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Modeling the electrical percolation of mixed carbonfillers in polymer blends

0008-6223/$ - see front matter Crown Copyright � 2014 Published by Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.carbon.2014.01.001

* Corresponding author.E-mail address: guozx@mail.tsinghua.edu.cn (Z.-X. Guo).

Zhuo-Yue Xiong, Bo-Yuan Zhang, Li Wang, Jian Yu, Zhao-Xia Guo *

Key Laboratory of Advanced Materials (MOE), Department of Chemical Engineering, Tsinghua University, Beijing 100084, PR China

A R T I C L E I N F O A B S T R A C T

Article history:

Received 9 July 2013

Accepted 3 January 2014

Available online 10 January 2014

A model based on excluded volume theory is proposed for describing the electrical perco-

lation of mixed carbon fillers in polymer blends by adjusting the unit volume of the previ-

ous model concerning mixed carbon-filler-filled single polymer systems. An equation

capable of predicting the percolation threshold from those of individual carbon fillers in

the single matrix polymer is developed from the model and further corrected to suit the

actual situation. The corrected equation fits the experimental results obtained from

multi-walled carbon nanotubes/carbon black-filled polybutylene terephthalate/styrene–

acrylonitrile (SAN), polycarbonate (PC)/SAN and PC/acrylonitrile–butadiene–styrene blends

well. The model and equation show clearly the advantages of using both mixed fillers and

polymer blends, and can provide an important theoretical basis for designing the struc-

tures and predicting the electrical properties of conductive polymer composites.

Crown Copyright � 2014 Published by Elsevier Ltd. All rights reserved.

the advantages of using both polymer blends and mixed fill-

1. Introduction

Polymer blends that contain carbon-based fillers [1–32], such

as carbon black (CB), graphite and carbon nanotubes (CNTs),

have been of interest for more than two decades because of

their significantly lower electrical percolation thresholds

compared to the corresponding single polymer-filler compos-

ites. The work reported so far involves primarily a single type

of carbon-filler filled polymer blends.

Using mixed carbon fillers to prepare conductive polymer

composites has received considerable attention in recent

years [9–26]. The electrical and thermal conductivities and

the mechanical properties of the materials are significantly

improved [9–11], sometimes exceeding the additive effect of

each individual filler. The electrical percolation threshold of

mixed fillers usually lies between those of the two individual

fillers [9,12–18] because of the formation of co-supporting

conductive networks of the two types of carbon fillers, indi-

cating that the use of mixed carbon fillers can provide a good

balance between the resulting properties and cost. Adding

mixed carbon fillers to polymer blends could capitalize on

ers and could thus be a good strategy for preparing high

-performance conductive materials. A suitable electrical per-

colation model or equation is certainly needed for predicting

the electrical percolation threshold and for guiding the design

of such complicated multicomponent materials.

There are numerous electrical percolation models for car-

bon filler-filled polymers, including both numerical simula-

tions and analytical models [18,33–43]. A single type carbon

filler (such as CNTs, CB and graphite) in a single polymer ma-

trix has been frequently modeled, however only few reports

[18,42,43] involve mixed carbon filler-filled single polymer

matrices. Li and Chou [42] have proposed a versatile

simulation method for modeling percolation in a single ma-

trix containing multiple fillers with arbitrary shapes by

approximating any given shape with polygons. Rahatekar

et al. [43] have used a dissipative particle dynamics method

to simulate mixtures of conducting fibers and spheres, as well

as mixtures of fibers of different aspect ratios. Our research

group [18] has previously reported an analytical percolation

model for mixed carbon filler-filled single polymer systems

234 C A R B O N 7 0 ( 2 0 1 4 ) 2 3 3 – 2 4 0

by extending the excluded volume theory. An equation is de-

duced from the model, predicting the electrical percolation

threshold of mixed carbon fillers in a polymer matrix using

those of individual carbon fillers in the same polymer. The

equation is currently being used to calculate the theoretical

value of the percolation threshold [9,12–16], which is used to

determine if a synergistic effect of the two fillers exists by

comparing the theoretical value with the experimental value.

To the best of our knowledge, modeling the electrical percola-

tion of mixed carbon fillers in polymer blends is still lacking.

