© 2014 The Korean Society of Rheology and Springer 319

Korea-Australia Rheology Journal, Vol.26, No.3, pp.319-326 (August 2014)DOI: 10.1007/s13367-014-0036-y

www.springer.com/13367

New way to characterize the percolation threshold of polyethylene and

carbon nanotube polymer composites using Fourier transform (FT) rheology

Deepak Ahirwal1,2

, Humberto Palza3, Guy Schlatter

2 and Manfred Wilhelm

1,*1Karlsruhe Institute of Technology (KIT), Institute of Chemical Technology and Polymerchemistry, Engesserstr. 18,

76131 Karlsruhe, Germany2Institut de Chimie et Procédés pour l’Energie, l’Environnement et la Sant, ICPEES-UMR7515,Universit de Strasbourg,

CNRS, Institut Carnot MICA, Ecole Européenne de Chimie, Polymères et Matériaux, 25 rue Becquerel,67087 Strasbourg, cedex 2, France

3Departamento de Ingenieria Quimica y Biotecnologia, Facultad de Ciencias Fisicas y Matematicas, Universidad de Chile, Beauchef 861, Casilla 277, Santiago, Chile

(Received January 28, 2014; final revision May 9, 2014; accepted May 23, 2014)

In this article, a new way to characterize the percolation threshold of polymer nanocomposites made ofpolyethylene (PE) with single and multi walled carbon nanotubes (SWCNTs and MWCNTs) is presented.Small and large oscillatory shear (SAOS and LAOS) experiments were performed to characterize the degreeof dispersion and percolation threshold. The analysis of the stress response in the LAOS regime as a func-tion of the applied deformation amplitude and frequency was performed using Fourier Transform (FT)-Rhe-ology. The zero strain intrinsic nonlinear parameter, Q0(ω), was calculated by extrapolation of I3/1(γ0, ω) andwas, used to quantify the nonlinearity measured by FT-Rheology. Interestingly, a drop in Q0 as a functionof the CNT weight fraction at a fixed frequency was found that was below the percolation threshold. Thiswas followed by, a steep rise in Q0 above the percolation threshold. Therefore, the new method based onthis observation that is proposed and described with this article has the potential to lead to a better under-standing of structure-property relationships in polymer nanocomposites.

Keywords: FT-Rheology, nanocomposites, carbon nanotubes, percolation threshold, LAOS

1. Introduction

Polymer nanocomposites based on carbon nanotubes

(CNTs) (Iijima, 1991) attracted enormous interest due to

their exceptional electrical (Baughman et al., 2002),

mechanical (Ajayan et al., 2000; Kharchenko et al.,

2004) and thermal (Huxtable et al., 2003) properties at a

relatively small concentration in the polymer matrix.

Due to the improvements in these properties relative to

the neat polymer, polymer nanocomposites based on

CNTs have found manufacturing applications in elec-

trostatic painting (Ramasubramaniam et al., 2003), pro-

tective coatings for electronic components (Terrones,

2003), radiation shielding (Kota et al., 2007) and flam-

mability reduction (Kashiwagi et al., 2002). The addition

of CNTs into a polymer matrix has a significant effect on

the processing conditions (e.g. mixing, moulding, extru-

sion, etc.) because of the substantial changes that occur

in the viscoelastic properties of polymer nanocomposite

materials. To optimize the processing conditions, it is

important to understand the changes in the flow prop-

erties that occur up upon addition of CNTs into a poly-

mer matrix. The important parameters which play a

significant role in this are the chemical composition of

the polymer, the aspect ratio of the CNTs, the orientation

of the CNTs, the type of CNT (e.g. single and multi

walled CNTs), the polymer-filler interactions and the

degree of dispersion in the polymer matrix.

