobbie and K. B. Migler, Physical Review Letters, 1999, 82, 5393-5396.;

68.) M. Smoluchowski, Z Physical Chemistry, 1917, 92, 129-168.;

69.) K. Weissenberg, Nature, 1949, 159, 310-311.;

70.) a S. B. Kharchenko, J. F. Douglas, J. Obrzut, E. A. Grulke and K. B. Migler, Nature Materials,

2004, 3, 564-568; b D. H. Xu, Z. G. Wang and J. F. Douglas, Macromolecules, 2008, 41,

815-825.;

71.) R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymer Liquids Volume 1: Fluid

Mechanics, John Wiley and Sons, 1987.;

72.) P. A. Janmey, M. E. McCormick, S. Rammensee, J. L. Leight, P. C. Georges and F. C.

Mackintosh, Nature Materials, 2007, 6, 48-51.;

73.) N. Minagawa and J. L. White, Journal of Applied Polymer Science, 1976, 20, 501-523.;


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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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Rheology of polymer carbon nanotubes composites

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