It is of practical importance to predict the electrical perco-

lation thresholds of complicated systems using those of sim-

ple systems because the latter is likely known. Using our

previous model [18], the electrical percolation threshold of

mixed carbon fillers in a polymer blend cannot be correctly

predicted from those of individual carbon fillers in a single

polymer where the conductive filler is selectively localized

(usually the matrix phase) due to the presence of the other

immiscible polymer phase. The objective of this work is to

build up an electrical percolation model for mixed carbon fil-

ler-filled polymer blend by extending the excluded volume the-

ory and deduce an equation capable of predicting the

percolation threshold from those of individual carbon fillers

in a single polymer. The idea for building up the model is to ad-

just the unit volume by deducting the volume fraction occu-

pied by the non-carbon filler-containing polymer phase. The

reliability of the equation is tested by comparison with the

experimental data.

2. Experimental

2.1. Materials

Multi-walled carbon nanotubes (MWCNTs), with an average

diameter of 10 nm and length of several micrometers, were

supplied by Tsinghua University, China. CB (VXC-605), with

a primary particle diameter of 25 nm, a dibutyl phthalate

(DBP) absorption of 1.48 cm3/g, and an iodine absorption of

90 mg/g, were purchased from Cabot (Shanghai). Polycarbon-

ate (PC, FN2500A), with a density of 1.20 g/cm3 and a melt flow

index of 8.0 g/10 min, was kindly provided by the Idemitsu

Corporation, Japan. Acrylonitrile–butadiene-styrene (ABS,

PA757), with a rubber content of 10% and a melt flow index

of 17.0 g per 10 min, was purchased from the Chi Mei Corpo-

ration. Styrene–acrylonitrile (SAN), with a density of 1.06 g/

cm3, a melt flow index of 44.2 g/10 min and an AN content

of 24 wt.%, and polyoxymethylene (POM, F2002), with a den-

sity of 1.41 g/cm3 and a melt flow index of 9.0 g per 10 min,

were kindly provided by the Mitsubishi Corporation, Japan.

Polybutylene terephthalate (PBT, 301G20), with a density of

1.54 g/cm3, and glass fiber (GF, ECS 305K), with an average

length of 4.5 mm, were purchased from the Hua Hui Plastic

Corporation and the Chongqing Polycomposites International

Corporation, China, respectively.

2.2. Preparation of the composites

All of the composites were prepared by melt mixing in one

step in a torque rheometer (MR-200A Rheometer, Harbin

Hapro Electrical Technology Co., Ltd.) at 250 �C. The mixing

time was 10 min, and the rotation rate was 60 rpm. The re-

ported percentages refer to weight percentages.

2.3. Characterization

Volume resistivity measurements were performed on com-

pression-molded films. For samples with higher electrical

resistivities (>108 X cm), disk samples with a diameter of

75 mm and a thickness of 0.38 mm were prepared and tested

using the ZC-36 resistivity test (Shanghai Cany Precision

Instrument). For more conductive samples (<108 X cm), disk

samples with a diameter of 30 mm and a thickness of

2.5 mm were prepared and tested using the KDY-1 resistivity

test (Guangzhou Kunde Technology Ltd.).

To investigate the phase morphology of the samples, all of

the samples for field emission scanning electron microscopy

(FESEM) (JEOL model JSM-7401 instrument) were cryo-frac-

tured in liquid nitrogen, and the MWCNT/CB-filled PBT/SAN

samples were etched with acetone at room temperature for

4 h to remove the SAN phase.

To investigate the dispersion and localization of the car-

bon fillers, transmission electron microscopy (TEM) (Hitachi

H-800 electron microscope operating at 200 kV) measure-

ments were performed. Ultrathin sections were cut using an

ultratome (EM UC6, Leica) in combination with a diamond

knife.

3. Results and discussion

3.1. Electrical percolation in polymer blends

In Fig. 1, the electrical resistivity is plotted as a function of the

filler content for PBT/SAN, PC/SAN and PC/ABS blends filled

with either a single conductive filler (MWCNTs or CB) or

MWCNT/CB-mixed conductive fillers.

The percolation threshold of MWCNTs (0.4–0.6 wt.%) is al-

ways less than that of CB (6–11 wt.%) in the same polymer

blend because of their high conductivity and high aspect ra-

tio, which is more favorable for the formation of conductive

paths. The percolation threshold (P 0) of an individual filler in

a polymer blend is always less than in the corresponding ma-

trix polymer (P) (Table 1), which indicates the advantage of

using polymer blends. For example, the percolation thresh-

olds of MWCNTs and CB in PBT/SAN are 0.54 and 3.96 wt.%,

respectively, while those in PBT are 0.97 and 7.28 wt.%. In

Fig. 1, all of the curves with mixed fillers are located in the

middle, indicating that to gain the same level of conductivity,

the need for CB is dramatically reduced when a small amount

of MWCNTs is used because of the formation of co-support-

ing conductive networks of MWCNTs and CB. This observa-

tion is consistent with mixed filler-filled single polymers

[9,18,22,26].