To characterize the state of nanoparticle dispersion in a

polymer matrix, the percolation scaling law presents a the-

oretical basis for calculating the properties of composites

containing impenetrable particles (Sahimi, 1994; Vigolo et

al., 2005). The scaling law P (φ–φc)δ (Kota et al., 2007;

Huang et al., 2011), is valid for polymer composites

where P is a generalized material property, φ is the filler

volume fraction, δ is the critical scaling exponent and φc

is the volume fraction at the percolation threshold con-

centration. The P could be, for example, a rheological

property (G’, η, δ0) or an electrical property (e.g. DC con-

ductivity, σDC). Two material properties are widely used

with the percolation theory to describe the dispersion state

of CNTs in a polymer matrix: the electrical conductivity

(σ) and the dynamic elastic shear modulus [G’(ω)]. The

effect of CNT on the linear viscoelastic properties of var-

ious polymers has been addressed in the literature: poly-

styrene (PS) (Kota et al., 2007), polycarbonate (PC) (Sun

et al., 2010), poly(methyl methacrylate) (PMMA) (Du et

al., 2004), polyethylene (PE) (McNally et al., 2005) and

polypropylene (PP) (Lee et al., 2008). Winter et al. (2001)

described the rheological changes with increasing CNT

∝

*Corresponding author: Manfred.Wilhelm@kit.edu

Deepak Ahirwal, Humberto Palza, Guy Schlatter and Manfred Wilhelm

320 Korea-Australia Rheology J., Vol. 26, No. 3 (2014)

content within the framework of a liquid-like to solid-like

transition. Furthermore, Du et al. (2001) provided a per-

spective for both the rheological and conductivity per-

colation threshold by describing the network formation

between the polymer matrix and CNTs, and between the

CNTs themselves using percolation theory. However, most

industrial processes operate at large deformation ampli-

tudes where knowledge of the nonlinear viscoelastic

behavior is important. Furthermore, under these large

deformation amplitudes, there can be changes to the per-

colation behavior due to orientation and stretching of the

polymer and orientation of the CNTs, which can affect the

conductivity as previously reported (Alig et al., 2007;

Skipa et al., 2010; Alig et al., 2008a; Alig et al., 2008b).

Therefore, to optimize industrial processes and tailor

material properties, an understanding of polymer com-

posite flow behavior under large amplitude oscillatory

shear is necessary.

Recently, Large Amplitude Oscillatory Shear (LAOS)

was used to characterize the nonlinear behavior of a vari-

ety of soft materials (Hyun et al., 2007; Ewoldt et al.,

2007; Hyun and Wilhelm, 2009; Hyun et al., 2011; Rein-

heimer et al., 2012; Wilhelm et al., 2012; Kempf et al.,

2013; Ahirwal et al., 2014). The interest in LAOS flow for

characterization can be attributed to the development of

several methods to analyze the complex periodic flow

response, such as Fourier Transform (FT) rheology (Wil-

helm, 2002), Stress Decomposition (SD) (Cho et al.,

2005), which, was further developed to describe higher

order moduli in a meaningful way using Chebyshev poly-

nomials (Ewoldt et al., 2008), and the sequence of Phys-

ical Process (SPP) framework (Rogers and Lattinga,

2012). In particular, most attention has been focused on

FT-Rheology, where the stress response is analyzed in

Fourier space. Generally, the third order higher Fourier

harmonic relative to the fundamental Fourier harmonic, I3/1,

is used to quantify the nonlinear behavior in LAOS flow.

In the LAOS regime, for a range of applied deformation

amplitudes, a square scaling, , was observed both

experimentally (Hyun and Wilhelm, 2009; Reinheimer et

al., 2012; Kempf et al., 2013; Ahirwal et al., 2014) and in

simulations of various nonlinear constitutive equations

[Giesekus model (Gurnon and Wagner, 2013), Pom-Pom

model (Hyun et al., 2013) and the Molecular Stress Func-

tion (MSF) model (Wagner et al. 2011)]. Motivated by

this square scaling, Hyun and Wilhelm (2009) introduced

a new nonlinear parameter . In

the limit of small deformation amplitudes, the Q-param-

eter converges to a constant value, which they called the

zero strain intrinsic nonlinearity, .