3.2. Morphology of the blends

The above-mentioned conductive filler-filled polymer blends

have similar structures. They are all immiscible blends with

a sea-island phase morphology, and the majority of the

Fig. 1 – Electrical resistivity as a function of the conductive filler content for (a) PBT/SAN (7:3); (b) PC/SAN (7:3); (c) PC/ABS (7:3);

and (d) POM/20 wt.% GF. (A color version of this figure can be viewed online.)

C A R B O N 7 0 ( 2 0 1 4 ) 2 3 3 – 2 4 0 235

conductive fillers are selectively localized in the matrix phase.

As an example, FESEM micrographs of cryo-fractured surfaces

showing the phase morphology and TEM micrographs of PBT/

SAN blends filled with MWCNT/CB showing the localization

of conductive fillers are given in Fig. 2(a–c). This type of struc-

ture resembles conductive filler-filled polymer/inert filler sys-

tems [6], in which the inert filler occupies a certain space and

the conductive filler can only diffuse into the polymer, as

shown in Fig. 2(d–f) for MWCNT/CB-filled POM/20 wt.% GF.

Therefore, POM/20 wt.% GF is taken as a model for polymer

blends, and POM/20 wt.% GF filled with various amounts of

either individual or mixed conductive fillers are investigated.

Similar to the cases of polymer blends (Fig. 1(a–c)), the electri-

cal resistivity-conductive filler content curve for MWCNT/CB

mixed conductive fillers is located in the middle (Fig. 1(d)),

indicating the formation of co-supporting networks of

MWCNTs and CB. An electrical percolation model is then pro-

posed for MWCNT/CB-filled POM/20 wt.% GF in the next sec-

tion by extending the excluded volume theory to represent

the electrical percolation model for mixed filler-filled polymer

blends.

3.3. Modeling the electrical percolation

The excluded volume theory [44,45] is originally proposed for

one type of object in a homogeneous matrix. According to the

excluded volume theory, the number of objects per unit

volume at percolation qp is inversely proportional to the

excluded volume Vhexi of the object, i.e.,

qp /1

Vhexið1Þ

By introducing a proportionality constant k, the inverse

relationship between the number of objects Nc per unit

volume Vunit at percolation and the excluded volume of the

object can be written as follows:

Nc

Vunit¼ k

1Vhexi

ð2Þ

That is:

Vunit ¼ NcVhexi

kð3Þ

Eq. (3) means: Vunit can be divided into Nc much smaller

equivalent volumes, and each small volume is Vhexi=k. When

Table 1 – Percolation thresholds of single carbon fillers in polymer blends (P 0) and in the corresponding matrix polymers (P) aswell as the measured and calculated X values.a

Polymer blend Filler P0 (wt.%) P (wt.%) Xm Xm Xc Xm=Xc

PBT/SAN (7:3) CNT 0.54 0.97 0.56 0.55 0.66 83%CB 3.96 7.28 0.54

PC/SAN (7:3) CNT 0.56 0.93 0.60 0.62 0.68 91%CB 6.76 10.64 0.64

PC/ABS (7:3) CNT 0.44 0.93 0.47 0.45 0.67 67%CB 4.48 10.64 0.42

POM/20 wt.% GF CNT 0.74 0.96 0.77 0.79 0.87 91%CB 5.40 6.74 0.80

a Xm ¼ P0=P, Xm is the average value of Xm, and Xc is the theoretical value calculated using excluded volume theory.

236 C A R B O N 7 0 ( 2 0 1 4 ) 2 3 3 – 2 4 0

all the small volumes are filled with conductive fillers, perco-

lation occurs.

For a system containing two types of conductive fillers

CNTs and CB in a homogeneous matrix [18], the unit volume

is divided into two types of small volumes, VhCNTi=kCNT and

VhCBi=kCB, where VhCNTi and VhCBi are the excluded volumes of

one CNT and one CB object, and kCNT and kCB are the corre-

sponding constants for the CNT and CB objects, respectively.

When all of the small volumes are filled, percolation occurs.

In the current case (MWCNT/CB-filled POM/20 wt.% GF),

when 87% of the small volumes are filled, percolation occurs,

because 13% of the unit volume is occupied by GF (Fig. 3).

Therefore, we have:

0:87 Vunit ¼ N0CNT

VhCNTi

kCNTþN0CB

VhCBi

kCBð4Þ

where N0CNT and N0CB are the numbers of CNT and CB objects

required for percolation to occur. Because [18];

N0CNT ¼VCNT

/c;CNT

NCNT ð5Þ

N0CB ¼VCB

/c;CB

NCB ð6Þ

where VCNT and VCB are the actual volume fractions of CNT

and CB objects, respectively; /c,CNT and /c,CB are the percola-

tion concentrations of MWCNTs and CB, respectively, ex-

pressed in volume fraction if the unit volume is filled with

MWCNTs or CB alone; and NCNT and NCB are the numbers of

CNT and CB objects at the corresponding percolation concen-

trations, respectively.

Entering Eqs. (5) and (6) into Eq. (4):

0:87 Vunit ¼VCNT

/c;CNT

NCNTVhCNTi

kCNTþ VCB

/c;CB

NCBVhCBi

kCBð7Þ

Because [18],

NCNTVhCNTi

kCNT¼ Vunit ¼ NCB

VhCBi

kCBð8Þ

Therefore,

0:87 Vunit ¼VCNT

/c;CNT

Vunit þVCB

/c;CB

Vunit ð9Þ

That is:

0:87 ¼ VCNT

/c;CNT

þ VCB

/c;CB

ð10Þ

For multicomponent systems, such as polymer/inert filler/

two conductive fillers and polymer blends/two conductive

fillers, Eq. (10) can be generalized as follows:

VA

/c;A

þ VB

/c;B

¼ X ð11Þ

For practice, Eq. (11) can be written as:

mA

Pc;Aþ mB

Pc;B¼ X ð12Þ

where mA and mB are the weight fractions of fillers A and B,

respectively; Pc,A and Pc,B are the corresponding percolation

concentrations when A and B are used individually; and X is

the volume fraction of the continuous phase.

Eq. (12) can provide guidance for predicting the electrical

percolation threshold and the electrical properties of materi-

als. When mA=Pc;A þmB=Pc;B ¼ X, percolation occurs; when

mA=Pc;A þmB=Pc;B > X, the material is conductive with a rela-

tively low electrical resistivity. When mA is fixed, mB can be

calculated using the following equation:

mB ¼ X� mA

Pc;A

� �Pc;B ð13Þ

3.4. Correction of the X value

The experimental data shown in Fig. 1 are used to examine

Eq. (12). For the MWCNT/CB-filled POM/20 wt.% GF system,

the MWCNT content is fixed at 0.3 wt.%. When the CB content

is 4 wt.% or greater, mA=Pc;A þmB=Pc;B > 0:87, the materials are

conductive with a relatively low electrical resistivity (log

q < 4). According to Eq. (12), the CB content required for perco-

lation to occur is 3.8 wt.%, i.e., the total conductive filler con-

tent is 4.1 wt.%. The experimental data were fit using Eq. (12).

Similarly, the data for the MWCNT/CB-filled PBT/SAN and PC/

SAN blends were fit using Eq. (12). However, the data for

MWCNT/CB-filled PC/ABS blends exhibited a substantial devi-

ation. According to Eq. (12), the CB content required for perco-

lation to occur is 4.9 wt.%, whereas the actual amount

required is approximately 2 wt.%, which is considerably lower

than the predicted amount.

Note that the conditions for using Eq. (12) are as follows:

(1) the dispersion and localization of one type of filler is not

affected by the other [18], (2) the conductive fillers are selec-

tively localized in the matrix phase of a polymer blend, and

(3) their dispersion state does not change when their matrix

Fig. 2 – FESEM (a, d–f) and TEM (b, c) micrographs for (a–c) MWCNT/CB filled PBT/SAN (7:3) and (d–f) MWCNT/CB filled POM/

20 wt.% GF. The sample was etched by acetone to remove the SAN phase in (a).