Since then, Q0, has been used to characterize a variety of

complex fluids, e.g. polymer melts (Vittorias et al., 2011;

Kempf et al., 2013; Ahirwal et al., 2014), dilute emulsions

(Reinheimer et al., 2011; Reinheimer et al., 2012), micelle

systems (Gurnon and Wagner, 2013) and polymer com-

posites (Lim et al., 2013; Hassanabadi et al., 2013).

In this article, PE/MWCNTs and PE/SWCNTs nano-

composites were used for the rheological investigation. In

the LAOS regime, a minimum in the zero strain intrinsic

nonlinearity, Q0(ω), as a function of the weight fraction of

CNTs (φw) at a fixed frequency, ω, was observed. Palza et

al. (2011) observed qualitatively similar behavior for the

electrical conductivity measured under LAOS. The

asymptotic scaling (percolation scaling law) was valid for

both the linear viscoelastic material parameter (G’) and the

nonlinear viscoelastic material parameter (Q0). It was con-

firmed that the percolating networks of CNT-polymer and

CNT-CNT have a significantly higher influence on the

properties measured in the nonlinear viscoelastic regime

compared to those measured in the linear viscoelastic

regime. This was verified by calculation of the percolation

scaling exponent, δ, which had a higher value for the

intrinsic nonlinearity, Q0 (nonlinear viscoelastic regime),

relative to that for the elastic modulus, G’ (linear vis-

coelastic regime).

2. Materials and Experimental protocols

2.1. Materials and preparation of nanocompositesMWCNTs were bought from Bayer Material Science

(Baytubes C150P), while SWCNTs were provided by

Prof. Mr. Kappes (KIT) and were prepared by laser vapor-

ization as described in detail elsewhere (Lebedkin et al.,

2002). The PE polymer is a LLDPE ethylene copolymer

with 7 mol% of 1-octene (short chain branching) from

Dow (commercial name: Engage-8450). The following

data is provided by the supplier for the PE: Mw = 100 kg/

mol, Mn = 45 kg/mol and Tm (melting temperature) =

100oC. A detailed description of the polymer nanocom-

posite preparation method is given elsewhere (Palza et al.,

2011).

2.2. Experimental protocols for rheological measure-ments

The rheological measurements were carried out on an

ARES-G2 rotational rheometer from TA Instruments.

Oscillatory shear measurements in the linear regime were

performed using the cone and plate geometry (13 mm and

25 mm, α = 0.1 rad) at 140oC. This particular temperature

(140oC) was chosen to compliment previous results by

Palza et al. (2011) on the conductivity of PE, PE/

MWCNTs and PE/SWCNTs that were measured at this

temperature. Nonlinear were also performed on the same

ARES-G2 rheometer at 140oC using an implemented FT-

Rheology module. Strain amplitudes over the range γ0 =

0.001 – 1.5 were applied at various frequencies. The sam-

ples were press-molded under vacuum at 180oC into 13

and 25 mm discs for the experiments.

I3 1⁄ γ02

∝

Q γ0 ω T, ,( ) I3 1⁄ γ0 ω T, ,( ) γ02

⁄=

Q0 ω T,( ) Q γ0 ω T, ,( )γ0

0→lim=

New way to characterize the percolation threshold of polyethylene and carbon nanotube polymer composites......

Korea-Australia Rheology J., Vol. 26, No. 3 (2014) 321

3. Results and Discussions

Palza et al., (2011) reported on the experimental mea-

surement of electrical conductivity for the PE/MWCNTs

and PE/SWCNTs composites at 140oC under oscillatory

shear flow. The electrical percolation threshold for the PE/

MWCNTs composites was found to lie between a

MWCNT concentration of 1-3 wt%, whereas, for the PE/

SWCNTs composites, percolation was not attained (Fig.

1). Furthermore, a minimum was observed in the electrical

conductivity as a function of the weight fraction of

MWCNTs in the composites. Below the electrical per-

colation threshold, a drop in the DC conductivity upon

addition of MWCNTs was explained by the scavenger

effect of MWCNTs (on impurities and additives), which

helps to reduce the contribution from impurities to the

electrical conductivity (e.g. Al or Ti from the Ziegler-

Natta polymerization process). In this work, we focus

instead on the mechanical characterization of these com-

posites using FT-Rheology.

The linear viscoelastic property, G’(ω), of the PE/

MWCNTs and PE/SWCNTs composites is presented in

Fig. 2. The concentration at which the linear viscoelastic

properties of the nanocomposites change from liquid-like

to solid-like behavior is called the percolation threshold.

Structurally, at the percolation threshold, the earliest net-

work spanning path is developed. Therefore, the state of

dispersion can be quantified through the percolation

threshold ( , in weight %). For example, under small

amplitude oscillatory shear (SAOS), the low frequency

storage (G’) and loss (G’’) moduli exhibit terminal behav-

ior with G’ ~ ω2 and G’’ ~ ω, respectively, for the simple

Maxwell model. Incorporation of CNTs in the polymer

matrix gradually transforms the liquid-like terminal

behavior into solid-like non-terminal behavior (i.e. both G’

and G’’ become independent of ω as ω → 0). The per-

colation threshold for the PE/MWCNTs composites is

= 1.0 wt% because a substantial change in the scaling expo-

nent of the elastic modulus, G’, was observed (Fig. 2a)

above this concentration. Specifically, the scaling exponent

below the percolation threshold is 1.06 and then changes

exponentially from 1.06 to 0 above the percolation thresh-

old. However, for the PE/SWCNTs composites, the per-

colation threshold could not be reached (Fig. 2b).

The nonlinear viscoelastic properties of the PE/

MWCNTs and PE/SWCNTs composites were investigated

under LAOS flow. The resulting shear stress responses

were analyzed using FT-Rheology. To quantify the non-

linear viscoelasticity of complex fluids, the ratio of the 3rd

to the 1st harmonic, I3/1(γ0, ω), is frequently used in FT-

Rheology (Schlatter et al., 2005; Vittorias and Wilhelm

2007; Hyun et al., 2011; Hyun and Wilhelm 2009; Vit-

torias et al., 2011; Kempf et al., 2013; Ahirwal et al.,

2014). Fig. 3 shows I3/1(γ0) as a function of the applied

deformation amplitude, γ0, and the weight fraction of

MWCNTs for the PE/MWCNTs composites. The square

scaling, I3/1 , was observed for the neat PE polymer

and the PE/MWCNTs composites at limited deformation

φw

c

φw

c

γ02

∝Fig. 1. The influence of CNT content on the DC conductivity

(σDC ~ |σ*| at 0.1 Hz) of PE composites in the melt state at T =

140oC. The data is reproduced from Palza et al. (2011).

Fig. 2. The dynamics frequency spectrum of PE/MWCNTs and PE/SWCNTs composites at 140oC, (a) elastic modulus, G’(ω, φw), vs.

ω for PE/MWCNTs and (b) elastic modulus, G’(ω, φw), vs. ω for PE/SWCNTs.

Deepak Ahirwal, Humberto Palza, Guy Schlatter and Manfred Wilhelm

322 Korea-Australia Rheology J., Vol. 26, No. 3 (2014)

amplitudes (the points which are overlapping with the dot-

ted line in the Fig. 3). Interestingly, I3/1(γ0) at concentra-

tions below the percolation threshold (φw = 1.0 wt%) were

lower relative to neat PE and the square scaling was

observed over a wide range of deformation amplitudes, ã0

= 0.2−1.1. However, at concentrations above the perco-

lation threshold (φw = 3.0 and 12.0 wt%), the square scal-

ing was observed only over a very narrow range of

deformation amplitudes (for φw = 3.0, γ0 = 0.05−0.08 and

for φw= 12.0, γ0 = 0.01 − 0.02). The appearance of more

pronounced nonlinearities at much smaller applied defor-

mation amplitudes (relative to neat PE) could be attributed

to the formation of PE-MWCNT and MWCNT-MWCNT

networks above the percolation threshold for the PE/

MWCNTs composites.