C A R B O N 7 0 ( 2 0 1 4 ) 2 3 3 – 2 4 0 237

changes from a single polymer to a polymer blend. In prac-

tice, the first condition can be easily satisfied [25], but the

localization and dispersion states of the conductive fillers in

polymer blends are often not as ideal as in the case of POM/

20 wt.% GF. Some of the conductive fillers may enter the dis-

persed phase, and the dispersion state of the conductive fill-

ers may change because of the presence of a second

polymer. The former increases the X value, whereas the latter

affects the average excluded volume of the conductive fillers.

Therefore, the X value must be corrected by taking these

changes into consideration. Because all these changes can

be reflected in the percolation threshold, we propose that

X ¼P0c;APc;A

or X ¼P0c;BPc;B

ð14Þ

where P0c;A and P0c;B are the percolation thresholds of A and B

when used alone in the polymer blend. In our investigation,

we found that P0c;A=Pc;A is almost the same as P0c;B=Pc;B for all

three polymer blends (Table 1), suggesting that the two differ-

ent types of conductive fillers have similar behavior with re-

spect to the changes in their localization and dispersion,

which is presumably due to their carbon-based nature.

Therefore, Eq. (12) can be written as follows:

mA

Pc;Aþ mB

Pc;B¼ X ¼

P0c;APc;A¼

P0c;BPc;B

ð15Þ

This equation indicates that the percolation threshold and

electrical properties of two conductive filler-filled polymer

blends can be predicted as long as three parameters are

known, Pc,A, Pc,B and P0c;A or P0c;B, i.e., the percolation thresholds

of the two conductive fillers used alone in the single matrix

polymer and that of any one type of the conductive fillers

used in the polymer blend. Eq. (15) means: To get the compos-

ite material conductive, it is not necessary to build up con-

ductive paths in the whole blend matrix. In an ideal case,

when the conductive paths are built up in the matrix phase,

the material is conductive. In a non-ideal case, the proportion

that needs to form the conductive paths can be experimen-

tally determined by the ratio of the percolation threshold of

any one type of the conductive fillers in the polymer blend

to that in the single matrix polymer.

For the MWCNT/CB-filled PC/ABS system, the measured X

value is 0.45. According to Eq. (15), when 0.2 wt.% MWCNTs is

used, the calculated CB content required for percolation to

Fig. 3 – Schematic illustrations of the excluded volume for mixed carbon fillers-filled POM/20 wt.% GF system in (a) the actual

state and (b) an extreme state. The black dots, black lines and gray sticks represent carbon black, MWCNTs and GF,

respectively.

238 C A R B O N 7 0 ( 2 0 1 4 ) 2 3 3 – 2 4 0

occur is 2.5 wt.%. When the CB content is 3 wt.% or greater,

mA=Pc;A þmB=Pc;B > 0.45, the materials are conductive. The

data shown in Fig. 1(c) were fit with Eq. (15). Considering

the possible experimental error, the measured X values for

the PET/SAN and PC/SAN systems do not exhibit considerable

differences compared to the volume fractions of the continu-

ous phase (i.e., the calculated values) (Table 1). The data

shown in Fig. 1(a) & (b) were fit with Eq. (15).

In the case of the PC/ABS system, the considerably smaller

actual X value compared to the calculated value indicates an

improved dispersion of the conductive fillers. Because ABS

melts before PC during compounding, most of the conductive

fillers first enter the ABS phase and then migrate to the PC

phase through a thermodynamic driving force [7]. The migra-

tion rate will be inhibited to some extent by the presence of

rubber in ABS compared to SAN, and the dispersion is there-

fore likely to improve due to the time difference for individual

conductive objects to reach the PC phase.

Eq. (15) is very useful because one often has a library of

percolation threshold data of relatively simple systems, such

as a single conductive filler-filled single polymers or polymer

blends. The percolation thresholds of multicomponent sys-

tems containing two conductive fillers and two polymers

can be easily predicted using Eq. (15). For example, the perco-

lation thresholds of CNTs and CB in PBT are 0.97 and

7.28 wt.%, respectively (Table 1), and that of CNTs in PBT/PC

(7:3) blend is 0.58 wt.%, then we can use mA/0.97 + mB/

7.28 = 0.6 (i.e., 0.58/0.97) to predict the percolation threshold

of MWCNT/CB mixed filler-filled PBT/PC (7:3) blend. When

0.2% CNTs are used, the prediction for CB is 2.87 wt.% to get

percolation to occur. When 0.2 wt.% CNTs and 3 wt.% CB are

used together in PBT/PC (7:3) blend, log q = 3.2, confirming

the prediction. Furthermore, Eq. (15) can be generalized to

two carbon filler-filled polynary polymer blends if the disper-

sion and localization of one type filler is not affected by the

other and if most of the conductive fillers selectively localize

in the continuous phase. To demonstrate this viewpoint, we

have investigated PET/PBT (25:5) and PC/PET/PBT (70:25:5)