It has been well established that the Q0(ω)−ω master

curve is extremely sensitive in the detection of different

relaxation processes relative to the dynamics frequency

curve (G’ and G’’ vs. ω) (Hyun and Wilhelm, 2009; Kempf

et al., 2013).

The two previous studies (Lim et al., 2013; Hassanabadi

et al., 2013) that were performed on polymer composites

under LAOS flow only investigated the influence of the

percolating network and were performed at low frequen-

cies (Lim et al., 2013 at ω = 1 rad/s and Hassanabadi et

al., 2013 at ω/2π = 0.05 Hz). However, no studies have

yet focused on the determination of the percolation thresh-

old using the Q0(ω)−ω master curve. This method has the

potential to provide us with a new and sensitive way to

detect the percolation threshold because, at the percolation

threshold, the polymer composites make a transition from

liquid-like to solid-like behavior. For the first time, an

investigation was made to measure the percolation thresh-

old in the nonlinear viscoelastic regime.

Fig. 4 shows the Q0(ω)−ω master curve for the PE/

MWCNT and the PE/SWCNT composites at 140oC. The

intrinsic nonlinearity, Q0(ω), increased monotonically with

increasing frequency, ω/2π, below the percolation thresh-

old for the PE/MWCNT and PE/SWCNT composites.

However, above the percolation threshold, the Q0(ω)

curve displayed a monotonic decrease with increasing fre-

quency. Furthermore, it is interesting to note that, upon

addition of CNTs to the PE polymer matrix, Q0(φw)

decreased monotonically at concentrations below the per-

colation threshold for any fixed frequency. For example,

Fig. 5 shows Q0(φw) as a function of the MWCNT weight

fraction at a fixed frequency, ω/2π = 0.1 Hz. This specific

frequency was chosen because it is well known that addi-

tion of nanoparticles into a polymer matrix significantly

affects the behavior at low frequencies (terminal regime).

Furthermore, Plaza et al. also used 0.1 Hz for dielectric

measurements of the conductivity of PE/MWCNTs com-

posites. Therefore, our rheological data at 0.1 Hz will be

correlated with their conductivity measurements. A sig-

nificant drop (60% decrease in Q0 relative to neat PE) was

observed for Q0(φw) until the percolation threshold was

reached. This was followed by a steep rise in Q0(φw) with

concentration above this threshold. It is important to note

that there is almost no literature available that focuses spe-

Fig. 3. Fourier intensity of the third harmonic relative to the fun-

damental harmonic, I3/1(γ0), as a function of the applied defor-

mation amplitude, γ0, at ω/2π = 0.1 Hz and 140oC for different

weight fractions of MWCNTs in the PE/MWCNTs nanocom-

posites.

Fig. 4. The intrinsic nonlinearity, Q0(ω), as a function of the applied angular frequency, ω, and the weight fraction of CNTs in the PE/

CNT composites at 140oC, (a) Q0(ω) vs. ω for PE/MWCNT composites and (b) Q0(ω) vs. ω for PE/SWCNT composites.

New way to characterize the percolation threshold of polyethylene and carbon nanotube polymer composites......

Korea-Australia Rheology J., Vol. 26, No. 3 (2014) 323

cifically on the rheological changes caused by CNT dis-

persion in a polymer matrix below the percolation

threshold. Of special interest to note is that the trend

observed for Q0(φw) as a function of the weight fraction of

MWCNTs, φw, is the same as the trend seen in the elec-

trical conductivity measurements in LAOS flow (Palza et

al., 2011). The following two important observations, (i)

Q0(ω) monotonically increases below the percolation

threshold and decreases above the percolation threshold,

and (ii) the appearance of a minimum in Q0(φw), are

explained in the following paragraphs.

To explain both observations, single mode Maxwell

(Dealy and Larson, 2011) and Pom-Pom model (McLeish

and Larson, 1998) simulations were performed where the

backbone relaxation, τ, was set to 1 sec and the modulus

was 105 Pa. The details of the Pom-Pom equations that

were used in our simulations can be found elsewhere

(Kempf et al., 2013). Fig. 6 shows the dynamic frequency

spectrum simulated using the Maxwell model and the non-

linear master curve, Q0(ω), simulated using the Pom-Pom

model for different branching degrees, q (Fig. 6a-6b). The

qualitative changes in the Q0(ω) master curve for different

branching degrees, q, were similar except for the case of

q = 1 (linear molecule) where a maximum was observed

Fig. 5. The intrinsic nonlinearity, Q0(φw), as a function of the

weight fraction of MWCNTs for PE/MWCNT composites at ω/

2π= 0.1 Hz and T=140oC where the percolation threshold is

= 1.0 wt%.φw

c

Fig. 6. The single mode Maxwell and Pom-Pom model simulations used to explain the experimental observations for the PE/MWCNT

composites at G0 = 105 Pa, τ = 1 sec, Sb = 10, where Sb is the normalized molecular weight of the backbone relative to the entanglement

molecular weight, and Sa = 4, where Sa is the normalized molecular weight of an arm relative to the entanglement molecular weight:

(a) the dynamic frequency spectrum obtained using the Maxwell model simulation, (b) the nonlinear master curve Q0(ω) obtained using

the Pom-Pom model at various number of branches, q, (c) schematic representation of the polymer composite system with increasing

weight fraction of CNTs and (d) the intrinsic nonlinearity, Q0, in the terminal regime as a function of increasing degree of branching, q.

Deepak Ahirwal, Humberto Palza, Guy Schlatter and Manfred Wilhelm

324 Korea-Australia Rheology J., Vol. 26, No. 3 (2014)

at the reptation relaxation frequency, ωd = 1/τ. From the

dynamic frequency spectrum, it is evident that, below ω <

ωd, the viscous response dominates and the polymer flow

behavior is liquid-like and, above ω > ωd, the elastic

response becomes dominant and the polymer behaves

more solid-like.

It is interesting to note in Fig. 6 that, in the liquid-like

regime, Q0(ω) is a monotonically increasing function of

the angular frequency, ω, and that, in the solid-like regime,

it is instead a monotonically decreasing function of the

angular frequency, ω. A similar trend for Q0(ω) was

observed in the experimental results for the PE/MWCNT

and PE/SWCNT composites as shown in Fig. 4. For the

simple case of a linear polymer, the transition from the liq-

uid-like regime to the solid-like regime can be monitored

by measuring the crossover in the dynamic frequency

spectrum (G’ and G’’ vs. ω), but for branched polymers it

is not necessarily true that the transition from liquid-like

to solid-like flow behavior occurs at the crossover. This

result was explained by Kempf et al. (2013) for the case

of comb polymers where the Q0(ω) master curve was used

to obtain the reptation relaxation time, τ. Using our sim-

ulations and experimental results on PE/MWCNT com-

posites and results from previous studies (Kempf et al.,

2013; Hyun and Wilhelm, 2009), we conclude that the

Q0(ω) master curve can be used to determine the per-

colation threshold or the transition from liquid-like to

solid-like flow behavior.

To explain the second observation, we first assume that

the stress response is a function of the polymer-polymer

and polymer-CNT interactions and that the CNT-CNT

interactions are negligible below the percolation threshold

(Fig. 6c). This assumption is reasonable because we know

that below this threshold there are negligible CNT-CNT

interactions. The decrease in Q0(φw) can be explained by

an increase in polymer chain stretching due to polymer-

CNT interactions. This result is, in a way, similar to the

effect of an increase in branching in a polymer because

addition of more CNTs to the polymer matrix causes the

weight fraction of the “branched” structure due to poly-

mer-CNT interactions to increase. The decrease in Q0 is,

therefore, a consequence of an increase in the average

polymer chain stretching, which is a result of the increas-

ing branch-like structure. This observation is supported by

a recently proposed analytical solution of the MSF model

(Wagner et al., 2011) using Q0(ω) (α – β) where α is the

orientation parameter and β is the stretching parameter.