blends filled with a single carbon filler (either MWCNT or

CB) and PC/PET/PBT (70:25:5) blend filled with MWCNT/CB

mixed fillers, and found that the equation mA/1.59 + mB/

7.01 = 0.57 (Pc,A = 1.59, Pc,B = 7.01, and X = 0.57) can be used to

predict the percolation threshold of MWCNT/CB-filled PC/

PET/PBT (70:25:5) blend where the carbon fillers are selectively

localized in the continuous polyester phase.

Eq. (15) is especially useful if a series of polymer blends

that have the same matrix polymer are being investigated,

where Pc,A and Pc,B are constant. For example, we already

know that for PC-based polymer blends, Pc,A = 0.93 wt.% and

Pc,B = 10.64 wt.%, and we also know that the percolation

threshold for the CB-filled PC/PPO (7:3) blend is

P0c;B = 8.19 wt.%. Therefore, we can predict the percolation

threshold of the MWCNT/CB-filled PC/PPO blend using Eq.

(15). If 0.5 wt.% MWCNT is to be used, the calculated CB con-

tent for percolation to occur is 2.5 wt.%. The actual log value

of the electrical resistivity is 3.71, indicating that the material

is conductive when 0.5 wt.% MWCNT and 3 wt.% CB are com-

bined together in the PC/PPO blend, which is consistent with

the predicted value.

4. Conclusions

An electrical percolation model based on excluded volume

theory is proposed for two conductive carbon filler-filled poly-

mer blends. An equation is developed from the model for pre-

dicting the electrical percolation threshold, in which the

advantage of using polymer blends as the matrix is clearly ex-

pressed. The model and equation are generally applicable to

any multicomponent system that contains two types of car-

bon-based conductive fillers as long as the dispersion and

localization of one type of filler is not affected by the other

and most of the conductive fillers selectively localize in one

continuous polymer phase. This work can provide an

important theoretical basis for designing the structures and

predicting the electrical properties of conductive polymer

composites.

Acknowledgments

We thank Beijing Key Laboratory of Green Reaction Engineer-

ing and Technology, Department of Chemical Engineering,

Tsinghua University, for kindly providing the MWCNTs. This

C A R B O N 7 0 ( 2 0 1 4 ) 2 3 3 – 2 4 0 239

work was supported by the National Natural Science Founda-

tion of China (No. 50973053) and the Specialized Research

Fund for the Doctoral Program of Higher Education (No.

20090002110072).

R E F E R E N C E S

[1] Sumita M, Sakata K, Asai S, Miyasaka K, Nakagawa H.Dispersion of fillers and the electrical conductivity ofpolymer blends filled with carbon black. Polym Bull1991;25(2):265–71.

[2] Li YJ, Shimizu H. Conductive PVDF/PA6/CNTsnanocomposites fabricated by dual formation ofcocontinuous and nanodispersion structures.Macromolecules 2008;41(14):5339–44.

[3] Le HH, Schoss M, Ilisch S, Gohs U, Heinrich G, Pham T, et al.CB filled EOC/EPDM blends as a shape-memory material:manufacturing, morphology and properties. Polymer2011;52(25):5858–66.

[4] Thongruang W, Balik CM, Spontak RJ. Volume-exclusioneffects in polyethylene blends filled with carbon black,graphite, or carbon fiber. J Polym Sci Part B Polym Phys2002;40(10):1013–25.

[5] Xiong ZY, Sun Y, Wang L, Guo ZX, Yu J. Electricalconductivities of carbon nanotube-filled polycarbonate/polyester blends. Sci China Ser B 2012;55(5):808–13.

[6] Bao HD, Guo ZX, Yu J. Effect of electrically inert particulatefiller on electrical resistivity of polymer/multi-walled carbonnanotube composites. Polymer 2008;49(17):3826–31.

[7] Sun Y, Guo ZX, Yu J. Effect of ABS rubber content on thelocalization of MWCNTs in PC/ABS blends and electricalresistivity of the composites. Macromol Mater Eng2010;295(3):263–8.