Furthermore, this is consistent with our simulations as

shown in Fig. 6d. In the terminal regime, it was seen that,

as the branching parameter q increased, a decrease in Q0

was detected. The significant increase above the perco-

lation threshold can be explained by increasing contri-

butions from the CNT-CNT interactions.

It is well known that, above the percolation threshold,

the rheological and electrical properties show an asymp-

totic scaling P ~ where P represents the material

property and δ is a scaling exponent. The value of δ pro-

vides information on how strong the influence of the per-

colating network is on the rheological and electrical

properties. Fig. 7 shows the normalized electrical con-

ductivity (data taken from Palza et al., 2009) and material

parameters in the linear viscoelastic regime (G’) and the

nonlinear viscoelastic regime (Q0) as a function of φw. The

calculated scaling exponent, d, for the electrical conduc-

tivity is . For the elastic modulus, G’, it is

and for the intrinsic nonlinearity, Q0, it is

. The percolating network has the highest

effect on the electrical conductivity, σDC, followed by the

intrinsic nonlinearity, Q0 (nonlinear viscoelastic regime)

and then the elastic modulus, G’ (linear viscoelastic

regime). Interestingly, the influence of the percolating net-

work on Q0 is about a factor of 10 higher than its influence

on G’. This implies that network formation has a signif-

icantly higher effect on the parameters measured in the

nonlinear viscoelastic regime compared to those measured

in the linear viscoelastic regime. Therefore, we conclude

that Q0 is a more suitable material parameter for the detec-

tion of the percolation threshold than the elastic modulus,

G’. The zero strain intrinsic nonlinearity, Q0, could be

potentially useful in the characterization and process opti-

mization of polymer/CNT composites because of its

higher sensitivity compared to other rheological tools

available in (mostly) the linear viscoelastic regime.

4. Conclusions

The electrical and rheological percolation threshold for

the PE/MWCNT composites was found to be between 1-

3 wt% of MWCNTs. However, for the PE/SWCNT com-

∝

φw

φw

c

∼( )δ

δ 4.3 0.3±=

δ 1.7 0.7±=

δ 3.0 1.0±=

Fig. 7. The effect of the percolating network on the conductivity

(σDC) and the linear (G’) and nonlinear (Q0) viscoelastic prop-

erties for the PE/MWCNT composites at ω/2π = 0.1 Hz and T =

140oC.

New way to characterize the percolation threshold of polyethylene and carbon nanotube polymer composites......

Korea-Australia Rheology J., Vol. 26, No. 3 (2014) 325

posites, no percolation threshold was detected up to 1 wt%

of SWCNTs. A new rheological method was proposed to

detect the percolation threshold in the nonlinear viscoelas-

tic regime using the nonlinear Q0(ω) master curve for con-

ductive or nonconductive particles. A minimum was

observed in Q0(φw) as a function of the MWCNT weight

fraction, w. The decrease in Q0(φw) below the percolation

threshold was explained by increased stretching of the

polymer chains due to interactions between the PE and

CNTs. The asymptotic variation of the material param-

eters in the linear viscoelastic regime and in the nonlinear

viscoelastic regime were observed to follow the scaling

. The calculated scaling exponent in the lin-

ear viscoelastic regime (using G’) was and in

the nonlinear viscoelastic regime (using Q0) was

. The higher scaling exponent in the nonlinear

viscoelastic regime relative to that in the linear viscoelas-

tic regime implies that network formation has a more sig-

nificant effect on parameters measured in the nonlinear

viscoelastic regime.

Acknowledgments

Financial support received from the Region Alsace,

France and the Karlsruhe Institute of Technology, Ger-

many for the doctoral fellowship of Mr. Ahirwal is

acknowledged. We are thankful to Prof. Mr. Kappes (KIT)

for providing us with SWCNTs for our investigation. We

are also thankful to Dimitri Merger and Dr. Kübel for

proofreading this manuscript and for constructive discus-

sions.

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