[8] Sun Y, Jia MY, Guo ZX, Yu J, Nagai S. Effect of Styrene–acrylonitrile on the electrical resistivity of polycarbonate/multiwalled carbon nanotube composites. J Appl Polym Sci2011;120(6):3224–32.

[9] Wen M, Sun XJ, Su L, Shen JB, Li J, Guo SY. The electricalconductivity of carbon nanotube/carbon black/polypropylenecomposites prepared through multistage stretchingextrusion. Polymer 2012;53(7):1602–10.

[10] King J, Via MD, Mills OP, Alpers DS, Sutherland JC, Bogucki GR.Effects of multiple carbon fillers on the electrical and thermalconductivity and tensile and flexural modulus ofpolycarbonate-based resins. J Compos Mater2012;46(3):331–50.

[11] Cheng HF, Sahoo NG, Pan YZ, Li L, Chan SH, Zhao JH, et al.Complementary effects of multiwalled carbon nanotubesand conductive carbon black on polyamide 6. J Polym Sci PolPhys 2010;48(11):1203–12.

[12] Zhang SM, Lin L, Deng H, Gao X, Bilotti E, Peijs T, et al.Synergistic effect in conductive networks constructed withcarbon nanofillers in different dimensions. Express PolymLett 2012;6(2):159–68.

[13] Dang ZM, Shehzad K, Zha JW, Mujahid A, Hussain T, Nie J,et al. Complementary percolation characteristics of carbonfillers based electrically percolative thermoplastic elastomercomposites. Compos Sci Technol 2011;72(1):28–35.

[14] Socher R, Krause B, Hermasch S, Wursche R, Potschke P.Electrical and thermal properties of polyamide 12 compositeswith hybrid fillers systems of multiwalled carbon nanotubesand carbon black. Compos Sci Technol 2011;71(8):1053–9.

[15] Sumfleth J, Buschhorn ST, Schulte K. Comparison ofrheological and electrical percolation phenomena in carbon

black and carbon nanotube filled epoxy polymers. J Mater Sci2011;46(3):659–69.

[16] Sumfleth J, Adroher XC, Schulte K. Synergistic effects innetwork formation and electrical properties of hybrid epoxynanocomposites containing multi-wall carbon nanotubesand carbon black. J Mater Sci 2009;44(12):3241–7.

[17] Yin HB, Bao HD, Li J, Guo ZX, Yu J. Electrical properties ofmultiwalled carbon nanotube/carbon black hybrid filler filledpolyoxymethylene composites. Acta Polym Sin2010;9:1152–6.

[18] Sun Y, Bao HD, Guo ZX, Yu J. Modeling of the electricalpercolation of mixed carbon fillers in polymer-basedcomposites. Macromolecules 2009;42:459–63.

[19] Zha JW, Li WK, Liao RJ, Bai JB, Dang ZM. High performancehybrid carbon fillers/binary-polymer nanocomposites withremarkably enhanced positive temperature coefficient effectof resistance. J Mater Chem A 2013;1(3):843–51.

[20] Li Q, Basavarajaiah S, Kim NH, Heo SB, Lee JH. Synergy effectof hybrid fillers on the positive temperature coefficientbehavior of polypropylene/ultra-high molecular weightpolyethylene composites. J Appl Polym Sci2010;116(1):116–24.

[21] Yan N, Xia HS, Zhan YH, Fei GX, Chen C. Co-compatibilisingeffect of carbon nanotubes and liquid isoprene rubber oncarbon black filled natural rubber/polybutadiene rubberblend. Plast Rubber Compos 2012;41(9):365–72.

[22] Hu HQ, Zhao L, Liu JQ, Liu Y, Cheng JM, Luo J, et al. Enhanceddispersion of carbon nanotube in silicone rubber assisted bygraphene. Polymer 2012;53(15):3378–85.

[23] Dinesh P, Renukappa NM, Siddaramaiah, Rajan JS. Electricalproperties and EMI shielding characteristics of multiwalledcarbon nanotubes filled carbon black-high densitypolyethylene nanocomposites. Compos Interface2012;19(2):121–33.

[24] Yu K, Zhang ZC, Liu YJ, Leng JS. Carbon nanotube chains in ashape memory polymer/carbon black composite: tosignificantly reduce the electrical resistivity. Appl Phys Lett2011;98(7):074102.

[25] Bilotti E, Zhang H, Deng H, Zhang R, Fu Q, Peijs T.Controlling the dynamic percolation of carbon nanotubebased conductive polymer composites by addition ofsecondary nanofillers: the effect on electrical conductivityand tuneable sensing behaviour. Compos Sci Technol 2013;74:85–90.

[26] Ma PC, Liu MY, Zhang H, Wang SQ, Wang R, Wang K, et al.Enhanced electrical conductivity of nanocompositescontaining hybrid fillers of carbon nanotubes and carbonblack. Acs Appl Mater Inter 2009;1(5):1090–6.

[27] Gubbels F, Jerome R, Teyssie Ph, Vanlathem E, Deltour R,Calderone A, et al. Selective localization of carbon black inimmiscible polymer blends: a useful tool to designelectrical conductive composites. Macromolecules 1994;27(7):1972–4.

[28] Goldel A, Marmur A, Kasaliwal GR, Potschke P, Heinrich G.Shape-dependent localization of carbon nanotubes andcarbon black in an immiscible polymer blend during meltmixing. Macromolecules 2011;44(15):6094–102.

[29] Zonder L, Ophir A, Kenig S, McCarthy S. The effect of carbonnanotubes on the rheology and electrical resistivity ofpolyamide 12/high density polyethylene blends. Polymer2011;52(22):5085–91.

[30] Xiong ZY, Wang L, Sun Y, Guo ZX, Yu J. Migration of MWCNTsduring melt preparation of ABS/PC/MWCNT conductivecomposites via PC/MWCNT masterbatch approach. Polymer2013;54:447–55.

[31] Goldel A, Kasaliwal G, Potschke P. Selective localization andmigration of multiwalled carbon nanotubes in blends of

240 C A R B O N 7 0 ( 2 0 1 4 ) 2 3 3 – 2 4 0

polycarbonate and poly(styrene–acrylonitrile). MacromolRapid Commun 2009;30(6):423–9.

[32] Goldel A, Kasaliwal GR, Potschke P, Heinrich G. The kineticsof CNT transfer between immiscible blend phases duringmelt mixing. Polymer 2012;53(2):411–21.

[33] Li J, Ma PC, Chow WS, To CK, Tang BZ, Kim JK. Correlationsbetween percolation threshold, dispersion state, and aspectratio of carbon nanotubes. Adv Funct Mater2007;17(16):3207–15.

[34] Ounaies Z, Park C, Wise KE, Siochi EJ, Harrison JS. Electricalproperties of single wall carbon nanotube reinforcedpolyimide composites. Compos Sci Technol2003;63(11):1637–46.

[35] Berhan L, Sastry AM. Modeling percolation in high-aspect-ratio fiber systems. I. Soft-core versus hard-core models. PhysRev E 2007;75:041120.

[36] Berhan L, Sastry AM. Modeling percolation in high-aspect-ratio fiber systems. II. The effect of waviness on thepercolation onset. Phys Rev E 2007;75:041121.

[37] White SI, DiDonna BA, Mu MF, Lubensky TC, Winey KI.Simulations and electrical conductivity of percolatednetworks of finite rods with various degrees of axialalignment. Phys Rev B 2009;79:024301.

[38] Lu WB, Chou TW, Thostenson ET. A three-dimensional modelof electrical percolation thresholds in carbon nanotube-based composites. Appl Phys Lett 2010;96(22):223106.

[39] Li JT, Zhang SL. Finite-size scaling in stick percolation. PhysRev E 2009;80(4):040104.

[40] Hu N, Masuda Z, Yan C, Yamamoto G, Fukunaga H, Hashida T.The electrical properties of polymer nanocomposites withcarbon nanotube fillers. Nanotechnology 2008;19:215701.

[41] Celzard A, McRae E, Deleuze C, Dufort M, Furdin G, MarecheJF. Critical concentration in percolating systems containing ahigh-aspect-ratio filler. Phys Rev B 1996;53(10):6209–14.

[42] Li CY, Chou TW. Continuum percolation of nanocompositeswith fillers of arbitrary shapes. Appl Phys Lett2007;90(17):174108.

[43] Rahatekar SS, Shaffer MSP, Elliott JA. Modelling percolation infibre and sphere mixtures: routes to more efficient networkformation. Compos Sci Technol 2010;70(2):356–62.

[44] Balberg I. Excluded-volume explanation of Archie’s law. PhysRev B 1986;33:3618–20.

[45] Balberg I, Binenbaum N, Wagner N. Percolation thresholds inthe three-dimensional sticks system. Phys Rev Lett1984;52:1465–8.